HPLC for Pharmaceutical Scientists 2007 (Part 6)

Chia sẻ: Big Big | Ngày: | Loại File: PDF | Số trang:17

Thêm vào BST

Báo xấu

103
lượt xem 19
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Size-exclusion chromatography (SEC) separates polymer molecules and biomolecules based on differences in their molecular size. The separation process in simpliﬁed form is based on the ability of sample molecules to penetrate inside the pores of packing material and is dependent on the relative size of analyte molecules and the respective pore size of the absorbent. The process also relies on the absence of any interactions with the packing material surface. Two types of SEC are usually distinguished: 1. Gel permeation chromatography (GPC)—separation of synthetic (organic-soluble) polymers. ...

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: HPLC for Pharmaceutical Scientists 2007 (Part 6)

6 SIZE-EXCLUSION CHROMATOGRAPHY Yuri Kazakevich and Rosario LoBrutto 6.1 SEPARATION OF THE ANALYTE MOLECULES BY THEIR SIZE Size-exclusion chromatography (SEC) separates polymer molecules and bio- molecules based on differences in their molecular size. The separation process in simpliﬁed form is based on the ability of sample molecules to penetrate inside the pores of packing material and is dependent on the relative size of analyte molecules and the respective pore size of the absorbent. The process also relies on the absence of any interactions with the packing material surface. Two types of SEC are usually distinguished: 1. Gel permeation chromatography (GPC)—separation of synthetic (organic-soluble) polymers. GPC is a powerful technique for polymer characterization using primarily organic solvents. 2. Gel ﬁltration chromatography (GFC)—separation of water-soluble biopolymers. GFC uses primarily aqueous solvents (typically for aqueous soluble polymers, proteins, etc.). Physical and chemical properties of polymers are dependent on their molec- ular weight and molecular weight distribution.The separation principle in SEC is based on the forced transport of the polymer molecules through the porous stationary-phase media under the conditions of suppressed interactions of the HPLC for Pharmaceutical Scientists, Edited by Yuri Kazakevich and Rosario LoBrutto Copyright © 2007 by John Wiley & Sons, Inc. 263
264 SIZE-EXCLUSION CHROMATOGRAPHY polymer analyte with the surface. The mobile-phase eluent is selected in such way that it interacts with the surface of packing material stronger than the polymer. Under these conditions, the smaller the size of the molecule, the more it is able to penetrate inside the pore space and the movement through the column is retarded. On the other hand, the bigger the molecular size, the higher the probability the molecule will travel around the particles of the packing material and, thus, is eluted earlier. The molecules are separated in order of decreasing molecular weight, with the largest molecules eluting from the column ﬁrst and smaller molecules eluting last (Figure 6-1). Molecules larger than the pore size do not enter the pores and elute together as the ﬁrst peak in the chromatogram and this is called total exclu- sion volume which deﬁnes the exclusion limit for a particular column. Mole- cules that can enter the pores diffuse into the internal pore structure of the gel to an extent depending on their size and the pore size distribution of the Figure 6-1. Illustrative description of separation in SEC.
SEPARATION OF THE ANALYTE MOLECULES BY THEIR SIZE 265 gel. The molecules will have an average residence time in the particles that depends on the molecules size and shape in the particular mobile phase.There- fore, different molecules have different total residence times in the column. This portion of a chromatogram is called the selective permeation region (the effective volume in which separation can occur). Molecules that are smaller than the pore size can enter all pores, have the longest residence time on the column, and will elute all together as the last peak in the chromatogram. This last peak in the chromatogram determines the total permeation limit for a par- ticular column. The largest elution volume (retention volume) in any given SEC column is equal to the total mobile-phase volume in the column (known as the void volume, V0). The exclusion range indicates the molecular weight of solutes above which all solutes having a molecular weight greater than the exclusion limit. These analytes will elute at the same retention time as a single peak. A speciﬁc column can be used for separation of solutes with molecular weights that are within the molecular weight window between the exclusion and permeation limits (Figure 6-2). Separation process in SEC is based on the actual size of the molecules, which in turn reﬂects the molecular weight of the polymer. The resulting SEC Figure 6-2. Elution of analytes in SEC. (Reprinted with permission from reference 1.)
266 SIZE-EXCLUSION CHROMATOGRAPHY chromatogram reﬂects the size distribution of the polymer sample, and its rela- tionship with the molecular weight distribution which lays the foundation of the SEC theory. 6.2 MOLECULAR SIZE AND MOLECULAR WEIGHT A polymer molecule in solution has a certain shape that strongly depends on the type of polymer, type of solvent, temperature, and other conditions. Usually a polymer forms some kind of globular species whose size is depen- dent on the degree of solvation by solvent molecules. This globe could be described by its volume (v) and hydrodynamic radius (R). Hydrodynamic radius (radius of gyration) of the polymer in the solution could be expressed in the form 1 R =  pM[h] 3 3 (6-1) 4  where [η] is intrinsic viscosity and M is molecular weight. For some polymers that are not ﬂexible, the effective R is used to represent the radius of the sphere. The parameter R is equivalent to the mechanical behavior of the polymer in solution. Viscosity is the simplest parameter of the polymer solution. From the Stokes and Einstein equations, the volume of the equivalent sphere is proportional to the product of intrinsic viscosity and polymer molecular weight: [h]M = 2.5N Av (6-2) where [η] is intrinsic viscosity, M is molecular weight, NA is Avogadro’s number, and v is the volume of the equivalent sphere. As one could see, the intrinsic vis- cosity is an important parameter related to the molecular weight of the polymer and its molecular volume. By deﬁnition, intrinsic viscosity is a limit of the ratio of the speciﬁc viscosity of the polymer solution to its concentration at c → 0, or it is the y-intercept of the dependence of ηsp versus concentration. Polymer molecules of a different nature but with the same molecular weight usually have different hydrodynamic radii. This is due to the differences in coil ﬂexibility, intramolecular interactions, and, most importantly, the differences in their interactions with the solvent. This essentially means that if two differ- ent polymers analyzed at identical SEC conditions show similar peaks with identical elution volume, it does not conﬁrm that the molecular weights of these polymers are identical. It only indicates that at the given conditions the gyration radii of the molecules are the same, causing similar elution. The nature of the solvent also has a signiﬁcant effect on the polymer conformation and thus on its gyration radius and molecular volume. If the
SEPARATION MECHANISM 267 solvent–polymer interactions are favorable or essentially prevailing over the interactions between different segments of the same polymer, then we can expect a high degree of solvation and the polymer globe will swell. For instance, if polystyrene is dissolved in toluene, due to the similar nature of the solvent and polystyrene monomer, toluene will solvate polymer molecules and their gyration radius increases. On the contrary, if the same polystyrene is dissolved in tetrahydrofuran (THF), then interactions between polystyrene segments prevail over the interactions with THF. As a consequence, the size of the polymer globe in THF is relatively small, especially in comparison to that in toluene. From equation (6-2), one can conclude that intrinsic viscosity is propor- tional to the polymer molecular volume. On the other hand, the effective mol- ecular volume is also the function of the molecular weight and the type of used solvent (or the nature of the solvent–polymer and polymer–polymer interac- tions). The intrinsic viscosity is an exponential function of the molecular weight with ﬁxed coefﬁcients for any speciﬁc polymer and solvent. [h] = K ⋅ M α (6-3) This expression is known as the Mark–Houwink equation, and K and α are constants for any given pair of polymer and solvent. These constants are tabulated and could be found for most known polymers in reference 2. 6.3 SEPARATION MECHANISM Eluent ﬂow through the chromatographic column packed with porous packing material has a velocity distribution depending on the pathway. Flow around the adsorbent particles is the fastest. Flow through the pore space is much slower. Since the smallest molecules can penetrate all of the pores, they can be distributed in the whole liquid volume of the column and their average migration speed is therefore the slowest. Molecules of intermediate size may penetrate into the pore space but may not come close to the pore walls, so their center of mass will be allocated closer to the center of the pores where ﬂow velocity is higher. Their average migration speed is higher. The biggest molecules experience steric hindrance in permeation inside the packing pore space and move through the column primarily around the particles with the fastest possible speed. As a result, the biggest molecules come out of the column ﬁrst, and the smallest ones come out last. Obviously, all molecules that are not able to penetrate into the pore space, move with the same velocity. Retention volume of all these molecules is the same and is called exclusion volume, also known as total exclusion. The total exclusion volume is a characteristic of a particular column which determines its upper separation limit.
268 SIZE-EXCLUSION CHROMATOGRAPHY 6.4 CALIBRATION SEC calibration establishes the relationship of a particular elution volume with speciﬁc molecular weight of the polymer (Figure 6-3) [3]. For calibration the elution volume of the solutions of polymer standards with known narrow molecular weight distributions are measured. An example of a separation is shown in Figure 6-4 [4]. In SEC, hydrodynamic volume of the polymer mole- cules is being measured rather than the actual mass of a particular species. The hydrodynamic volume is the space a particular polymer molecule occupies when it is in solution. The molecular weight can be approximated from SEC data from the relationship between molecular weight and hydrodynamic volume for particular known standards. However, the relationship between hydrodynamic volume and molecular weight is not the same for all polymers, so only an approximate measurement can be obtained. A series of commercially available polystyrene standards can be used for calibration. The elution volume (elution time multiplied by ﬂow rate) corre- sponding to a particular peak in the chromatogram is related to the molecu- lar weight of a particular polystyrene. After assignment of the molecular weight for each component to its elution volume, the logarithms of the mole- cular weight of the standards are plotted against their elution volumes in order to construct a calibration curve (Figure 6-3). Each combination of column, polymer, and solvent has its own calibration curve. The same polymer molecules could have different sizes in different solvents, and two molecules of different polymers might have the same size despite their Figure 6-3. Calibration curves for a set of AquaGel (Polymer Laboratories) columns designed for the separation of water soluble polymers. Calibration using PEO and PEG standards. (Reprinted from reference 3, with permission from Polymer Laboratories Inc.)
CALIBRATION 269 Figure 6-4. Example of a separation of calibration mixture of polystyrene standards. (Reprinted from reference 4, with permission from Phenomenex.) different molecular weight. So the calibration curve for the certain polymer is valid only if the standards used are of the same nature and used eluent was of the same type. If two different polymers in the same solvent have the same intrinsic viscosity, then their molecular weights are related as K1 ⋅ M1 1 = K 2 ⋅ M 2 2 a a (6-4) This makes it possible to use standards of one polymer for characterization of another if the corresponding Mark–Houwink constants are known. For most known polymers, Mark–Houwink constants are tabulated. For example, a polystyrene (PS)-based calibration could be used for characterization of polymethylmethacrylate (PMMA). Retention volume in SEC is proportional to the size of the polymer molecules in solution. In addition, as discussed in Section 6.2, equation (6-2), the product of the intrinsic viscosity of the polymer and its molecular weight is proportional to the hydrodynamic molecular volume. These relationships allowed Benoit et al. [5, 6] to introduce a universal molecular weight calibration.
270 SIZE-EXCLUSION CHROMATOGRAPHY In conventional molecular weight calibration, the dependence of the reten- tion volumes of a series of narrow molecular weight distribution standards with known average masses is plotted against the logarithms of their molecu- lar weights. In universal calibration, the logarithm of the product of the intrin- sic viscosity [η] and molecular weight M (essentially hydrodynamic volume) is plotted against retention volume. Hydrodynamic molecular volume is directly related to the retention volume if only the steric separation mecha- nism is involved. Benoit et al. [5, 6] found that plots of the logarithm of hydro- dynamic volumes versus corresponding retention volumes for a series of narrow standards of different polymers in different solvents resulted in a single calibration curve, as shown in Figure 6-5. The combination of the differential refractive index (RI) detector and on-line viscometer allows the direct use of the universal calibration and thus true molecular weight determination. The RI detector is concentration- sensitive, and the viscometer records speciﬁc viscosity. The ratio of the speciﬁc viscosity to the concentration is equivalent to intrinsic viscosity (as discussed in Section 6.1), and the continuous dependence of this ratio versus the reten- tion volume could be related to the universal calibration curve, thus allowing the correlation of each point on the chromatogram with the true molecular weight. More detailed information on this type of GPC analysis can be found in the book by Yau et al., Modern Size-Exclusion Liquid Chromatography [7]. Figure 6-5. Universal Benoit calibration. (Reprinted from reference 5, with permission.)
COLUMNS 271 6.5 COLUMNS Polymer-based packing materials are the main type of adsorbent used in size-exclusion HPLC. Most SEC analyses of synthetic organic polymers are performed on rigid or semirigid crosslinked polystyrene gel materials (styrene–divinylbenzene copolymers with different degree of crosslinkage). These materials require careful selection of separation conditions and solvent, since they can shrink and swell in different solvents and temperatures, thus changing their separation power. A major requirement of size-exclusion separation is the complete absence of any interactions between analyzed com- ponents and the surface of packing material in the column. Polymeric pack- ings do not have any active surface groups and could be synthesized with controlled porosity. For the separation of biopolymers in aqueous media, gel- ﬁltration chromatography is used with stationary phases that have a mildly hydrophilic surface, which is usually required to avoid noticeable hydropho- bic interactions with the surface. These types of stationary phases for GFC include (a) dextrans, agaroses, polyacrylamide, or mixtures of these compo- nents which are suitable for low- or medium-pressure chromatography and (b) porous silica-based media which are more suitable for higher-pressure applications. The separation range in size-exclusion chromatography for a particular column is relatively narrow, and it lies between the total volume of the liquid phase in the column (void volume) and the exclusion volume, Ve. The differ- ence between these two volumes is the total pore volume of the packing mate- rial in the column. Indeed, if some molecules of studied polymers are small enough to penetrate inside all pores of the packing material, they will elute with the column void volume. On the other hand, polymers with signiﬁcant molecular size that cannot penetrate inside the particles will all travel together around the particles and elute early with exclusion volume. To obtain wider separation range, longer and wider column dimensions are used. The greater the amount of packing material in the column, the higher the total pore volume and the higher the difference between void and exclu- sion volumes. Another important parameter is the ability of the column to discriminate different ranges of molecular masses. This range is dependent on the size of the pores of packing material and pore size distribution. Column manufac- turers usually provide the standard calibration curves of polystyrene standards in THF for each column, as shown in Figure 6-6 [8]. One of the most important requirements for the GPC column is the absence of the speciﬁc interactions with the studied polymer. The more inert the surface of the packing material, the better. Early applications of GPC separa- tion were sometimes performed on porous glass particles with controlled porosity [9]. The ease of the manufacturing of the controlled porosity par- ticles had determined this choice, but it was not always possible to ﬁnd an
272 SIZE-EXCLUSION CHROMATOGRAPHY Figure 6-6. Typical calibration curves for a set of SEC columns. (Reprinted from reference 8, with permission.) eluent that would suppress interactions of the polymer with the glass surface. Modern GPC columns are primarily made of styrene–divinylbenzene copolymer. Increased technological advances allow preparation of the rigid porous particles with relatively narrow pore size distribution from this copolymer. The presence of the aromatic rings in the body of the packing material is prone to π–π-type interactions, which could be a problem for the separation of polymers with signiﬁcant aromaticity, like polyimids. Traditional reversed-phase columns appear to offer the most inert surface since the alkyl-type bonded ligands at high bonding density can only partici- pate in weak dispersive interactions, which could be suppressed by practically any organic solvent. An example of the calibration curve made on a commer- cial reversed-phase column is shown in Figure 6-7. For the analysis of water-soluble polymers (such as surfactants, oligosac- charides, PEGs, lignosulfonates, polyacrylates, polysaccharides, PVA, cellulose derivatives, PEO, polyacrylic acids, polyacrylamides, hyaluronic acids, CMC, starches, gums) and for separations of oligomers and small molecules, columns that are comprised of macroporous material with hydrophilic functionalities may be used. The requirement for these columns in SEC mode is to eliminate or minimize ionic and hydrophobic effects that make aqueous SEC (otherwise known as GFC) very demanding. The interaction of analytes with neutral, ionic, and hydrophobic moieties must be suppressed. It is often necessary to modify the eluent (addition of salt) in order to avoid sample-to-sample and sample-to-column interactions that can result in poor aqueous SEC separa- tions and low recoveries.
MOLECULAR WEIGHT DISTRIBUTION 273 Figure 6-7. Calibration curve for low-molecular-weight region. Column: Prodigy- ODS2 (pore size 150 Å). Effective separation region for polystyrenes in THF from 100 to 50,000 Da. 6.6 MOLECULAR WEIGHT DISTRIBUTION It is impossible to ﬁnd a sample of a synthetic polymer in which all the chains have the same molecular weight. Instead, a distribution of molecular weights is reported. Some of the polymer chains will be much larger than the others, and some will be signiﬁcantly smaller. The largest number of similar chains will be populated around a central point of the distribution, the highest point on the curve called the “molecular weight distribution.” All synthetic polymers consist of the mixture of the molecules of the same nature but different size (length, or number of repeat units), so we have to deal with average molecular weights when we work with polymers. The average molecular weight can be calculated in different ways, and each approach has its own value. In the Mark–Houwink equation (6-3) the intrin- sic viscosity is related to the molecular weight; this is the viscous average weight. Two other important average molecular weights are number average (MN) and weight average (MW). The ﬁrst one is the weight calculated by average number of the molecules Mn = ∑N Mi i = ∑j i (6-5) ∑N i j ∑M i i where Ni is the number of the molecules in fraction i (or slice of GPC chromatogram), Mi is the molecular weight in that fraction, and φi is the area
274 SIZE-EXCLUSION CHROMATOGRAPHY fraction on the chromatogram from a concentration sensitive detector (RI). This molecular weight is usually measured by osmometry. Weight average molecular weight of the polymer is usually measured by light scattering and is deﬁned as Mw = ∑M ⋅ N i 2 i = ∑M j i i (6-6) ∑M N i i ∑j i The ratio of weight average to number average molecular weight is called the polydispersity, rd. The wider is the molecular weight distribution the higher the polydispersity value. The rd for unimolecular polymer is equal to 1. All natural polymers, such as peptides, DNA, and saccharides, have polydispersity equal to 1. 6.7 EFFECT OF ELUENT For the method development of the SEC separation, the main attention should be focused on the suppression of analyte interactions with the surface of packing material. This usually requires a careful selection of the SEC column. In GPC, commercially available synthetic organic polymer columns are usually packed with styrene–divinylbenzene copolymer particles, which are only capable of weak dispersive interactions. Any possible analyte interactions with the surface could be suppressed by using a strong solvent, which will be preferentially adsorbed on the packing material surface. Selection of such a solvent is limited since the polymer solubility in that particular solvent needs to be considered. Tetrahydrofuran is the most common solvent used for most GPC separations, although for polyimids and other high-temperature polymers the use of special solvents such as n-methylpyrrolidone may be necessary. The development of GFC methods generally has similar considerations, with much higher requirements for the packing material. GFC deals with water-soluble polymers; thus the main solvent is water, with some additives. The suppression of possible surface interactions has to be done with these additives and with careful selection of the packing material. 6.8 EFFECT OF TEMPERATURE Temperature of the system is also an important parameter since the viscosity of the polymer solutions has signiﬁcant dependence on the temperature. Refractive index and viscometric detectors are the most common detectors used in GPC, and their responses are both dependent on the temperature. Both of these detectors have relatively low sensitivity, resulting in the neces- sity to use concentrated polymer solutions, which in turn increases the sample
SOLVING MASS BALANCE ISSUES 275 viscosity. It is highly recommended to stabilize the temperature of the whole system and work at elevated temperatures. At higher temperature the solu- tion viscosity decreases, which allows working with lower backpressures. The lower backpressure also leads to a more stable liquid pump operation and a more stable ﬂow of the mobile phase. Even a slight variation in the mobile phase ﬂow rate results in the signiﬁcant error in the molecular weight deter- mination. Typical requirements for the ﬂow stability: GPC system should possess a ﬂow variation of less than 0.2%. 6.9 DETECTORS A refractive index (RI) detector is probably the most widely used in GPC. The main advantage of an RI detector is that any polymer solution will generate a response. This detector has several disadvantages: • Low sensitivity (2–3 orders of magnitude lower than UV) • Very sensitive to pressure, ﬂow, and temperature ﬂuctuations Two specialized detectors have been developed with speciﬁc applications for SEC. The ﬁrst is the laser light-scattering detector, which has been around for almost 20 years but now has new electronics and computer data acquisition capabilities. Substitution of bulky He–Ne gas lasers with small, inexpensive diode lasers has greatly reduced the size and cost of laser light-scattering detectors, and the development of reference ﬂow viscometers has provided similar size and cost advantages for viscometer detectors. Each detector measures a different and complementary variable. The light- scattering detector gives a response that is proportional to molecular weight and concentration. Likewise, the viscometer detector response is proportional to the intrinsic viscosity and concentration. 6.10 SOLVING MASS BALANCE ISSUES SEC can be used to solve mass balance issues that may be encountered in reversed-phase chromatography applications—for example, if the area% does not agree with the assay% value for pharmaceutical analysis of a particular active pharmaceutical ingredient or intermediate. The discrepancy in mass balance could be due to species that are not observed at the wavelength of detection using RPLC; the sample could contain salt, water, residual solvents, and even polymers. These polymer species may not have been eluted using the RPLC conditions. GPC may be employed to elucidate if high-molecular- weight species are present in the sample (leading to discrepancy in mass balance). The higher-molecular-weight species may or may not have a UV chromophore, so detection with UV and/or RI should be employed. This
276 SIZE-EXCLUSION CHROMATOGRAPHY technique is most suitable for quality control and polymer screening, as well as for troubleshooting during the analysis of pharmaceutical samples. 6.11 AQUEOUS SEC APPLICATIONS For aqueous SEC, ionic interactions need to be suppressed and the eluent usually modiﬁed by the addition of salt and/or the adjustment of pH. For water-soluble polymers with hydrophobic character, the addition of a weak organic solvent (methanol) is sometimes required to inhibit hydrophobic interactions. A general approach for the separation of the aqueous polymeric samples is shown in Figure 6-8. In biological separations, the choice of mobile phase is critical and must consider both column performance and the maintenance of biological function of the sample and the aim of separation. Figure 6-9 represents a preparative separation of a mixture of proteins at pH 7 in phosphate buffer. Typical buffers used for classical protein handling are acceptable for gel ﬁltration. These include denaturing buffers (6 M guanidine hydrochloride or 8 M urea) and those containing nonionic detergents (Tween-20, Brij-35 at concentrations from 0.01 to 0.1% v/v). All silica-based gel ﬁltration columns possess a slight negative charge. This charge is likely due to unreacted Figure 6-8. Principles of the selection of acceptable column type and mobile-phase type for the GFC separation of water-soluble polymers. (Reprinted from reference 3, with permission.)
AQUEOUS SEC APPLICATIONS 277 Figure 6-9. Separation of proteins on Zorbax GF-250 and GF-450 preparative columns. (Reprinted from reference 10, with permission.) silanol groups on the silica surface. The separation of biomacromolecules at typical buffer concentrations of 0.1–0.5 M is recommended. Typical buffers include sodium phosphate, Tris-HCl, and sodium acetate. The addition of 50–100 mM NaCl may also be added to suppress any undesired ionic interac- tions between the charged solute and the stationary phase. Also, the pH of the mobile phase is very important to suppress nonideal ion-exchange interactions of the biomacromolecules with the stationary phase. If ion-exchange effects are observed, they can potentially be suppressed by either changing the pH or increasing the ionic strength of the buffer. Using buffers with pH values above 8.5 does cause a slow base-catalyzed dissolution of the silica packing, but can be tolerated for short periods at the expense of reduced column life. Other types of GFC columns are chemically modiﬁed crosslinked polystyrene gels with sulfo groups on the surface. They are suitable for the
278 SIZE-EXCLUSION CHROMATOGRAPHY Figure 6-10. Example of a separation of dietary ﬁber mix on GPC column from Shodex. (Reprinted from reference 4, with permission from Phenomenex.) separation of water-soluble polymers like polysaccharides, starches, and cellu- loses. An example of such a separation is shown in Figure 6-10. Some other references that review the details of size-exclusion chromatography are avail- able [11–13]. REFERENCES 1. L. Snyder and J. J. Kirkland, (eds.), Introduction to Modern Liquid Chromatogra- phy, 2nd ed. John Wiley & Sons, New York, 1979. 2. J. Brandrup, E. H. Immergut, E. A. Grulke, A. Abe, and D. R. Bloch, Polymer Hand- book, 4th ed., John Wiley & Sons, New York, 2003. 3. www.polymerlabs.com 4. www.phenomenex.com
REFERENCES 279 5. H. Benoit, Z. Grubisic, P. Rempp, D. Decker, and J.-G. Zillox, Liquid-phase chro- matographic study of branched and linear polystyrenes of known structure, J. Chim. Phys. 63 (1966), 1507–1514. 6. Z. Grubisic, P. Rempp, and H. J. Benoit, Universal calibration for gel permeation chromatography, Polymer Sci. B, Polymer Lett. 5 (1967), 753–759. 7. W. W. Yau, J. J. Kirkland, and D. D. Bly, Modern Size-Exclusion Liquid Chro- matography, Wiley-Interscience, New York, 1979. 8. Waters Catalog, Waters Inc., Milford, MA, 2006. 9. B. G. Belenkii and L. Z. Vilenchik, Modern Liquid Chromatography of Macro- molecules, Journal of Chromatography Library, vol. 25, Elsevier Amsterdam, 1983. 10. Agilent Zorbax PrepHT GF-250 datasheet, Agilent Technologies, 2004. 11. Chi-San Wu, Handbook of Size Exclusion Chromatography and Related Techniques, Marcel Dekker, 2004. 12. J. Wen, T. Arakawa, and J. S. Philo, Review: Size exclusion chromatography with on-line light scattering, absorbance, and refractive index detectors for studying proteins and their interactions, Anal. Biochem. 240 (1996), 155. 13. M.-I. Aguilar, HPLC of Peptides and Proteins, Vol. 251, Humana Press, Totowa, NJ, 2004.