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Polydispersity in size-exclusion chromatography: A stochastic approach
Journal of Chromatography A, 1365 (2014) 156–163
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Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Polydispersity in size-exclusion chromatography: A stochastic approach夽 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 15 July 2014 Received in revised form 5 September 2014 Accepted 8 September 2014 Available online 17 September 2014 Keywords: Polydispersity Stochastic theory Size-exclusion chromatography
a b s t r a c t We investigate the impact of polydispersity of the sample molecules on the separation process and on the efficiency of size-exclusion chromatography. Polydispersity was integrated into the molecular (stochastic) model of chromatography; the characteristic function, the band profile and the most important moments of the elution profiles were calculated for several kind of pore structures. We investigated the parameters affected by polydispersity on the separation for a number of pore shapes. Our results demonstrate that even a small distribution in the molecular size (i.e. polydispersity) can contribute substantially to the total width of the chromatographic peak. The pure effect of polydispersity can only be investigated via mathematical modeling, because its contribution to an experimental chromatogram cannot be separated from other band-broadening effects. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Size-exclusion chromatography (SEC) is often used for determining the molecular size or molecular weight on the basis of the retention of the molecules in porous stationary phases, than for separating polymers from each other [1]. SEC is defined as the differential elution of solutes from the porous stationary phase caused by different degrees of steric exclusion of the solutes from the pore volume due to the differential molecular size of solutes [2]. The determination of the molecular weight (Mw ) can be achieved by a calibration process, where a number of special standard polymers with well-known molecular weight and size are eluted, when an accurate calibration curve is drawn by their retention volume or retention time (ln Mw vs. K, where K is the partition coefficient). The retention properties of the unknown sample are then compared with the calibration data to determine the size or weight of the investigated molecule. Even though the standards used for the calibration process have well-defined molecular weights, they do have a distribution in their molecular weight and manufacturers use a quantity
夽 Presented at the 41st International Symposium on High Performance Liquid Phase Separations – HPLC 2014, 10–15 May 2014, New Orleans, Louisiana, USA. ∗ 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). http://dx.doi.org/10.1016/j.chroma.2014.09.023 0021-9673/© 2014 Elsevier B.V. All rights reserved.
to characterize the fitness-to-purpose of these polymers, polydispersity (P). The polydispersity of a polymer sample is defined as the ratio of the weight- and number-averaged relative molecular weights of the polymer sample; P = Mw /Mn [3]. Although there is a new recommended IUPAC terminology for polydispersity [4] where ‘degree-of-polymerization dispersity’ and shortly the word ‘dispersity’ is recommended for use, we rather use the term polydispersity because this is still the common way polymer scientist refer to the distribution of polymer molecules. The knowledge and the effect of polydispersity – that it increases the peak width and leads to an apparent increase in the height equivalent to a theoretical plate (HETP) and a decrease in the number of theoretical plates (N) – is almost of the same age as SEC [5–8]. It is very difficult, if feasible at all, to obtain experimentally the real polydispersity by size-exclusion chromatography. However, the first equations that described the contribution of polydispersity to the HETP value greatly underestimated the effect of polydispersity [5,6], it was proven that similarly to a new equation given by Knox [3], the data given by the manufacturers overestimate it as well [9,10]. The reduced plate height (h = H/dp , where H is the height equivalent to a theoretical plate and dp is the diameter of a particle) is a measure of the column zone-broadening and therefore of the chromatographic efficiency. Due to the additivity of variances it is possible to divide the plate height into contributors as h = hkin + hP ,
(1)
A. Sepsey et al. / J. Chromatogr. A 1365 (2014) 156–163
where hkin represents the kinetic contributions and hP represents the contribution of polydispersity to the total – apparent – reduced plate height. The elimination of the first term of Eq. (1) is experimentally impossible, because several contributing factors affect the band-broadening at the same time. Although recent improvements in experimental detection techniques in SEC made precise determination of molecular-weight-distribution possible, Netopilík showed that for samples with low polydispersity, the errors due to the band-broadening and due to the error in the interdetector volume increases as polydispersity decreases [11]. The effect of polydispersity is in fact so large that even with P = 1.01 it is difficult to make accurate determinations of hkin because polydispersity of even a relatively narrow polymer fraction will contribute substantially to the total width of the eluted peak [3]. These results show that there is a need for a theoretical model to demonstrate the limitations of the contribution of polydispersity. In this study, we integrate polydispersity into the the microscopic (stochastic) theory of size-exclusion chromatography. With this model it is possible to obtain the real contribution of polydispersity to the retention properties and efficiency. These results can be compared against experimental evaluations and we can get a clearer picture from the magnitude of the other contributors of the band-broadening as well.
157
uncertain which exact size parameter determines the separation [22]. The average number of the pore ingress steps (np ) and the sojourn time in the pores ( p ) can be written as np = nperm (1 − )me ,
(4)
and p = perm (1 − )mp ,
(5)
where me and mp are constants depending on the ingress and on the egress processes, respectively, nperm and perm are the number of pore ingress steps and the time that the molecule spends inside the pore for a totally permeable molecule. These expressions are valid if takes a value between 0 and 1, otherwise both of them are zero similarly to Eq. (2). As it was shown earlier [12] both the pore ingress and egress processes affect the selectivity of SEC and parameters me , mp and m are related: m = me + mp .
(6)
The relative contribution of the pore egress process to the overall size-exclusion effect (˛) was introduced for a former relation between these parameters [14]: mp . m
2. Theory
˛=
In ideal size-exclusion chromatography the separation is governed by entropy only, there is no interaction between the sample molecules and the stationary phase and the separation mechanisms other than the size-exclusion effect are omitted. SEC is indeed based on randomly occurring entrapment and release of molecules in the mesopores of the stationary phase [12–15]. The molecular or stochastic model of chromatography is more convenient to describe the size-exclusion effect than other theories of chromatography, because it uses random variables and probabilistic terms to characterize the separation process. This model was first introduced for adsorption chromatography where the time of the adsorption and the number of the adsorption–desorption steps were used as parameters [16]. This model is fully independent of the physico-chemical mechanisms responsible for the retention so it has been extended for several methods such as partition, ion-exchange or size-exclusion chromatography. Thus, for SEC it can be used by the average number of pore ingress and egress processes (np ), by the average time spent by a molecule inside the pores ( p ) and by the average time spent in the mobile phase ( m ). The stochastic theory of chromatography was adapted to SEC by Carmichael [17–19] assuming that the ingress steps are determined by a Poisson process and the egress process is described by an exponential distribution. A detailed description of the stochastic theory of SEC – using the characteristic function method [20] – was given recently [21]. Here we only give a brief summary of the important equations and related concepts. The partition coefficient in SEC is
The characteristic function is the Fourier transform of the elution profile [20] and contains all information about the separation process accounted for the stagnant zone processes:
KSEC =
(1 − )m
if 0 ≤ ≤ 1
0
if > 1
(2)
where m is a constant whose value depends on the pore shape, and is the ratio of the gyration radius of the molecule (rG ) to the radius of the pore opening (rp ) =
rG . rp
(3)
In general the gyration radius of the molecule determines whether or not the molecule can enter the pore. Though, it is still
(7)
(ω) = exp
np − np 1 − iωp
(8)
,
where i is the imaginary unit and ω is an auxiliary real variable (frequency). Because the characteristic function of a random variable and the probability density function of the random variable form a Fourier transform pair, the time-domain signal, i.e. the probability density function can be obtained from the characteristic function by inverse Fourier transform as
f (t) =
np −t/p −np e I1 tp
4np t p
,
(9)
where I1 is a modified Bessel function of the first kind and first order and t is the time. The kth moment of the chromatographic peak can be calculated from the characteristic function (Eq. (8)) by the moment theorem of the Fourier transform as
k = i
−k
dk (ω) dωk
.
(10)
ω=0
The stochastic theory can be extended for heterogeneous adsorption kinetics, when the stationary phase consists of more than one types of adsorption sites and for the case where the adsorption energy of the sites is determined by a distribution. Cavazzini et al. concluded [23] that the probability density function describing the peak shape can be written as the probabilityweighted convolution of the probability density functions of the different sites. The calculation of the moments of such a peak profile is however difficult – if it is possible at all – in time domain, so it cannot practically be used in this form. The characteristic function can be defined when the pore size of the stationary phase particles is described by a distribution, and also the moments can be evaluated in that case [21]. The validation of the stochastic theory of size-exclusion chromatography for wide pore size distribution was established by experimental data [24].
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Those results demonstrate that a distribution of the pore size has important consequences on the retention properties of a monodisperse (i.e. uniform) sample. It was already shown in the 1960s, however, that polydispersity has major influence on the band width thus on the chromatographic efficiency [3], this has not yet been taken into account in the stochastic theory of SEC. Integrating the effect of the polydispersity into the stochastic theory, we can theoretically investigate its effects on the retention properties. 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. In all cases rp was set to 12.434 nm because in one of our former works we obtained that value for an Aeris Widepore stationary phase [24]. The other parameters used for a certain calculation are indicated there. The moments were obtained using the equations given below and validated via integration of the peak profiles.
4.1.1. Log-normal distribution The log-normal distribution is defined as
(rG )
log−norm
To determine the molecular weight distribution of a sample, an accurate calibration process is needed where the knowledge of the accurate molecular weight and size of the standards is of a great importance. When the size-exclusion process is described at the molecular level by the stochastic theory, polydispersity must be incorporated. This will provide a closer understanding of the influence of polydispersity and one will get a clearer picture about its consequences on the separation process and on the retention properties. The concept for multiple-site adsorption [23] can be extended for a continuous distribution of sites in adsorption chromatography as well as for a continuous distribution of pore sizes in SEC [21]. Following the same notion, it is feasible to take polydispersity of the sample into account. When k types of molecules are present, each with a relative abundance of pj , (j = 1, . . ., k), the characteristic function is then written as
(ω) = exp
np ⎣−1 +
⎩
k j=1
⎤⎫ ⎬ 1 ⎦ , pj 1 − iωp,j ⎭
(11)
where p,j is the time spent by the jth molecule in a pore. Eq. (11) can be extended to a continuous distribution of molecular sizes. Because both the pore ingress and egress processes are influenced by polydispersity, the probability density function of the molecular size distribution, (rG ), should be introduced to the model. The term rG in Eq. (3) and so in Eqs. (4) and (5) should be replaced by a probability density function (PDF) of a distribution. Then the following characteristic function is derived:
(ω) = exp
rp
(rG )nG 0
−
log−norm
1 = nperm perm a
1 − 1 dr G , 1 − iωG
(12)
where nG and G represents the dependence of np and p on the gyration radius of the molecule. Polydispersity is described by three types of distributions in the literature: Poisson, log-normal, and normal distribution. Although referring to the central limit theorem for large numbers, both the Poisson and log-normal distributions can be approximated by normal distribution; here we show the characteristic function and the derived moments for both the log-normal and normal distributions.
2
2 2
,
(13)
,
2 2 = 2nperm perm a
log−norm
log−norm
where a
(14) ,
(15)
is determined by the following expression:
1 2 2 (−)k ek /2 2
log−norm
a
=
erfc
k
k=0
4.1. Stochastic theory extended for polydispersity
⎡
= √ exp 2 rG
ln rG − ln rG,0
where rG,0 and represent the mean and the standard deviation of the distribution, respectively. After substituting Eq. (13) into Eq. (12) and calculating the moments by Eq. (10), the following moments are derived:
4. Results and discussion
⎧ ⎨
1
k 2 + ln √ 2
,
(16)
where refers to the proper sum of me and mp for the different moments and thus strongly depends on the pore geometry. 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 log−norm conical or spherical. In this case a
is equal to KSEC . The second central moment can be calculated using = me + 2mp , and the third central moment by using = me + 3mp . Note that Eqs. (14)–(16) are identical to the ones we obtained for the size-exclusion process of monodisperse samples on stationary phase particles with log-normal pore size distribution (see Eqs. (22)–(25) in Ref. [21]). Thus, we can conclude that pore size distribution of the stationary phase and polydispersity have absolutely identical effect on the band profile as long as their distributions are identical. There is no way to tell their influence apart. 4.1.2. Normal distribution For further calculations we use the normal distribution in this work. In case of normal distribution, the following PDF is used:
(rG )
norm
1
exp = √ 2
−
rG − rG,0
2
2 2
,
(17)
where rG,0 and represents the mean and the standard deviation of the normal distribution, respectively. The characteristic function of the peak shape for normal pore size distribution will be obtained after integrating the distribution ((rG )norm ) into the characteristic function:
(ω) = exp
1 √ 2
rp
nG exp 0
1 − 1 dr G 1 − iωG
−
rG − rG,0
2
2 2
(18)
One should exercise caution when Eq. (18) is used, because as polydispersity increases, smaller fraction of the molecules can enter the pores. The fraction of polymer molecules that can enter the pores should be determined by integrating the PDF of the normal distribution between the proper limits. After taking this into account, a new equation was derived for the characteristic function:
A. Sepsey et al. / J. Chromatogr. A 1365 (2014) 156–163
(ω) = (1 − ) + exp
×
1 √ 2
rp
nG exp 0
−
rG − rG,0
2
the pores, i.e. if ≈ 1. In this case the following equations can be used:
2 2
1 − 1 dr G 1 − iωG
(19)
where represents the fraction of the molecules that are small enough to enter the pores:
=
1 √ 2
rp
exp 0
−
(rG − rG,0 )2 2 2
dr G
159
(20)
Eq. (19) is the characteristic function of the elution profile for a polydisperse sample at any value of . It contains all information for the separation system and the chromatogram can be calculated as its inverse Fourier transform. Eq. (19), however, cannot be evaluated analytically for the general case, only the calculation of the moments is possible. A simple 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 [14] that for the ingress process me > 0 in Eq. (4) and for the egress process mp < 0 in Eq. (5). To obtain KSEC and the moments of a peak profile, one needs to know the size parameter, and the relative standard deviation of the molecule size distribution, ır defined as the ratio of the standard deviation of the molecule size distribution to the mean of that distribution: ır = (21) rG,0
1 = nperm perm anorm ,
(25)
2 2 = 2nperm perm anorm ,
(26)
strongly depends on the molecular size and on the pore where anorm
shape and its expression becomes more complex as m increases. The proper equations for the studied types of geometries are summarized in Appendix A. Although anorm and thus the moments of the band profile
can only be calculated analytically when me + mp > 0 for 1 or me + 2mp > 0 for 2 . Nevertheless, one can always obtain the elution profile numerically with the inverse Fourier transform of Eq. (18) and calculate the moments by numerical integration of the peak profile. The effect of polydispersity becomes more complicated as the size ratio exceeds above = 0.5. In this case is getting smaller than cannot be used any more. The reason for this is that a 1, and anorm
fraction of the molecules is excluded from the pores and elutes as a split peak. Thus, new equations were derived for the first absolute and for the second central moments that incorporate the effect of . For 1 this means only a multiplication by . For the calculation of 2 , the solution is not so simple. The extension of these equations to the split-peak scenario is beyond the scope of this paper.
Subscript r in Eq. (21) shows that ır depends on the radius of the molecules. It was shown [13] that by combining the Kirkwood–Riseman relationship with the Stokes–Einstein equation, one gets a direct relation of the gyration radius to the molecular weight (Mw ) as rG =
B kB TMw , 4 K
(22)
where kB is the Boltzmann constant, T the temperature, the viscosity of the mobile phase, and B the exponent of the power law relating the diffusion coefficient of the polymer to the reciprocal of its molecular weight. Since K and B are tabulated for various polymer–solvent systems, rG can be used in all equations based on . The relation between polydispersity, P defined by Knox and the relative standard deviation of the distribution of molecular weights, ıM is relatively simple [25]: ı2M = P − 1.
Fig. 1. Effect of the relative standard deviation of the molecule size distribution (polydispersity), ır on the size distribution of the sample molecules. The average molecule size was set to 5 nm and the individual curves represent the percentage probability of the molecule sizes if ır is 0.1, 0.2 and 0.3, respectively.
(23)
The relationship between the weight and radius of polymer molecules is defined by Eq. (22). The typical values for parameter B in that equation are around B ≈ 0.5. Therefore molecular weight is roughly proportional with the square of the molecular radius for polymers: Mw ∝ rG2 . The distribution of molecular weights will be no longer normal, it will differ from the distribution of the radii. Monte Carlo simulation by generating one million normally distributed random numbers was carried out. The simulation shows that a relationship ıM ≈ 2ır can be established. Thus, polydispersity is related to the size distribution of the molecules as 4ı2r ≈ P − 1.
(24)
The moments of the elution profile will be obtained from Eq. (18). They are similar to the moments obtained for log-normal distribution (see Eqs. (14)–(15)), but only if all the molecules can enter
Fig. 2. The dependence of parameter on the relative standard deviation of the molecule size distribution (polydispersity), ır and on the ratio of the gyration radius of the sample molecule to the pore size, . At the proper values of , only the shown percentage of molecules can enter the pores.
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A. Sepsey et al. / J. Chromatogr. A 1365 (2014) 156–163
Fig. 3. The effect of the relative standard deviation of the molecule size distribution (polydispersity) (ır ) and the size of the molecule relative to the pore size () on the partition coefficient (KSEC ) at various pore geometries. 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).
4.2. Effect of polydispersity The molecular model of chromatography and polydispersity described by normal distribution was used to show the effect of polydispersity on the retention properties in size-exclusion chromatography. To properly understand the effect of polydispersity, all parameters that have impact on the band profile have to be taken into account. Thus, we calculated chromatograms by inverse Fourier transform for various parameter combinations to observe the caused effects. Manufacturers characterize their polymers by polydispersity index. For a monodisperse sample, P = 1, while for a narrowly distributed standard P = 1.05 is a typical pre-estimate. According to
the Monte Carlo simulation this latter polydispersity corresponds to ır = 0.113, which is a moderately broad distribution. In Fig. 1, the effect of the relative standard deviation, ır on the distribution of the size of the sample molecules is shown. The average molecule size was set to rG = 5 nm and the individual curves represent the percentage probability of the molecule sizes if ır = 0.1, 0.2 and 0.3. It can be concluded that even a narrow distribution has a large impact on the molecular size. If ır were set to zero, all the molecules would be of the same size and we would only see a high pulse at 5 nm. In the present study, the effect of polydispersity will not be investigated for ır > 0.3, which value corresponds to P = 1.32 according to the Monte Carlo simulation. Furthermore, Fig. 1 confirms that when ır > 0.3, a significant part of the the distribution
Fig. 4. The effect of the relative standard deviation of the molecule size distribution (polydispersity), ır on the calculated chromatograms for slit shaped pores, for cylindrical pores and for spherical or conical pores when the size parameter is = 0.5. In the inserts the first absolute moment and the second central moments of the chromatograms are plotted. Parameters used for the calculations: nperm = 1600, perm = 1, ˛ = −1.
A. Sepsey et al. / J. Chromatogr. A 1365 (2014) 156–163
161
Fig. 5. The effect of the size parameter () on the calculated chromatograms for slit shaped pores, for cylindrical pores and for spherical or conical pores when ır = 0.2. In the inserts the the first absolute moment and the second central moment of the chromatograms are plotted. Parameters used for the calculations: nperm = 2000, perm = 0.4, ˛ = −1.
Fig. 6. 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 = 0.5 and ır = 0.2. In the inserts the first absolute moment and the second central moment of the chromatograms are plotted. Parameters used for the calculations: nperm = 2000, perm = 0.4, ˛ = −1. Table 1 Relative standard deviation values calculated according to the polydispersity, P using Eq. (24) and Monte Carlo simulation, respectively. P
ır (Eq. (24))
ır (MC)
1.00 1.05 1.10 1.15 1.20 1.25 1.30
0 0.112 0.158 0.194 0.224 0.250 0.274
0 0.113 0.162 0.199 0.233 0.263 0.291
would span to negative sizes and the model would lose its physical background. Table 1 summarizes the ır values calculated using Eq. (24) for various polydispersities. It can be seen that these values are in great agreement with the ones obtained via the Monte Carlo simulation.
As polydispersity increases, a fraction of the molecules will be excluded from the pores, since the distribution will span beyond the pore size and the characteristic function that considers partial exclusion is given in Eq. (19). The fraction of the molecules that can actually enter the pores ( ) may significantly drop if > 0.5, depending the value of polydispersity. This effect is shown in Fig. 2. Every single line represents the percentage of the molecules that can enter the pore at the given value of for the range 0 < ır < 0.3. For the other values of , (0 < < 0.5), is essentially equal to 1. Fig. 3 shows the effect of polydispersity on the partition coefficient at various pore geometries. This can be achieved by depicting the four anorm functions (see Eqs. (28)–(31) in Appendix A) in case
of = 1, when the obtained expression is equal to KSEC . In Fig. 3a, m = 0, so no geometry is assumed. It can be seen that in the near monodisperse case (ır = 0.01) there is a sharp step at = 1. This
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A. Sepsey et al. / J. Chromatogr. A 1365 (2014) 156–163
means that the molecules that are smaller than the pore size can enter the pores while the molecules that are larger than the pores are all excluded. This sharp division gets blurred as we increase the value of ır , because there will be molecules that are either smaller or larger than the average molecule. These molecules behave differently because of their size. The same effect can be seen in Fig. 3b–d where slit shaped, cylindrical and conical or spherical pore shapes are expected, respectively. The exclusion limit tends to decrease as the pore geometry becomes more complex. Usually, it is assumed that polydispersity does not affect the retention time (the first absolute moment) of the band profile, and only its band-broadening effect is studied. Fig. 4 demonstrates the effect of the relative standard deviation of the molecular size, ır on the chromatogram and on its first absolute and second central moments. The ratio of the molecule size to the pore size was = 0.5 thus all the molecules can enter the pores according to Fig. 2 in the investigated range of ır . It can be noticed that – in contrast to general beliefs – the increase in polydispersity leads to a shift of retention time (except for m = 1) and also causes band-broadening. The extent of retention time shift and band-broadening is increasing with the value of the pore geometry constant (m), thus with the complexity of the pore geometry. The chromatograms calculated for slit shaped pores have relatively high retention time and peak width compared with the other geometries. Therefore, we can conclude that the higher the complexity of the pore geometry the more efficient the separation is. The effect of the size ratio parameter, on the chromatograms, as well as on their first absolute and second central moment can be seen in Fig. 5. Classical SEC behavior and properties are present: by increasing parameter , the molecules are more excluded from the pores and are eluted sooner. Because our model contains no other effect than polydispersity, we can say, that the effect of that is very significant. The observed trend in the second central moments is because of the excluded ratio of the molecules appear at zero time as a Dirac delta. The farther the observed peaks are from zero, the more their second central moment change because of the contribution of ; for slit shaped pores (m = 1) one can see a high broadening in the range 0.6 < < 0.8 and a quick decay to = 1. For m = 2 and m = 3, the observed peaks are near to the Dirac delta so we perceive band constriction. In Fig. 6, the effect of the relative contribution of the pore egress process to the overall size-exclusion effect (˛) on the chromatograms and on their first absolute and second central moments are showed. By calculating these, we used ır = 0.2 and = 0.5. The first absolute moments of the peaks do not change for the individual pore geometries. As the egress processes come to the foreground, the band-broadening becomes more significant. Similarly to the other cases presented earlier, the alteration can be best observed for the conical or spherical pore shapes.
5. Conclusions The stochastic theory of size-exclusion chromatography was extended for polydisperse samples i.e. where the size of the molecules is described by a distribution. The results presented verify previous observations and experiences: a significant bandbroadening and loss in the selectivity can be observed for polydisperse samples. By integrating polydispersity into the stochastic theory of chromatography we have the proper tools to investigate the effect of polydispersity and that of the pore geometry on the chromatograms and on the moments at the same time. Although the effect of zone broadening caused only by polydispersity could not be evaluated experimentally, its effect is very important and should be understood to optimize the calibration processes and the separations.
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. Appendix A. Parameter anorm depends on which moment is being calculated
and on the pore geometry, thus on the value of me and mp . The equations shown below do not include the impact of the fraction of the molecules that cannot enter the pores, i.e. they consider only the cases where split peak are not observed; < 0.6 (see Fig. 2). For the sake of brevity, we introduce the normalized variable 1− = √ 2ır
(27)
is used if m = me + mp = 0 in 1 , or if me + 2mp = 0 in 2 . anorm 0 anorm = 0
1 erfc − 2
(28)
anorm is used if m = me + mp = 1 in 1 , or if me + 2mp = 1 in 2 . 1
anorm = 1
1− erfc − 2
1 +√
exp −
2
(29)
is used if m = me + mp = 2 in 1 , or if me + 2mp = 2 in 2 . anorm 2 anorm 2
(1 − )2 = 2
2
2
2
+1 2
erfc −
1 +√
exp −
2
(30)
is used if m = me + mp = 3 in 1 , or if me + 2mp = 3 in 2 . anorm 3 anorm = 3
(1 − )3 2
2
2
2
+3 2
erfc −
2+1 + √ exp − 3
2
(31)
References [1] W.W. Yau, J.J. Kirkland, D.D. Bly, Modern Size-Exclusion Liquid Chromatography, Wiley, New York, 1979. [2] L. Hagel, J.-C. Janson, Size-exclusion chromatography, in: E. Heftmann (Ed.), Fundamentals and Applications of Chromatography and Related Differential Migration Methods Fundamentals and Techniques, Vol. 51, Part A of Journal of Chromatography Library, Elsevier, 1992, pp. 267–307. [3] J.H. Knox, F. McLennan, Allowance for polydispersity in the determination of the true plate height in gpc, Chromatographia 10 (1977) 75–78. [4] R.F.T. Stepto, Dispersity in polymer science, Pure Appl. Chem. 8 (2009) 351–353. [5] D.D. Bly, Determination of theoretical plates in gel permeation chromatography by using polydisperse materials (polymers), J. Polym. Sci. Part A-1 6 (1968) 2085–2089. [6] D.D. Bly, Properties and uses of a linear, semilog calibration in gel permeation chromatography, Anal. Chem. 41 (1969) 477–480. [7] K.K. Unger, R. Kern, M.C. Ninou, K.-F. Krebs, High-performance gel permeation chromatography with a new type of silica packing material, J. Chromatogr. A 99 (1974) 435–443. [8] K.K. Unger, R. Kern, Column parameters controlling resolution in highperformance gel permeation chromatography, J. Chromatogr. A 122 (1976) 345–354. [9] Y.V. Heyden, S.-T. Popovici, B. Staal, P.J. Schoenmakers, Contribution of the polymer standards’ polydispersity to the observed band broadening in sizeexclusion chromatography, J. Chromatogr. A 986 (2003) 1–15. [10] M. Netopilík, S. Podzimek, P. Kratochvíl, Estimation of width of narrow molecular-weight distributions by size-exclusion chromatography with concentration and light scattering detection, J. Chromatogr. A 922 (2001) 25–36. [11] M. Netopilík, Effect of local polydispersity in size exclusion chromatography with dual detection, J. Chromatogr. A 793 (1998) 21–30. [12] F. Dondi, A. Cavazzini, M. Remelli, A. Felinger, M. Martin, Stochastic theory of size exclusion chromatography by the characteristic function approach, J. Chromatogr. A 943 (2002) 185–207.
A. Sepsey et al. / J. Chromatogr. A 1365 (2014) 156–163 [13] L. Pasti, F. Dondi, M. Van Hulst, P.J. Schoenmakers, M. Martin, A. Felinger, Experimental validation of the stochastic theory of size exclusion chromatography: 1. Retention on single and coupled columns, Chromatographia 57 (2003) S171–S186. [14] A. Felinger, L. Pasti, F. Dondi, M. van Hulst, P.J. Schoenmakers, M. Martin, Stochastic theory of size exclusion chromatography: peak shape analysis on single columns, Anal. Chem. 77 (2005) 3138–3148. [15] A. Felinger, Molecular dynamic theories in chromatography, J. Chromatogr. A 1184 (2008) 20–41. [16] J.C. Giddings, H. Eyring, A molecular dynamic theory of chromatography, J. Phys. Chem. 59 (1955) 416–421. [17] J.B. Carmichael, Stochastic model for gel permeation separation of polymers, J. Polym. Sci. Part A-2 6 (1968) 517–527. [18] J.B. Carmichael, The relation between pore size distribution and polymer separation in gel permeation chromatography, Macromolecules 1 (1968) 526–529.
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[19] J.B. Carmichael, Theory of gel filtration separation of biopolymers assuming a Gaussian distribution of pore sizes, Biopolymers 6 (1968) 1497–1499. [20] F. Dondi, M. Remelli, The characteristic function method in the stochastic theory of chromatography, J. Phys. Chem. 90 (1986) 1885–1891. [21] A. Sepsey, I. Bacskay, A. Felinger, Molecular theory of size exclusion chromatography for wide pore size distributions, J. Chromatogr. A 1331 (2014) 52–60. [22] I. Teraoka, Calibration of retention volume in size exclusion chromatography by hydrodynamic radius, Macromolecules 37 (2004) 6632–6639. [23] A. Cavazzini, M. Remelli, F. Dondi, A. Felinger, Stochastic theory of multiple-site linear adsorption chromatography, Anal. Chem. 71 (1999) 3453–3462. [24] I. Bacskay, A. Sepsey, A. Felinger, Determination of the pore size distribution of HPLC stationary phases, J. Chromatogr. A 1339 (2014) 110–117. [25] S.-T. Popovici, W.T. Kok, P.J. Schoenmakers, Band broadening in size-exclusion chromatography of polydisperse samples, J. Chromatogr. A 1060 (2004) 237–252.
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Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Polydispersity in size-exclusion chromatography: A stochastic approach夽 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 15 July 2014 Received in revised form 5 September 2014 Accepted 8 September 2014 Available online 17 September 2014 Keywords: Polydispersity Stochastic theory Size-exclusion chromatography
a b s t r a c t We investigate the impact of polydispersity of the sample molecules on the separation process and on the efficiency of size-exclusion chromatography. Polydispersity was integrated into the molecular (stochastic) model of chromatography; the characteristic function, the band profile and the most important moments of the elution profiles were calculated for several kind of pore structures. We investigated the parameters affected by polydispersity on the separation for a number of pore shapes. Our results demonstrate that even a small distribution in the molecular size (i.e. polydispersity) can contribute substantially to the total width of the chromatographic peak. The pure effect of polydispersity can only be investigated via mathematical modeling, because its contribution to an experimental chromatogram cannot be separated from other band-broadening effects. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Size-exclusion chromatography (SEC) is often used for determining the molecular size or molecular weight on the basis of the retention of the molecules in porous stationary phases, than for separating polymers from each other [1]. SEC is defined as the differential elution of solutes from the porous stationary phase caused by different degrees of steric exclusion of the solutes from the pore volume due to the differential molecular size of solutes [2]. The determination of the molecular weight (Mw ) can be achieved by a calibration process, where a number of special standard polymers with well-known molecular weight and size are eluted, when an accurate calibration curve is drawn by their retention volume or retention time (ln Mw vs. K, where K is the partition coefficient). The retention properties of the unknown sample are then compared with the calibration data to determine the size or weight of the investigated molecule. Even though the standards used for the calibration process have well-defined molecular weights, they do have a distribution in their molecular weight and manufacturers use a quantity
夽 Presented at the 41st International Symposium on High Performance Liquid Phase Separations – HPLC 2014, 10–15 May 2014, New Orleans, Louisiana, USA. ∗ 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). http://dx.doi.org/10.1016/j.chroma.2014.09.023 0021-9673/© 2014 Elsevier B.V. All rights reserved.
to characterize the fitness-to-purpose of these polymers, polydispersity (P). The polydispersity of a polymer sample is defined as the ratio of the weight- and number-averaged relative molecular weights of the polymer sample; P = Mw /Mn [3]. Although there is a new recommended IUPAC terminology for polydispersity [4] where ‘degree-of-polymerization dispersity’ and shortly the word ‘dispersity’ is recommended for use, we rather use the term polydispersity because this is still the common way polymer scientist refer to the distribution of polymer molecules. The knowledge and the effect of polydispersity – that it increases the peak width and leads to an apparent increase in the height equivalent to a theoretical plate (HETP) and a decrease in the number of theoretical plates (N) – is almost of the same age as SEC [5–8]. It is very difficult, if feasible at all, to obtain experimentally the real polydispersity by size-exclusion chromatography. However, the first equations that described the contribution of polydispersity to the HETP value greatly underestimated the effect of polydispersity [5,6], it was proven that similarly to a new equation given by Knox [3], the data given by the manufacturers overestimate it as well [9,10]. The reduced plate height (h = H/dp , where H is the height equivalent to a theoretical plate and dp is the diameter of a particle) is a measure of the column zone-broadening and therefore of the chromatographic efficiency. Due to the additivity of variances it is possible to divide the plate height into contributors as h = hkin + hP ,
(1)
A. Sepsey et al. / J. Chromatogr. A 1365 (2014) 156–163
where hkin represents the kinetic contributions and hP represents the contribution of polydispersity to the total – apparent – reduced plate height. The elimination of the first term of Eq. (1) is experimentally impossible, because several contributing factors affect the band-broadening at the same time. Although recent improvements in experimental detection techniques in SEC made precise determination of molecular-weight-distribution possible, Netopilík showed that for samples with low polydispersity, the errors due to the band-broadening and due to the error in the interdetector volume increases as polydispersity decreases [11]. The effect of polydispersity is in fact so large that even with P = 1.01 it is difficult to make accurate determinations of hkin because polydispersity of even a relatively narrow polymer fraction will contribute substantially to the total width of the eluted peak [3]. These results show that there is a need for a theoretical model to demonstrate the limitations of the contribution of polydispersity. In this study, we integrate polydispersity into the the microscopic (stochastic) theory of size-exclusion chromatography. With this model it is possible to obtain the real contribution of polydispersity to the retention properties and efficiency. These results can be compared against experimental evaluations and we can get a clearer picture from the magnitude of the other contributors of the band-broadening as well.
157
uncertain which exact size parameter determines the separation [22]. The average number of the pore ingress steps (np ) and the sojourn time in the pores ( p ) can be written as np = nperm (1 − )me ,
(4)
and p = perm (1 − )mp ,
(5)
where me and mp are constants depending on the ingress and on the egress processes, respectively, nperm and perm are the number of pore ingress steps and the time that the molecule spends inside the pore for a totally permeable molecule. These expressions are valid if takes a value between 0 and 1, otherwise both of them are zero similarly to Eq. (2). As it was shown earlier [12] both the pore ingress and egress processes affect the selectivity of SEC and parameters me , mp and m are related: m = me + mp .
(6)
The relative contribution of the pore egress process to the overall size-exclusion effect (˛) was introduced for a former relation between these parameters [14]: mp . m
2. Theory
˛=
In ideal size-exclusion chromatography the separation is governed by entropy only, there is no interaction between the sample molecules and the stationary phase and the separation mechanisms other than the size-exclusion effect are omitted. SEC is indeed based on randomly occurring entrapment and release of molecules in the mesopores of the stationary phase [12–15]. The molecular or stochastic model of chromatography is more convenient to describe the size-exclusion effect than other theories of chromatography, because it uses random variables and probabilistic terms to characterize the separation process. This model was first introduced for adsorption chromatography where the time of the adsorption and the number of the adsorption–desorption steps were used as parameters [16]. This model is fully independent of the physico-chemical mechanisms responsible for the retention so it has been extended for several methods such as partition, ion-exchange or size-exclusion chromatography. Thus, for SEC it can be used by the average number of pore ingress and egress processes (np ), by the average time spent by a molecule inside the pores ( p ) and by the average time spent in the mobile phase ( m ). The stochastic theory of chromatography was adapted to SEC by Carmichael [17–19] assuming that the ingress steps are determined by a Poisson process and the egress process is described by an exponential distribution. A detailed description of the stochastic theory of SEC – using the characteristic function method [20] – was given recently [21]. Here we only give a brief summary of the important equations and related concepts. The partition coefficient in SEC is
The characteristic function is the Fourier transform of the elution profile [20] and contains all information about the separation process accounted for the stagnant zone processes:
KSEC =
(1 − )m
if 0 ≤ ≤ 1
0
if > 1
(2)
where m is a constant whose value depends on the pore shape, and is the ratio of the gyration radius of the molecule (rG ) to the radius of the pore opening (rp ) =
rG . rp
(3)
In general the gyration radius of the molecule determines whether or not the molecule can enter the pore. Though, it is still
(7)
(ω) = exp
np − np 1 − iωp
(8)
,
where i is the imaginary unit and ω is an auxiliary real variable (frequency). Because the characteristic function of a random variable and the probability density function of the random variable form a Fourier transform pair, the time-domain signal, i.e. the probability density function can be obtained from the characteristic function by inverse Fourier transform as
f (t) =
np −t/p −np e I1 tp
4np t p
,
(9)
where I1 is a modified Bessel function of the first kind and first order and t is the time. The kth moment of the chromatographic peak can be calculated from the characteristic function (Eq. (8)) by the moment theorem of the Fourier transform as
k = i
−k
dk (ω) dωk
.
(10)
ω=0
The stochastic theory can be extended for heterogeneous adsorption kinetics, when the stationary phase consists of more than one types of adsorption sites and for the case where the adsorption energy of the sites is determined by a distribution. Cavazzini et al. concluded [23] that the probability density function describing the peak shape can be written as the probabilityweighted convolution of the probability density functions of the different sites. The calculation of the moments of such a peak profile is however difficult – if it is possible at all – in time domain, so it cannot practically be used in this form. The characteristic function can be defined when the pore size of the stationary phase particles is described by a distribution, and also the moments can be evaluated in that case [21]. The validation of the stochastic theory of size-exclusion chromatography for wide pore size distribution was established by experimental data [24].
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Those results demonstrate that a distribution of the pore size has important consequences on the retention properties of a monodisperse (i.e. uniform) sample. It was already shown in the 1960s, however, that polydispersity has major influence on the band width thus on the chromatographic efficiency [3], this has not yet been taken into account in the stochastic theory of SEC. Integrating the effect of the polydispersity into the stochastic theory, we can theoretically investigate its effects on the retention properties. 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. In all cases rp was set to 12.434 nm because in one of our former works we obtained that value for an Aeris Widepore stationary phase [24]. The other parameters used for a certain calculation are indicated there. The moments were obtained using the equations given below and validated via integration of the peak profiles.
4.1.1. Log-normal distribution The log-normal distribution is defined as
(rG )
log−norm
To determine the molecular weight distribution of a sample, an accurate calibration process is needed where the knowledge of the accurate molecular weight and size of the standards is of a great importance. When the size-exclusion process is described at the molecular level by the stochastic theory, polydispersity must be incorporated. This will provide a closer understanding of the influence of polydispersity and one will get a clearer picture about its consequences on the separation process and on the retention properties. The concept for multiple-site adsorption [23] can be extended for a continuous distribution of sites in adsorption chromatography as well as for a continuous distribution of pore sizes in SEC [21]. Following the same notion, it is feasible to take polydispersity of the sample into account. When k types of molecules are present, each with a relative abundance of pj , (j = 1, . . ., k), the characteristic function is then written as
(ω) = exp
np ⎣−1 +
⎩
k j=1
⎤⎫ ⎬ 1 ⎦ , pj 1 − iωp,j ⎭
(11)
where p,j is the time spent by the jth molecule in a pore. Eq. (11) can be extended to a continuous distribution of molecular sizes. Because both the pore ingress and egress processes are influenced by polydispersity, the probability density function of the molecular size distribution, (rG ), should be introduced to the model. The term rG in Eq. (3) and so in Eqs. (4) and (5) should be replaced by a probability density function (PDF) of a distribution. Then the following characteristic function is derived:
(ω) = exp
rp
(rG )nG 0
−
log−norm
1 = nperm perm a
1 − 1 dr G , 1 − iωG
(12)
where nG and G represents the dependence of np and p on the gyration radius of the molecule. Polydispersity is described by three types of distributions in the literature: Poisson, log-normal, and normal distribution. Although referring to the central limit theorem for large numbers, both the Poisson and log-normal distributions can be approximated by normal distribution; here we show the characteristic function and the derived moments for both the log-normal and normal distributions.
2
2 2
,
(13)
,
2 2 = 2nperm perm a
log−norm
log−norm
where a
(14) ,
(15)
is determined by the following expression:
1 2 2 (−)k ek /2 2
log−norm
a
=
erfc
k
k=0
4.1. Stochastic theory extended for polydispersity
⎡
= √ exp 2 rG
ln rG − ln rG,0
where rG,0 and represent the mean and the standard deviation of the distribution, respectively. After substituting Eq. (13) into Eq. (12) and calculating the moments by Eq. (10), the following moments are derived:
4. Results and discussion
⎧ ⎨
1
k 2 + ln √ 2
,
(16)
where refers to the proper sum of me and mp for the different moments and thus strongly depends on the pore geometry. 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 log−norm conical or spherical. In this case a
is equal to KSEC . The second central moment can be calculated using = me + 2mp , and the third central moment by using = me + 3mp . Note that Eqs. (14)–(16) are identical to the ones we obtained for the size-exclusion process of monodisperse samples on stationary phase particles with log-normal pore size distribution (see Eqs. (22)–(25) in Ref. [21]). Thus, we can conclude that pore size distribution of the stationary phase and polydispersity have absolutely identical effect on the band profile as long as their distributions are identical. There is no way to tell their influence apart. 4.1.2. Normal distribution For further calculations we use the normal distribution in this work. In case of normal distribution, the following PDF is used:
(rG )
norm
1
exp = √ 2
−
rG − rG,0
2
2 2
,
(17)
where rG,0 and represents the mean and the standard deviation of the normal distribution, respectively. The characteristic function of the peak shape for normal pore size distribution will be obtained after integrating the distribution ((rG )norm ) into the characteristic function:
(ω) = exp
1 √ 2
rp
nG exp 0
1 − 1 dr G 1 − iωG
−
rG − rG,0
2
2 2
(18)
One should exercise caution when Eq. (18) is used, because as polydispersity increases, smaller fraction of the molecules can enter the pores. The fraction of polymer molecules that can enter the pores should be determined by integrating the PDF of the normal distribution between the proper limits. After taking this into account, a new equation was derived for the characteristic function:
A. Sepsey et al. / J. Chromatogr. A 1365 (2014) 156–163
(ω) = (1 − ) + exp
×
1 √ 2
rp
nG exp 0
−
rG − rG,0
2
the pores, i.e. if ≈ 1. In this case the following equations can be used:
2 2
1 − 1 dr G 1 − iωG
(19)
where represents the fraction of the molecules that are small enough to enter the pores:
=
1 √ 2
rp
exp 0
−
(rG − rG,0 )2 2 2
dr G
159
(20)
Eq. (19) is the characteristic function of the elution profile for a polydisperse sample at any value of . It contains all information for the separation system and the chromatogram can be calculated as its inverse Fourier transform. Eq. (19), however, cannot be evaluated analytically for the general case, only the calculation of the moments is possible. A simple 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 [14] that for the ingress process me > 0 in Eq. (4) and for the egress process mp < 0 in Eq. (5). To obtain KSEC and the moments of a peak profile, one needs to know the size parameter, and the relative standard deviation of the molecule size distribution, ır defined as the ratio of the standard deviation of the molecule size distribution to the mean of that distribution: ır = (21) rG,0
1 = nperm perm anorm ,
(25)
2 2 = 2nperm perm anorm ,
(26)
strongly depends on the molecular size and on the pore where anorm
shape and its expression becomes more complex as m increases. The proper equations for the studied types of geometries are summarized in Appendix A. Although anorm and thus the moments of the band profile
can only be calculated analytically when me + mp > 0 for 1 or me + 2mp > 0 for 2 . Nevertheless, one can always obtain the elution profile numerically with the inverse Fourier transform of Eq. (18) and calculate the moments by numerical integration of the peak profile. The effect of polydispersity becomes more complicated as the size ratio exceeds above = 0.5. In this case is getting smaller than cannot be used any more. The reason for this is that a 1, and anorm
fraction of the molecules is excluded from the pores and elutes as a split peak. Thus, new equations were derived for the first absolute and for the second central moments that incorporate the effect of . For 1 this means only a multiplication by . For the calculation of 2 , the solution is not so simple. The extension of these equations to the split-peak scenario is beyond the scope of this paper.
Subscript r in Eq. (21) shows that ır depends on the radius of the molecules. It was shown [13] that by combining the Kirkwood–Riseman relationship with the Stokes–Einstein equation, one gets a direct relation of the gyration radius to the molecular weight (Mw ) as rG =
B kB TMw , 4 K
(22)
where kB is the Boltzmann constant, T the temperature, the viscosity of the mobile phase, and B the exponent of the power law relating the diffusion coefficient of the polymer to the reciprocal of its molecular weight. Since K and B are tabulated for various polymer–solvent systems, rG can be used in all equations based on . The relation between polydispersity, P defined by Knox and the relative standard deviation of the distribution of molecular weights, ıM is relatively simple [25]: ı2M = P − 1.
Fig. 1. Effect of the relative standard deviation of the molecule size distribution (polydispersity), ır on the size distribution of the sample molecules. The average molecule size was set to 5 nm and the individual curves represent the percentage probability of the molecule sizes if ır is 0.1, 0.2 and 0.3, respectively.
(23)
The relationship between the weight and radius of polymer molecules is defined by Eq. (22). The typical values for parameter B in that equation are around B ≈ 0.5. Therefore molecular weight is roughly proportional with the square of the molecular radius for polymers: Mw ∝ rG2 . The distribution of molecular weights will be no longer normal, it will differ from the distribution of the radii. Monte Carlo simulation by generating one million normally distributed random numbers was carried out. The simulation shows that a relationship ıM ≈ 2ır can be established. Thus, polydispersity is related to the size distribution of the molecules as 4ı2r ≈ P − 1.
(24)
The moments of the elution profile will be obtained from Eq. (18). They are similar to the moments obtained for log-normal distribution (see Eqs. (14)–(15)), but only if all the molecules can enter
Fig. 2. The dependence of parameter on the relative standard deviation of the molecule size distribution (polydispersity), ır and on the ratio of the gyration radius of the sample molecule to the pore size, . At the proper values of , only the shown percentage of molecules can enter the pores.
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Fig. 3. The effect of the relative standard deviation of the molecule size distribution (polydispersity) (ır ) and the size of the molecule relative to the pore size () on the partition coefficient (KSEC ) at various pore geometries. 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).
4.2. Effect of polydispersity The molecular model of chromatography and polydispersity described by normal distribution was used to show the effect of polydispersity on the retention properties in size-exclusion chromatography. To properly understand the effect of polydispersity, all parameters that have impact on the band profile have to be taken into account. Thus, we calculated chromatograms by inverse Fourier transform for various parameter combinations to observe the caused effects. Manufacturers characterize their polymers by polydispersity index. For a monodisperse sample, P = 1, while for a narrowly distributed standard P = 1.05 is a typical pre-estimate. According to
the Monte Carlo simulation this latter polydispersity corresponds to ır = 0.113, which is a moderately broad distribution. In Fig. 1, the effect of the relative standard deviation, ır on the distribution of the size of the sample molecules is shown. The average molecule size was set to rG = 5 nm and the individual curves represent the percentage probability of the molecule sizes if ır = 0.1, 0.2 and 0.3. It can be concluded that even a narrow distribution has a large impact on the molecular size. If ır were set to zero, all the molecules would be of the same size and we would only see a high pulse at 5 nm. In the present study, the effect of polydispersity will not be investigated for ır > 0.3, which value corresponds to P = 1.32 according to the Monte Carlo simulation. Furthermore, Fig. 1 confirms that when ır > 0.3, a significant part of the the distribution
Fig. 4. The effect of the relative standard deviation of the molecule size distribution (polydispersity), ır on the calculated chromatograms for slit shaped pores, for cylindrical pores and for spherical or conical pores when the size parameter is = 0.5. In the inserts the first absolute moment and the second central moments of the chromatograms are plotted. Parameters used for the calculations: nperm = 1600, perm = 1, ˛ = −1.
A. Sepsey et al. / J. Chromatogr. A 1365 (2014) 156–163
161
Fig. 5. The effect of the size parameter () on the calculated chromatograms for slit shaped pores, for cylindrical pores and for spherical or conical pores when ır = 0.2. In the inserts the the first absolute moment and the second central moment of the chromatograms are plotted. Parameters used for the calculations: nperm = 2000, perm = 0.4, ˛ = −1.
Fig. 6. 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 = 0.5 and ır = 0.2. In the inserts the first absolute moment and the second central moment of the chromatograms are plotted. Parameters used for the calculations: nperm = 2000, perm = 0.4, ˛ = −1. Table 1 Relative standard deviation values calculated according to the polydispersity, P using Eq. (24) and Monte Carlo simulation, respectively. P
ır (Eq. (24))
ır (MC)
1.00 1.05 1.10 1.15 1.20 1.25 1.30
0 0.112 0.158 0.194 0.224 0.250 0.274
0 0.113 0.162 0.199 0.233 0.263 0.291
would span to negative sizes and the model would lose its physical background. Table 1 summarizes the ır values calculated using Eq. (24) for various polydispersities. It can be seen that these values are in great agreement with the ones obtained via the Monte Carlo simulation.
As polydispersity increases, a fraction of the molecules will be excluded from the pores, since the distribution will span beyond the pore size and the characteristic function that considers partial exclusion is given in Eq. (19). The fraction of the molecules that can actually enter the pores ( ) may significantly drop if > 0.5, depending the value of polydispersity. This effect is shown in Fig. 2. Every single line represents the percentage of the molecules that can enter the pore at the given value of for the range 0 < ır < 0.3. For the other values of , (0 < < 0.5), is essentially equal to 1. Fig. 3 shows the effect of polydispersity on the partition coefficient at various pore geometries. This can be achieved by depicting the four anorm functions (see Eqs. (28)–(31) in Appendix A) in case
of = 1, when the obtained expression is equal to KSEC . In Fig. 3a, m = 0, so no geometry is assumed. It can be seen that in the near monodisperse case (ır = 0.01) there is a sharp step at = 1. This
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means that the molecules that are smaller than the pore size can enter the pores while the molecules that are larger than the pores are all excluded. This sharp division gets blurred as we increase the value of ır , because there will be molecules that are either smaller or larger than the average molecule. These molecules behave differently because of their size. The same effect can be seen in Fig. 3b–d where slit shaped, cylindrical and conical or spherical pore shapes are expected, respectively. The exclusion limit tends to decrease as the pore geometry becomes more complex. Usually, it is assumed that polydispersity does not affect the retention time (the first absolute moment) of the band profile, and only its band-broadening effect is studied. Fig. 4 demonstrates the effect of the relative standard deviation of the molecular size, ır on the chromatogram and on its first absolute and second central moments. The ratio of the molecule size to the pore size was = 0.5 thus all the molecules can enter the pores according to Fig. 2 in the investigated range of ır . It can be noticed that – in contrast to general beliefs – the increase in polydispersity leads to a shift of retention time (except for m = 1) and also causes band-broadening. The extent of retention time shift and band-broadening is increasing with the value of the pore geometry constant (m), thus with the complexity of the pore geometry. The chromatograms calculated for slit shaped pores have relatively high retention time and peak width compared with the other geometries. Therefore, we can conclude that the higher the complexity of the pore geometry the more efficient the separation is. The effect of the size ratio parameter, on the chromatograms, as well as on their first absolute and second central moment can be seen in Fig. 5. Classical SEC behavior and properties are present: by increasing parameter , the molecules are more excluded from the pores and are eluted sooner. Because our model contains no other effect than polydispersity, we can say, that the effect of that is very significant. The observed trend in the second central moments is because of the excluded ratio of the molecules appear at zero time as a Dirac delta. The farther the observed peaks are from zero, the more their second central moment change because of the contribution of ; for slit shaped pores (m = 1) one can see a high broadening in the range 0.6 < < 0.8 and a quick decay to = 1. For m = 2 and m = 3, the observed peaks are near to the Dirac delta so we perceive band constriction. In Fig. 6, the effect of the relative contribution of the pore egress process to the overall size-exclusion effect (˛) on the chromatograms and on their first absolute and second central moments are showed. By calculating these, we used ır = 0.2 and = 0.5. The first absolute moments of the peaks do not change for the individual pore geometries. As the egress processes come to the foreground, the band-broadening becomes more significant. Similarly to the other cases presented earlier, the alteration can be best observed for the conical or spherical pore shapes.
5. Conclusions The stochastic theory of size-exclusion chromatography was extended for polydisperse samples i.e. where the size of the molecules is described by a distribution. The results presented verify previous observations and experiences: a significant bandbroadening and loss in the selectivity can be observed for polydisperse samples. By integrating polydispersity into the stochastic theory of chromatography we have the proper tools to investigate the effect of polydispersity and that of the pore geometry on the chromatograms and on the moments at the same time. Although the effect of zone broadening caused only by polydispersity could not be evaluated experimentally, its effect is very important and should be understood to optimize the calibration processes and the separations.
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. Appendix A. Parameter anorm depends on which moment is being calculated
and on the pore geometry, thus on the value of me and mp . The equations shown below do not include the impact of the fraction of the molecules that cannot enter the pores, i.e. they consider only the cases where split peak are not observed; < 0.6 (see Fig. 2). For the sake of brevity, we introduce the normalized variable 1− = √ 2ır
(27)
is used if m = me + mp = 0 in 1 , or if me + 2mp = 0 in 2 . anorm 0 anorm = 0
1 erfc − 2
(28)
anorm is used if m = me + mp = 1 in 1 , or if me + 2mp = 1 in 2 . 1
anorm = 1
1− erfc − 2
1 +√
exp −
2
(29)
is used if m = me + mp = 2 in 1 , or if me + 2mp = 2 in 2 . anorm 2 anorm 2
(1 − )2 = 2
2
2
2
+1 2
erfc −
1 +√
exp −
2
(30)
is used if m = me + mp = 3 in 1 , or if me + 2mp = 3 in 2 . anorm 3 anorm = 3
(1 − )3 2
2
2
2
+3 2
erfc −
2+1 + √ exp − 3
2
(31)
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