Tao

Aa

b

c

a

ARRAA

KEFFIIIIPP

1

tecmanleacil

K

0d

Journal of Chromatography A, 1216 (2009) 1067–1083

Contents lists available at ScienceDirect

Journal of Chromatography A

journa l homepage: www.e lsev ier .com/ locate /chroma

heoretical study of the accuracy of the pulse method, frontal analysis,nd frontal analysis by characteristic points for the determinationf single component adsorption isotherms

nna Andrzejewskaa,b, Krzysztof Kaczmarski c, Georges Guiochona,b,∗

Department of Chemistry, University of Tennessee, Knoxville, TN 37996-1600, USADivision of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6120, USADepartment of Chemical and Process Engineering, Rzeszow University of Technology, Rzeszow 35959, Poland

r t i c l e i n f o

rticle history:eceived 5 August 2008eceived in revised form 22 October 2008ccepted 1 December 2008vailable online 13 December 2008

eywords:CPACProntal analysis

a b s t r a c t

The adsorption isotherms of selected compounds are our main source of information on the mechanismsof adsorption processes. Thus, the selection of the methods used to determine adsorption isotherm dataand to evaluate the errors made is critical. Three chromatographic methods were evaluated, frontal anal-ysis (FA), frontal analysis by characteristic point (FACP), and the pulse or perturbation method (PM),and their accuracies were compared. Using the equilibrium-dispersive (ED) model of chromatography,breakthrough curves of single components were generated corresponding to three different adsorptionisotherm models: the Langmuir, the bi-Langmuir, and the Moreau isotherms. For each breakthroughcurve, the best conventional procedures of each method (FA, FACP, PM) were used to calculate the cor-responding data point, using typical values of the parameters of each isotherm model, for four different

sotherm accuracysotherm determinationsotherm modelingsotherm precisionerturbation methodulse method

values of the column efficiency (N = 500, 1000, 2000, and 10,000). Then, the data points were fitted toeach isotherm model and the corresponding isotherm parameters were compared to those of the ini-tial isotherm model. When isotherm data are derived with a chromatographic method, they may sufferfrom two types of errors: (1) the errors made in deriving the experimental data points from the chro-matographic records; (2) the errors made in selecting an incorrect isotherm model and fitting to it theexperimental data. Both errors decrease significantly with increasing column efficiency with FA and FACP,but not with PM.

. Introduction

Preparative liquid chromatography is now playing an impor-ant role in the pharmaceutical industry. It permits the selectivextraction and the purification of a wide variety of active pharma-eutical ingredients. For obvious economical reasons this processust be conducted at finite concentrations, not at infinite dilution

s analytical chromatography. Detailed knowledge of the process isow required by regulators. This implies that the adsorption equi-

ibrium isotherms or relationships between the concentrations ofach component in the two phases of the system at equilibrium,

t constant temperature and pressure must be measured for theompounds of interest. These data are critical for the understand-ng of adsorption processes taking place at the solid–gas [1,2] andiquid–solid [3,4] interfaces which are used in preparative chro-

∗ Corresponding author at: Department of Chemistry, University of Tennessee,noxville, TN 37996-1600, USA. Fax: +1 865 974 2667.

E-mail address: guiochon@utk.edu (G. Guiochon).

021-9673/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.chroma.2008.12.021

© 2008 Elsevier B.V. All rights reserved.

matography. An accurate determination of a set of adsorption datain a sufficiently wide range of concentration is required for theproper modeling of the nonlinear chromatographic process and forits optimization. The purpose of this work is to assess and comparethe precision of the common chromatographic methods used forthis purpose.

For the measurement of the adsorption excess, several chro-matographic methods can be used. These dynamic methods arefaster and more accurate than static ones [5], so they have becomethe most popular. The best known are frontal analysis (FA), elution(ECP) and frontal analysis (FACP) by characteristic points, the pulseor perturbation method (PM), and the inverse method (IM).

Frontal analysis is considered as the most accurate of thesechromatographic techniques. In this method, a solution of knownconcentration of the studied compound is percolated through a

column packed with the selected adsorbent. Periodically, the con-centration of this solution is changed abruptly at the column inletand the breakthrough curve is recorded [6–9]. The mass of studiedcompound adsorbed at equilibrium is calculated from the retentiontime of the breakthrough curve. There are three ways to estimate

1 omato

tearoapttebcceTeas

aatcETiataacbtcibntttLpwte5ffisEsmcifdo

o[bcuov

068 A. Andrzejewska et al. / J. Chr

his retention time: from the inflection point of the curve, from thelution time of the half-height of the plateau (the middle point)nd using the equal area method that, according to the Maxwellule, determines the time location such that the areas on both sidesf this time are equal to each other. The last method is based onrigorous application of the mass balance of the solute and, in

rinciple, is the only method can and should be used. The firstwo methods may cause large systematic errors, particularly whenhe breakthrough curve is unsymmetrical and tends slowly towardquilibrium at high concentrations. Using empirical relationshipsetween the apparent axial dispersion coefficient and the con-entration and calculating breakthrough curves for series of inputoncentrations with the equilibrium-dispersive (ED) model, Sajonzt al. [10] modeled this process of isotherm data determination.hey found that the first two methods of determination of thelution time of the breakthrough curve cause significant system-tic errors when the apparent axial dispersion coefficient dependstrongly on the concentration.

The second chromatographic method of estimating thedsorbed amount at equilibrium uses the profile of the rear bound-ry recorded in FA, when a stream of the pure mobile phase is usedo purge the highest concentration solution equilibrated with theolumn. This is frontal analysis by characteristic point (FACP) [11].ach point of the diffuse boundary giving one point of the isotherm.he method of elution by characteristic point (ECP) [11,12] is sim-lar to FACP. The disadvantage of these two methods is that themount adsorbed at equilibrium with the mobile phase concentra-ion is calculated from an equation derived from the ideal model,ssuming that the column efficiency is infinite. Therefore, the ECPnd the FACP methods should be used only with highly efficientolumns, for which the contribution of the column efficiency toand broadening is negligible. Moreover, the data points close tohe top of the profile in ECP and those that are far along the tail,lose to the baseline in both methods, are more affected than thosen the middle by the sources of band broadening. They should note used in the determination of the isotherm. Unfortunately, it isot possible to reject the data points close to the baseline, sincehe integration of the profile has to be made from c to 0. Thus,he methods suffer from a systematic error. Guan et al. [13] usedhe ED model to derive band profiles corresponding to a knownangmuir isotherm, added a noise sequence to simulate the elutionrocess more realistically, calculated the isotherm, and compared itith the initial one. They found significant differences between the

rue and the experimental isotherms at low and moderate columnfficiencies, concluding that the column efficiency should exceed000 theoretical plates, the signal-to-noise 500, and the loadingactor about 0.2. Similar studies were made by Ravald et al. [14]or the bi-Langmuir isotherm. They found that the way of weight-ng of bi-Langmuir function during the nonlinear regression affectstrongly the model error. Another important source of error in theCP method is due to the need to inject large volumes of sampleolutions, which tend to tail considerably. The theory of the ECPethod assumes that the injection profiles are rectangular, so it

annot account for such profiles. Samuelsson and Fornstedt [15]nvestigated this problem lately. They introduced a new techniqueor ECP (the “cut-injection technique”) allowing a more accurateetermination of the adsorption isotherms by overcoming this seri-us problem.

The two types of pulse methods (PM) used are elution of a pulsen a plateau and elution of an isotopic (or possibly enantiomeric16]) pulse on a plateau. The former method was first suggested

y Reilley et al. [17]. A steady stream of solution of the studiedompound in the mobile phase is pumped through the columnntil equilibrium is reached. Then, a small pulse of the componentr of pure mobile phase is injected. The pulse velocity is theelocity associated with the plateau concentration. Its retention

gr. A 1216 (2009) 1067–1083

time permits the derivation of the slope of the isotherm at theplateau concentration. The isotherm is determined by repeating theprocedure multiple times while progressively increasing the solu-tion concentration and integrating numerically the plot of dq/dcvs. c.

The method of elution of an isotopic pulse on a plateau wasdeveloped by Helfferich and Peterson [18]. If labeled and unla-beled molecules are injected simultaneously on a concentrationplateau, the unlabeled molecules travel at the velocity associ-ated with the plateau concentration (see above) while the labeledmolecules travel at the velocity associated with the concentra-tion shock, �q/�C. Thus, the injection of a mixture of labeledand unlabeled molecules provides the simultaneous determina-tion of a point of the isotherm and its tangent. The retention timeof the injected molecules, the invisible tracer peak (also calledthe mass peak [19]), can be used for adsorption isotherm deter-mination, a method which is called the tracer pulse method. Itwas used in single-component gas chromatography with radio-metric [20] and mass spectrometric [21] detection of an isotopictracer. The PM theory was extended to encompass the multicom-ponent case and was recently validated for the determination ofthe adsorption isotherm parameters from binary mixtures [22].When the excess sample is injected, the adsorption equilibriumis disturbed. Plateau molecules are displaced from the station-ary phase, and a perturbation wave propagates quickly along thecolumn. The injected molecules migrate at a lower velocity thanthis wave and coelute with a plateau deficiency zone, which isnot detected. The phenomenon was called a paradox [18] sinceit was unexpected that two zones containing identical moleculescould have different velocities. It was first studied systematicallyby Samuelsson [19]. Arnell [23] and Samuelsson [24] validatedthe use of the tracer-pulse method for the determination of real,nonparametric, multicomponent adsorption isotherms in liquidchromatography.

In the inverse method (IM), an optimization algorithm permitsthe determination of the best values of the coefficients of isothermequations that minimize the differences between the measuredprofile of a high-concentration band and the profiles calculatedfrom these isotherms and the best value of the column efficiency.Using a satisfactory isotherm equation and a robust minimizationalgorithm permit the determination of the best set of isothermparameters [25–27]. With this method, an adsorption isothermmodel is needed a priori while in the other methods (e.g., FA, FACP,ECP, or PM) the isotherm data points are obtained directly from theexperimental data and model fitting is done only afterward [26].This is the main source of difficulty with this heavy computationalapproach. Kaczmarski [28] lately studied the possible problemsregarding IM. He compared the results of isotherm parameter esti-mation obtained with the inverse method when the ED model wassolved with either the Rouchon or the OCFE scheme. He provedthat the optimal separation conditions calculated with the Rou-chon method can differ significantly from those derived with theOCFE method. The accuracy of the Rouchon method decreases withincreasing number of analyzed components, due to errors made inthe approximation of the apparent axial dispersion by numericaldiffusion. This means that this method may not give most reliableestimates of parameters for multi-component isotherm, even forvery efficient columns. This can also lead to errors made in theadsorption mechanism. The advantage of the method is its simplic-ity, the short calculation times, and the small amount of chemicalsneeded. Thus, it can be used for preliminary studies of the adsorp-

tion mechanism or for the determination of the influence of thetemperature, the mobile phase composition, or the pressure on theparameters of an isotherm model since the nature of the isothermfunction will not be affected by moderate changes of these param-eters.

omato

ioerbaadc

melsptiTmetaioeita

cpcmcdodstImtppToitbttsam

ifiet[taoa

A. Andrzejewska et al. / J. Chr

The information obtainable from instrumental signals is alwaysnfluenced by the signal noise. Noise has a significant impactn the accuracy and the precision of analytical results. Thus, annhancement of the signal-to-noise ratio is an important step toeduce the uncertainty of the analytical data. Noise can be causedy: thermal fluctuations or by the quantum nature of light, matter,nd charges, minor temperature changes, electromagnetic radi-tions, frequency and harmonics of the electric power, etc. Theisturbance caused by these noise sources can be minimized butannot be eliminated totally [29].

Fluctuations of the experimental conditions during a series ofeasurements causes fluctuations in the signals recorded, hence

rrors in the isotherm parameters. The distribution of the ana-yte between the stationary and the mobile phases depends on thetationary phase selected, the mobile phase composition, the tem-erature and, possibly the flow velocity. The latter may influencehe data in two different ways. First, a systematic error can be madef the isotherm data are measured before equilibrium is reached.he use of the integration of the FA breakthrough curve to deter-ine the amount of solute adsorbed should eliminate this source of

rror. Also, flow rate fluctuations are associated with fluctuations ofhe pressure gradient along the column. Although this effect is usu-lly minor, there are cases in which the pressure dependence of thesotherm is sufficient to cause additional errors [30]. There are a fewther sources of errors, arising not from the limited stability of thexperimental conditions but from measurement errors made on thentermediate data needed to calculate the isotherm data points, e.g.,he data used in FA to calculate the mass of adsorbed componentnd its concentration in the adsorbed phase.

In this work, we evaluated the results obtained with the threelassical chromatographic methods of measurement of single com-onent isotherm data: frontal analysis (FA), frontal analysis byharacteristic point (FACP), and the pulse method (PM) and deter-ined the errors made. The aim of this work was to check and

ompare the precision of these three methods. The following proce-ure was applied. In a first step, chromatographic signals like thosebtained in FA, FACP, and PM were generated using the equilibrium-ispersive model of chromatography, as explained later. Theseignals were handled like conventional experimental data and usedo calculate the isotherm data points obtained with each method.n the next step, these data points were fitted to a series of isotherm

odels (Langmuir, bi-Langmuir, or Moreau) and used to selecthe best model and to estimate the best values of these isothermarameters. Finally, the isotherm parameters obtained were com-ared to the initial ones, those used in the ED model calculations.he chromatographic methods studied generated two major typesf errors: (1) the method errors taking place when deriving thesotherm data points from the band profiles; (2) the model errorsaking place during the determination of the best model parametersy fitting the measured isotherm data to the different adsorp-ion isotherm equations. The values of the errors generated inhe reconstruction of the isotherm equations inform on the pos-ible errors made during the analysis of the actual experimentaldsorption processes when implementing the FA, FACP, and PMethods.The contributions of these sources of errors add to another,

mportant error made in the determination of the best isothermor a given compound in a chromatographic system, the error maden the determination of the hold-up time of the column used. Thisrror may affect the numerical estimates of the model parame-ers [31] or even the very choice of the adsorption isotherm model

32]. Samuelsson et al. [32] showed that a small underestimate ofhe column hold-up time may lead to assume wrongly that thedsorption of the studied compound is heterogeneous while a slightverestimate may lead to the incorrect assumption of multi-layerdsorption.

gr. A 1216 (2009) 1067–1083 1069

2. Theory

This study applies a principle that is somewhat similar to theone used in earlier investigations [13]. It simulates the acquisi-tion process of experimental data points by calculating the exactprofiles of a series of breakthrough curves of a hypothetical com-pound, using the basic principle of chromatography and a suitableisotherm model [33]. Breakthrough curves were generated usingthe equilibrium-dispersive model and were calculated with anorthogonal collocation on finite element algorithm (OCFE) [33–36]for the three adsorption models used. These calculations were per-formed as explained earlier in [34]. The set of differential equationsobtained from the partial differential equation of the ED model afterapplication of the OCFE method was solved with the VODE solver,with a relative error of 1 × 10−6 and an absolute error of 1 × 10−8.The algorithm for no stiff equation was used. The number of sub-domains considered in the OCFE method was set in such a way thatthere was no visible oscillation on the concentration profiles. Thenumber of collocation points for each subdomain was equal to 3.Each set of calculations with specific adsorption isotherm parame-ters was performed for four different column efficiencies (N = 500,1000, 2000, and 10,000). Then, the calculated curves are used in thesame fashion as conventional experimental records and the proce-dures traditionally used in chromatography to derive isotherm datapoints are applied to them. Finally, the best isotherm was selectedon a statistical basis and the best-fit parameters of the adsorptionisotherm equation were derived.

This section covers the method used to calculate the break-through profiles and the procedures used to derive the data pointsfrom the breakthrough profiles.

2.1. Calculation of the profiles breakthrough curves

Several mathematical models could be used to describe themigration of a sample band along a chromatographic column[33,37,38]. The simplest one is the equilibrium-dispersive (ED)model. This model is based on the ideal model of chromatographybut it takes the finite column efficiency into account by introduc-ing in the calculations a numerical dispersion that simulates theactual band dispersion due to the combination of axial dispersionand mass transfer resistance [33]. Accordingly, in the ED model ofchromatography, the differential mass balance equation is written:

∂c

∂t+ F

∂q

∂t+ u

∂c

∂z= Da

∂2c

∂z2(1)

where F is the phase ratio (volume of the stationary phase dividedby the volume of the mobile phase or F = (1 − �)/�, with � the totalporosity of the column), u is the linear mobile phase velocity, c and qare the solute concentrations in the mobile and the stationary phaseat position z in the column and time t. The surface concentration,q, is related to the mobile phase concentration by the adsorptionisotherm equation (see Section 2.2). The apparent axial dispersioncoefficient, Da, is related to the column efficiency by the followingequation:

Da = HL

2t0= uL

2N(2)

where H and N are the height equivalent to a theoretical plate(HEPT) and the number of theoretical plates in the column, respec-tively, L is the column length, and t0 is the hold-up time (t0 = L/u).In accordance with the ED model assumptions, the apparent axial

dispersion coefficient does not depend on the concentration of thecompound examined.

The initial and boundary conditions used to integrate Eq. (1)correspond to a column that initially contains only the pure mobilephase (no solute, c(t = 0, z) = 0), and a step injection of a solution

1 omato

oi

c

Ic

u

Ta

Tb(

2

eichaiattit

q

wreppu

eumseiiiaobclcce

q

wacc

070 A. Andrzejewska et al. / J. Chr

f concentration c0 at the inlet of the column. The initial conditions

(0, z) = 0 (3)

n this work the Danckwerts boundary condition was used. At theolumn inlet (t > 0 and z = 0) it is defined as

c − Da∂c

∂z= uc0 (4)

he boundary condition at the column outlet (t > 0 and z = L) isssumed as follows:

∂c

∂z= 0 (5)

he ED differential mass balance equation was solved numerically,y mean of an orthogonal collocation on finite element algorithmOCFE) [33–36].

.2. Single-component adsorption equilibrium isotherm models

Adsorption isotherms are the critical physicochemical prop-rty that accounts for the liquid–solid equilibria of any compoundn HPLC. Isotherms can be broadly sorted out into three kinds,orresponding respectively to an ideal, homogeneous surface, toeterogeneous surfaces on which adsorbate molecules do not inter-ct, and to homogeneous surfaces on which adsorbate moleculesnteract. To investigate the influence of the type of isotherm on theccuracy of the measurements while limiting the importance ofhe investigation, we selected three different isotherm equations,he Langmuir isotherm, the bi-Langmuir isotherm and the Moreausotherm, which are simple models corresponding to these threeypes of adsorption behavior, respectively.

The Langmuir isotherm equation is

= qsbc

1 + bc(6)

here qs is the monolayer saturation capacity and b is the equilib-ium constant at infinite dilution, which is related to the adsorptionnergy, �a, through the relationship b = b0e(�a/RT), where b0 is there-exponential factor that can be derived from the molecularartition functions in the bulk and the adsorbed phases, R is theniversal gas constant, T is the absolute temperature.

However, the surface of actual adsorbents are energetically het-rogeneous. Particularly, adsorption data measured on adsorbentssed in HPLC are usually not well accounted for by the Lang-uir model. Often in RPLC, two or even more kinds of adsorption

ites are observed and generally these sites behave independently,xhibiting different chemistry, corresponding to different chem-cal environment for the adsorbed molecules. One type of sitess hydrophobic and related to the bonded alkyl chains, anothers hydrophilic and related to residual silanol groups [39–42]. Thedsorption of enantiomers onto enantioselective stationary phasesften exhibits also two kinds of different adsorption sites, one typeeing non-selective, the other enantioselective [43–48]. In theseases, the Langmuir isotherm seems to account well for the equi-ibrium behavior of the solutes involved on each type of sites. In allases, the total amount adsorbed is equal to the sum of the amountsorresponding to each term of the bi- or multi-Langmuir isothermquation:

= qs1b1c + qs2

b2c(7)

1 + b1c 1 + b2c

here the subscripts 1 and 2 correspond to the concentrationsdsorbed on the two different types of adsorption sites. In mostases, the coefficient b1 of one type of site is larger than b2 and theoefficient qs1 smaller than qs2. The monolayer capacity of the most

gr. A 1216 (2009) 1067–1083

abundant type of sites is much larger than that of the less abundantones and the equilibrium constant on these sites lower.

Another important reason why the adsorption behavior ofsolutes deviates from Langmuir isotherm behavior is that theadsorbed molecules may interact. Interactions between adsor-bate molecules can be accounted for using one of many differentisotherm models. We selected for this purpose the Moreau isothermmodel. This is the simplest isotherm model for homogeneous adsor-bent surface on which adsorbate–adsorbate interactions take place[49]. The isotherm equation is

q = qsbc + Ib2c2

1 + 2bc + Ib2c2(8)

where qs, b, and I are the monolayer saturation capacity, the equi-librium constant at infinite dilution, and the adsorbate–adsorbateinteraction parameter, respectively. The Moreau model was suc-cessfully used to account for the adsorption behavior of propranololfrom a water–methanol solution onto C18 bonded silica surfaces[50].

2.3. Calculation of the amounts adsorbed

Three methods were used to derive the equilibrium constants,the frontal analysis (FA), the frontal analysis by characteristic points(FACP), and the perturbation (PM) method.

2.3.1. Frontal analysisThe breakthrough curves that are recorded during the FA exper-

iments provide the values of the stationary phase concentration, q,in equilibrium with the inlet concentration, c0. It can be estimatedusing one of three different procedures: from the retention time ofthe inflection point of the curve, from that of the half-height of theplateau (the middle point), or with the equal area method. Thesevalues are derived from the retention time of the breakthroughcurve:

q = c0

F· tR − t0

t0(9)

The retention time, tR, of a breakthrough curve is the importantparameter in the above equation. It can be estimated using one ofthree different procedures. In the present study, these three meth-ods were used, evaluated, and compared.

2.3.2. Frontal analysis by characteristic pointsIn FACP, the amount adsorbed at equilibrium is derived from the

profile of the rear boundary of the breakthrough curve recorded,during the return to initial conditions (i.e., to c(t) = 0 in the feed)for the most concentrated solution used in FA. The stationary phaseconcentration is calculated by integration of the area under the partof the breakthrough curve from the point corresponding to the con-centration c in the mobile phase until the end of the curve (c = 0 inthe eluent):

q(c) = 1F

∫ c

0

t − timp − t0

t0dc (10)

where timp is the time of injection. Eq. (10) is based on the idealmodel that assumes that the column efficiency is infinite. The influ-ence of the actual column efficiency on the profile of the rearboundary of the breakthrough curve is the main disadvantage of theFACP method. To investigate the influence of the column efficiencyon the adsorption parameters, the calculations were performed for

four different values of the column efficiency.

2.3.3. Perturbation methodThe measurements of the velocity and the retention time of the

perturbation peak permit the derivation of the slope of the isotherm

omato

a

u

Tetcmw

2

upeitFwtafaMiDgemi

2

pto

mbieaccsacwppdfimb

1vompd

i

equation (qs1 = 25 mM, b1 = 0.2 mM−1). For the second type of

A. Andrzejewska et al. / J. Chr

t the plateau concentration, through the equations:

z = u

1 + F(dq/dc)|c=c0

, tR(c0) = t0

(1 + F

dq

dc|c=c0

)(11)

he set of the retention times of pulses measured with a columnquilibrated with solutions of progressively increasing concentra-ions allows the calculation of the isotherm slopes at these differentoncentrations. This set of retention times constitutes the funda-ental data set needed for determining the adsorption isotherm,hich is obtained by numerical integration of dq/dc vs. c.

.4. Scatchard data plot

Scatchard plots are plots of q/c versus q [51]. These plots aresed as a first check of the nature of a new isotherm because theyrovide a rapid, easy selection among the possible isotherm mod-ls and may validate the isotherm equation used [42]. When thesotherm data follow the Langmuir model, the Scatchard plot ofhe data is a straight line with a negative slope (q/c = qsb − bq).or a bi-Langmuir model, the Scatchard plot has two asymptotesith negative slopes, corresponding to the two Langmuir terms of

he equation, and it is convex downward. This shape characterizesdsorption models that take the heterogeneity of the adsorbent sur-ace into account. A convex upward Scatchard plot corresponds tonon-ideal homogeneous adsorption isotherm equation, like theoreau model. As will be shown later, the shape of Scatchard plot

s only one of the factors that characterize the adsorption isotherm.epending on the actual column efficiency and on the chromato-raphic method used to measure the adsorption data, importantrrors could be made in the choice of the particular adsorptionodel by relying only on the Scatchard plot, which might lead to

ncorrect explanations of the adsorption mechanism.

.5. Determination of the adsorption parameters

Once the isotherm data have been determined, the isothermarameters are estimated by fitting these data to the three adsorp-ion models selected in this work: the Langmuir, the bi-Langmuir,r the Moreau model.

For this fitting procedure, a non-linear regression analysis of theodels was used. The fitting was done by mean of ORIGIN software

ased on the Levenberg–Marquard’s algorithm [52], which min-mizes the sum-of-squares of the vertical distances between thexperimental data and the model curve. It is the most widely usedlgorithm in nonlinear least squares fitting. Points further from theurve contribute more to the sum-of-squares. Points close to theurve contribute little, which is expected from the nature of thecatter of the experimental data. In many experimental cases, theverage absolute value of the distance of the data points from theurve corresponding to the true isotherm is expected to be largerhen the fitted function is also larger. The high concentration dataoints, which have the largest value for their absolute scatter willrovide the larger contribution to the sum-of-squares and thusominate the results of the calculations of the best isotherm coef-cients. To restore an equal weighting to all the data points, oneay rather choose an appropriate weighing method, as described

elow.The weighting method most often used consists in weighing by

/Y2 (making W = 1/Y2 the weight of each data point). It is con-enient to think of this method as minimizing the sum-of-squaresf the relative distances of the data from the theoretical curve. This

ethod is appropriate when a constant relative error of the data

oints is expected. Minimizing the sum of the squares of the relativeistances restores equal weighing to all data points.

Weighing by 1/Y is, admittedly, a compromise between min-mizing the actual sum-of-distances squared and minimizing the

gr. A 1216 (2009) 1067–1083 1071

relative sum-of-distances squared. A particular case in which the1/Y weighing is appropriate is when the Y values follow a Poissondistribution, in which case, the standard error made on a value isequal to the square root of that value. Therefore, dividing the dis-tance between the data point and the curve by its square root, andsquaring the result gives the correct weighing. Often, however, non-linear regression is carried out without weighing. In this work, wefollowed the two procedures in order to illustrate how the influ-ence of weighting can affect the results of the fitting and the finalconclusions.

2.6. Fisher parameter

In order to decide which of several possible models is thebest isotherm model or, as we did in this work, to compare theperformance of the several methods of adsorption isotherm deter-mination, the statistical Fisher parameter, Fp, is useful [53]:

Fp = ND − P

ND − 1·

ND∑i=1

(q′exp′,i − q′exp′ )2

ND∑i=1

(q′exp′,i − qtrue,i)2

(12)

In this equation, ND stands for the number of data points and Pfor the number of model parameters. The subscript ′exp′,i denotesthe experimental data (i.e., in this study, the data derived by theparticular chromatographic method selected) of the solid phaseconcentration of the adsorbate in equilibrium with a given liquidphase concentration, ci, q′exp′,i is the mean value of the adsorptiondata, the subscript true, i represents the true adsorption isothermdata, which is derived from the equation of the isotherm modelselected (the model with the parameters assumed in advance,which served to derive the band profiles). A higher value of Fp

suggests a better method and/or a better fit to the experimentaldata. Fp increases when (ND − P) rewards a method with more datepoints, �(q′exp′,i − qtrue,i)

2 decreases with increasing goodness of

the fit and �(q′exp′,i − q′exp′,i)2 increases with increasing width of

the data range. The Fisher parameter assumes that the residuals arenormally distributed. To determine if a method or model providesdata that are statistically more accurate than another, the ratio ofthe two Fisher parameters corresponding to the data obtained withthis method or model is calculated and compared to critical F-ratiosfound in most statistical books.

3. Results and discussion

3.1. Calculation of band profiles

Breakthrough curves were calculated with the ED model asexplained above, using three adsorption models, the Langmuir, thebi-Langmuir, and the Moreau models. The ‘true’ values of the modelparameters were assumed as follows:

1. For the Langmuir model, the monolayer capacity is qs = 25 mMand the equilibrium constant, b = 0.2 mM−1,

2. For the bi-Langmuir model, the isotherm parameters for theadsorption sites of type 1 were taken the same as in the Langmuir

adsorption sites, qs2 = 5 mM and b2 = 1 mM−1,3. For the Moreau model, the monolayer capacity and the

equilibrium constant remain the same as in the Langmuircase (qs = 25 mM, b = 0.2 mM−1); the interaction parameter isI = 1.35

1072 A. Andrzejewska et al. / J. Chromatogr. A 1216 (2009) 1067–1083

Table 1Experimental system parameters.

L = 10.00 cm The column lengthI.D. = 4.00 mm The inner diameter of the columnFv = 1.00 ml/min The flow ratet0 = 60 s The hold-up timeu0 = 0.1326 cm/s The superficial velocityεFc

i(e

tIwTi

Fac1s

= 0.7958 The total porosity= 0.2566 The phase ratio

stock = 0.1–100 mM The concentration range of stock solutions

Each set of calculations was made with these adsorptionsotherm parameters and for four different column efficienciesN = 500, 1000, 2000, and 10,000). The experimental system param-ters are listed in Table 1[50].

The injection time (the impulse time), timp, was taken as 50 s forhe Langmuir model and 60 s for the two other adsorption models.

n the PM method, the very short impulse of pure mobile phaseas assumed to last 0.3 s (injection of 5 �L of pure mobile phase).

ypical examples of calculated FA breakthrough curves are shownn Fig. 1. These profiles were calculated using the Langmuir model

ig. 1. Typical profiles of frontal analysis. Profiles calculated with the Langmuirdsorption isotherm and the ED model (qs = 25 mM, b = 0.2 mM−1). Column effi-iency, N = 2000. Concentration range of the stock solutions is (a) from 0.1 mM tomM and (b) from 1 mM to 10 mM. Other parameters used in the calculations are

hown in Table 1.

Fig. 2. Typical normalized profiles obtained with the pulse method. Langmuir

adsorption isotherm and ED model (qs = 25 mM, b = 0.2 mM−1) were used forthe profile calculations. Column efficiency, N = 10,000. Concentration range of thestock solutions, from 0.1 mM to 100 mM; volume of mobile phase injected, 5 �L(timp = 0.3 s). Other parameters used in the calculations are shown in Table 1.

for a column efficiency, N = 2000, with concentrations of the stocksolutions ranging from 0.1 to 1 mM (Fig. 1a) and from 1 mM to10 mM (Fig. 1b). As expected, the breakthrough curves calculatedfor the lowest concentrations are symmetrical, contrasting with theprofiles calculated for highly concentrated solutions. The more con-centrated the solution, the closer the shock front is to the hold-uptime.

Fig. 2 shows typical examples of PM band profiles, calculated alsowith the Langmuir isotherm model, but with a number of theoreti-cal plates equal to N = 10,000. The perturbations were obtained byinjection of 5 �L of pure mobile phase into a column equilibratedwith a solution of a given concentration. This injection disturbsthe equilibrium conditions, generating a perturbation wave thatmigrates along the column. When this wave reaches the columnoutlet, a negative peak is registered. The wave velocity depends onthe stock concentration and so does the elution time of the pertur-bation. With the convex upward isotherms used here, the higherthe concentration of the stock solution, the shorter is the retentiontime of the perturbation.

Fig. 3 a–c shows FA breakthrough curves calculated for the Lang-muir, the bi-Langmuir, and the Moreau isotherm equations and forthe four column efficiencies selected, respectively. The return fromplateau to baseline, also shown in the figures, is used to calculatethe FACP data. These profiles were calculated assuming that theconcentration of the stock solution was 5 mM in all cases. The dif-ferences between the shapes of the three profiles show that whenthe adsorbent surface is heterogeneous (bi-Langmuir isotherm),the profile exhibit a more significant tailing effect than when it ishomogeneous. This is due to the presence of the adsorption sitesof the second type. Ravald et al. [50] came to the same conclusionin their similar study of equilibrium isotherms derived from calcu-lated ECP band profiles. In all the calculated profiles in the Fig. 3a–c,one should also notice the influence of the column efficiency. Thedata points that are close to the top of the profiles and those closeto the baseline are more strongly affected by the different sourcesof band broadening than those that are in the middle.

Similarly to the FA and FACP generated profiles, the PM signalswere calculated using the same adsorption models and for the samefour values of the column efficiencies. Fig. 4 illustrates the depen-dencies of the pulse retention time on the column efficiency andon the pulse size. In the main figure, 5 �L pulses of the pure mobile

A. Andrzejewska et al. / J. Chromatogr. A 1216 (2009) 1067–1083 1073

Fig. 3. FA breakthrough curves for different column efficiencies, demonstrating thedifference between (a) the Langmuir, (b) the bi-Langmuir, and (c) the Moreau adsorp-tion model used. The concentration of the solution injected is the same in all casesand equal to c = 5 mM. The other parameters used to calculate the curves are col-lected in Table 1.

Fig. 4. PM breakthrough curves for the Moreau adsorption model, demonstratingthe difference in shapes and retention times, depending on the column efficiencies.The concentration of the stock solution is the same in all cases and equal to cstock =

10 mM. Insert: Dependence of the retention times on the injection size: (1) 1 �Linjection of pure mobile phase, (2) 2 �L injection of pure mobile phase, (3) 3 �Linjection, (4) 4 �L injection, and (5) 5 �L injection. Column efficiency, N = 10,000.Other parameters used to calculate the curves are collected in Table 1.

phase were injected on columns equilibrated with a 10 mM stocksolution. There was no significant difference between the behav-iors of the curves derived for the three different adsorption models.Therefore, the Moreau isotherm was chosen to draw the figure.The elution peak becomes less symmetrical and the pulse reten-tion time increases with increasing column efficiency. The retentiontime also increases with increasing pulse size, as illustrated in theinsert graph of Fig. 4, which shows the pulse profiles for N = 10,000and the pulse volume increasing from 1 to 5 �L. At high column effi-ciencies, peaks are narrower, hence taller. The retention behaviorat high concentrations is not linear, which is why the peak shapeis not Gaussian and pulse retention times increase with increasingpulse size (hence also, with increasing column efficiency). There-fore, a significant error would be made if the retention times of toolarge pulses were to be measured. We chose an injection volumeVimp = 5 �L in order to be as close to experimental conditions aspossible.

3.2. Calculation of experimental isotherm data points from thecalculated band profiles

Most often, experimentalists using FA to measure isotherm datapoints make the mistake of collecting an insufficient number ofdata points. To avoid this mistake, we calculated 32 different break-through curves and used both FA and PM to calculate the adsorptiondata for every column efficiency and adsorption model used. Thestock solution concentration was increased by 0.1 mM in the range0.1–1 mM (10 profiles), by 1 mM in range 2–10 mM (9 profiles),by 5 mM in range 15–50 mM (8 profiles), and by 10 mM in range60–100 mM (5 profiles). Each FA or PM breakthrough curve givesone isotherm point.

In FACP, the amount adsorbed at equilibrium is derived fromthe profile of the rear boundary of the most concentrated solu-tion used in FA, namely cstock = 100 mM. Since the integration ofthe rear profile must be made starting from c = 0 to c, the inflec-

tion point of this part of curve was used as the upper boundary ofthe integration. Because the influence of the column efficiency issignificant at higher concentrations, this decision avoids the inclu-sion in the acquired set of isotherm data points that would be too

1074 A. Andrzejewska et al. / J. Chromato

Fig. 5. FA isotherms calculated with the Langmuir case with three different meth-ods: (a) equal area, (b) middle point, and (c) inflection point. The dashed linerepresents the true Langmuir isotherm (qs = 25 mM, b = 0.2 mM−1). The calcula-tions were carried for four column efficiencies: N = 500, 1000, 2000, and 10,000.

gr. A 1216 (2009) 1067–1083

erroneous. For the high concentrations points, the apparent adsorp-tion constant decreases with increasing bulk concentration. Wegeneralize the end of integration which is now the same for dif-ferent efficiencies.

Fig. 5 compares the experimental isotherm data points that werederived from the FA breakthrough curves using the three differentdefinitions of the retention time of these curves: the equal areamethod (Fig. 5a), the middle point method (Fig. 5b), and the inflec-tion point method (Fig. 5c). This figure shows isotherms that werederived from the FA breakthrough profiles calculated with the Lang-muir isotherm. The same general trend of the data behavior wasobserved in the case of the other two adsorption isotherm models.Obviously, the inflection point method gives the largest error forsmall column efficiencies while the best results, at first glance, areobtained with the equal area method in the whole efficiency range.As discussed in the next section, for efficiency larger than 2000, the

middle point and inflection point methods give smaller errors. Inthis case these methods should be used.

This type of error is called ‘a method error’. Fig. 5 shows also thatthe distance between the true and the best isotherm derived from

Fig. 6. Comparison of the experimental isotherms generated by the FACP and PMmethods (a) FACP isotherms calculated for a bi-Langmuir adsorption model. Thedashed line represents the true bi-Langmuir isotherm (qs1 = 25 mM, b1 = 0.2 mM−1,qs2 = 5 mM, b2 = 1 mM−1), (b) PM isotherms calculated for the Moreau adsorp-tion model. The dashed line represents the true Moreau isotherm (qs = 25 mM,b = 0.2 mM−1, I = 1.35). The calculations were carried for four column efficiencies:N = 500, 1000, 2000, and 10,000.

A. Andrzejewska et al. / J. Chromatogr. A 1216 (2009) 1067–1083 1075

Table 2Fisher parameter table.

Efficiency Model

FA FACP PM

Method: EqAr Method: MidP Method: InflP

Langmuir N = 500 3,511.04 203.29 23.48 26.43 472.46N = 1000 4,459.87 768.65 100.69 64.44 152.89N = 2000 17,130.42 3,225.74 451.87 217.20 70.19N = 10,000 14,397.98 163,531.50 32,102.79 2,413.71 19.19

Bi-Langmuir N = 500 4,453.34 362.21 43.51 37.45 259.27N = 1000 6,294.24 1,449.84 173.63 108.96 157.41N = 2000 6,883.07 6,390.96 952.98 412.74 78.28N = 10,000 13,628.01 834,033.98 38,326.16 5,569.69 22.32

M 213816

3,46278,061

Fpi[

tm

Fc

oreau N = 500 3,024.81N = 1000 4,349.99N = 2000 7,958.59N = 10,000 8,720.73 2

A measurements increases with decreasing number of theoreticallates. The true isotherm, in this case the Langmuir isotherm, is

ntermediate between the different experimental ones. Sajonz et al.45] came to the same findings.

In contrast, all experimental FACP isotherms, independently ofhe column efficiency assumed and of the adsorption isotherm

odel used, are overestimated compared to the true isotherm

ig. 7. Dependence of the FA method error (100%(qFA − qtrue)/qtrue) on the mobile phaseolumn efficiencies (a) N = 500, (b) N = 1000, (c) N = 2000, and (d) N = 10,000.

.90 26.83 34.70 510.39

.76 113.42 111.25 147.75

.32 485.86 345.41 69.38

.91 25,891.13 3,681.08 18.37

(Fig. 6a). The FACP adsorption data points presented in Fig. 6a werecalculated for the case of a heterogeneous surface (bi-Langmuir

isotherm model) but similar results were also found for the othertwo adsorption isotherm models (Langmuir and Moreau).

It is worth noting, among the significant differences betweenthe results in Figs. 5 and 6, that the sequences of the FA and FACPisotherms (Fig. 6a) are the same but are also the opposite of the

concentration for a homogeneous surface (Langmuir model) calculated for four

1 omatogr. A 1216 (2009) 1067–1083

suiwL

3

ftbhftIltPetslr

3

tmtbtwtttsrimemtrmN

PeeaeFaadmeio

pnew

076 A. Andrzejewska et al. / J. Chr

equence of the PM isotherms (Fig. 6b). The more efficient the col-mn, the larger the error made with the PM method, due to the

nfluence of the size of the injected pulse. Similar observationsere noticed in the case of the other two adsorption models, the

angmuir and the bi-Langmuir isotherms.

.3. Fisher parameter

The statistical Fisher parameter, Fp, is used to evaluate the dif-erent methods of adsorption data determination [53]. Accordingo the Fisher test, the method that gives experimental data thatetter correlate with the true isotherm is the one that exhibits theighest F-value (see data in Table 2). Comparing these data, we see,

or example, that the higher the number of theoretical plates ofhe column, the larger is the Fisher parameter for FA or for FACP.n contrast, the results are opposite with PM, in which case thearger the column efficiency, the smaller the Fp factor. This meanshat increasing the column efficiency decreases the accuracy of theM method. This paradoxical result is easily explained by the influ-nce of the pulse volume injected on the method accuracy. Amonghe methods investigated, the most accurate one is frontal analy-is using the equal area retention time definition and a moderate orow efficiency column. For highly efficient column the middle pointetention time definition seems to be better.

.4. Calculation of the method error

Fig. 7 compares the method errors made in FA as a function ofhe higher concentration used, when the amount adsorbed is esti-

ated using one of the three definitions of the retention time. Allhe results shown in Fig. 7 are calculated for the Langmuir modelut they do not differ qualitatively much when the bi-Langmuir orhe Moreau isotherm are used. As noted earlier, this error decreasesith increasing column efficiency [45]. The results obtained with

he equal area-based method gives results which are very close tohe true ones while the largest error is made when using the reten-ion time of the inflection point. Finally, the method error dependstrongly on the largest value of the mobile phase concentrationeached. It increases with increasing the compound concentrationn the solution. At low-concentrations (inserts) the middle point

ethod gives more accurate results at the highest two columnfficiencies. In the case of a 10 000 theoretical plates column, theiddle point method gives best results in the whole concentra-

ion range. This suggests that, in order to achieve the most accurateesults, one should better use this particular method for the esti-ation of the retention times of the FA breakthrough curves whenreaches or exceeds 10,000 (Fig. 7d).Fig. 8 shows the method errors made with FACP (Fig. 8a) and

M (Fig. 8b) when using the bi-Langmuir and the Moreau isothermquations, respectively. Similarly to FA, FACP generates smallerrrors at high column efficiencies, in contrast to the PM method,s explained earlier. The figure shows also the dependence of theserrors on the solute concentration. The profiles of the curves inigs. 7 and 8a and b are very different. The FACP method error hasminimum and a maximum while the similar PM curves reachplateau. There is not much difference between the error curveserived with a given method nor with a given chromatographicethod for the various adsorption models. This means that a mod-

rate degree of surface heterogeneity of the adsorbent or smallnteractions between adsorbate molecules does not affect the sizef the method error made.

In summary, the method error strongly depends on the mobilehase concentration, on the column efficiency, and on the tech-ique used for the estimation of the amounts adsorbed atquilibrium. The general trend is that all FA methods give some-hat underestimated adsorption data, in contrast with FACP. With

Fig. 8. Dependence of (a) the FACP and (b) the PM method error on the mobile phaseconcentration for the bi-Langmuir and the Moreau model, respectively, calculatedfor four column efficiencies N = 500, 1000, 2000, and 10,000.

both methods, the higher the column efficiency, the smaller theerror. The conclusion regarding PM is opposite. With this methodthe maximum deviation of the pulse signal is significant comparedto the plateau concentration, leading to nonlinear behavior of theisotherm. This explains the surprising apparent influence of the col-umn efficiency on the accuracy. In all cases showed so far, there isnot much difference between the magnitude of the method errorsobserved with the three different adsorption models studied.

3.5. Estimation of the best-fit Langmuir, bi-Langmuir and Moreauisotherm parameters from experimental data points

Having acquired isotherms data points, the next step in the studyof an adsorption or retention mechanism is usually the determina-tion of the best isotherm model, including the calculation of itsparameters. As explained earlier, the Scatchard plots [51] may beused as a first step for the selection of one or a few isothermsamong the many possible models [42]. Fig. 9 shows the exper-imental isotherm data points derived from the three adsorption

models selected, using the three different chromatographic tech-niques chosen. The Scatchard plots of these data are given as inserts.The column efficiency was assumed to be 2000. For the Lang-muir isotherms (Fig. 9a), the Scatchard plots should theoreticallybe straight lines, but only the one obtained in the case of FA used

A. Andrzejewska et al. / J. Chromatogr. A 1216 (2009) 1067–1083 1077

Fig. 9. Comparison of the experimental isotherms generated by the FA, FACP and PMmethods for a column efficiency with N = 2000, for (a) the Langmuir model (qs =25 mM, b = 0.2 mM−1), (b) the bi-Langmuir model (qs1 = 25 mM, b1 = 0.2 mM−1,qs2 = 5 mM, b2 = 1 mM−1) and (c) the Moreau model (qs = 25 mM, b = 0.2 mM−1,I = 1.35). The dashed line represents the true isotherm. Inserts show Scatchard plots.

Fig. 10. Dependence of (a) the FA equal area, (b) the FACP and (c) the PM model erroron the mobile phase concentration for a homogeneous surface (Langmuir), a het-erogeneous surface (bi-Langmuir) and the Moreau isotherm models, respectively,calculated for four column efficiencies N = 500, 1000, 2000, and 10,000. W = 1/Y2.The inserts in (b) and (c) represent the model errors calculated for different weight-ing method for N = 10,000.

1 omato

wprsmwieomF

cmipemotmto

�

wppmnpwwt

mpsrc

ctm(ttfitqtswaldrctp

F

078 A. Andrzejewska et al. / J. Chr

ith the equal area method is really linear. This result confirms ourrevious assertion that this particular method is the most accu-ate. Scatchard plot obtained with the FACP data applied to theame breakthrough curves (still Fig. 9a) suggests that a bi-Langmuirodel should be considered. The same plot made with the PM dataould rather point to a Moreau isotherm. Both choices would be

ncorrect because a Langmuir isotherm was used to calculate thexperimental signals. Similar errors would be made on the basisf the Scatchard plots obtained from the other two adsorptionodels (see bi-Langmuir isotherm, Fig. 9b and Moreau isotherm,

ig. 9c).Because we know what was the isotherm equation used to cal-

ulate the experimental points, we used exactly the same adsorptionodel to fit the data. This advantage of hindsight is not available

n practice when we need to determine the proper isotherm in aractical case. In the present case, the isotherm parameters arestimated by fitting the adsorption data to the correct adsorptionodel. For the fitting procedure, a non-linear regression analysis

f the models was used. As mentioned in previous section, the fit-ing was based on the Levenberg–Marquard’s algorithm [52], which

inimizes the sum-of-squares of the vertical distances betweenhe experimental data and the model calculations. The followingbjective functions were minimized:

=

√√√√ 1ND − P

ND∑i=1

(q′exp′,i − qth,i

(q′exp′,i)a

)2

, a = 0, 0.5, 1 (13)

here ND and P are the number of data points and of modelarameters, respectively. q′exp′,i and qth,i are the experimental dataoints and the corresponding theoretical values calculated from theodel, respectively. The values of the exponent a in the denomi-

ator of the sum-of-squares denotes the weighing method. If thearameter a is equal to 0 that means that the regression was doneithout any weighing (W = 1). When a is equal to 0.5 or 1 theeights of the nonlinear regression are W = 1/Y or 1/Y2, respec-

ively.Since we tried to evaluate and compare various chromatographic

ethods and the most common ways of derivation of the isothermarameters, we first did the calculations with a non-linear regres-ion analysis that minimizes the residual sum of the squares of theelative differences between the experimental data and the modelalculations (W = 1/Y2).

Fig. 10 compares the differences between the adsorbed amountsalculated from the fitted isotherms, qmodel , and the experimen-al values generated by three chromatographic techniques: the FA

ethod, according to the equal area definition of the retention timeqFAEqAr

, Fig. 10a), FACP (qFACP , Fig. 10b) and PM (qPM , Fig. 10c). As inhe previous section, the errors showed in Fig. 10 are calculated inhe case of FA for the Langmuir isotherm model, in the case of FACPor the bi-Langmuir model and in the case of the PM for the Moreausotherm. The general trend exhibited by the curves is essentiallyhe same in each case for two other models. The adsorbed amounts,model , reported in Fig. 10 were calculated by fitting the data usinghe relative weighing method, W = 1/Y2. In the case of FA, veryimilar results were obtained with no weighing, W = 1, or witheighing the data by W = 1/Y . The only difference was observed

t low concentrations, in which case, the errors were larger with theater two weights. The model error of FA with equal area methodoes not exceed 1.5%, which is a very good result, while for two otheretention time definitions the error is significantly higher. For low

olumn efficiency the error is even seven times larger in the case ofhe middle point method and twenty times higher for the inflectionoint method.

The curves in Fig. 10b have more oscillations than those inig. 10a. Another significant difference is that in the fitting of the

gr. A 1216 (2009) 1067–1083

FACP or PM data, the errors made increase with increasing columnefficiency, even though the FACP experimental adsorbed amountsbecome closer and closer to those calculated with the true bi-Langmuir isotherm when the efficiency increases (Fig. 6a). Thissuggests that the nonlinear regression gives incorrect results in thiscase. Very similar conclusions were derived by Ravald et al. [14] intheir analysis of the ECP model errors. Ravald found that fitting thedata obtained with a weight 1/Y gave in general the most success-ful determination of the bi-Langmuir isotherm parameters in thelow and medium concentration ranges. This result confirms ourindependent findings (see insert in Fig. 10b representing the modelerrors calculated with different weighing methods for N = 10,000)since the processes used to derive the ECP and FACP experimen-tal surface concentrations are the same. The weight 1/Y turns outto be the best to obtain estimates of the Moreau isotherm parame-ters from the FACP data as well. For bulk concentrations higher then30 mM, however, it is even better to apply no weighing because thisgives the smallest model error in this concentration range. Actu-ally, from the insert in Fig. 10c, it seems that one should not use anyweighing method when fitting the PM adsorption data because thisprovides the smallest model error.

3.6. Comparison of the best-fit model parameters with the trueones. Accuracy of the model parameters determination

The data in Table 3 show the difference between the true andthe best-fitted isotherm parameters calculated for the four differ-ent column efficiencies and the three adsorption models and therelative errors made on the isotherm parameters. The parameterswere determined with the relative weighing method, W = 1/Y2.The worst results were obtained for the bi-Langmuir model. Thedifferences between the true parameters (assumed at the verybeginning of the calculations when determining the profiles of thebreakthrough curves) and the parameters obtained by fitting theexperimental isotherm data are as high as 300%. This means thatlarge errors could be made in the experimental estimates of theisotherm parameters of heterogeneous surfaces, particularly in thevalues of the surface capacities of the high-energy, low-surface areatype of adsorption sites and of their adsorption constant. There-fore, we need to compare the accuracy of the results given by thedifferent methods.

For the Langmuir isotherm, all methods provide reasonablyaccurate results. It is only unfortunate that Langmuir isothermsare so rarely found, particularly in RPLC. The data in Table 3 showthat FACP provides the least accurate measurements of the charac-teristics of the retention process onto heterogeneous surfaces. Forlow column efficiencies (N less than 2000), the measured valuesof the surface capacity of the sites of type 2, qs2, is almost threetimes larger than the true value; for the highest number of theoret-ical plates (N = 10,000) it is still twice larger. In compensation, themeasured adsorption constant, b2, is only half as large as it shouldbe. The main reason for this result is that the least square fittingmethod is only a local optimization method. There may occur adanger of local minimum of the objective function. This can beavoided by applying different initial parameters. For the Moreauisotherm, the pulse method gives very important errors on the val-ues of the interaction parameter, errors which do not decrease withincreasing column efficiency. Even for the other two parameters,the experimental result deviates too much from the true values tobe acceptable.

It should be emphasized that for all three adsorption models

the most accurate method of measuring the isotherm parame-ters is frontal analysis. Except for the highest column efficiency,the most accurate results are obtained when using the equalarea method to estimate the retention time of the breakthroughcurves. For a column efficiency of 10 000 theoretical plates, the

A. Andrzejewska et al. / J. Chromatogr. A 1216 (2009) 1067–1083 1079

Table 3Values of the relative errors (perror = 100%(pmodel − ptrue)/ptrue , where p = qs, b, qs1, b1, qs2, b2, I) made on the isotherm parameters derived from the difference between thetrue and the best-fit isotherm parameters estimated for four different column efficiencies and three adsorption models.

Method Model Langmuir Bi-Langmuir Moreau

qs error (%) berror (%) qs1 error (%) b1 error (%) qs2 error (%) b2 error (%) qs error (%) berror (%) Ierror (%)

N = 500EqAr −0.88 0.99 1.976 6.015 −15.94 9.92 −1.148 0.79 2.94

FA MidP −3.13 4.10 6.00 21.06 −49.45 45.64 −4.12 2.89 12.93InflP −9.28 13.08 7.63 51.68 −88.88 290.84 −10.97 7.88 38.79

FACP 12.90 −4.00 12.20 −51.13 100.82 −21.96 31.04 −15.78 −40.62PM 10.46 −34.28 8.03 −0.69 −18.26 −80.14 −4.63 −37.08 219.61

N = 1000EqAr −0.66 0.78 1.13 4.20 −10.59 6.14 −1.02 0.73 2.62

FA MidP −1.58 2.05 3.64 10.98 −28.19 19.72 −2.13 1.49 6.48InflP −4.41 5.88 1.83 29.16 −63.80 78.20 −5.62 3.99 17.87

FACP 8.97 −6.72 −29.26 −79.21 313.60 −58.08 13.72 −8.90 −15.45PM 13.94 −36.16 8.96 −3.66 −4.48 −80.73 −1.36 −38.18 201.00

N = 2000EqAr −0.40 0.44 1.27 4.26 −10.89 6.28 −0.56 0.36 1.60

FA MidP −0.76 0.98 1.94 5.39 −14.54 8.88 −1.04 0.72 3.12InflP −2.06 2.64 4.25 13.46 −33.71 25.08 −2.62 1.81 8.20

FACP 7.70 −8.48 2.85 69.76 289.02 −99.08 −5.74 −7.94 73.44PM 17.40 −38.00 11.98 −6.25 −2.10 −81.25 1.51 −39.37 191.90

N = 10,000EqAr −0.39 0.48 0.88 2.93 −7.68 4.44 −0.65 0.45 1.73

FA MidP −0.08 0.11 0.19 0.41 −1.22 0.65 −0.07 0.05 0.32InflP −0.14 0.16 0.50 1.00 −3.12 1.79 −0.17 0.12 0.50

FACP 6.73 −10.74 −6.20 50.18 2.42 −69.97 −17.38 −11.26 192.95PM 28.00 −43.11 20.75 −12.60 2.49 −82.52 10.83 −42.94 162.08

During the fitting procedure the weighting factor was W = 1/Y2.

Table 4Values of the relative errors (perror = 100%(pmodel − ptrue)/ptrue , where p = qs, b, qs1, b1, qs2, b2, I) made on the isotherm parameters derived from the difference between thetrue and the best-fit isotherm parameters, calculated for four different column efficiencies and three adsorption models.

Method Model Langmuir Bi-Langmuir Moreau

qs error (%) berror (%) qs1 error (%) b1 error (%) qs2 error (%) b2 error (%) qs error (%) berror (%) Ierror (%)

N = 500EqAr −1.1 2.2 7.0 11.8 −42.1 51.6 −1.4 −0.6 7.3

FA MidP −4.2 10.7 12.0 31.5 −81.3 367.0 −4.9 −3.8 35.6InflP −11.2 32.3 9.4 62.1 −99.4 −1523.3 −12.0 −11.1 121.3

FACP 27.2 −28.8 −4.1 −67.7 213.9 −46.6 37.5 −15.4 −54.9PM 3.3 −11.7 17.3 17.5 −79.7 −94.4 1.5 −19.6 30.5

N = 1000EqAr −0.9 1.9 4.8 8.2 −29.3 28.7 −1.2 −0.3 5.7

FA MidP −2.2 5.5 10.4 20.0 −63.6 132.2 −2.6 −2.0 17.5InflP −5.8 15.2 11.9 39.5 −88.9 800.0 −6.5 −5.0 49.7

FACP 15.9 −18.8 −19.6 −64.9 228.7 −49.8 22.9 −8.5 −43.3PM 6.8 −15.8 19.0 16.5 −53.5 −98.0 5.6 −19.8 15.9

N = 2000EqAr −0.5 0.9 5.3 8.6 −31.5 32.2 −0.7 −0.7 18.6

FA MidP −1.1 2.7 7.4 11.8 −43.1 55.1 −1.3 −1.0 8.4InflP −2.8 7.0 10.9 22.8 −68.4 164.1 −3.2 −2.7 22.6

FACP 9.3 −11.8 −23.1 −54.2 196.8 −47.0 14.0 −4.2 −32.4PM 10.2 −19.2 21.1 14.6 −24.9 −98.7 8.9 −21.3 11.2

N = 10,000EqAr −0.5 1.1 3.9 6.1 −23.1 21.2 −0.8 −0.4 4.4

FA MidP −0.1 0.4 1.0 1.2 −5.2 3.9 −0.1 −0.2 1.0InflP −0.2 0.6 3.5 4.1 −18.6 17.2 −0.2 −0.04 1.0

FACP 2.7 −4.0 −11.7 −23.4 82.5 −29.6 4.3 −0.7 −13.3PM 20.7 −28.1 22.6 12.6 23.1 −97.8 20.3 −25.1 −7.9

During the fitting procedure, there was no weighting assumed (W = 1).

1 omatogr. A 1216 (2009) 1067–1083

mcm

rtaw

tssdiwidf(oa

ttttmimsooagcir

3a

pmdtctrir

omt

•••

oouFd

Fig. 11. Dependence of (a) the FA (middle point method), (b) the FACP, and (c) thePM model errors (100%(qmodel − qtrue)/qtrue) on the mobile phase concentration for

080 A. Andrzejewska et al. / J. Chr

iddle point definition of the retention time of the breakthroughurve causes the least errors in the values of the best-fit para-eters.All the isotherm parameters in Table 3 were calculated with the

elative weighing method, using a weight of W = 1/Y2. The rela-ive errors of the best-fit parameters calculated without applyingweighing are reported in Table 4. The results obtained are muchorse than the previous ones in almost all cases.

Another very important decision that can affect significantlyhe values of the adsorption parameters is whether a constrainthould be imposed or not to the fitted parameters. With no con-traint applied, a negative value can be found for a parameter, asemonstrated in the case of the lowest column efficiency, N = 500

n the case of the parameter b2 for the bi-Langmuir model measuredith the FA- inflection point method. This is result is physically

mpossible and this value should be rejected. This suggests that theetermination of the isotherm parameters for heterogeneous sur-

aces should not be carried out without applying some constrainte.g., the isotherm parameters should be larger than or equal to 0)r without properly weighing the data points during their fitting ton isotherm model.

In the case of the FA data, the errors are much larger than inhe previous case for all adsorption models considered. The generalrend is the same, however. The most accurate parameter values arehose determined by FA, using the equal area retention time defini-ion. Similarly, for N = 10,000, the middle point definition gives the

ost accurate. In contrast with what happens with the Langmuirsotherm used with the relative weighing method, the calculations

ade with FACP data in the case of the bi-Langmuir isotherm givemaller differences between the true parameters and the fittednes. For PM, the best results for the bi-Langmuir isotherm arebtained while a weighing factor equal to W = 1/Y2. The PM datacquired in the cases of the Langmuir and the Moreau isothermsive more accurate parameter values when the calculations arearried out without any weighing. The parameters of bi-Langmuirsotherm models seem to be the most difficult to determine accu-ately. Errors made in their estimations are the largest.

.7. Influence of the number of experimental data points onccuracy of the adsorption isotherm parameters determination

A very common mistake made during the estimation of isothermarameters consists in acquiring an inadequate number of experi-ental data points. The concentration range of the data sometimes

oes not cover properly the most important domains of the adsorp-ion isotherm. An insufficient number of data points may beollected in the low-concentration region because of, e.g. a limi-ation of a detector detection limits or in the high-concentrationegion, due e.g. an insufficient solubility of the compound stud-ed [54]. The importance of acquiring data points in these oppositeanges has been investigated.

From FA and PM breakthrough curves, we derived 32 pointsf the adsorption data for every column efficiency and adsorptionodel used. We investigated three following cases of data reduc-

ion:

removing the smallest concentration data (0.1–1 mM)leaving every third point of the dataremoving the largest concentration data (1–100 mM)

In FACP, each point of the rear part of the 100 mM curve gives

ne isotherm data point. The concentration at the inflection pointf this part of the curve, 75% of it, and 125% of it were successivelysed as the upper boundaries of the adsorption data determination.or the high concentration points, the apparent adsorption constantecreases with increasing bulk concentration.

the Langmuir, the bi-Langmuir and the Moreau models, respectively, calculated forfour column efficiencies N = 500, 1000, 2000, and 10,000. The inserts represent theerrors found when using the whole range of data for the fit.

Fig. 11 shows the method errors made for different range ofadsorption data acquisition. It is important to notice that in caseof FA (Fig. 11a) the smallest errors are made when the last 13 dataare rejected. The middle point method error of Langmuir isotherm

A. Andrzejewska et al. / J. Chromato

Fig. 12. Dependence of (a) the FA, (b) the FACP, and (c) the PM parameter errors(100%(pmodel − ptrue)/ptrue) on the column efficiency for the Langmuir, the bi- Lang-muir and the Moreau models, respectively.

gr. A 1216 (2009) 1067–1083 1081

is shown as the example but we observed the same trend for otherFA isotherms. A removal of the small concentration data (0.1–1 mM)or using only two thirds of the isotherm data points do not affectsignificantly the errors made.

Fig. 11c shows the influence of the number of fitted data on thePM error. The Moreau isotherm error has been presented but as inprevious cases the general behavior is the same for the two otheradsorption models. The smallest error is made when fitting theisotherm without the first nine data points in the small concen-tration region. With an increase of the bulk concentration, the bestresults are obtained when fitting the isotherm data without the last13 data points.

In the case of the FACP method, the best results are obtainedwhen 125% of the inflection concentration is taken as the inte-gration boundary. By doing so, we included data representing anincreasing underestimate of the adsorbed amount with increas-ing bulk concentration. Since FACP overestimates the adsorbedamounts, this reduces the errors made.

Fig. 12 compares the error made on the best values of the param-eters obtained, after data acquisition and using the full standardrange of data. In the case of FA (Fig. 12a), the best values of the Lang-muir parameters are much closer to the true ones when the last 13points are omitted from the data fitted. The general trend is thesame, no matter how many data points we took into account. Thesmallest errors are made with the equal area method for two lowestefficiencies (500 and 1000) while for higher two ones, the mid-dle point method gives better results. The most erroneous resultswere obtained with the inflection point method. In Fig. 12b onecan see the difference in the bi-Langmuir parameter errors for theisotherms fitted with the inflection point and with 125% of it as theintegration boundary. Errors made on all the parameters estimatedfrom fitting with the integration boundary at 125% of the inflec-tion point of the isotherm are smaller at all efficiencies. In the caseof PM (Fig. 12c), the smallest error is made when one removes thelow concentration data. The difference is significant. Similar resultswere obtained for the other adsorption models.

4. Conclusions

The selection of the chromatographic method used to acquireadsorption isotherm data of single component, the way in whichthese data are derived from the recorded signals, and the methodselected for fitting the data obtained to an adsorption model equa-tion considerably affect the accuracy of the model coefficients,hence the conclusions that can be drawn regarding the adsorptionmechanism investigated. One of the aims of our work was to assessand compare the accuracy of the three classical dynamic techniquesof adsorption data determination: frontal analysis, frontal analysisby characteristic points, and the pulse methods, by simulating theirmeasurement processes under such experimental conditions thatthe true values of these parameters of the adsorption model areknown. Thus, the difference between the results of the measure-ment process and the known, true values of the model parametersindicates the accuracy of the measurement process.

Our results show that the accuracies of the three methods stud-ied are markedly different. They show also that, whatever themethod used, the accuracy strongly depends on the column effi-ciency. The accuracy of the FA method is the best, although thismethod tends to underestimate slightly the adsorption data. TheFACP method tends to be somewhat less accurate and to overes-timate the adsorption data. With both the FA and FACP methods,

the higher the efficiency, the smaller the systematic method error.The influence of the column efficiency on the accuracy of the PMis more complex. At constant pulse size, the pulse height increaseswith increasing column efficiency, causing the perturbation of theequilibrium to become more important and the method error to

1 omato

grSottne

ttrfd((ooalwitorWommt

idFtrtdrcpoao

ogac

N

a

bbbbccDF

F

[

[[[

082 A. Andrzejewska et al. / J. Chr

row. However, the PM method is accurate only if the perturbationemains linear, which is possible only if this perturbation is small.o, accurate results with the PM method require that small pulsesf mobile phase be injected onto the concentration plateaus. Onhe other hand, the precision of the measurements of the retentionimes of pulses depends on the ratio of the pulse height and the sig-al noise on the plateaus. A compromise is necessary. The randomrror caused by the noise will be discussed later.

The accuracy of the isotherm coefficients obtained by fittinghe adsorption data points to a model equation depends much onhe objective function used in the regression. We carried out theegression calculations with the data obtained with the three dif-erent methods of adsorption data measurements applied to threeifferent isotherm models, by applying three objective functions1) no weighing of the data (W = 1); (2) their statistical weighingW = 1/Y); (3) and their relative weighing (W = 1/Y2). The resultsf these calculations showed that the model error depends stronglyn the weighing selected. For the data obtained with the FA methodnd for all three adsorption isotherm models, this errors dependsittle on the weight applied to the data. Significant differenceas only noticed for measurements made at low-concentrations,

n which case the method W = 1/Y2 gave smaller errors. Whenhe isotherm data were derived by FACP, larger differences werebserved between the true parameters and those afforded by theegression when the data were weighed by either W = 1/Y or

= 1/Y2) than when they were not weighed. Similar results werebtained with the PM method. Therefore, when these last two chro-atographic methods are used, the nonlinear regression should beade with no weighing of the data, to maximize the accuracy of

he isotherm parameters.The fitting of the adsorption data and the accuracy of the

sotherm parameters derived are very sensitive to the number ofata points recorded by the specific chromatographic method used.rontal analysis gives good values of adsorption coefficients, upo moderate concentrations. On account of the limitation of theetention time estimation methods for high bulk concentrations,hose breakthrough curves should not be taken into account in theetermination of the adsorption isotherm data. In contrast, betteresults are obtained with the pulse method, when the minor con-entration data are rejected. Use of frontal analysis by characteristicoint exposes scientists to the risk of overestimating the amountf adsorbed solute adsorbed. In order to find better values of thedsorption parameters, one should end the integration of the rearf the breakthrough curve slightly above its inflection point.

More studies will be carried out to investigate the influencef different degrees of noise introduced in theoretical chromato-raphic signals. Also, it would be worthwhile, to study the accuracynd precision of adsorption isotherm determination for multi-omponent systems.

omenclature

exponent in the denominator of the sum-of-squares of thevertical distances between the experimental data and themodel calculations denoting the weighting methodadsorption equilibrium constant at infinite dilution

0 pre-exponential factor in adsorption equilibrium constant1 equilibrium constant of adsorption on site type 12 equilibrium constant of adsorption on site type 2

solute concentration in mobile phase

0 feed or initial concentrationa apparent axial dispersion coefficient

phase ratio (volume of the stationary phase divided byvolume of the mobile phase)

p Fisher parameter

[[

[[[

gr. A 1216 (2009) 1067–1083

H height equivalent to a theoretical plate (HETP)I adsorbate-adsorbate interaction in stationary phaseL column lengthN number of theoretical plates (efficiency) or number of

data points fittedND number of data points fittedP number of model parametersq solute concentration in stationary phaseq′ exp,i′ experimental solute concentration in stationary phase in

equilibrium with a given liquid phase concentration ci

qth,i theoretical values of adsorption data calculated from themodel

q′true,i′ true adsorption isotherm data, derived from the equationof the isotherm model selected initially

qs monolayer saturation capacityqs1 monolayer saturation capacity of the adsorption site type

1qs2 monolayer saturation capacity of the adsorption site type

2R universal gas constantt timet0 hold-up timetR retention timeT absolute temperatureu linear chromatographic mobile phase velocityuz velocity associated with a concentrationW weighting methodz distance along the column length� total porosity of the column�a adsorption energy� function minimized in Marquard’s algorithm

Acknowledgments

This work was supported in part by grant CHE-06-08659 of theNational Science Foundation and by the cooperative agreementbetween the University of Tennessee and the Oak Ridge NationalLaboratory.

References

[1] B.L. Karger, L. Snyder, C. Horvath, An Introduction to Separation Science, Wiley,New York, 1973.

[2] C.J. King, Separation Processes, McGraw-Hill, New York, 1980.[3] G.D. Parfitt, Adsorption from Solution at the Solid/Liquid Interface, Academic

Press, London, 1983.[4] R.H. Ottawil, Adsorption from Solution, Academic Press, London, 1983.[5] J. Oscik, Adsorption, PWN, Warsaw, 1982.[6] G. Schay, G. Szekely, Acta Chim. Hung. 5 (1954) 167.[7] D.H. James, C.S.G. Phillips, J. Chem. Soc. (1954) 1066.[8] Y.A. Eltekov, Y.V. Kazakevich, A.V. Kiselev, A. Zhukov, Chromatographia 20

(1985) 525.[9] Y.A. Eltekov, Y.V. Kazakevitch, J. Chromatogr. 35 (1986) 213.

[10] P. Sajonz, G. Zhong, G. Guiochon, J. Chromatogr. A 731 (1996) 1.[11] E. Glueckauf, Trans. Faraday Soc. 51 (1955) 1540.12] E. Cremer, H. Huber, Angew. Chem. 73 (1961) 461.

[13] H. Guan, B. Stanley, G. Guiochon, J. Chromatogr. A 659 (1994) 27.[14] L. Ravald, T. Fornstedt, J. Chromatogr. A 908 (2001) 111.[15] J. Samuelsson, T. Fornstedt, Anal. Chem., doi:10.1021/ac8010999.[16] P. Forssén, J. Lindholm, T. Fornstedt, J. Chromatogr. A 991 (2003) 31.[17] C.N. Reilley, G.P. Hildebrand, J.W.A. Jr., Anal. Chem. 34 (1962) 1198.[18] F.G. Helfferich, D.L. Peterson, J. Chem. Educ. 41 (1964) 410.[19] J. Samuelsson, P. Forssén, M. Stefansson, T. Fornstedt, Anal. Chem. 76 (2004)

953.20] D.L. Peterson, F. Helfferich, R. Carr, AIChE J. 12 (1966) 903.21] J.F. Parcher, M.I. Selim, Anal. Chem. 51 (1979) 2154.22] J. Lindholm, P. Forssén, T. Fornstedt, Anal. Chem. 76 (2004) 4856.

23] R. Arnell, T. Fornstedt, Anal. Chem. 78 (2006) 4615.24] J. Samuelsson, R. Arnell, J.S. Diesen, J. Tibbelin, A. Paptchikhine, T. Fornstedt,

P.J.R. Sjöberg, Anal. Chem. 80 (2008) 2105.25] E.V. Dose, G. Guiochon, Anal. Chem. 62 (1990) 816.26] E.V. Dose, S. Jacobson, G. Guiochon, Anal. Chem. 63 (1991) 833.27] J. Jönsson, P. Lövkvist, J. Chromatogr. A 408 (1987) 1.

omato

[

[

[[[[

[[

[

[

[

[[[[[[[[[

[[

A. Andrzejewska et al. / J. Chr

28] K. Kaczmarski, M. Mazzotti, G. Storti, M. Morbidelli, Comput. Chem. Eng. 21(1997) 641.

29] A. Felinger, Data Analysis and Signal Processing in Chromatography, Elsevier,1998.

30] F. Gritti, G. Guiochon, J. Chromatogr. A 1043 (2004) 159.31] P. Sajonz, J. Chromatogr. A 1050 (2004) 129.32] J. Samuelsson, P. Sajonz, T. Fornstedt, J. Chromatogr. A 1189 (2008) 19.33] G. Guiochon, A. Felinger, S. Shirazi, A. Katti, Fundamentals of Preparative and

Nonlinear Chromatography, Elsevier, 2006.34] K. Kaczmarski, J. Chromatogr. A 1176 (2007) 57.35] V. Villadsen, M. Michelsen, Solution of Differential Equation Model by Polyno-

mial Approximation, Physical & Chemical Engineering Science, Prentice Hall,Englewood Cliffs, NJ, 1978.

36] J. Berninger, R.D. Whitley, X. Zhang, N.-H.L. Wang, Comput. Chem. Eng. 15 (1991)749.

37] D.M. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, NewYork, 1984.

38] M. Suzuki, Adsorption Engineering, Chemical Engineering Monographs, Else-vier, Amsterdam, 1990.

[[

[[[

gr. A 1216 (2009) 1067–1083 1083

39] F. Gritti, G. Gotma, J.B. Stanley, G. Guiochon, J. Chromatogr. A 988 (2003) 185.40] F. Gritti, G. Guiochon, Anal. Chem. 75 (2003) 5726.41] F. Gritti, A. Felinger, G. Guiochon, J. Chromatogr. A 1017 (2003) 45.42] F. Gritti, G. Guiochon, J. Chromatogr. A 1028 (2004) 105.43] S. Jacobson, S. Golshan-Shirazi, G. Guiochon, J. Am. Chem. Soc. 112 (1990) 6492.44] S. Jacobson, S. Golshan-Shirazi, G. Guiochon, AIChE J. 37 (1991) 836.45] T. Fornstedt, P. Sajonz, G. Guiochon, J. Am. Chem. Soc. 119 (1997) 1254.46] T. Fornstedt, P. Sajonz, G. Guiochon, Chirality 10 (1998) 375.47] T. Fornstedt, G. Gotmar, M. Andersson, G. Guiochon, J. Am. Chem. Soc. 121 (1999)

1164.48] G. Gotmar, T. Fornstedt, G. Guiochon, Anal. Chem. 72 (2000) 3908.49] M. Moreau, P. Valentin, C. Vidal-Madjar, B.C. Lin, G. Guiochon, J. Colloid Interface

Sci. 141 (1991) 127.

50] F. Gritti, G. Guiochon, J. Chromatogr. A 1047 (2004) 33.51] J.D. Andrade, Surface and Interfacial Aspects of Biomedical Polymers, Plenum

Press, New York, NY, 85.52] D. Marquardt, J. Soc. Indust. Appl. Math. 11 (1963) 431.53] I. Quinones, G. Guiochon, J. Chromatogr. A 796 (1998) 15.54] F. Gritti, G. Guiochon, J. Chromatogr. A 1097 (2005) 98.