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Molecular theory of size exclusion chromatography for wide pore size distributions
Journal of Chromatography A, 1331 (2014) 52–60
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Molecular theory of size exclusion chromatography for wide pore size distributions Annamária Sepsey a , Ivett Bacskay b , Attila Felinger a,b,∗ a b
MTA–PTE Molecular Interactions in Separation Science Research Group, Ifjúság útja 6, H-7624 Pécs, Hungary Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary
a r t i c l e
i n f o
Article history: Received 14 November 2013 Received in revised form 7 January 2014 Accepted 9 January 2014 Available online 16 January 2014 Keywords: Pore size distribution Stochastic theory Size exclusion chromatography
a b s t r a c t Chromatographic processes can conveniently be modeled at a microscopic level using the molecular theory of chromatography. This molecular or microscopic theory is completely general; therefore it can be used for any chromatographic process such as adsorption, partition, ion-exchange or size exclusion chromatography. The molecular theory of chromatography allows taking into account the kinetics of the pore ingress and egress processes, the heterogeneity of the pore sizes and polymer polydispersion. In this work, we assume that the pore size in the stationary phase of chromatographic columns is governed by a wide lognormal distribution. This property is integrated into the molecular model of size exclusion chromatography and the moments of the elution profiles were calculated for several kinds of pore structure. Our results demonstrate that wide pore size distributions have strong influence on the retention properties (retention time, peak width, and peak shape) of macromolecules. The novel model allows us to estimate the real pore size distribution of commonly used HPLC stationary phases, and the effect of this distribution on the size exclusion process. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Modern porous or core–shell stationary phases may exhibit a momentous pore size distribution. Experimental data confirm that the size of mesopores can cover a rather wide range [1,2]. The nature and the breadth of pore size distributions have significant impact on the mass-transfer properties of stationary phases. The separation of macromolecules is particularly influenced by the pore size distribution, since their hindered diffusion in the pore network gives a critical contribution to band broadening. Size exclusion chromatography (SEC) is one of the most widely used techniques to determine the molecular size distribution of polymers of any kind. The separation mechanism relies on the size and shape of sample molecules relative to the size and shape of the pores in the stationary phase particles. Because we do not exactly know the structure and the dimensions of the porous media, the determination of the molecular mass relies on a calibration step based on the behavior of well-known monodisperse polymers in the columns containing porous stationary phase particles. The study of the pore structure of the stationary phases used in liquid chromatography has been of great interest among
∗ Corresponding author at: Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary. Tel.: +36 72 501500x24582; fax: +36 72 501518. E-mail address: [email protected] (A. Felinger). 0021-9673/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chroma.2014.01.017
chromatographers in the last decades [3–6]. Kubín has modeled pore size irregularity with a diffusion model, assuming that molecules can penetrate into the porous particles to a distance that depends on the size of the molecules [6]. There are a number of methods for determining relevant information about the porous media such as low-temperature nitrogen adsorption, mercury intrusion, microscopy and solute exclusion. These techniques are either too expensive and/or they destroy the chromatographic column (so they cannot be used for any further analysis), or they do not give relevant information about all fine details. The influence of pore size distribution on separation efficiency can conveniently be studied with inverse size exclusion chromatography. Inverse size exclusion chromatography is used to derive information about the structure of the pores of the packing material from the retention data of a series of known analytes, for instance, polymers of narrow molecular mass distribution and known average molecular mass [7]. In this study, we develop a model that integrates the pore size distribution into the microscopic theory of size exclusion chromatography. With this model one is able to determine the influence of the breadth of pore size distribution on retention properties and efficiency. The molecular, or stochastic theory of chromatography is a microscopic model introduced by Giddings and Eyring in 1955 [8]. That theory uses random variables and probabilistic terms to describe the migration of the molecules along the chromatographic
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
column. The stochastic approaches may become very complex to work with for situations except the most simple case of adsorption, so the use of this theory was not convenient until the characteristic function (CF) approach was introduced to this field [9]. The use of characteristic functions made the stochastic theory of chromatography simpler even for complicated cases such as heterogeneous adsorption chromatography. The stochastic theory seems rather suitable for describing size exclusion chromatography among the basic theories of chromatography. Size exclusion chromatography is indeed based on the random migration of molecules, where randomly occurring entrapment and release of molecules in the mesopores of the stationary phase builds up the separation process [10–13].
2. Theory 2.1. Classical size exclusion considerations Size exclusion chromatography (SEC) has been investigated from many point of views, which led to an asystematic nomenclature. To eliminate the misapprehensions, all techniques that are founded on the size exclusion separation mechanism, such as gel filtration, molecular sieve chromatography, gel chromatography and gel-permeation, are parts of SEC. However the only difference between the individual ones is in the samples and solutions used. Size exclusion chromatography is a well-known separation method where the main retention mechanism is the size exclusion effect. However there are a lot of other mechanisms responsible for the migration of the molecules along the column, such as hydrodynamic and stress-induced diffusion, the polarization effect, multipath, enthalpic and soft-body interactions and the so-called wall effect, they can be ignored in almost every case. In an ideal case there is no interaction between the molecules and the stationary phase particles or the mobile phase particles. In SEC, the sample components are separated according to their size and molecular mass. The separation is governed by entropy only; the retention depends on the relative penetration of the sample molecules to the pores. The stationary phase of a SEC column is always a mechanically stable porous media that can be built up on a rigid carrier such as silica or the whole stationary phase is made from this porous material. The molecules traveling along the column in the mobile phase can enter the pores if the size of the pore is larger than that of the dimensions of the molecule. However it is still uncertain which exact size parameter determines the separation [14]. In general it is accepted that the gyration radius or diameter of the molecule is used to determine whether or not the molecule can enter the pore. The mobile phase in the pores is stagnant; the molecules can migrate in the pores only by diffusion. Two molecule sizes have special significance in SEC: the size of the completely permeable particle and the size of the barely excluded particle. The completely permeable particle (indicated by a subscript perm in the equations) is small enough to visit all the pores so that both the stagnant mobile phase in the pores of the stationary phase and the moving zone of the mobile phase between the stationary phase particles is completely accessible for it. If the molecule is too large to enter the pores, it will be excluded (indicated by a subscript excl in the equations). These molecules can only wander in the moving mobile phase and have access only to the interstitial volume of the mobile phase between the stationary phase particles. The excluded molecules elute at the void volume (V0 ). According to this mechanism, we can obtain information of the size, shape, aggregation state or kinetics of the ligand–polymer binding of the molecules investigated.
53
The partition coefficient can easily be calculated by means of the retention times of the above mentioned and the unknown particles using the following equation: KSEC =
t − texcl , tperm − texcl
(1)
where the numerator stands for time spent by the investigated molecule in the pores of the stationary phase particles and the denominator indicates the residence time spent by the completely permeable particle in the pores. The partition coefficient can be rewritten as tp KSEC = (2) tp,perm where the subscript p indicates the time spent in the pores by the molecules. The partition coefficient strongly depends on the ratio of the size of the migrating particle to the size of the pore. The partition coefficient is in a widely used retention model [10,11,15] defined using the size of the molecule investigated and the size of the pore of 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 size parameter 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 on the 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 chromatographic process is modeled at a molecular level was developed in 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 by a Poisson process and the time that a molecule spends bound to the stationary phase (residence or sojourn time) is determined by an exponential distribution. The stochastic theory is completely independent of the physical–chemical mechanisms responsible for the retention; therefore it can be used in any field of chromatography. Accordingly, the model has been extended and improved for several chromatographic methods such as adsorption, partition, ion-exchange or size exclusion chromatography (adapted to SEC by Carmichael [16–19]). The real breakthrough in developing the theory was the introduction of the characteristic function approach. Later the effects of the mobile phase dispersion were introduced (stochastic-dispersive model) and they involved an increasing number of parameters specified in the description of the system. A detailed description of the stochastic theory of SEC via the characteristic function method was introduced by Dondi et al. [10]. The simplest theory of size exclusion chromatography assumes that a molecule of a certain size enters and leaves the pores n times on average during the migration along the column and spends p time on average in a single pore. After leaving a pore, the molecule spends m time on average in the mobile phase before entering another pore. All these variables are random quantities, therefore each molecule has an individual path while migrating along the column. However, the molecules of the same size behave in a similar manner, because of the anomalies in the retention paths of the
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A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
molecules we observe a nearly Gaussian curve as chromatographic peak. The observed retention time of a peak is the mean of the distribution of the individual retention times and the width of the peak is described by the standard deviation. The average number of the pore ingress steps (if takes a value between 0 and 1) can be written as np = nperm (1 − )me ,
(5)
and the residence time in the pores will be: p = perm (1 − )mp ,
(6)
where me and mp are constants depending on the ingress and the egress processes, respectively. Both np and p take the value 0 if > 1. As Dondi et al. showed earlier [10], parameters me , mp and m are related as m = me + mp
(7)
Thus it can be seen that both the pore ingress and egress processes affect the selectivity of SEC. The relationship between m, me , and mp can be re-expressed when we introduce parameter ˛ to characterize the relative contribution of the pore egress process to the overall size exclusion effect [12]: mp ˛= . m
(8)
The characteristic function is the main equation of the stochastic theory. It is the Fourier transform of the elution profile and it contains all the information about the separation process. In this simple case the following characteristic function describes the system the best:
(ω) = exp
np
1 −1 1 − iωp
,
(9)
where i is the imaginary unit, np is the average number of the pore ingress and egress steps, p is the average time spent by a molecule in a single pore and ω is an auxiliary real variable (frequency). In this case the probability density function (the time-domain signal) can be written as the inverse Fourier transform of the characteristic function as:
f (t) =
np −t/p −np e I1 tp
4np t p
,
(10)
where I1 is a modified Bessel function of the first kind and first order. There is a simple relationship between the characteristic function and the moments about the origin. The kth moment of the chromatographic peak can be calculated from the characteristic function (Eq. (9)) by the moment theorem of the Fourier transform as
k = i−k
dk (ω) dt
k
.
(11)
ω=0
It is well-known that the first moment is the mean residence time and the second central moment is the variance of the observed chromatographic peak. By calculating these moments using the equation above we obtain: 1 = np p
(12)
and 2 = 2np p2 .
(13)
The third central moment gives information about the peak symmetry: 3 = 6np p3 .
(14)
The above equations are only valid if we assume that the pore size in the stationary phase particles is uniform. For heterogeneous kinetics, the simplest stochastic model cannot be employed. Cavazzini et al. extended the stochastic model of chromatography to the case where the stationary phase consists of more than one types of adsorption sites and for the case when the adsorption energy of the sites is determined by a distribution [20]. It was assumed that if there are several types of adsorption sites in the column, the molecules bind with a certain probability to each of them. Thus, the probability density function describing the peak shape can be obtained as the probability-weighted convolution of the probability density functions of the different sites. For example, for two different adsorption sites the following characteristic function was obtained: (ω) = exp
n1
1 −1 1 − iω1
exp
n2
1 −1 1 − iω2
(15)
The corresponding peak shape is:
f (t) =
n1 −t/1 −n1 e I1 t1
∗
n2 −t/2 −n2 e I1 t2
4n1 t 1
4n2 t 2
,
(16)
where the subscripts refer to the respective sites and the sign * stands for convolution. The calculation of the moments and that of the 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 package Mathematica 9 (Wolfram Research). The elution profiles were obtained via numerical inverse Fourier transform using 1024 points. Except for the case where we illustrate the effect of parameter , in all other cases perm , nperm and rp,0 were arbitrarily set to 1 s, 2000 and 12.434 nm, respectively. To illustrate the effect of changing the parameter on the elution profiles we used perm = 0.1 s. 4. Results and discussion In size exclusion chromatography, the most important factor is the pore size of the stationary phase, since the separation is based on the size of the analyte molecules relative to the pore size. In ideal SEC, because there is no physico-chemical interaction between the sample molecules and the stationary phase surface, the type of the silica and the chemical modification has no effect on the retention and on the selectivity. The size of the stationary phase particles of the modern HPLC columns varies in a quite thin range, and it has been recently demonstrated that there is no evident correlation between the particle size distribution and column efficiency [21]. The effect of the structure of the stationary phase particle (e.g. non-porous, fully porous particles, core–shell) on the separation efficiency is quite significant, because diffusion within the particles has a strong impact on the brand broadening in all modes of HPLC. The size exclusion effect on non-porous particles is nonexistent, only the hydrodynamic effect is present. The fullyporous particles have a large pore volume where the molecules can diffuse and macromolecules may spend long time there, thus slow pore diffusion gives rise to band broadening of the observed peaks. In the core–shell particles the pore volume is more limited and the diffusion times are shorter [22].
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
55
Results obtained with low-temperature nitrogen adsorption confirmed that the presently commercially available HPLC columns do not have a uniform pore size, but contain pores in a relatively wide pore size range. A packing material which is marketed as a 200-A˚ pore size stationary phase will definitely contain pores of 100 and of 300 A˚ as well. This is an important aspect, because molecules of a given size are not equally likely to enter each pore. Molecules that are small enough to enter the pores of 200 A˚ may be excluded ˚ This ultimately leads to the distortion in from the pores of 100 A. the peak shapes in practice. The pore size distribution (PSD) of the porous stationary phase particles has been already investigated and modeled in the early 1980s by Knox and Scott [3]. The stochastic theory describes the chromatographic process at the molecular level so it is obvious to introduce the pore size distribution into that in order to obtain more relevant information about the retention properties. The concept introduced by Eq. (15) can be extended to multisite heterogeneous adsorption and the following characteristic function is obtained when m type of sites are present, each with a relative abundance of pj , (j = 1, . . ., m) [20]:
Eq. (21) is the characteristic function of the peak shape for lognormal pore size distribution. It is important to note that both the number of pore entries and the individual sojourn times of the molecules in a pore depend on the size of the molecule relative to the size of the pore. The respective relationships are given by Eqs. (5) and (6). The above characteristic function describes the peak shape in Fourier domain for size exclusion chromatography when the pore sizes are not uniform. The peak shape itself can be calculated as the inverse Fourier transform of (ω). Unfortunately, Eq. (21) cannot be evaluated analytically for the general case. Nevertheless, the calculation of the moments is possible. An analytical expression of the moments can only be obtained if parameters me and mp are both integers. Intuition suggests and extensive data processing of SEC data confirms [12] that for the ingress process me > 0 in Eq. (5) and for the egress process mp < 0 in Eq. (6). The first absolute moment as well as the second and the third central moments of the elution profile will be obtained from Eq. (21) using Eq. (11) as
⎧ ⎡ ⎤⎫ m ⎬ ⎨ 1 ⎦ (ω) = exp n ⎣−1 + pj 1 − iωj ⎭ ⎩
1 = nperm perm a,
(22)
2 2 = 2nperm perm a,
(23)
3 a, 3 = 6nperm perm
(24)
(17)
j=1
This equation can be extended for a continuous distribution of sites. When size exclusion chromatography is modeled and both the pore ingress and egress processes are influenced by the pore size distribution, the following characteristic function is obtained:
∞
(ω, rp ) = exp
(rp , rp,0 , )np rG
1 −1 1 − iωp
dr p
(rp , rp,0 , ) = √ exp 2 rp
−
ln rp − ln rp,0
(18)
2
,
2 2
(19)
where rp,0 and represents the maximum and width of the lognormal distribution, respectively. One should note that the true first absolute and second central moments of the lognormal distribution (Eq. (19)) are 1,lognorm = rp,0 e
2 /2
2 2,lognorm = rp,0 e
,
2
2
e − 1
(20)
For the sake of simplicity, however, further on we refer to rp,0 and as the mean and the standard deviation of the lognormal distribution, respectively. We obtain the characteristic function in the case of lognormal pore size distribution when we combine Eqs. (18) and (19): (ω) = exp
×
√
1 2
∞
rG
np (rp ) exp rp
1 −1 1 − iωp (rp )
−
ln rp − ln rp,0
2
2 2
dr p
k2 2 1 = (−)k e 2 2
a = KSEC
k=0
If all the pores were of the same size, one would obtain the characteristic function and the moments of the band profile as written by Eqs. (9)–(11). However, when the pore size is governed by a distribution, the probability density function of the pore size distribution, (rp , rp,0 , ), is included in the model. As a consequence, we replace the term rp in Eq. (4) by a probability density function (PDF). If we assume a lognormal distribution, which is shown by the experimental evidence to describe the pore size distribution of the HPLC packing materials, the following probability density function is used: 1
where parameter a is actually equal to the KSEC .
(21)
k
erfc
k 2 + ln √ 2
.
(25)
Parameter a – and thus KSEC – strongly depends on the pore shape. 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, the pore is either conical or spherical. The second central moment can be calculated using = me + 2mp , and the third central moment by using = me + 3mp . Parameter a can only be calculated by Eq. (25) when > 0. For instance, when mp = −3 and me = 6, and thus m = 3, in the calculation of the third central moment = −3, and the analytical calculation of that moment is not possible with the equations written above, nevertheless, the first and the second moments can be still evaluated. The first three moments can be calculated in all the cases when ˛ > −0.5. If ˛ < −0.5, for the calculation of the third moment one would get < 0 and it is not possible to use Eqs. (22)–(25). One can always obtain the elution profile with the inverse Fourier transform of Eq. (21) and calculate the moments by numerically integrating the peak profile. The equations above demonstrate that pore size distribution will have important consequences on retention time and peak shape. By plotting parameter a when m = 0 against and we obtain the relative pore accessibility (see Fig. 1). One can see in that figure that in the case of monopores ( = 0), there is a sharp distinction between the molecules that visit the pores and the ones that 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. However, when a range of pore sizes are present in the stationary phase, ( > 0), there are pores that are accessible for the molecules larger than rp,0 too. The broader the pore size distribution, the smoother is 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 or spherical pore geometry), see Fig. 2. To exploit the effect of the pore size distribution on the elution profile, we calculated various chromatograms by changing the
56
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
Fig. 1. Relative pore accessibility. The effect of the breadth of the PSD () and the effect of the size parameter () on the size-exclusion process.
standard deviation of the pore size distribution and the size of the solute molecule relative to the pore size. The effect of increasing the breadth of the distribution () on the chromatograms can be seen in Fig. 3 for a relatively large molecule, when the relative sizing parameter is = 0.95. This figure illustrates the positive effect of the PSD on the chromatographic process, where the solid lines represent = 0.1, the long-dashed lines represent = 0.5 and the short-dashed lines stand for = 1. Depending on the pore shape, the retention times of the profiles vary in a quite wide range. However, it can be seen in all cases, that the retention time increases and the skew of the elution profile decreases as increases. If the sample molecules are very large, they can only in the rarest case get inside a pore. However once they enter a pore, they cannot escape from it and the molecule remains trapped at the same position as time passes (n is very small and is very
1.0
1.0
a. σ =1
0.6 0.4
0 0.2 0.4 0.6 0.8 1
0.2
σ=1
0.4 0.2
0.5
1.0
ρ
1.5
2.0
1.0
2.5
0 0.2 0.4 0.6 0.8 1
0.6 σ =1
0.4 0.2
σ =1
0.5
1.0
ρ
1.5
2.0
1.0
2.5
d.
0.8
KSEC
KSEC
0.0 0.0
c.
0.8
0.0 0.0
0 0.2 0.4 0.6 0.8 1
0.6
σ =0
0.0 0.0
b.
0.8
KSEC
0.8
KSEC
large). Most of the molecules, however, cannot enter a pore and for this reason they are unretained and elute at the void time (t0 ). This is best illustrated by the green solid line that stands for conical or spherical pore shapes (m = 3) and relatively small variance ( = 0.1) in Fig. 3. This is also demonstrated for the case of cylindrical pores, too, where we obtain an exponential decreasing line as the elution profile when the pore size distribution is rather narrow. As the standard deviation of the pore size is increased, the elution profile becomes a Gaussian peak and if = 1 one obtains the most symmetric profile for every kind of pore shape. For smaller molecules ( = 0.1–0.5), the trends when is increased are rather different depending on the pore shapes. In case of slit shaped pores, has a more dominant effect on the KSEC value than in the other two cases (cylindrical or conical pores) and it was also demonstrated in Fig. 2. For example if the particular molecular size is = 0.5 the retention time increases when m = 2 or 3 and decreases when m = 1 as the value of is increasing. The chromatograms become more asymmetrical when increases. The increase of asymmetry is most significant for m = 3. For smaller values (such as = 0.2) the retention time will decrease in the case of all kind of pore shapes. The effect of the relative size of the molecules, , on the elution profile is of course most important for the retention time, i.e. for the first moment. The larger the , the less included is the molecule. By calculating the skew of the peak profiles, we can see that the peaks become more asymmetrical while increases. Chromatograms are plotted in Fig. 4 for different pore shapes and molecular sizes. The retention times of the observed chromatogram changes considerably. We can conclude that the pore size distribution has little influence on the behavior of the small molecules while its effect on the large molecules ( > 0.5) is intensive. The larger the molecule, the harder it can enter the pores, but once it enters a pore it gets stuck there, remains at the same position as time passes and will not emerge from the pore for a wile. That is why the peaks of large
0 0.2 0.4 0.6 0.8 1
0.6 σ =1
0.4 0.2
σ =0
0.5
1.0
ρ
1.5
2.0
2.5
0.0 0.0
σ =0
0.5
1.0
ρ
1.5
2.0
2.5
Fig. 2. The influence of the pore shape parameter (m) and the size of the molecule relative to the pore size () on the partition coefficient when (a) the pore geometry is not included (m = 0 i.e. relative pore accessibility), (b) slit shaped pores (m = 1), (c) cylindrical pores (m = 2) and (d) conical or spherical pores (m = 3).
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
57
20 m=3
15
m=3
Intensity
m=2
m=2
0.4
0.6
10
0.8
σ
m=2
m=2
0.10 0.05
m=1
0.2
m=3
m=1
0.15
1.0 0.5
m=2
0.20
m=3
1.5
μ1
Skew
2.0
1.0
0.00
m=3
0.2
0.4
0.6
σ
0.8
m=1
m=1 m=1
m m m m m m m m m
5
0 0.0
0.1
0.2 Normalized retention time
1, 1, 1, 2, 2, 2, 3, 3, 3,
0.1 0.5 1 0.1 0.5 1 0.1 0.5 1
0.3
0.4
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 size parameter is = 0.95. In the inserts the skew and the first absolute moment of the chromatograms are plotted.
m=3
0.8
m=2
Skew
60
m=3
0.6
40 m=2
Μ1
150
20
Intensity
100
m=1
0
m=3
0.2
m=1
0.2
0.4
m=2
0.6 Ρ
0.8
m=1
m=1 m=2
0.4 m=3
1.0
0.2
m=3 m=2
0.4
0.8
m m m m m m m m m
0. 2
0. 4
0.6
1.0
m=1
50
0 0. 0
0.6 Ρ
0.8
1. 0
1, 1, 1, 2, 2, 2, 3, 3, 3,
0.1 0.5 1 0.1 0.5 1 0.1 0.5 1
1.2
1.4
Normalized retention time Fig. 4. The effect of the size parameter () on the calculated chromatograms for slit shaped pores, for cylindrical pores and for spherical or conical pores when the breadth of the PSD is = 0.5. In the inserts the skew and the first absolute moment of the chromatograms are plotted.
molecules (i.e. with high values) become extremely broad and skewed. Parameter ˛ expresses the relation of the pore ingress and egress processes (see Eq. (8)). We calculated elution profiles for various ˛ values and for the mentioned pore shapes. The observed profiles are demonstrated in Fig. 5 where the relative size parameter () was set to 0.95 and the variance of the PSD was = 0.5. In this case the size of the sample molecules is very close to the pore size, however the effects of changing the parameter ˛ is very meaningful. As it could be expected, the retention time (the first moment) does not depend on ˛. The second central moment and the skew of the observed profiles increase as ˛ decreases, and this is demonstrated in the subfigures of Fig. 5. In the case of slit shaped pores (m = 1), the relation of the ingress and egress processes does not affect the peak symmetry as much as in the other cases where the peaks become more asymmetrical as ˛ increases. The same tendency could be
Table 1 The effect of the relative contribution of the pore egress process to the overall size exclusion 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 central moments (1 , 2 and 3 ) in case of slit shaped pores (m = 1). m
˛
mp
me
me + mp
me + 2mp
me + 3mp
1 1 1 1 1 1 1 1 1 1
−0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1
−0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
1 1 1 1 1 1 1 1 1 1
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
m=3
Skew
30
25
m=2
3.5 3.0 2.5 2.0 1.5 1.0 0.5
m=3
µ2
58
m=2 m=1
1.0
0.8
Intensity
20
0.6 α
0.4
0.0005 0.0004 0.0003 0.0002 0.0001
m=1 m=2 m=3
1.0
0.2
0.8
0.6 α
m=1
m m m m m m m m m
15
10
0.4
1, 1, 1, 2, 2, 2, 3, 3, 3,
0.2
0.1 0.5 1 0.1 0.5 1 0.1 0.5 1
5
0 0.00
0.05
0.10
0.15
0.2 0
0.25
Normalized retention time Fig. 5. The effect of the relative contribution of the pore egress process to the overall size exclusion effect (˛) on the calculated chromatograms for slit shaped pores, for cylindrical pores and for spherical or conical pores when the size parameter is = 0.95, and the breadth of the PSD is = 0.5. In the inserts the skew and the second central moment of the chromatograms are plotted.
observed for all value of : the profiles become more asymmetrical as ˛ decreases. From the results presented in Figs. 2–5 we can conclude that the effect of the parameters studied (i.e. , , ˛) is the most intensive in case of conical or spherical pores and the least intensive in case of slit shaped pores. We investigated how the peak resolution and the efficiency of the separation are affected by the pore size distribution. The analytical calculation of the relative resolution is only possible if the first absolute and the second central moments of the peak are calculated for integer me and mp values. There are only a few situations this calculation can be done. Table 1 helps the reader to consider
ρ1 0.2, ρ2 0.3 ρ1 0.5, ρ2 0.6 ρ1 0.8, ρ2 0.9
Relative resolution
Relative resolution
ρ1 0.1, ρ2 0.2 ρ1 0.4, ρ2 0.5 ρ1 0.7, ρ2 0.8
a. 1. 2 1. 0 0. 8 0. 6 0. 4 0.0
whether the moments and the resolution can analytically be calculated, or not. The change of the resolution and the number of theoretical plates with the breadth of pore size distribution are reported in Figs. 6 and 7. Several important conclusions can be drawn from the data presented in the figures. The curves can be divided into two cases based on the molecule size. In the first one, the molecules are small enough to separate them by size exclusion chromatography and neither the relative resolution nor the number of theoretical plates is affected by the pore size distribution. This can be seen in Figs. 6 and 7, where the plots referring to the small molecules hardly
0. 2
0. 4
0.6
0. 8
1.0
ρ1 0.3, ρ2 0.4 ρ1 0.6, ρ2 0.7 ρ1 0.9, ρ2 0.95
b.
0.9 0.8 0.7 0.6
1.0
0. 0
0.2
0.4
2. 0 c. 1. 8 1. 6 1. 4 1. 2 1. 0 0. 8 0. 6 0. 0
0. 2
0. 4
0.6
σ
0. 8
1.0
0.6
0.8
1. 0
σ Relative resolution
Relative resolution
σ
0. 6
0. 8
1.0
12 d. 10 8 6 4 2 0 0.0
0.2
0.4
σ
Fig. 6. Relative resolution for calculated chromatograms of differing size parameters (). The pore shape parameter (m) and the relative contribution of the pore egress process to the overall size exclusion effect (˛) were varied as (a) m = 1 and ˛ = −1, (b) m = 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 ρ2 ρ2 ρ2 ρ2 ρ2
a.
1500
500 0 0. 0
0. 2
0.4
σ
0.6
0. 8
1. 0
c.
1200
1400 1200 1000 800 600 400 200 0 0. 0
b.
1000
80 0
800
60 0
600
0. 4
σ
0.6
0.8
1.0
d.
40 0
400
20 0
200
0
0 0. 0
0.2
1000
N
N
0.2 0.4 0.6 0.8 1
ρ2 ρ2 ρ2 ρ2 ρ2
N
N
1000
0.1 0.3 0.5 0.7 0.9
59
0.2
0.4
σ
0.6
0. 8
1. 0
0.0
0.2
0.4
σ
0.6
0.8
1.0
Fig. 7. Number of theoretical plates for calculated chromatograms of differing size parameters (). The pore shape parameter (m) and the relative contribution of the pore egress process to the overall size exclusion effect (˛) were varied as (a); m = 1 and ˛ = −1, (b); m = 2 and ˛ = −0.5, (c); m = 2 and ˛ = −1 and (d); m = 3 and ˛ = −1.
show any change as increases, while the relative resolution and the number of theoretical plates for the larger and especially for the largest molecules significantly increase as increases. The effect is stronger as the value of m increases (i.e. in the case of cylindrical and spherical or conical pores relative to slit shaped pores). This observation drives us to more general conclusions in the field of chromatography, because the size exclusion effect is not only in size exclusion chromatography important, but also in other chromatographic methods, such as reversed phase or HILIC separations. In those cases, the hindered pore diffusion of macromolecules is very important, and the size exclusion process becomes more important than the other effects of the retention.
5. Conclusions The stochastic theory of size exclusion chromatography was extended to wide pore size distribution for a number of various pore geometries (slit shaped, cylindrical, and conical or spherical). The statistical moments of the peak profiles can easily be calculated by assuming a pore shape. The calculation of the chromatograms is feasible by inverse Fourier transform of the characteristic function. The trend we observe for the calculated chromatograms emphasizes the significance of pore size distribution: the PSD has strong influence on the retention properties (retention time, peak width, and peak shape) of macromolecules. Eq. (25) summarizes how pore size distribution affects the partition coefficient in size exclusion chromatography. That equation can serve as the basis for the experimental determination of the width of pore size distribution when KSEC is plotted against the gyration radius of polymer molecules. This is going to be exploited in a forthcoming study [23]. However, inverse size exclusion chromatography (ISEC) is used to derive information about the pore structure of the packing material from the retention data of series of known analytes, is was shown that it does not give appropriate characterization of the irregularly shaped porous materials [24]. We truly believe that our model – which contains information about both the pore geometry and the distribution of the pore sizes – is usable to develop ISEC and
so to obtain relevant information from the pore structure by nondestructive ISEC measurements. The experimental chromatograms may be characterized by pore geometry contributions and so the effect of different types of pores could be investigated. For the separation of macromolecules, the wide pore size distribution will increase retention and efficiency. Therefore in all modes of liquid chromatography, the efficient separation of macromolecules calls for a broad pore size distribution. Acknowledgements This research was realized in the frames of TÁMOP 4.2.4. A/211-1-2012-0001 “National Excellence Program – Elaborating and operating an inland student and researcher personal support system.” The project was subsidized by the European Union and co-financed by the European Social Fund. The work was supported in part by the grants TÁMOP-4.2.2. A11/1/KONV-2012-0065 and OTKA K 106044. References [1] F. Gritti, I. Leonardis, J. Abia, G. Guiochon, J. Chromatogr. A 1217 (2010) 3819. [2] B.M. Wagner, S.A. Schuster, B.E. Boyes, J.J. Kirkland, J. Chromatogr. A 1264 (2012) 22. [3] J.H. Knox, H.P. Scott, J. Chromatogr. 316 (1984) 311. [4] J.H. Knox, H.J. Ritchie, J. Chromatogr. 387 (1987) 65. [5] Y. Yao, A.M. Lenhoff, J. Chromatogr. A 1037 (2004) 273. [6] M. Kubin, J. Chromatogr. 108 (1975) 1. [7] I. Halász, K. Martin, Angew. Chem. Int. Ed. Engl. 17 (1978) 901. [8] J.C. Giddings, H. Eyring, J. Phys. Chem. 59 (1955) 416. [9] F. Dondi, M. Remelli, J. Phys. Chem. 90 (1986) 1885. [10] F. Dondi, A. Cavazzini, M. Remelli, A. Felinger, M. Martin, J. Chromatogr. A 943 (2002) 185. [11] L. Pasti, F. Dondi, M. Van Hulst, P.J. Schoenmakers, M. Martin, A. Felinger, Chromatographia 57 (2003) S171. [12] A. Felinger, L. Pasti, F. Dondi, M. van Hulst, P.J. Schoenmakers, M. Martin, Anal. Chem. 77 (2005) 3138. [13] A. Felinger, J. Chromatogr. A 1184 (2008) 20. [14] I. Teraoka, Macromolecules 37 (2004) 6632. [15] W.W. Yau, J.J. Kirkland, D.D. Bly, Modern Size-Exclusion Liquid Chromatography, Wiley, New York, 1979. [16] J.B. Carmichael, J. Polym. Sci. A-2 6 (1968) 517. [17] J.B. Carmichael, Macromolecules 1 (1968) 526.
60 [18] [19] [20] [21]
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60 J.B. Carmichael, Polym. Prepr. 9 (1968) 572. J.B. Carmichael, Biopolymers 6 (1968) 1497. A. Cavazzini, M. Remelli, F. Dondi, A. Felinger, Anal. Chem. 71 (1999) 3453. F. Gritti, T. Farkas, J. Heng, G. Guiochon, J. Chromatogr. A 1218 (2011) 8209.
[22] A. Felinger, J. Chromatogr. A 1218 (2011) 1939. [23] I. Bacskay, A. Sepsey, A. Felinger, J. Chromatogr. A (2014), JCA-14-62. [24] D. Lubda, W. Lindner, M. Quaglia, C. du Fresne von Hohenesche, K.K. Unger, J. Chromatogr. A 1083 (2005) 14.
Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Molecular theory of size exclusion chromatography for wide pore size distributions Annamária Sepsey a , Ivett Bacskay b , Attila Felinger a,b,∗ a b
MTA–PTE Molecular Interactions in Separation Science Research Group, Ifjúság útja 6, H-7624 Pécs, Hungary Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary
a r t i c l e
i n f o
Article history: Received 14 November 2013 Received in revised form 7 January 2014 Accepted 9 January 2014 Available online 16 January 2014 Keywords: Pore size distribution Stochastic theory Size exclusion chromatography
a b s t r a c t Chromatographic processes can conveniently be modeled at a microscopic level using the molecular theory of chromatography. This molecular or microscopic theory is completely general; therefore it can be used for any chromatographic process such as adsorption, partition, ion-exchange or size exclusion chromatography. The molecular theory of chromatography allows taking into account the kinetics of the pore ingress and egress processes, the heterogeneity of the pore sizes and polymer polydispersion. In this work, we assume that the pore size in the stationary phase of chromatographic columns is governed by a wide lognormal distribution. This property is integrated into the molecular model of size exclusion chromatography and the moments of the elution profiles were calculated for several kinds of pore structure. Our results demonstrate that wide pore size distributions have strong influence on the retention properties (retention time, peak width, and peak shape) of macromolecules. The novel model allows us to estimate the real pore size distribution of commonly used HPLC stationary phases, and the effect of this distribution on the size exclusion process. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Modern porous or core–shell stationary phases may exhibit a momentous pore size distribution. Experimental data confirm that the size of mesopores can cover a rather wide range [1,2]. The nature and the breadth of pore size distributions have significant impact on the mass-transfer properties of stationary phases. The separation of macromolecules is particularly influenced by the pore size distribution, since their hindered diffusion in the pore network gives a critical contribution to band broadening. Size exclusion chromatography (SEC) is one of the most widely used techniques to determine the molecular size distribution of polymers of any kind. The separation mechanism relies on the size and shape of sample molecules relative to the size and shape of the pores in the stationary phase particles. Because we do not exactly know the structure and the dimensions of the porous media, the determination of the molecular mass relies on a calibration step based on the behavior of well-known monodisperse polymers in the columns containing porous stationary phase particles. The study of the pore structure of the stationary phases used in liquid chromatography has been of great interest among
∗ Corresponding author at: Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary. Tel.: +36 72 501500x24582; fax: +36 72 501518. E-mail address: [email protected] (A. Felinger). 0021-9673/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chroma.2014.01.017
chromatographers in the last decades [3–6]. Kubín has modeled pore size irregularity with a diffusion model, assuming that molecules can penetrate into the porous particles to a distance that depends on the size of the molecules [6]. There are a number of methods for determining relevant information about the porous media such as low-temperature nitrogen adsorption, mercury intrusion, microscopy and solute exclusion. These techniques are either too expensive and/or they destroy the chromatographic column (so they cannot be used for any further analysis), or they do not give relevant information about all fine details. The influence of pore size distribution on separation efficiency can conveniently be studied with inverse size exclusion chromatography. Inverse size exclusion chromatography is used to derive information about the structure of the pores of the packing material from the retention data of a series of known analytes, for instance, polymers of narrow molecular mass distribution and known average molecular mass [7]. In this study, we develop a model that integrates the pore size distribution into the microscopic theory of size exclusion chromatography. With this model one is able to determine the influence of the breadth of pore size distribution on retention properties and efficiency. The molecular, or stochastic theory of chromatography is a microscopic model introduced by Giddings and Eyring in 1955 [8]. That theory uses random variables and probabilistic terms to describe the migration of the molecules along the chromatographic
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
column. The stochastic approaches may become very complex to work with for situations except the most simple case of adsorption, so the use of this theory was not convenient until the characteristic function (CF) approach was introduced to this field [9]. The use of characteristic functions made the stochastic theory of chromatography simpler even for complicated cases such as heterogeneous adsorption chromatography. The stochastic theory seems rather suitable for describing size exclusion chromatography among the basic theories of chromatography. Size exclusion chromatography is indeed based on the random migration of molecules, where randomly occurring entrapment and release of molecules in the mesopores of the stationary phase builds up the separation process [10–13].
2. Theory 2.1. Classical size exclusion considerations Size exclusion chromatography (SEC) has been investigated from many point of views, which led to an asystematic nomenclature. To eliminate the misapprehensions, all techniques that are founded on the size exclusion separation mechanism, such as gel filtration, molecular sieve chromatography, gel chromatography and gel-permeation, are parts of SEC. However the only difference between the individual ones is in the samples and solutions used. Size exclusion chromatography is a well-known separation method where the main retention mechanism is the size exclusion effect. However there are a lot of other mechanisms responsible for the migration of the molecules along the column, such as hydrodynamic and stress-induced diffusion, the polarization effect, multipath, enthalpic and soft-body interactions and the so-called wall effect, they can be ignored in almost every case. In an ideal case there is no interaction between the molecules and the stationary phase particles or the mobile phase particles. In SEC, the sample components are separated according to their size and molecular mass. The separation is governed by entropy only; the retention depends on the relative penetration of the sample molecules to the pores. The stationary phase of a SEC column is always a mechanically stable porous media that can be built up on a rigid carrier such as silica or the whole stationary phase is made from this porous material. The molecules traveling along the column in the mobile phase can enter the pores if the size of the pore is larger than that of the dimensions of the molecule. However it is still uncertain which exact size parameter determines the separation [14]. In general it is accepted that the gyration radius or diameter of the molecule is used to determine whether or not the molecule can enter the pore. The mobile phase in the pores is stagnant; the molecules can migrate in the pores only by diffusion. Two molecule sizes have special significance in SEC: the size of the completely permeable particle and the size of the barely excluded particle. The completely permeable particle (indicated by a subscript perm in the equations) is small enough to visit all the pores so that both the stagnant mobile phase in the pores of the stationary phase and the moving zone of the mobile phase between the stationary phase particles is completely accessible for it. If the molecule is too large to enter the pores, it will be excluded (indicated by a subscript excl in the equations). These molecules can only wander in the moving mobile phase and have access only to the interstitial volume of the mobile phase between the stationary phase particles. The excluded molecules elute at the void volume (V0 ). According to this mechanism, we can obtain information of the size, shape, aggregation state or kinetics of the ligand–polymer binding of the molecules investigated.
53
The partition coefficient can easily be calculated by means of the retention times of the above mentioned and the unknown particles using the following equation: KSEC =
t − texcl , tperm − texcl
(1)
where the numerator stands for time spent by the investigated molecule in the pores of the stationary phase particles and the denominator indicates the residence time spent by the completely permeable particle in the pores. The partition coefficient can be rewritten as tp KSEC = (2) tp,perm where the subscript p indicates the time spent in the pores by the molecules. The partition coefficient strongly depends on the ratio of the size of the migrating particle to the size of the pore. The partition coefficient is in a widely used retention model [10,11,15] defined using the size of the molecule investigated and the size of the pore of 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 size parameter 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 on the 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 chromatographic process is modeled at a molecular level was developed in 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 by a Poisson process and the time that a molecule spends bound to the stationary phase (residence or sojourn time) is determined by an exponential distribution. The stochastic theory is completely independent of the physical–chemical mechanisms responsible for the retention; therefore it can be used in any field of chromatography. Accordingly, the model has been extended and improved for several chromatographic methods such as adsorption, partition, ion-exchange or size exclusion chromatography (adapted to SEC by Carmichael [16–19]). The real breakthrough in developing the theory was the introduction of the characteristic function approach. Later the effects of the mobile phase dispersion were introduced (stochastic-dispersive model) and they involved an increasing number of parameters specified in the description of the system. A detailed description of the stochastic theory of SEC via the characteristic function method was introduced by Dondi et al. [10]. The simplest theory of size exclusion chromatography assumes that a molecule of a certain size enters and leaves the pores n times on average during the migration along the column and spends p time on average in a single pore. After leaving a pore, the molecule spends m time on average in the mobile phase before entering another pore. All these variables are random quantities, therefore each molecule has an individual path while migrating along the column. However, the molecules of the same size behave in a similar manner, because of the anomalies in the retention paths of the
54
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
molecules we observe a nearly Gaussian curve as chromatographic peak. The observed retention time of a peak is the mean of the distribution of the individual retention times and the width of the peak is described by the standard deviation. The average number of the pore ingress steps (if takes a value between 0 and 1) can be written as np = nperm (1 − )me ,
(5)
and the residence time in the pores will be: p = perm (1 − )mp ,
(6)
where me and mp are constants depending on the ingress and the egress processes, respectively. Both np and p take the value 0 if > 1. As Dondi et al. showed earlier [10], parameters me , mp and m are related as m = me + mp
(7)
Thus it can be seen that both the pore ingress and egress processes affect the selectivity of SEC. The relationship between m, me , and mp can be re-expressed when we introduce parameter ˛ to characterize the relative contribution of the pore egress process to the overall size exclusion effect [12]: mp ˛= . m
(8)
The characteristic function is the main equation of the stochastic theory. It is the Fourier transform of the elution profile and it contains all the information about the separation process. In this simple case the following characteristic function describes the system the best:
(ω) = exp
np
1 −1 1 − iωp
,
(9)
where i is the imaginary unit, np is the average number of the pore ingress and egress steps, p is the average time spent by a molecule in a single pore and ω is an auxiliary real variable (frequency). In this case the probability density function (the time-domain signal) can be written as the inverse Fourier transform of the characteristic function as:
f (t) =
np −t/p −np e I1 tp
4np t p
,
(10)
where I1 is a modified Bessel function of the first kind and first order. There is a simple relationship between the characteristic function and the moments about the origin. The kth moment of the chromatographic peak can be calculated from the characteristic function (Eq. (9)) by the moment theorem of the Fourier transform as
k = i−k
dk (ω) dt
k
.
(11)
ω=0
It is well-known that the first moment is the mean residence time and the second central moment is the variance of the observed chromatographic peak. By calculating these moments using the equation above we obtain: 1 = np p
(12)
and 2 = 2np p2 .
(13)
The third central moment gives information about the peak symmetry: 3 = 6np p3 .
(14)
The above equations are only valid if we assume that the pore size in the stationary phase particles is uniform. For heterogeneous kinetics, the simplest stochastic model cannot be employed. Cavazzini et al. extended the stochastic model of chromatography to the case where the stationary phase consists of more than one types of adsorption sites and for the case when the adsorption energy of the sites is determined by a distribution [20]. It was assumed that if there are several types of adsorption sites in the column, the molecules bind with a certain probability to each of them. Thus, the probability density function describing the peak shape can be obtained as the probability-weighted convolution of the probability density functions of the different sites. For example, for two different adsorption sites the following characteristic function was obtained: (ω) = exp
n1
1 −1 1 − iω1
exp
n2
1 −1 1 − iω2
(15)
The corresponding peak shape is:
f (t) =
n1 −t/1 −n1 e I1 t1
∗
n2 −t/2 −n2 e I1 t2
4n1 t 1
4n2 t 2
,
(16)
where the subscripts refer to the respective sites and the sign * stands for convolution. The calculation of the moments and that of the 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 package Mathematica 9 (Wolfram Research). The elution profiles were obtained via numerical inverse Fourier transform using 1024 points. Except for the case where we illustrate the effect of parameter , in all other cases perm , nperm and rp,0 were arbitrarily set to 1 s, 2000 and 12.434 nm, respectively. To illustrate the effect of changing the parameter on the elution profiles we used perm = 0.1 s. 4. Results and discussion In size exclusion chromatography, the most important factor is the pore size of the stationary phase, since the separation is based on the size of the analyte molecules relative to the pore size. In ideal SEC, because there is no physico-chemical interaction between the sample molecules and the stationary phase surface, the type of the silica and the chemical modification has no effect on the retention and on the selectivity. The size of the stationary phase particles of the modern HPLC columns varies in a quite thin range, and it has been recently demonstrated that there is no evident correlation between the particle size distribution and column efficiency [21]. The effect of the structure of the stationary phase particle (e.g. non-porous, fully porous particles, core–shell) on the separation efficiency is quite significant, because diffusion within the particles has a strong impact on the brand broadening in all modes of HPLC. The size exclusion effect on non-porous particles is nonexistent, only the hydrodynamic effect is present. The fullyporous particles have a large pore volume where the molecules can diffuse and macromolecules may spend long time there, thus slow pore diffusion gives rise to band broadening of the observed peaks. In the core–shell particles the pore volume is more limited and the diffusion times are shorter [22].
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
55
Results obtained with low-temperature nitrogen adsorption confirmed that the presently commercially available HPLC columns do not have a uniform pore size, but contain pores in a relatively wide pore size range. A packing material which is marketed as a 200-A˚ pore size stationary phase will definitely contain pores of 100 and of 300 A˚ as well. This is an important aspect, because molecules of a given size are not equally likely to enter each pore. Molecules that are small enough to enter the pores of 200 A˚ may be excluded ˚ This ultimately leads to the distortion in from the pores of 100 A. the peak shapes in practice. The pore size distribution (PSD) of the porous stationary phase particles has been already investigated and modeled in the early 1980s by Knox and Scott [3]. The stochastic theory describes the chromatographic process at the molecular level so it is obvious to introduce the pore size distribution into that in order to obtain more relevant information about the retention properties. The concept introduced by Eq. (15) can be extended to multisite heterogeneous adsorption and the following characteristic function is obtained when m type of sites are present, each with a relative abundance of pj , (j = 1, . . ., m) [20]:
Eq. (21) is the characteristic function of the peak shape for lognormal pore size distribution. It is important to note that both the number of pore entries and the individual sojourn times of the molecules in a pore depend on the size of the molecule relative to the size of the pore. The respective relationships are given by Eqs. (5) and (6). The above characteristic function describes the peak shape in Fourier domain for size exclusion chromatography when the pore sizes are not uniform. The peak shape itself can be calculated as the inverse Fourier transform of (ω). Unfortunately, Eq. (21) cannot be evaluated analytically for the general case. Nevertheless, the calculation of the moments is possible. An analytical expression of the moments can only be obtained if parameters me and mp are both integers. Intuition suggests and extensive data processing of SEC data confirms [12] that for the ingress process me > 0 in Eq. (5) and for the egress process mp < 0 in Eq. (6). The first absolute moment as well as the second and the third central moments of the elution profile will be obtained from Eq. (21) using Eq. (11) as
⎧ ⎡ ⎤⎫ m ⎬ ⎨ 1 ⎦ (ω) = exp n ⎣−1 + pj 1 − iωj ⎭ ⎩
1 = nperm perm a,
(22)
2 2 = 2nperm perm a,
(23)
3 a, 3 = 6nperm perm
(24)
(17)
j=1
This equation can be extended for a continuous distribution of sites. When size exclusion chromatography is modeled and both the pore ingress and egress processes are influenced by the pore size distribution, the following characteristic function is obtained:
∞
(ω, rp ) = exp
(rp , rp,0 , )np rG
1 −1 1 − iωp
dr p
(rp , rp,0 , ) = √ exp 2 rp
−
ln rp − ln rp,0
(18)
2
,
2 2
(19)
where rp,0 and represents the maximum and width of the lognormal distribution, respectively. One should note that the true first absolute and second central moments of the lognormal distribution (Eq. (19)) are 1,lognorm = rp,0 e
2 /2
2 2,lognorm = rp,0 e
,
2
2
e − 1
(20)
For the sake of simplicity, however, further on we refer to rp,0 and as the mean and the standard deviation of the lognormal distribution, respectively. We obtain the characteristic function in the case of lognormal pore size distribution when we combine Eqs. (18) and (19): (ω) = exp
×
√
1 2
∞
rG
np (rp ) exp rp
1 −1 1 − iωp (rp )
−
ln rp − ln rp,0
2
2 2
dr p
k2 2 1 = (−)k e 2 2
a = KSEC
k=0
If all the pores were of the same size, one would obtain the characteristic function and the moments of the band profile as written by Eqs. (9)–(11). However, when the pore size is governed by a distribution, the probability density function of the pore size distribution, (rp , rp,0 , ), is included in the model. As a consequence, we replace the term rp in Eq. (4) by a probability density function (PDF). If we assume a lognormal distribution, which is shown by the experimental evidence to describe the pore size distribution of the HPLC packing materials, the following probability density function is used: 1
where parameter a is actually equal to the KSEC .
(21)
k
erfc
k 2 + ln √ 2
.
(25)
Parameter a – and thus KSEC – strongly depends on the pore shape. 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, the pore is either conical or spherical. The second central moment can be calculated using = me + 2mp , and the third central moment by using = me + 3mp . Parameter a can only be calculated by Eq. (25) when > 0. For instance, when mp = −3 and me = 6, and thus m = 3, in the calculation of the third central moment = −3, and the analytical calculation of that moment is not possible with the equations written above, nevertheless, the first and the second moments can be still evaluated. The first three moments can be calculated in all the cases when ˛ > −0.5. If ˛ < −0.5, for the calculation of the third moment one would get < 0 and it is not possible to use Eqs. (22)–(25). One can always obtain the elution profile with the inverse Fourier transform of Eq. (21) and calculate the moments by numerically integrating the peak profile. The equations above demonstrate that pore size distribution will have important consequences on retention time and peak shape. By plotting parameter a when m = 0 against and we obtain the relative pore accessibility (see Fig. 1). One can see in that figure that in the case of monopores ( = 0), there is a sharp distinction between the molecules that visit the pores and the ones that 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. However, when a range of pore sizes are present in the stationary phase, ( > 0), there are pores that are accessible for the molecules larger than rp,0 too. The broader the pore size distribution, the smoother is 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 or spherical pore geometry), see Fig. 2. To exploit the effect of the pore size distribution on the elution profile, we calculated various chromatograms by changing the
56
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
Fig. 1. Relative pore accessibility. The effect of the breadth of the PSD () and the effect of the size parameter () on the size-exclusion process.
standard deviation of the pore size distribution and the size of the solute molecule relative to the pore size. The effect of increasing the breadth of the distribution () on the chromatograms can be seen in Fig. 3 for a relatively large molecule, when the relative sizing parameter is = 0.95. This figure illustrates the positive effect of the PSD on the chromatographic process, where the solid lines represent = 0.1, the long-dashed lines represent = 0.5 and the short-dashed lines stand for = 1. Depending on the pore shape, the retention times of the profiles vary in a quite wide range. However, it can be seen in all cases, that the retention time increases and the skew of the elution profile decreases as increases. If the sample molecules are very large, they can only in the rarest case get inside a pore. However once they enter a pore, they cannot escape from it and the molecule remains trapped at the same position as time passes (n is very small and is very
1.0
1.0
a. σ =1
0.6 0.4
0 0.2 0.4 0.6 0.8 1
0.2
σ=1
0.4 0.2
0.5
1.0
ρ
1.5
2.0
1.0
2.5
0 0.2 0.4 0.6 0.8 1
0.6 σ =1
0.4 0.2
σ =1
0.5
1.0
ρ
1.5
2.0
1.0
2.5
d.
0.8
KSEC
KSEC
0.0 0.0
c.
0.8
0.0 0.0
0 0.2 0.4 0.6 0.8 1
0.6
σ =0
0.0 0.0
b.
0.8
KSEC
0.8
KSEC
large). Most of the molecules, however, cannot enter a pore and for this reason they are unretained and elute at the void time (t0 ). This is best illustrated by the green solid line that stands for conical or spherical pore shapes (m = 3) and relatively small variance ( = 0.1) in Fig. 3. This is also demonstrated for the case of cylindrical pores, too, where we obtain an exponential decreasing line as the elution profile when the pore size distribution is rather narrow. As the standard deviation of the pore size is increased, the elution profile becomes a Gaussian peak and if = 1 one obtains the most symmetric profile for every kind of pore shape. For smaller molecules ( = 0.1–0.5), the trends when is increased are rather different depending on the pore shapes. In case of slit shaped pores, has a more dominant effect on the KSEC value than in the other two cases (cylindrical or conical pores) and it was also demonstrated in Fig. 2. For example if the particular molecular size is = 0.5 the retention time increases when m = 2 or 3 and decreases when m = 1 as the value of is increasing. The chromatograms become more asymmetrical when increases. The increase of asymmetry is most significant for m = 3. For smaller values (such as = 0.2) the retention time will decrease in the case of all kind of pore shapes. The effect of the relative size of the molecules, , on the elution profile is of course most important for the retention time, i.e. for the first moment. The larger the , the less included is the molecule. By calculating the skew of the peak profiles, we can see that the peaks become more asymmetrical while increases. Chromatograms are plotted in Fig. 4 for different pore shapes and molecular sizes. The retention times of the observed chromatogram changes considerably. We can conclude that the pore size distribution has little influence on the behavior of the small molecules while its effect on the large molecules ( > 0.5) is intensive. The larger the molecule, the harder it can enter the pores, but once it enters a pore it gets stuck there, remains at the same position as time passes and will not emerge from the pore for a wile. That is why the peaks of large
0 0.2 0.4 0.6 0.8 1
0.6 σ =1
0.4 0.2
σ =0
0.5
1.0
ρ
1.5
2.0
2.5
0.0 0.0
σ =0
0.5
1.0
ρ
1.5
2.0
2.5
Fig. 2. The influence of the pore shape parameter (m) and the size of the molecule relative to the pore size () on the partition coefficient when (a) the pore geometry is not included (m = 0 i.e. relative pore accessibility), (b) slit shaped pores (m = 1), (c) cylindrical pores (m = 2) and (d) conical or spherical pores (m = 3).
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
57
20 m=3
15
m=3
Intensity
m=2
m=2
0.4
0.6
10
0.8
σ
m=2
m=2
0.10 0.05
m=1
0.2
m=3
m=1
0.15
1.0 0.5
m=2
0.20
m=3
1.5
μ1
Skew
2.0
1.0
0.00
m=3
0.2
0.4
0.6
σ
0.8
m=1
m=1 m=1
m m m m m m m m m
5
0 0.0
0.1
0.2 Normalized retention time
1, 1, 1, 2, 2, 2, 3, 3, 3,
0.1 0.5 1 0.1 0.5 1 0.1 0.5 1
0.3
0.4
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 size parameter is = 0.95. In the inserts the skew and the first absolute moment of the chromatograms are plotted.
m=3
0.8
m=2
Skew
60
m=3
0.6
40 m=2
Μ1
150
20
Intensity
100
m=1
0
m=3
0.2
m=1
0.2
0.4
m=2
0.6 Ρ
0.8
m=1
m=1 m=2
0.4 m=3
1.0
0.2
m=3 m=2
0.4
0.8
m m m m m m m m m
0. 2
0. 4
0.6
1.0
m=1
50
0 0. 0
0.6 Ρ
0.8
1. 0
1, 1, 1, 2, 2, 2, 3, 3, 3,
0.1 0.5 1 0.1 0.5 1 0.1 0.5 1
1.2
1.4
Normalized retention time Fig. 4. The effect of the size parameter () on the calculated chromatograms for slit shaped pores, for cylindrical pores and for spherical or conical pores when the breadth of the PSD is = 0.5. In the inserts the skew and the first absolute moment of the chromatograms are plotted.
molecules (i.e. with high values) become extremely broad and skewed. Parameter ˛ expresses the relation of the pore ingress and egress processes (see Eq. (8)). We calculated elution profiles for various ˛ values and for the mentioned pore shapes. The observed profiles are demonstrated in Fig. 5 where the relative size parameter () was set to 0.95 and the variance of the PSD was = 0.5. In this case the size of the sample molecules is very close to the pore size, however the effects of changing the parameter ˛ is very meaningful. As it could be expected, the retention time (the first moment) does not depend on ˛. The second central moment and the skew of the observed profiles increase as ˛ decreases, and this is demonstrated in the subfigures of Fig. 5. In the case of slit shaped pores (m = 1), the relation of the ingress and egress processes does not affect the peak symmetry as much as in the other cases where the peaks become more asymmetrical as ˛ increases. The same tendency could be
Table 1 The effect of the relative contribution of the pore egress process to the overall size exclusion 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 central moments (1 , 2 and 3 ) in case of slit shaped pores (m = 1). m
˛
mp
me
me + mp
me + 2mp
me + 3mp
1 1 1 1 1 1 1 1 1 1
−0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1
−0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
1 1 1 1 1 1 1 1 1 1
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
A. Sepsey et al. / J. Chromatogr. A 1331 (2014) 52–60
m=3
Skew
30
25
m=2
3.5 3.0 2.5 2.0 1.5 1.0 0.5
m=3
µ2
58
m=2 m=1
1.0
0.8
Intensity
20
0.6 α
0.4
0.0005 0.0004 0.0003 0.0002 0.0001
m=1 m=2 m=3
1.0
0.2
0.8
0.6 α
m=1
m m m m m m m m m
15
10
0.4
1, 1, 1, 2, 2, 2, 3, 3, 3,
0.2
0.1 0.5 1 0.1 0.5 1 0.1 0.5 1
5
0 0.00
0.05
0.10
0.15
0.2 0
0.25
Normalized retention time Fig. 5. The effect of the relative contribution of the pore egress process to the overall size exclusion effect (˛) on the calculated chromatograms for slit shaped pores, for cylindrical pores and for spherical or conical pores when the size parameter is = 0.95, and the breadth of the PSD is = 0.5. In the inserts the skew and the second central moment of the chromatograms are plotted.
observed for all value of : the profiles become more asymmetrical as ˛ decreases. From the results presented in Figs. 2–5 we can conclude that the effect of the parameters studied (i.e. , , ˛) is the most intensive in case of conical or spherical pores and the least intensive in case of slit shaped pores. We investigated how the peak resolution and the efficiency of the separation are affected by the pore size distribution. The analytical calculation of the relative resolution is only possible if the first absolute and the second central moments of the peak are calculated for integer me and mp values. There are only a few situations this calculation can be done. Table 1 helps the reader to consider
ρ1 0.2, ρ2 0.3 ρ1 0.5, ρ2 0.6 ρ1 0.8, ρ2 0.9
Relative resolution
Relative resolution
ρ1 0.1, ρ2 0.2 ρ1 0.4, ρ2 0.5 ρ1 0.7, ρ2 0.8
a. 1. 2 1. 0 0. 8 0. 6 0. 4 0.0
whether the moments and the resolution can analytically be calculated, or not. The change of the resolution and the number of theoretical plates with the breadth of pore size distribution are reported in Figs. 6 and 7. Several important conclusions can be drawn from the data presented in the figures. The curves can be divided into two cases based on the molecule size. In the first one, the molecules are small enough to separate them by size exclusion chromatography and neither the relative resolution nor the number of theoretical plates is affected by the pore size distribution. This can be seen in Figs. 6 and 7, where the plots referring to the small molecules hardly
0. 2
0. 4
0.6
0. 8
1.0
ρ1 0.3, ρ2 0.4 ρ1 0.6, ρ2 0.7 ρ1 0.9, ρ2 0.95
b.
0.9 0.8 0.7 0.6
1.0
0. 0
0.2
0.4
2. 0 c. 1. 8 1. 6 1. 4 1. 2 1. 0 0. 8 0. 6 0. 0
0. 2
0. 4
0.6
σ
0. 8
1.0
0.6
0.8
1. 0
σ Relative resolution
Relative resolution
σ
0. 6
0. 8
1.0
12 d. 10 8 6 4 2 0 0.0
0.2
0.4
σ
Fig. 6. Relative resolution for calculated chromatograms of differing size parameters (). The pore shape parameter (m) and the relative contribution of the pore egress process to the overall size exclusion effect (˛) were varied as (a) m = 1 and ˛ = −1, (b) m = 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 ρ2 ρ2 ρ2 ρ2 ρ2
a.
1500
500 0 0. 0
0. 2
0.4
σ
0.6
0. 8
1. 0
c.
1200
1400 1200 1000 800 600 400 200 0 0. 0
b.
1000
80 0
800
60 0
600
0. 4
σ
0.6
0.8
1.0
d.
40 0
400
20 0
200
0
0 0. 0
0.2
1000
N
N
0.2 0.4 0.6 0.8 1
ρ2 ρ2 ρ2 ρ2 ρ2
N
N
1000
0.1 0.3 0.5 0.7 0.9
59
0.2
0.4
σ
0.6
0. 8
1. 0
0.0
0.2
0.4
σ
0.6
0.8
1.0
Fig. 7. Number of theoretical plates for calculated chromatograms of differing size parameters (). The pore shape parameter (m) and the relative contribution of the pore egress process to the overall size exclusion effect (˛) were varied as (a); m = 1 and ˛ = −1, (b); m = 2 and ˛ = −0.5, (c); m = 2 and ˛ = −1 and (d); m = 3 and ˛ = −1.
show any change as increases, while the relative resolution and the number of theoretical plates for the larger and especially for the largest molecules significantly increase as increases. The effect is stronger as the value of m increases (i.e. in the case of cylindrical and spherical or conical pores relative to slit shaped pores). This observation drives us to more general conclusions in the field of chromatography, because the size exclusion effect is not only in size exclusion chromatography important, but also in other chromatographic methods, such as reversed phase or HILIC separations. In those cases, the hindered pore diffusion of macromolecules is very important, and the size exclusion process becomes more important than the other effects of the retention.
5. Conclusions The stochastic theory of size exclusion chromatography was extended to wide pore size distribution for a number of various pore geometries (slit shaped, cylindrical, and conical or spherical). The statistical moments of the peak profiles can easily be calculated by assuming a pore shape. The calculation of the chromatograms is feasible by inverse Fourier transform of the characteristic function. The trend we observe for the calculated chromatograms emphasizes the significance of pore size distribution: the PSD has strong influence on the retention properties (retention time, peak width, and peak shape) of macromolecules. Eq. (25) summarizes how pore size distribution affects the partition coefficient in size exclusion chromatography. That equation can serve as the basis for the experimental determination of the width of pore size distribution when KSEC is plotted against the gyration radius of polymer molecules. This is going to be exploited in a forthcoming study [23]. However, inverse size exclusion chromatography (ISEC) is used to derive information about the pore structure of the packing material from the retention data of series of known analytes, is was shown that it does not give appropriate characterization of the irregularly shaped porous materials [24]. We truly believe that our model – which contains information about both the pore geometry and the distribution of the pore sizes – is usable to develop ISEC and
so to obtain relevant information from the pore structure by nondestructive ISEC measurements. The experimental chromatograms may be characterized by pore geometry contributions and so the effect of different types of pores could be investigated. For the separation of macromolecules, the wide pore size distribution will increase retention and efficiency. Therefore in all modes of liquid chromatography, the efficient separation of macromolecules calls for a broad pore size distribution. Acknowledgements This research was realized in the frames of TÁMOP 4.2.4. A/211-1-2012-0001 “National Excellence Program – Elaborating and operating an inland student and researcher personal support system.” The project was subsidized by the European Union and co-financed by the European Social Fund. The work was supported in part by the grants TÁMOP-4.2.2. A11/1/KONV-2012-0065 and OTKA K 106044. References [1] F. Gritti, I. Leonardis, J. Abia, G. Guiochon, J. Chromatogr. A 1217 (2010) 3819. [2] B.M. Wagner, S.A. Schuster, B.E. Boyes, J.J. Kirkland, J. Chromatogr. A 1264 (2012) 22. [3] J.H. Knox, H.P. Scott, J. Chromatogr. 316 (1984) 311. [4] J.H. Knox, H.J. Ritchie, J. Chromatogr. 387 (1987) 65. [5] Y. Yao, A.M. Lenhoff, J. Chromatogr. A 1037 (2004) 273. [6] M. Kubin, J. Chromatogr. 108 (1975) 1. [7] I. Halász, K. Martin, Angew. Chem. Int. Ed. Engl. 17 (1978) 901. [8] J.C. Giddings, H. Eyring, J. Phys. Chem. 59 (1955) 416. [9] F. Dondi, M. Remelli, J. Phys. Chem. 90 (1986) 1885. [10] F. Dondi, A. Cavazzini, M. Remelli, A. Felinger, M. Martin, J. Chromatogr. A 943 (2002) 185. [11] L. Pasti, F. Dondi, M. Van Hulst, P.J. Schoenmakers, M. Martin, A. Felinger, Chromatographia 57 (2003) S171. [12] A. Felinger, L. Pasti, F. Dondi, M. van Hulst, P.J. Schoenmakers, M. Martin, Anal. Chem. 77 (2005) 3138. [13] A. Felinger, J. Chromatogr. A 1184 (2008) 20. [14] I. Teraoka, Macromolecules 37 (2004) 6632. [15] W.W. Yau, J.J. Kirkland, D.D. Bly, Modern Size-Exclusion Liquid Chromatography, Wiley, New York, 1979. [16] J.B. Carmichael, J. Polym. Sci. A-2 6 (1968) 517. [17] J.B. Carmichael, Macromolecules 1 (1968) 526.
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