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Journal of Chromatography A, 1194 (2008) 205–212
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Journal of Chromatography A
journa l homepage: www.e lsev ier .com/ locate /chroma
Impact of an error in the column hold-up time for correct adsorption isothermdetermination in chromatographyII. Can a wrong column porosity lead to a correct prediction of
overloaded elution profiles?Jorgen Samuelssona, Jia Zangb, Anne Murungab, Torgny Fornstedta,∗, Peter Sajonzb,∗
wedenellence
as de mowithre usethat
experdal eitraryof thehere
a Department of Physical and Analytical Chemistry, BMC Box 577, SE-751 23 Uppsala, Sb Merck Research Laboratories, P.O. Box 2000, Separation and Purification Center of Exc
a r t i c l e i n f o
Article history:Received 9 December 2007Received in revised form 17 April 2008Accepted 21 April 2008Available online 25 April 2008
Keywords:Column porosityHold-up volumeVoid volumeFrontal analysisEquilibrium isothermIdeal modelAdsorption energy distribution
a b s t r a c t
The adsorption isotherm wfrontal analysis in staircasadsorption isotherm data,generated parameters weexperiments. It was foundwell with the value foundanalysis confirmed a bimopredicted with a quite arbmodel is chosen insteadpreparative separations w
Prediction of band profiles
1. Introduction
In preparative chromatography the hold-up time (or volume) isan essential parameter for computer-assisted optimizations of theexperimental conditions to get maximal throughput and productyield. However, the determination of the column hold-up volumeis not trivial and its proper estimation has remained an impor-tant issue for many years [1–3]. More recently Gritti et al. madea more stringent definition of the hold-up volume [4]. All meth-ods for determination of the hold-up volume contain sources oferrors; therefore the value obtained from the hold-up volume willbe different depending on which method was used. For example,recently a 14% difference was found between the unretained markerthiourea and the pyconometry method for determining the hold-upvolume in the same chromatographic system [5].
In a recent study [6] it was found based on computer-generateddata that an error in the hold-up volume results in serious errors
∗ Corresponding authors. Tel.: +1 732 5948430; fax: +1 732 5943887.E-mail addresses: [email protected] (T. Fornstedt),
Peter [email protected] (P. Sajonz).
0021-9673/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.chroma.2008.04.053
, Rahway, NJ 07065, USA
etermined for phenol in methanol/water on a C-8 stationary phase usingde, assuming different total column porosities, from 1 to 87%. Each set ofa certain column porosity, was fitted to various adsorption models and thed to calculate overloaded elution band profiles that were compared with
the bi-Langmuir model had an optimum fit for a porosity that correspondsimentally. The adsorption energy distribution (AED) calculations and errornergy distribution. It was also found that band profiles can be accuratelychosen porosity, under prerequisite that a wrong but flexible adsorptioncorrect one. The latter result is very useful for quick optimizations of
the exact value of the column porosity is not available.© 2008 Elsevier B.V. All rights reserved.
in the adsorption isotherm coefficients and that the error increases
for larger degrees of nonlinearity of the chromatographic system.This result was later confirmed with experimental data [5]. Seidel-Morgenstern showed that a wrong porosity could predict quitesatisfactorily peak profiles, at least for low concentrations [7]; how-ever, as this was not the main topic of the paper, the effect was notsystematically investigated. In a more recent study we investigatedthe importance of a small error in the hold-up time on the choiceof adsorption isotherm model that can fit to computer-generatedadsorption isotherm data [8]. Our results showed that data from atrue Langmuir or a true bi-Langmuir model used with an under-estimated hold-up time have a better fit to a more heterogeneousmodel while for an overestimated hold-up time models describ-ing false adsorption processes such as multi-layer adsorption orsolute–solute interactions are assumed. Scatchard plots and calcu-lations of the adsorption energy distribution (AED) confirmed thedeviations from the Langmuir behavior [8].In this paper we present experimental adsorption isotherm datadetermined by frontal analysis in the staircase mode for phenol ona C-8 stationary phase and methanol/water as mobile phase. Thepurpose of the study is twofold. Firstly, we will systematically inves-tigate if wrong column porosities can still lead to correct predictions
mato
206 J. Samuelsson et al. / J. Chroof overloaded elution profiles. Secondly, we will examine if it is pos-sible to estimate the columns porosity only from the measurementof equilibrium isotherm data, assuming that the true adsorptionisotherm model is known.
2. Experimental
2.1. Chemicals and materials
The water used was distilled and purified by a Hydro Systempurchased from Hydro Service & Supplies Inc. (Garfield, NJ, USA).HPLC grade methanol and phenol were obtained from Fluka (Buchs,Switzerland). Thiourea was purchased from Aldrich (St. Louis, MO,USA).
2.2. Instrumentation and HPLC method conditions
An Agilent 1100 Series HPLC system from Agilent Technologies(Palo Alto, CA, USA) was used for all experiments. This system isequipped with an auto injector containing a sample tray cooler,a multi-solvent delivery system and a temperature controlledcolumn compartment set at 25.0 ◦C. The detector-wavelengthsmonitored were 254, 292 and 260 nm, respectively. The flowrate used for all experiments was 1.0 mL/min. The mobile phaseused was 40/60 (v/v) methanol/water. The mobile phase wasalso used as diluent for sample preparations. The column wasan Advantage ARMOR ADV5218 C-8 (nominal particle size: 5 �m;4.6 mm × 250 mm I.D.) obtained from Analytical Sales and Services(Pompton Plains, NJ, USA).
2.3. Procedures
The retention volumes of the peaks were determined at themaximum peak height. The retention data for the elution exper-iments were corrected for the extra-column volume, determinedto 0.06 mL, by replacing the column with a zero dead volume unionand injecting small volumes (5 �L) of thiourea. The column platenumber was determined to n = 9508 from the width at half-heightof the peak resulting from a 5 �L injection of a diluted phenol solu-tion. For this experiment the column was placed as close to theinjection valve as possible, to minimize the extra-column volume.The column plate number of phenol was used later for the numer-ical calculation of overloaded elution profiles.
The concentration steps generating the frontal chromatograms
were obtained in a stepwise manner using one solvent channelwith mobile phase and another with a bulk concentration of phe-nol (80.96 g/L) in mobile phase. The retention times of the frontalchromatograms were calculated at the half-heights of each stepin the staircase. We validated the retention times from the half-height method with the area method as reference. The averagedifference between the two methods was only 0.4% which con-firmed that it is acceptable to acquire the adsorption data by thehalf-height method for our experimental system. The so-called gra-dient delay volume, i.e. the extra-column volume for the frontalanalysis experiments, was determined to 1.08 mL by performing aconcentration step with thiourea as sample. The retention timesfor the breakthrough curve experiments were corrected for thisdelay volume. It was tested if this gradient delay volume had aneffect on the retention times of the breakthrough curves, in the fol-lowing way. With the inert marker thiourea, the column hold-uptime was calculated by either performing conventional injectionsor by performing small steps with the pump and then correct forthe gradient delay volume. The value of the hold-up time variedless then 2% by using these two ways. For this reason we canconclude that the gradient delay volume is not too big for thegr. A 1194 (2008) 205–212
purpose of acquiring accurate retention times of the breakthroughcurves.
It has to be noted that even if the experimental setup that isused in this study is suitable for the purpose of accurate isothermmeasurements it is not useful for the measurement of kinetic data.The reason for this is because band broadening effects occur due tothe large extra-column volume which compared to the column voidvolume is quite significant. For the accurate measurement of kineticdata (and also isotherm data) a system that uses two large injectionloops with two sample switching valves that are connected close tothe column inlet (as described in Jacobson et al. [9]) should be used.Such an experimental setup not only minimizes the extra-columnvolume and thus the band broadening effects, but also minimizesthe amount of sample that is required for a complete set of frontalanalysis experiments.
The plateaus of the frontal analysis experiments data were usedto calculate a fourth degree polynomial calibration curve convert-ing the absorbance units from the detector to concentration units.All calculations were performed using Matlab version 7.0 (Math-Works Inc., Natick, MA, USA).
In this study the column porosity (ε) has been used rather thanthe hold-up time because it is a dimensionless parameter (definedas ε= V0/Vg, where Vg is the geometrical column volume and V0 isthe hold-up volume). A porosity of 0% represents a column com-pletely filled with stationary phase and 100% is an empty column.In reality, the column porosity will vary within narrower limits [10].
The raw adsorption isotherm data were fitted to the n-Langmuirisotherm equation using a nonlinear fitting procedure and theMarquardt-Levenberg algorithm with at least 1000 different ini-tial guesses selected randomly over all possible solutions. Ann-Langmuir adsorption isotherm model describes the relationshipbetween the solute concentration in the stationary and mobilephases. The equation for the model is written as
q =n∑
i=1
aiC
1 + KiC(1)
where ai and Ki are numerical coefficients. Three different modelswere considered in this study, i.e. n = 1 (Langmuir), n = 2 (bi-Langmuir), and n = 3 (tri-Langmuir). The ratios ai/Ki representthe monolayer solute saturation capacities qs,i of each individualadsorption site in the column.
The AEDs were calculated from the raw adsorption isothermdata [11,12]. The Langmuir model was used for the local adsorp-
tion isotherm and the AED integral was solved with an iterativealgorithm (expectation-maximization method) [13]. All calcula-tions were carried out by expanding the integration limits between0.1/Cmax to 10/Cmin, where Cmin and Cmax are the minimumand maximum concentration used in determining the adsorptionisotherm. The expansion is necessary to promote conversion ofadsorption sites with energy near the integration limits.3. Results and discussion
3.1. Calculation of adsorption isotherm data from frontal analysisexperiments
Frontal analysis experiments for the purpose of adsorptionisotherm determination are carried out by a series of concentra-tion step experiments. These experiments can be a series of stepsfrom 0 to Ci, or successive steps from 0 to C1, then C1 to C2 etc.The first method is known as frontal analysis in the step-seriesmode, the second one as frontal analysis in the staircase mode [14].The advantage of the staircase method is that the column does nothave to be re-equilibrated to C = 0 after reaching the plateau of each
mato
J. Samuelsson et al. / J. ChroTable 1Experimental retention times of breakthrough curves for phenol at various soluteconcentration steps
Ci (g/L) Ci+1 (g/L) VR (mL)
0 4.05 10.6134.05 8.10 8.4718.10 12.14 7.182
12.14 16.19 6.44216.19 24.29 5.67524.29 32.38 4.98632.38 40.48 4.56540.48 48.58 4.22948.58 64.77 3.89464.77 80.96 3.649
Ten concentration steps were made in staircase mode. The retention volumes wereobtained from the half-height of each concentration step.
new concentration step. In this study we use the staircase mode.The experimental retention volumes for the breakthrough curvesof phenol on the C-8 stationary phase are shown in Table 1. Phenolwas used as solute because it is highly soluble in methanol/waterand the adsorption isotherm data has been shown to fit well to an
n-Langmuir isotherm model [5]. The solute concentrations in thestationary phase are calculated from the breakthrough curves bythe integrated mass balance equation:qi+1 = qi +Ci+1 − Ci�
VR,i+1 − V0
V0(2)
where � is the phase ratio (i.e. the stationary phase volume dividedwith the mobile phase volume in the column) VR,i+1 is the break-through retention volume of the step, qi and qi+1 are the initial andfinal solute concentrations in the stationary phase in equilibriumwith Ci and Ci+1, respectively.
The calculated adsorption isotherms for column porosities rang-ing from 1 to 87% using Eq. (2) is presented in Fig. 1a and thecorresponding Scatchard plots in Fig. 1b. Higher values of the col-umn porosity were not used because at values higher than 87% theadsorption isotherm data become unrealistic, since the concentra-tion in the stationary phase starts to decline at higher mobile phaseconcentrations. It is obvious that such a porosity value cannot beused because solute concentrations in the stationary phase shouldnot decrease with increasing mobile phase concentration at least
Fig. 1. Experimental adsorption isotherm data (a) and corresponding Scatchardplots (b) for phenol on a C-8 stationary phase in methanol/water using differentvalues of the total column porosity from 1 to 87%. For experimental data, see Table 1and Section 2.
gr. A 1194 (2008) 205–212 207
not for any n-Langmuir adsorption model. This recently discoveredphenomenon is exemplified on an adsorption isotherm of phenolon a ODS column in a recent study by Gritti and Guiochon at themobile phase concentration 200 g/L (see Fig. 5C in Ref. [5]).
In Fig. 1a it can be seen that the adsorption isotherm plots barelychange for lower porosity values between 1 and 50%, and that theadsorption isotherms are nearly linear. At higher porosities the non-linearity of the adsorption isotherm increases and the changes inthe adsorption isotherm data are quite apparent. This is a directconsequence of the decreasing amount of the stationary phase atincreasing column porosity. The Scatchard plots (Fig. 1b) are con-cave for porosities up to at least 70% indicating a heterogeneousadsorption process that can be found for example for adsorptionisotherm models such as the bi-Langmuir and the Toth. For higherporosities (80 and 87%) the Scatchard plots are convex and adsorp-tion models that describe such Scatchard plots are e.g. the Jovanovicor Moreau adsorption models; this is in line with our previousstudy [8]. Since the value of the true hold-up volume is unknownwe cannot draw any absolute conclusions about the true adsorp-tion process, based on only the experimental determined Scatchardplots.
3.2. Fitting of the adsorption isotherm data to adsorptionisotherm models
The adsorption isotherms calculated with various porosity val-ues were fitted to n-Langmuir models using a nonlinear fittingalgorithm with the sum of least squares (S2) as an objective func-tion (see Eq. (3)). The Fisher parameters were also calculated (seeEq. (4)). The equations for the sum of least squares and the Fisherparameter are written as
S2 =∑
(qi − qfit)2 (3)
Fisher = (n− l)∑ (qi − q)2
(n− 1)∑
(qi − qfit)2
(4)
where qi and qfit are the experimentally determined and calculated(fitted) adsorption isotherm data points, respectively, and q is theaverage value of the adsorption isotherm data. The parameter nis the number of data points, and l is the number of adjustableparameters in the adsorption isotherm model. A higher value ofthe Fisher parameter suggests a better fit to the experimental data.To determine if one model is significantly better in describing the
adsorption process, the ratio of two Fisher parameters are taken andcompared against critical F-ratios that are found in most statisticalbooks e.g. see Ref. [15].In Fig. 2a a plot of the sum of least squares vs. porosity is shownfor porosity values from 1 to 87% whereas Fig. 2b shows the cor-responding Fisher parameter vs. porosity. It is noteworthy that theleast squares vs. porosity plots for the Langmuir and bi-Langmuirmodels each go through a different local minimum at 76 and 69%,respectively. The bi-Langmuir and tri-Langmuir do not go throughany local minimum, but instead have the minimum at porosities of1%. The curves for bi-Langmuir and tri-Langmuir merge at higherporosities values, and above 65% there is no difference betweenthem, indicating that the extra variables in the tri-Langmuir donot contribute any further to the quality of the fitting. Further-more it is quite interesting that all three curves merge at higherporosity values (above 76%, after passing the local minimum forthe Langmuir model), indicating that none of the n-Langmuir mod-els describe well the high porosity data. For porosities above 60%the tri-Langmuir model did not convert well during the nonlinearregression, probably because the model has six adjustable parame-ters that are fitted to the adsorptions isotherms data, and between
208 J. Samuelsson et al. / J. Chromatogr. A 1194 (2008) 205–212
-Lan
gmu
iran
dtr
i-La
ngm
uir
adso
rpti
onis
oth
erm
,raw
dat
ap
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nte
din
Tabl
e1
Site
2Si
te3
]q s
[g/L
]a
K[L
/g]
q s[g
/L]
aK
[L/g
]q s
[g/L
]O
verl
ap[%
]
621
1–
––
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–10
718
0–
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–4
48
189
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756
204
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915
219
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986
276
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81
661
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0550
35.0
––
–98
422
03.
64
0.05
9561
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––
998
169
3.58
0.10
733
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––
985
219
8.07
5.46
×10
251.
48
×10
−25
––
–98
627
65.
016.
81×
1066
7.35
×10
−67
––
–81
050
1.09
0.10
610
.30.
711.
25×
10−0
55.
68
×10
0498
510
22.
074
40.
113
18.4
0.42
2.91
×10
−04
1.43
×10
0498
816
93.
580.
107
33.6
7.25
7.6
4×
1011
9.4
9×
10−1
298
521
90.
360
7.94
×10
074.
53×
10−0
918
.35.
14×
1053
3.56
×10
−53
976
276
1.43
6.92
×10
072.
07×
10−0
87.
424.
01×
1053
1.85
×10
−53
80
Fig. 2. Sum of least squares (a) and the Fisher parameters (b) plotted vs. porosity forthe Langmuir, bi-Langmuir and tri-Langmuir models.
70 and 80% the Scatchard plot is nearly linear. The Fisher parame-ter vs. porosity plots Fig. 2b show more clearly the same behavioras the least squares plots, however there is a maximum insteadof a minimum. The models can be compared by taking the ratiobetween the calculated Fisher parameters and compare this withcritical F-ratios. The F-ratios show that the tri-Langmuir model isnever significantly better in describing the adsorption process thanthe bi-Langmuir model. The bi-Langmuir model is however signif-icantly better than the Langmuir model in describing adsorptionprocess for porosities smaller than 74%. Linear regression of theScatchard plot (linear Scatchard plots are only obtained for theLangmuir model) shows that the most linear Scatchard plot occursat a porosity of 78% (R = −0.9994), with the parameters a = 9.62 andK = 0.045 L/g. This porosity value is very close to the porosity of 76%that was obtained from the minimum of the least square fit.
Adsorption isotherm coefficients (a and K) and saturation capac-ities (qs) for porosities from 1 to 87% are presented in Table 2. Itcan be observed that the second adsorption site (second Langmuirterm) of the bi-Langmuir model is negligibly small for porosities of80 and 87%. For these cases, the bi-Langmuir model can be simpli-fied to the Langmuir model. For the tri-Langmuir model the third
site is negligibly small, for porosities of 69–87% and its second sitefor porosities of 80 and 87%. The third site of the tri-Langmuir modelhas also unrealistic high capacity, for porosities of 1 and 50%. Thisis in line with our previous study for the case of an underestimatedhold-up time [8]. In that study we found that for underestimatedhold-up time the true adsorption model was best described if anextra linear term is added to it (this will be discussed later in Section3.5.2). Because in this case we used n-Langmuir models, the extralinear term will theoretically have an infinite capacity for the low-est energy site, as noted in the tri-Langmuir case. This is a goodindication that the bi-Langmuir model describes the separationsystem.3.3. Prediction of band profiles
If the adsorption isotherm data are used only for the chro-matographic scale-up, i.e. for the prediction of overloaded elutionprofiles, then it is irrelevant if the adsorption isotherm data and thehold-up volume are correct. It is only important that chromato-graphic band profiles can be accurately predicted. To investigatehow good the n-Langmuir models predict band profiles, a 100 �L Ta
ble
2Fi
tted
adso
rpti
onis
oth
erm
dat
afo
rd
iffe
ren
tp
oros
itie
sto
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gmu
ir,b
i
Mod
elPo
rosi
ty[%
]Si
te1
aK
[L/g
Lan
gmu
ir1
2.23
0.01
050
3.91
0.02
169
6.39
0.03
376
8.4
80.
041
8010
.40.
047
8717
.00.
061
Bi-
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ir1
0.98
80.
001
501.
180.
005
693.
850.
022
8010
.40.
047
8717
.00.
061
Tri-
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gmu
ir1
1.20
520.
024
502.
499
80.
024
693.
850.
022
8010
.40.
047
8717
.00.
061
J. Samuelsson et al. / J. Chromato
model, and at 69% for the bi-Langmuir model. The interesting
Fig. 3. Comparison of experimental overloaded elution profiles and calculationsusing various porosities. The porosities used for (a) Langmuir were 1 (black solidline), 50 (dashed black line), 69 (dotted black line), 76 (thick black line) and 87%(gray line) and for (b) bi-Langmuir and (c) tri-Langmuir 1 (black solid line), 50(dashed black line), 69 (dotted black line), 80 (thick black line) and 87% (grayline).
injection of 80.96 g/L phenol was used. Porosity values from 1 to87% were used for the calculations. The calculated band profilesare presented in Fig. 3. We also calculated the overlap between theexperimental and predicted band profiles in order to quantitativelyevaluate the quality of fit.
In Fig. 3a the band profiles that are obtained with the Lang-muir adsorption isotherm model are plotted. We can see thatthey do not predict the experimental profiles very well, not evenat the optimum porosity of 76% (the minimum in least squaresplot for Langmuir model). The band profiles obtained with thebi-Langmuir model in Fig. 3b represent the experimental datamuch better. The bi-Langmuir model is more suited for the systemstudied, with good agreement of the experimental and calculatedband profiles for the optimum porosity of 69% (minimum of leastsquares plot for bi-Langmuir model), and also for lower poros-ity values. In Fig. 3c band profiles from the tri-Langmuir modelare plotted. We cannot observe any visual improvement as com-pared to the bi-Langmuir model. The overlap indicates that theLangmuir model is better described at a porosity of 80% with an
overlap of 98% as compared to the porosity 76% with an overlapof 91% (see Table 2). The bi-Langmuir and tri-Langmuir modelsdo not have any optimum porosity but fails to predict good pro-files between the porosity of 74 and 79%. Therefore the profilesfor 80% porosity are plotted in Figs. 3b–c instead of 76% as in theLangmuir case Fig. 3a. The predicted overlaps are around 98% forbi-Langmuir and tri-Langmuir model at porosities between 1 and74%.It is very interesting to observe that a very large underestima-tion of the porosity still results in a satisfactory prediction of theband profile for the bi-Langmuir and tri-Langmuir models. This is,however, not the case for an overestimation, where quite signif-icant errors are noticeable already at small errors in the hold-uptime. These errors are obviously due to the fact that the n-Langmuirmodel is unsuitable if the porosity is chosen too large, i.e. the modeldoes not fit the data well. The agreement gets successively bet-ter if the model restrictions loosen, i.e. if more parameters areintroduced. While this is a logical consequence it still leads to theconclusion that the adsorption model depends on the hold-up vol-ume.
gr. A 1194 (2008) 205–212 209
Table 3Experimentally determined hold-up volume values with corresponding porosities
Method Hold-up time,V0 (mL)
Total columnporosity, ε (%)
Fisher parameter, F
Langmuir fit 3.16 76 (L) 3.87 × 1006
Langmuir Scatchard 3.26 78 (L) 2.37 × 1006
Langmuir overlap 3.32 80 (L) 1.02 × 1006
Bi-Langmuir fit 2.87 69 (BL) 8.74 × 1007
Bi-Langmuir errora 2.89 70 (BL) 8.61 × 1007
Thiourea injection 2.77 67 (L) 4.88 × 1005
Estimationb 2.5 60 (L) 2.32 × 1005
(L) for Langmuir model and (BL) for bi-Langmuir model.a Error function for adsorption isotherm models see Eq. (5) in Section 3.5.2.b 1 mL void volume/cm column length.
3.4. Comparison of hold-up volume values
The optimum curve fit found with the Fisher parameter plots inFig. 2, and the calculated overlap from the overloaded elution pro-file fittings can be used for the estimation of the column hold-upvolume, provided that the model chosen represents the experi-mental data appropriately. In Table 3 the results for the porosityvalues obtained from the Fisher parameter for the Langmuir and bi-Langmuir together with the values that are obtained from injectionof non-retained markers is presented. Interestingly the experimen-tal hold-up volume value obtained using the inert marker thioureaagrees well with the value obtained from the optimum of the bi-Langmuir model. The Langmuir model on the other side is notable to predict profiles at the experimentally determined hold-upvolume. Instead, the determined Langmuir adsorption isotherm issignificant better at a porosity of 76% corresponding to a hold-upvolume of 3.16 mL which is a 14% overestimation as compared tothe experimental thiourea injection (at 2.77 mL), cf. Table 3. Thebi-Langmuir model is therefore a more realistic representation ofthe experimental reality. At the porosity 69% (showing an over-lap of 98%, see Table 2) the corresponding hold-up volume valueis 2.87 mL which corresponds only to a 3% overestimation of thethiourea injection.
3.5. Verification of the results
So far the experimental data have been analysed using Lang-muir, bi-Langmuir and tri-Langmuir adsorption isotherm models.The Fisher parameter has a maximum at 76% for the Langmuir
finding is that the data is significantly better described with the bi-Langmuir model compared to the Langmuir for porosities smallerthan 74%. To further investigate which model and porosity describesthe system best; we calculated the AEDs followed by an erroranalysis. With AED calculations for different porosities we candetect whether the AED is unimodal or multimodal. The erroranalysis is used to verify if the porosity affects the determinedadsorption isotherm in a similar way as in our previous study[8].
3.5.1. Adsorption energy distributionFig. 4 shows the AEDs calculated using 250 grid points and
500 000 iterations for column porosities 1, 50, 60, 69, 76 and 87%.Similar trends can be seen here as in our previous study, with anextra site at low adsorption energy at porosity of 60% and lower.One can also note that the AED for 69% is extremely wide andseams to be tailing toward higher energy values. This could be asign of heterogeneity in the AED (e.g. Langmuir–Freundlich adsorp-tion isotherm) or of unresolved sites with small adsorption energydifference. In order to test if this is due to unresolved extra adsorp-
210 J. Samuelsson et al. / J. Chromato
Fig. 4. Adsorption energy distribution calculated for different porosities using500 000 iterations and 250 grid points.
tion sites, 10 million iterations were performed at this stage for
porosities between 69 and 76% (see Fig. 5).In Fig. 5 we can see that the AED is bimodal, and for a porosityof 69% the energy difference between the sites is 4.4 kJ/mol. Thisresult is in line with reported values for the adsorption of phenol onreversed phase columns [16]. The energy difference increases withincreasing porosity because the second site moves faster towardhigher adsorption energies than the first site. At a porosity of 72%the energy difference is 5.2 kJ/mol, and at 74% it has increased to10.3 kJ/mol. At even higher porosities the second site is unresolved.To resolve this site, adsorption isotherm data points from lowerconcentrations are needed. Unresolved high energy sites have pre-viously been noted for overestimate porosities with too few lowconcentration data points (see Ref. [5], Fig. 9B). If the AED was onlycalculated for a porosity corresponding to 76% the second adsorp-tion site could be mistaken as a calculation artefact, and the modelcould be assumed to be unimodal. This is an explanation why thebi-Langmuir model fails to predict good data at porosity of 74–79%.At porosities above 79% the model has converted into the Langmuirmodel and the AED calculation is unimodal for this data set. In ourprevious study (see Fig. 4 in Ref. [8]) we conclude that the second
Fig. 5. Adsorption energy distribution calculated with 10 million iterations using250 grid points for various values of the column porosity. The bottom figure is azoom-in of the top plot.
gr. A 1194 (2008) 205–212
site in a bi-Langmuir model moved toward higher energy valuesif the hold-up volume is overestimated. In this case the AED doesgive us a conclusive answer that the true adsorption isotherm hasat least a bimodal AED, and will therefore be better described witha bi-Langmuir model than a Langmuir model.
3.5.2. Error analysisError analysis of the determined adsorption isotherms due to
wrong hold-up time was already investigated in our previous study[8]. However, now we use the dimensionless porosity instead of thehold-up volume.
q = 1 − ε0
1 − ˛ε0q0 + ε0 − ˛ε0
1 − ˛ε0C, (5)
where q and q0 are the predicted and true adsorption isotherm rawdata, respectively. ˛ is a coefficient with the following properties:if ˛= 1 the true porosity (ε0) is assumed, if ˛> 1 we have over-estimated the porosity, and for ˛< 1 we have underestimated theporosity. Notice that ˛ε0 is the assumed porosity. For justificationof Eq. (5) see Ref. [8] Section 3.1.6.
By defining two constants and ω:
= 1 − ε0
1 − ˛ε0; ω = ε0 − ˛ε0
1 − ˛ε0. (6)
Eq. (5) could be simplified to Eq. (7):
q = q0 +ωC, (7)
The adsorption isotherms in Fig. 1 are more linear at low porosities.The reason for this is that is smaller than one, and this will leadto a reduction of the contribution of the true nonlinear adsorp-tion isotherm to the observed adsorption isotherm. Furthermorethe extra linear constant ω is positive and will have an increas-ingly large contribution on the observed adsorption isotherm whenthe porosity is decreasing, especially at high concentrations. Foroverestimated porosities, is larger than 1 and ω is negative. Theinfluence on the Scatchard plot is evident, where a negative ω willlead to a convex plot, and a positive ω will lead to a concave plot.
In Fig. 2 all three S2 and Fisher parameter curves merge at higherporosity values, indicating that none of the Langmuir models areproperly describing high porosity data. In the previous study theMoreau model was used for an overestimated hold-up volume witha Langmuir model as true adsorption isotherm. It was shown thatthe true adsorption isotherm modified as in Eq. (7) describes all sit-uations excellent. The reason for this is that none of the adsorption
isotherm parameters were allowed to be negative.For an underestimated column porosity the true model shouldnot be able to predict underestimated porosity as good as if we usea model that is more complicated like for example the bi-Langmuir(if the true model is a the Langmuir). The reason for this is that theextra linear term could be estimated by adding an extra Langmuir(Langmuir becomes bi-Langmuir etc.) and the equilibrium constant(K) for the extra site should be extremely low so that KC � 1. For thiscase the extra Langmuir term can be approximated with a linearadsorption isotherm ωC. This trend is evident in Table 2 for the tri-Langmuir model at the porosity 1 and 50%, were the third site has avery low equilibrium constant, and extremely high capacity. The tri-Langmuir model can better describe low porosity data as comparedto the bi-Langmuir model. For this reason we could not find anyglobal optimum in Fig. 2 for the tri-Langmuir model because theextra Langmuir mode compensates for the linear error-term ωC.
As a guideline we could state that modifying the true adsorptionisotherm model with Eq. (7) will lead to perfect model predic-tion, independent of selected porosity. This could be very useful forprocess chromatography and separation systems were the experi-mental estimation of porosity is very difficult. To test this, the raw
mato
J. Samuelsson et al. / J. Chroadsorption isotherm data for porosity values from 1 to 87% were fit-ted to a bi-Langmuir model that was modified according to Eq. (7).The modified bi-Langmuir has a higher Fisher parameter as com-pared to the bi-Langmuir model at all porosities except between 68and 70% which is due to the fact that it contains one more adjustableparameter, thus leading to a larger l value (see Eq. (4)). The modi-fied model is significantly better at porosities larger than 74%. Theaverage fitted parameters for bi-Langmuir parameters are a1 = 4.20,K1 = 0.0243 L/g, a2 = 3.39, K2 = 0.117 L/g and porosity of 69.6%. Thedifference in adsorption energy between the sites is 3.8 kJ/mol.The model used to predict chromatographic band profiles and theoverlap were the same (99%) for all porosities between 1 and 87%,indicating a near perfect model agreement.
Because the adsorption energy between the sites is rather smalla Langmuir model could describe the separation system rathergood, but not excellent. To investigate if the experimental datacould be described with a Langmuir adsorption isotherm with anextra linear term, to introduce some flexibility into the adsorptionisotherm, we fitted the data to Langmuir model modified with Eq.(7) to the data. The model parameters were a = 8.51, K = 0.0417 L/gand porosity of 76.1%. The modified Langmuir model was also usedto predict profiles and with an average overlap of 91% for porositiesbetween 1 and 87%.
A more engineering approach would have been to present thedata in this study as independent on porosity. This could be doneby studying the adsorbed amount Q instead of the stationary phaseconcentration q; utilizing the relation q = Q/Va. The problem withthe error in the determination of the porosity still remains becausethe determined stationary phase volume is dependent on the esti-mated porosity. To study this, we define the true adsorbed amount(Q0) and determined adsorbed amount (Q) as:
q = Q
(1 − ˛ε0)Vg, q0 = Q0
(1 − ε0)Vg. (8)
By inserting Eq. (8) into Eq. (7), we get the relationship of observedamount to true adsorbed amount:
Q = Q0 + Vgε0(1 − ˛)C, (9)
Eq. (9) is a simpler expression than Eq. (7) because it does not con-tain the constant before Q0. However, the result will still be that weneed to add an extra linear term to compensate for the error in theporosity.
4. Conclusion
An underestimation of the column porosity has little or no effecton the prediction of band profiles while an overestimation leads toquite large errors. This is an important finding and useful for situa-tions when no suitable hold-up volume marker is readily available.If the adsorption model is flexible then the error in the hold-up vol-ume is quite insignificant. We advice that, in a process-scale unitthe true (or assumed) adsorption isotherm is used, but added withan extra linear term that is allowed to be negative (cf. Eq. (7)). Withsuch an adsorption isotherm band profiles can be accurately pre-dicted with an arbitrary chosen porosity. With this modificationwe can predict all experimental data and get an excellent overlapbetween experimental and predicted profiles for all porosities inan industrial process-scale setting.
In our previous study we showed that the adsorption energydistribution produces an extra low energy site for underestimatedporosity if the monolayer saturation capacity is high enough. Inthis investigation we noted that we could loose the high energyabsorption site due to an overestimated porosity because then theestimated adsorption energy increases, and if we are not careful
gr. A 1194 (2008) 205–212 211
then the high energy site may be unresolved. This may lead to asimpler model that is found in the investigation. For these reasons itis important to test different porosities to investigate the possibilityto resolve the low energy site, or eventually gain back the highenergy site.
It is obvious that an adsorption isotherm that better representsthe experimental data also results in a better fit or more accurateprediction of chromatographic elution band profiles. It is howeversurprising that the adsorption isotherm model does not have tobe the correct one. The experimental adsorption data fit better toanother wrong adsorption isotherm model with a wrong porosity.In this case the porosity or the hold-up volume just becomes anadditional parameter in the fitting problem, besides the coefficientsin the isotherm model.
In this study we examined if it is feasible to estimate the columnhold-up volume from the measurement of equilibrium isothermdata assuming the adsorption isotherm model is known. From atheoretical point of view this is possible but more practical it couldbe hard to know the adsorption isotherm in advance. The resultindicates at least that a model that describes the system better(probably bi-Langmuir in this case) has an optimum porosity thatis closer to the experimental determined one than a model thatdoes not describe the adsorption process as good (Langmuir in thiscase). Thus, the determined porosity could be used as an extra toolto indicate that the correct model is selected.
5. Nomenclature
ai, Ki numerical coefficients of the Langmuir isothermC solute concentration in the mobile phase (g/L)Ci solute concentration in the mobile phase of the ith con-
centration step (g/L)Ci+1 solute concentration in the mobile phase of the [i + 1]st
concentration step (g/L)q solute concentration in the stationary phase (g/L)qfit fitted stationary phase concentrationqi solute concentration in the stationary phase of the ith
concentration step (g/L)qs solute saturation capacity (g/L)qi+1 solute concentration in the stationary phase of the [i + 1]th
concentration step (g/L)q0 true stationary phase concentrationVg total (geometrical) volume of the empty column (mL)
VR retention volume (mL)VR i+1 retention volume of the [i + 1]st concentration step (mL)V0 column hold-up volume (mL)Greek symbols˛ error coefficientε total column porosityε0 true total column porosity� volume ratio of stationary to mobile phase , ω error coefficients in Eq. (7)
Acknowledgment
The work was supported by a grant (TF) from the SwedishResearch Council (VR) for the project “Fundamental Studies onMolecular Interactions Aimed at Preparative Separations andBiospecific Measurements”.
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