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Journal of Chromatography A, 1331 (2014) 52– 60

Contents lists available at ScienceDirect

Journal of Chromatography A

j our nal homep age: www.elsev ier .com/ locate /chroma

olecular theory of size exclusion chromatography for wideore size distributions

nnamária Sepseya, Ivett Bacskayb, Attila Felingera,b,∗

MTA–PTE Molecular Interactions in Separation Science Research Group, Ifjúság útja 6, H-7624 Pécs, HungaryDepartment of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary

r t i c l e i n f o

rticle history:eceived 14 November 2013eceived in revised form 7 January 2014ccepted 9 January 2014vailable online 16 January 2014

eywords:

a b s t r a c t

Chromatographic processes can conveniently be modeled at a microscopic level using the moleculartheory of chromatography. This molecular or microscopic theory is completely general; therefore it canbe used for any chromatographic process such as adsorption, partition, ion-exchange or size exclusionchromatography. The molecular theory of chromatography allows taking into account the kinetics of thepore ingress and egress processes, the heterogeneity of the pore sizes and polymer polydispersion. Inthis work, we assume that the pore size in the stationary phase of chromatographic columns is governed

ore size distributiontochastic theoryize exclusion chromatography

by a wide lognormal distribution. This property is integrated into the molecular model of size exclusionchromatography and the moments of the elution profiles were calculated for several kinds of pore struc-ture. Our results demonstrate that wide pore size distributions have strong influence on the retentionproperties (retention time, peak width, and peak shape) of macromolecules. The novel model allows usto estimate the real pore size distribution of commonly used HPLC stationary phases, and the effect ofthis distribution on the size exclusion process.

. Introduction

Modern porous or core–shell stationary phases may exhibit aomentous pore size distribution. Experimental data confirm that

he size of mesopores can cover a rather wide range [1,2]. Theature and the breadth of pore size distributions have significant

mpact on the mass-transfer properties of stationary phases. Theeparation of macromolecules is particularly influenced by the poreize distribution, since their hindered diffusion in the pore networkives a critical contribution to band broadening.

Size exclusion chromatography (SEC) is one of the most widelysed techniques to determine the molecular size distribution ofolymers of any kind. The separation mechanism relies on the sizend shape of sample molecules relative to the size and shape of theores in the stationary phase particles. Because we do not exactlynow the structure and the dimensions of the porous media, theetermination of the molecular mass relies on a calibration stepased on the behavior of well-known monodisperse polymers in

he columns containing porous stationary phase particles.

The study of the pore structure of the stationary phasessed in liquid chromatography has been of great interest among

∗ Corresponding author at: Department of Analytical and Environmental Chem-stry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624écs, Hungary. Tel.: +36 72 501500x24582; fax: +36 72 501518.

E-mail address: felinger@ttk.pte.hu (A. Felinger).

021-9673/$ – see front matter © 2014 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.chroma.2014.01.017

© 2014 Elsevier B.V. All rights reserved.

chromatographers in the last decades [3–6]. Kubín has modeledpore size irregularity with a diffusion model, assuming thatmolecules can penetrate into the porous particles to a distance thatdepends on the size of the molecules [6].

There are a number of methods for determining relevant infor-mation about the porous media such as low-temperature nitrogenadsorption, mercury intrusion, microscopy and solute exclusion.These techniques are either too expensive and/or they destroy thechromatographic column (so they cannot be used for any furtheranalysis), or they do not give relevant information about all finedetails.

The influence of pore size distribution on separation efficiencycan conveniently be studied with inverse size exclusion chromatog-raphy. Inverse size exclusion chromatography is used to deriveinformation about the structure of the pores of the packing materialfrom the retention data of a series of known analytes, for instance,polymers of narrow molecular mass distribution and known aver-age molecular mass [7].

In this study, we develop a model that integrates the pore sizedistribution into the microscopic theory of size exclusion chro-matography. With this model one is able to determine the influenceof the breadth of pore size distribution on retention properties andefficiency.

The molecular, or stochastic theory of chromatography is amicroscopic model introduced by Giddings and Eyring in 1955[8]. That theory uses random variables and probabilistic terms todescribe the migration of the molecules along the chromatographic

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A. Sepsey et al. / J. Chrom

olumn. The stochastic approaches may become very complex toork with for situations except the most simple case of adsorption,

o the use of this theory was not convenient until the characteristicunction (CF) approach was introduced to this field [9]. The use ofharacteristic functions made the stochastic theory of chromatog-aphy simpler even for complicated cases such as heterogeneousdsorption chromatography.

The stochastic theory seems rather suitable for describing sizexclusion chromatography among the basic theories of chromatog-aphy. Size exclusion chromatography is indeed based on theandom migration of molecules, where randomly occurring entrap-ent and release of molecules in the mesopores of the stationary

hase builds up the separation process [10–13].

. Theory

.1. Classical size exclusion considerations

Size exclusion chromatography (SEC) has been investigatedrom many point of views, which led to an asystematic nomen-lature. To eliminate the misapprehensions, all techniques that areounded on the size exclusion separation mechanism, such as gelltration, molecular sieve chromatography, gel chromatographynd gel-permeation, are parts of SEC. However the only differenceetween the individual ones is in the samples and solutions used.

Size exclusion chromatography is a well-known separationethod where the main retention mechanism is the size exclusion

ffect. However there are a lot of other mechanisms responsi-le for the migration of the molecules along the column, such asydrodynamic and stress-induced diffusion, the polarization effect,ultipath, enthalpic and soft-body interactions and the so-calledall effect, they can be ignored in almost every case. In an ideal case

here is no interaction between the molecules and the stationaryhase particles or the mobile phase particles.

In SEC, the sample components are separated according to theirize and molecular mass. The separation is governed by entropynly; the retention depends on the relative penetration of the sam-le molecules to the pores. The stationary phase of a SEC column

s always a mechanically stable porous media that can be built upn a rigid carrier such as silica or the whole stationary phase isade from this porous material. The molecules traveling along the

olumn in the mobile phase can enter the pores if the size of theore is larger than that of the dimensions of the molecule. How-ver it is still uncertain which exact size parameter determines theeparation [14]. In general it is accepted that the gyration radiusr diameter of the molecule is used to determine whether or nothe molecule can enter the pore. The mobile phase in the poress stagnant; the molecules can migrate in the pores only by diffu-ion.

Two molecule sizes have special significance in SEC: the sizef the completely permeable particle and the size of the barelyxcluded particle. The completely permeable particle (indicated by

subscript perm in the equations) is small enough to visit all theores so that both the stagnant mobile phase in the pores of the sta-ionary phase and the moving zone of the mobile phase betweenhe stationary phase particles is completely accessible for it. If the

olecule is too large to enter the pores, it will be excluded (indi-ated by a subscript excl in the equations). These molecules cannly wander in the moving mobile phase and have access only tohe interstitial volume of the mobile phase between the stationary

hase particles. The excluded molecules elute at the void volumeV0). According to this mechanism, we can obtain information ofhe size, shape, aggregation state or kinetics of the ligand–polymerinding of the molecules investigated.

r. A 1331 (2014) 52– 60 53

The partition coefficient can easily be calculated by means of theretention times of the above mentioned and the unknown particlesusing the following equation:

KSEC = t − texcl

tperm − texcl, (1)

where the numerator stands for time spent by the investigatedmolecule in the pores of the stationary phase particles and thedenominator indicates the residence time spent by the completelypermeable particle in the pores. The partition coefficient can berewritten as

KSEC = tp

tp,perm(2)

where the subscript p indicates the time spent in the pores by themolecules.

The partition coefficient strongly depends on the ratio of thesize of the migrating particle to the size of the pore. The partitioncoefficient is in a widely used retention model [10,11,15] definedusing the size of the molecule investigated and the size of the poreof the stationary phase used as follows:

KSEC ={

(1 − �)m if 0 ≤ � ≤ 1

0 if � > 1(3)

where m is a constant whose value depends on the pore shape,and � is the size of the molecule relative to the pore size. The sizeparameter � can be defined as

� = rG

rp(4)

where rG stands for the gyration radius of the molecule while rp

indicates the radius of the pore opening. The retention depends onthe hydrodynamic radius or the gyration radius of the molecules,which can be changed by the hydration state and the shape.

2.2. Stochastic theory of SEC

The stochastic theory of chromatography, in which the chro-matographic process is modeled at a molecular level was developedin 1955 by Giddings and Eyring for adsorption chromatography[8]. The theory assumes that, if we ignore the axial dispersion,the number of adsorption and desorption steps is determined bya Poisson process and the time that a molecule spends bound tothe stationary phase (residence or sojourn time) is determined byan exponential distribution. The stochastic theory is completelyindependent of the physical–chemical mechanisms responsible forthe retention; therefore it can be used in any field of chromatog-raphy. Accordingly, the model has been extended and improvedfor several chromatographic methods such as adsorption, parti-tion, ion-exchange or size exclusion chromatography (adapted toSEC by Carmichael [16–19]). The real breakthrough in develop-ing the theory was the introduction of the characteristic functionapproach. Later the effects of the mobile phase dispersion wereintroduced (stochastic-dispersive model) and they involved anincreasing number of parameters specified in the description of thesystem. A detailed description of the stochastic theory of SEC via thecharacteristic function method was introduced by Dondi et al. [10].

The simplest theory of size exclusion chromatography assumesthat a molecule of a certain size enters and leaves the pores n timeson average during the migration along the column and spends �p

time on average in a single pore. After leaving a pore, the moleculespends �m time on average in the mobile phase before entering

another pore. All these variables are random quantities, thereforeeach molecule has an individual path while migrating along thecolumn. However, the molecules of the same size behave in a sim-ilar manner, because of the anomalies in the retention paths of the

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4 A. Sepsey et al. / J. Chrom

olecules we observe a nearly Gaussian curve as chromatographiceak. The observed retention time of a peak is the mean of the dis-ribution of the individual retention times and the width of the peaks described by the standard deviation.

The average number of the pore ingress steps (if � takes a valueetween 0 and 1) can be written as

p = nperm(1 − �)me , (5)

nd the residence time in the pores will be:

p = �perm(1 − �)mp , (6)

here me and mp are constants depending on the ingress and thegress processes, respectively. Both np and �p take the value 0 if

> 1.As Dondi et al. showed earlier [10], parameters me, mp and m

re related as

= me + mp (7)

Thus it can be seen that both the pore ingress and egress pro-esses affect the selectivity of SEC. The relationship between m, me,nd mp can be re-expressed when we introduce parameter toharacterize the relative contribution of the pore egress process tohe overall size exclusion effect [12]:

= mp

m. (8)

The characteristic function is the main equation of the stochasticheory. It is the Fourier transform of the elution profile and it con-ains all the information about the separation process. In this simplease the following characteristic function describes the system theest:

(ω) = exp

{np

[1

1 − iω�p− 1

]}, (9)

here i is the imaginary unit, np is the average number of the porengress and egress steps, �p is the average time spent by a moleculen a single pore and ω is an auxiliary real variable (frequency). Inhis case the probability density function (the time-domain signal)an be written as the inverse Fourier transform of the characteristicunction as:

(t) =√

np

t�pe−t/�p−np I1

(√4npt

�p

), (10)

here I1 is a modified Bessel function of the first kind and firstrder.

There is a simple relationship between the characteristic func-ion and the moments about the origin. The kth moment of thehromatographic peak can be calculated from the characteristicunction (Eq. (9)) by the moment theorem of the Fourier transforms

k = i−k

[dk�(ω)

dtk

]ω=0

. (11)

It is well-known that the first moment is the mean residenceime and the second central moment is the variance of the observedhromatographic peak. By calculating these moments using thequation above we obtain:

1 = np�p (12)

nd

′2 = 2np�2

p . (13)

he third central moment gives information about the peak sym-etry:

′3 = 6np�3

p . (14)

. A 1331 (2014) 52– 60

The above equations are only valid if we assume that the poresize in the stationary phase particles is uniform.

For heterogeneous kinetics, the simplest stochastic model can-not be employed. Cavazzini et al. extended the stochastic model ofchromatography to the case where the stationary phase consists ofmore than one types of adsorption sites and for the case when theadsorption energy of the sites is determined by a distribution [20].It was assumed that if there are several types of adsorption sites inthe column, the molecules bind with a certain probability to eachof them. Thus, the probability density function describing the peakshape can be obtained as the probability-weighted convolution ofthe probability density functions of the different sites. For exam-ple, for two different adsorption sites the following characteristicfunction was obtained:

�(ω) = exp{

n1

[1

1 − iω�1− 1

]}exp

{n2

[1

1 − iω�2− 1

]}(15)

The corresponding peak shape is:

f (t) =(√

n1

t�1e−t/�1−n1 I1

(√4n1t

�1

))

∗(√

n2

t�2e−t/�2−n2 I1

(√4n2t

�2

)), (16)

where the subscripts refer to the respective sites and the sign *stands for convolution. The calculation of the moments and that ofthe peak profile is difficult in time domain – if it is possible at all,so it cannot practically be used in this form.

3. Experiments

All experiments were carried out with the software packageMathematica 9 (Wolfram Research). The elution profiles wereobtained via numerical inverse Fourier transform using 1024points. Except for the case where we illustrate the effect of param-eter �, in all other cases �perm, nperm and rp,0 were arbitrarily setto 1 s, 2000 and 12.434 nm, respectively. To illustrate the effect ofchanging the parameter � on the elution profiles we used �perm = 0.1s.

4. Results and discussion

In size exclusion chromatography, the most important factor isthe pore size of the stationary phase, since the separation is basedon the size of the analyte molecules relative to the pore size.

In ideal SEC, because there is no physico-chemical interactionbetween the sample molecules and the stationary phase surface,the type of the silica and the chemical modification has no effecton the retention and on the selectivity. The size of the stationaryphase particles of the modern HPLC columns varies in a quite thinrange, and it has been recently demonstrated that there is no evi-dent correlation between the particle size distribution and columnefficiency [21]. The effect of the structure of the stationary phaseparticle (e.g. non-porous, fully porous particles, core–shell) on theseparation efficiency is quite significant, because diffusion withinthe particles has a strong impact on the brand broadening in allmodes of HPLC. The size exclusion effect on non-porous particlesis nonexistent, only the hydrodynamic effect is present. The fully-porous particles have a large pore volume where the molecules can

diffuse and macromolecules may spend long time there, thus slowpore diffusion gives rise to band broadening of the observed peaks.In the core–shell particles the pore volume is more limited and thediffusion times are shorter [22].

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A. Sepsey et al. / J. Chrom

Results obtained with low-temperature nitrogen adsorptiononfirmed that the presently commercially available HPLC columnso not have a uniform pore size, but contain pores in a relativelyide pore size range. A packing material which is marketed as a

00-A pore size stationary phase will definitely contain pores of 100nd of 300 A as well. This is an important aspect, because moleculesf a given size are not equally likely to enter each pore. Moleculeshat are small enough to enter the pores of 200 A may be excludedrom the pores of 100 A. This ultimately leads to the distortion inhe peak shapes in practice.

The pore size distribution (PSD) of the porous stationary phasearticles has been already investigated and modeled in the early980s by Knox and Scott [3].

The stochastic theory describes the chromatographic processt the molecular level so it is obvious to introduce the pore sizeistribution into that in order to obtain more relevant informationbout the retention properties. The concept introduced by Eq. (15)an be extended to multisite heterogeneous adsorption and theollowing characteristic function is obtained when m type of sitesre present, each with a relative abundance of pj, (j = 1, . . ., m) [20]:

(ω) = exp

⎧⎨⎩n

⎡⎣−1 +

m∑j=1

pj1

1 − iω�j

⎤⎦⎫⎬⎭ (17)

This equation can be extended for a continuous distribution ofites. When size exclusion chromatography is modeled and bothhe pore ingress and egress processes are influenced by the poreize distribution, the following characteristic function is obtained:

(ω, rp) = exp

[∫ ∞

rG

�(rp, rp,0, �)np

(1

1 − iω�p− 1

)drp

](18)

If all the pores were of the same size, one would obtain the char-cteristic function and the moments of the band profile as writteny Eqs. (9)–(11). However, when the pore size is governed by aistribution, the probability density function of the pore size dis-ribution, �(rp, rp,0, �), is included in the model. As a consequence,e replace the term rp in Eq. (4) by a probability density function

PDF). If we assume a lognormal distribution, which is shown by thexperimental evidence to describe the pore size distribution of thePLC packing materials, the following probability density function

s used:

(rp, rp,0, �) = 1√2�rp

exp

(−(

ln rp − ln rp,0)2

2�2

), (19)

here rp,0 and � represents the maximum and width of the lognor-al distribution, respectively. One should note that the true first

bsolute and second central moments of the lognormal distributionEq. (19)) are

1,lognorm = rp,0e�2/2, �′2,lognorm = r2

p,0e�2(

e�2 − 1)

(20)

For the sake of simplicity, however, further on we refer to rp,0nd � as the mean and the standard deviation of the lognormalistribution, respectively.

We obtain the characteristic function in the case of lognormalore size distribution when we combine Eqs. (18) and (19):[

1∫ ∞

np(rp)( (

ln rp − ln rp,0)2)

(ω) = exp √2� rG

rpexp −

2�2

×(

11 − iω�p(rp)

− 1

)drp

](21)

r. A 1331 (2014) 52– 60 55

Eq. (21) is the characteristic function of the peak shape for log-normal pore size distribution. It is important to note that both thenumber of pore entries and the individual sojourn times of themolecules in a pore depend on the size of the molecule relativeto the size of the pore. The respective relationships are given byEqs. (5) and (6).

The above characteristic function describes the peak shape inFourier domain for size exclusion chromatography when the poresizes are not uniform. The peak shape itself can be calculated as theinverse Fourier transform of �(ω).

Unfortunately, Eq. (21) cannot be evaluated analytically for thegeneral case. Nevertheless, the calculation of the moments is possi-ble. An analytical expression of the moments can only be obtainedif parameters me and mp are both integers. Intuition suggests andextensive data processing of SEC data confirms [12] that for theingress process me > 0 in Eq. (5) and for the egress process mp < 0 inEq. (6).

The first absolute moment as well as the second and the thirdcentral moments of the elution profile will be obtained from Eq.(21) using Eq. (11) as

�1 = nperm�perma, (22)

�′2 = 2nperm�2

perma, (23)

�′3 = 6nperm�3

perma, (24)

where parameter a is actually equal to the KSEC.

a = KSEC = 12

∑k=0

(−�)kek2�2

2

(

k

)erfc

(k�2 + ln �√

2�

). (25)

Parameter a – and thus KSEC – strongly depends on the poreshape. For the calculation of �1, = me + mp should be used and if = 1, the pore is slit shaped, if = 2 it is cylindrical and if = 3, thepore is either conical or spherical. The second central moment canbe calculated using = me + 2mp, and the third central moment byusing = me + 3mp.

Parameter a can only be calculated by Eq. (25) when > 0. Forinstance, when mp = −3 and me = 6, and thus m = 3, in the calculationof the third central moment = −3, and the analytical calculationof that moment is not possible with the equations written above,nevertheless, the first and the second moments can be still eval-uated. The first three moments can be calculated in all the caseswhen > −0.5. If < −0.5, for the calculation of the third momentone would get < 0 and it is not possible to use Eqs. (22)–(25).

One can always obtain the elution profile with the inverseFourier transform of Eq. (21) and calculate the moments by numer-ically integrating the peak profile.

The equations above demonstrate that pore size distributionwill have important consequences on retention time and peakshape. By plotting parameter a when m = 0 against � and � weobtain the relative pore accessibility (see Fig. 1). One can see inthat figure that in the case of monopores (� = 0), there is a sharpdistinction between the molecules that visit the pores and the onesthat are excluded. All the molecules that are smaller than � = 1, i.e.molecules for which rG < rp can visit all the pores. On the other hand,every molecule for which � > 1 is excluded from all the pores. How-ever, when a range of pore sizes are present in the stationary phase,(� > 0), there are pores that are accessible for the molecules largerthan rp,0 too. The broader the pore size distribution, the smootheris the transition between inclusion and exclusion. For the effect of� and � on the partition coefficient in case of m=1 (slit shaped pore

geometry), m=2 (cylindrical pore geometry) and m=3 (conical orspherical pore geometry), see Fig. 2.

To exploit the effect of the pore size distribution on the elu-tion profile, we calculated various chromatograms by changing the

56 A. Sepsey et al. / J. Chromatogr

Fe

ss

cwtwrowtittt

the large molecules (� > 0.5) is intensive. The larger the molecule,

Fi

ig. 1. Relative pore accessibility. The effect of the breadth of the PSD (�) and theffect of the size parameter (�) on the size-exclusion process.

tandard deviation of the pore size distribution and the size of theolute molecule relative to the pore size.

The effect of increasing the breadth of the distribution (�) on thehromatograms can be seen in Fig. 3 for a relatively large molecule,hen the relative sizing parameter is � = 0.95. This figure illus-

rates the positive effect of the PSD on the chromatographic process,here the solid lines represent � = 0.1, the long-dashed lines rep-

esent � = 0.5 and the short-dashed lines stand for � = 1. Dependingn the pore shape, the retention times of the profiles vary in a quiteide range. However, it can be seen in all cases, that the retention

ime increases and the skew of the elution profile decreases as �ncreases. If the sample molecules are very large, they can only in

he rarest case get inside a pore. However once they enter a pore,hey cannot escape from it and the molecule remains trapped athe same position as time passes (n is very small and � is very

00.20.40.60.81

a.

0.0 0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

ρ

K SEC

00.20.40.60.81

c.

0.0 0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

ρ

K SEC

σ = 0

σ = 1

σ = 0

σ = 1

ig. 2. The influence of the pore shape parameter (m) and the size of the molecule relativncluded (m = 0 i.e. relative pore accessibility), (b) slit shaped pores (m = 1), (c) cylindrical

. A 1331 (2014) 52– 60

large). Most of the molecules, however, cannot enter a pore andfor this reason they are unretained and elute at the void time (t0).This is best illustrated by the green solid line that stands for con-ical or spherical pore shapes (m = 3) and relatively small variance(� = 0.1) in Fig. 3. This is also demonstrated for the case of cylindri-cal pores, too, where we obtain an exponential decreasing line asthe elution profile when the pore size distribution is rather narrow.As the standard deviation of the pore size is increased, the elutionprofile becomes a Gaussian peak and if � = 1 one obtains the mostsymmetric profile for every kind of pore shape.

For smaller molecules (� = 0.1–0.5), the trends when � isincreased are rather different depending on the pore shapes. Incase of slit shaped pores, � has a more dominant effect on the KSECvalue than in the other two cases (cylindrical or conical pores) andit was also demonstrated in Fig. 2. For example if the particularmolecular size is � = 0.5 the retention time increases when m = 2or 3 and decreases when m = 1 as the value of � is increasing. Thechromatograms become more asymmetrical when � increases. Theincrease of asymmetry is most significant for m = 3. For smaller �values (such as � = 0.2) the retention time will decrease in the caseof all kind of pore shapes.

The effect of the relative size of the molecules, �, on the elutionprofile is of course most important for the retention time, i.e. for thefirst moment. The larger the �, the less included is the molecule. Bycalculating the skew of the peak profiles, we can see that the peaksbecome more asymmetrical while � increases. Chromatograms areplotted in Fig. 4 for different pore shapes and molecular sizes. Theretention times of the observed chromatogram changes consid-erably. We can conclude that the pore size distribution has littleinfluence on the behavior of the small molecules while its effect on

the harder it can enter the pores, but once it enters a pore it getsstuck there, remains at the same position as time passes and willnot emerge from the pore for a wile. That is why the peaks of large

00.20.40.60.81

b.

0.0 0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

ρ

K SEC

00.20.40.60.81

d.

0.0 0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

ρ

K SEC

σ = 1

σ = 0

σ = 1

σ = 1

e to the pore size (�) on the partition coefficient when (a) the pore geometry is not pores (m = 2) and (d) conical or spherical pores (m = 3).

A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52– 60 57

m 1, 0.1m 1, 0.5m 1, 1m 2, 0.1m 2, 0.5m 2, 1m 3, 0.1m 3, 0.5m 3, 1

0.0 0. 1 0. 2 0.3 0.40

5

10

15

20

Normalized retention time

Intensity

0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

σ

Skew

0.2 0. 4 0. 6 0. 80.00

0.05

0.10

0.15

0.20

σ

μ1

m=3

m=2

m=1 m=3

m=2

m=1

m=1m=1m=1

m=2m=2

m=2m=3

m=3

m=3

Fig. 3. The effect of the breadth of the PSD (�) on the calculated chromatograms for slit shaped pores, for cylindrical pores and for spherical or conical pores when the sizeparameter is � = 0.95. In the inserts the skew and the first absolute moment of the chromatograms are plotted.

m 1, 0. 1m 1, 0. 5m 1, 1m 2, 0. 1m 2, 0. 5m 2, 1m 3, 0. 1m 3, 0. 5m 3, 1

0. 0 0. 2 0. 4 0.6 0.8 1. 0 1.2 1.40

50

100

150

Normalized retention time

Intensity

0.2 0.4 0. 6 0.8 1. 00

20

40

60

Ρ

Skew

0.2 0.4 0. 6 0.8 1.0

0.2

0.4

0.6

0.8

Ρ

Μ1

m=3

m=2

m=1

m=1

m=2

m=3

m=3m=3

m=3

m=2m=2

m=2

m=1m=1

m=1

F t shaped pores, for cylindrical pores and for spherical or conical pores when the breadtho romatograms are plotted.

ms

pvawteAnoirsm

Table 1The effect of the relative contribution of the pore egress process to the overall sizeexclusion effect (˛) on the value of the pore ingress and egress depending constants(mp and me) and on the value of the calculated first absolute, second and third centralmoments (�1, �′

2 and �′3) in case of slit shaped pores (m = 1).

m mp me me + mp me + 2mp me + 3mp

1 −0.1 −0.1 1.1 1 0.9 0.81 −0.2 −0.2 1.2 1 0.8 0.61 −0.3 −0.3 1.3 1 0.7 0.41 −0.4 −0.4 1.4 1 0.6 0.21 −0.5 −0.5 1.5 1 0.5 01 −0.6 −0.6 1.6 1 0.4 −0.21 −0.7 −0.7 1.7 1 0.3 −0.41 −0.8 −0.8 1.8 1 0.2 −0.61 −0.9 −0.9 1.9 1 0.1 −0.8

ig. 4. The effect of the size parameter (�) on the calculated chromatograms for slif the PSD is � = 0.5. In the inserts the skew and the first absolute moment of the ch

olecules (i.e. with high � values) become extremely broad andkewed.

Parameter expresses the relation of the pore ingress and egressrocesses (see Eq. (8)). We calculated elution profiles for various ˛alues and for the mentioned pore shapes. The observed profilesre demonstrated in Fig. 5 where the relative size parameter (�)as set to 0.95 and the variance of the PSD was � = 0.5. In this case

he size of the sample molecules is very close to the pore size, how-ver the effects of changing the parameter is very meaningful.s it could be expected, the retention time (the first moment) doesot depend on ˛. The second central moment and the skew of thebserved profiles increase as decreases, and this is demonstratedn the subfigures of Fig. 5. In the case of slit shaped pores (m = 1), the

elation of the ingress and egress processes does not affect the peakymmetry as much as in the other cases where the peaks becomeore asymmetrical as increases. The same tendency could be

1 −1 −1 2 1 0 −1

58 A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52– 60

m=2

m=1

m 1, 0.1m 1, 0.5m 1, 1m 2, 0. 1m 2, 0.5m 2, 1m 3, 0.1m 3, 0. 5m 3, 1

0.00 0.05 0.10 0.15 0.2 0 0.250

5

10

15

20

25

30

Normalized retention time

Intensity

1.0 0.8 0.6 0.4 0. 2

0.51.01.52.02.53.03.5

α

Skew

1.0 0.8 0.6 0.4 0.2

0.00010.00020.00030.00040.0005

α

µ 2

m=2

m=3 m=1

m=3

m=1

m=2

m=3

F rall sic .95, am

oa

eio

tiafc

Fp

ig. 5. The effect of the relative contribution of the pore egress process to the oveylindrical pores and for spherical or conical pores when the size parameter is � = 0oment of the chromatograms are plotted.

bserved for all value of �: the profiles become more asymmetricals decreases.

From the results presented in Figs. 2–5 we can conclude that theffect of the parameters studied (i.e. �, �, ˛) is the most intensiven case of conical or spherical pores and the least intensive in casef slit shaped pores.

We investigated how the peak resolution and the efficiency ofhe separation are affected by the pore size distribution. The analyt-

cal calculation of the relative resolution is only possible if the firstbsolute and the second central moments of the peak are calculatedor integer me and mp values. There are only a few situations thisalculation can be done. Table 1 helps the reader to consider

0.0 0. 2 0. 4 0.6 0. 8 1.00. 4

0. 6

0. 8

1. 0

1. 2

σ

Relativeresolution

0. 0 0. 2 0. 4 0.6 0. 8 1.00. 60. 81. 01. 21. 41. 61. 82. 0

σ

Relativeresolution

ρ1 0.1, ρ2 0.2 ρρ1 0.4, ρ2 0.5 ρρ1 0.7, ρ2 0.8 ρ

a.

c.

ig. 6. Relative resolution for calculated chromatograms of differing size parameters (�rocess to the overall size exclusion effect (˛) were varied as (a) m = 1 and = −1, (b) m =

ze exclusion effect (˛) on the calculated chromatograms for slit shaped pores, fornd the breadth of the PSD is � = 0.5. In the inserts the skew and the second central

whether the moments and the resolution can analytically be calcu-lated, or not.

The change of the resolution and the number of theoreticalplates with the breadth of pore size distribution are reported inFigs. 6 and 7.

Several important conclusions can be drawn from the data pre-sented in the figures. The curves can be divided into two casesbased on the molecule size. In the first one, the molecules are

small enough to separate them by size exclusion chromatographyand neither the relative resolution nor the number of theoreticalplates is affected by the pore size distribution. This can be seen inFigs. 6 and 7, where the plots referring to the small molecules hardly

0. 0 0.2 0.4 0. 6 0. 8 1.0

0.60.70.80.91.0

σ

Relativeresolution

0.0 0.2 0.4 0.6 0.8 1. 0024681012

σ

Relativeresolution

1 0.2, ρ2 0.3 ρ1 0.3, ρ2 0.41 0.5, ρ2 0.6 ρ1 0.6, ρ2 0.71 0.8, ρ2 0.9 ρ1 0.9, ρ2 0.95

b.

d.

). The pore shape parameter (m) and the relative contribution of the pore egress 2 and = −0.5, (c) m = 2 and = −1 and (d) m = 3 and = −1.

A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52– 60 59

0. 0 0. 2 0.4 0.6 0. 8 1. 00

500

1000

1500

σ

N

0. 0 0.2 0. 4 0.6 0.8 1.00

200400600800100012001400

σ

N

0. 0 0. 2 0.4 0.6 0. 8 1. 00

20040060080010001200

σ

N

0.0 0.2 0.4 0.6 0.8 1.0

0

20 0

40 0

60 0

80 0

1000

σ

N

ρ2 0.1 ρ2 0.2ρ2 0.3 ρ2 0.4ρ2 0.5 ρ2 0.6ρ2 0.7 ρ2 0.8ρ2 0.9 ρ2 1a. b.

c. d.

F parame = −1,

stlsa

fioctii

5

epTbfTsia

tcwgi

trsima

[

[

[

[

ig. 7. Number of theoretical plates for calculated chromatograms of differing sizegress process to the overall size exclusion effect (˛) were varied as (a); m = 1 and ˛

how any change as � increases, while the relative resolution andhe number of theoretical plates for the larger and especially for theargest molecules significantly increase as � increases. The effect istronger as the value of m increases (i.e. in the case of cylindricalnd spherical or conical pores relative to slit shaped pores).

This observation drives us to more general conclusions in theeld of chromatography, because the size exclusion effect is notnly in size exclusion chromatography important, but also in otherhromatographic methods, such as reversed phase or HILIC separa-ions. In those cases, the hindered pore diffusion of macromoleculess very important, and the size exclusion process becomes moremportant than the other effects of the retention.

. Conclusions

The stochastic theory of size exclusion chromatography wasxtended to wide pore size distribution for a number of variousore geometries (slit shaped, cylindrical, and conical or spherical).he statistical moments of the peak profiles can easily be calculatedy assuming a pore shape. The calculation of the chromatograms iseasible by inverse Fourier transform of the characteristic function.he trend we observe for the calculated chromatograms empha-izes the significance of pore size distribution: the PSD has strongnfluence on the retention properties (retention time, peak width,nd peak shape) of macromolecules.

Eq. (25) summarizes how pore size distribution affects the par-ition coefficient in size exclusion chromatography. That equationan serve as the basis for the experimental determination of theidth of pore size distribution when KSEC is plotted against the

yration radius of polymer molecules. This is going to be exploitedn a forthcoming study [23].

However, inverse size exclusion chromatography (ISEC) is usedo derive information about the pore structure of the packing mate-ial from the retention data of series of known analytes, is was

hown that it does not give appropriate characterization of therregularly shaped porous materials [24]. We truly believe that our

odel – which contains information about both the pore geometrynd the distribution of the pore sizes – is usable to develop ISEC and

[[

[[

eters (�). The pore shape parameter (m) and the relative contribution of the pore (b); m = 2 and = −0.5, (c); m = 2 and = −1 and (d); m = 3 and = −1.

so to obtain relevant information from the pore structure by non-destructive ISEC measurements. The experimental chromatogramsmay be characterized by pore geometry contributions and so theeffect of different types of pores could be investigated.

For the separation of macromolecules, the wide pore size dis-tribution will increase retention and efficiency. Therefore in allmodes of liquid chromatography, the efficient separation of macro-molecules calls for a broad pore size distribution.

Acknowledgements

This research was realized in the frames of TÁMOP 4.2.4. A/2-11-1-2012-0001 “National Excellence Program – Elaborating andoperating an inland student and researcher personal support sys-tem.” The project was subsidized by the European Union andco-financed by the European Social Fund.

The work was supported in part by the grants TÁMOP-4.2.2. A-11/1/KONV-2012-0065 and OTKA K 106044.

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