DILUTE SOLUTION THERMODYNAMICS,
3.9 GEL PERMEATION CHROMATOGRAPHY
While use of the viscosity-average molecular weight of a polymer in cali-brating K and a in equation (3.97), MVvalues usually are not initially known.
The calibration problem may be alleviated by one or more of the following methods:
1. Use of fractionated polymers.
2. Use of polymers prepared with narrow molecular weight distributions, such as via anionic polymerization.
3. Use of weight-average molecular weight of the polymer, since it is closer than the number-average molecular weight.
3.8.5 Example Calculation Involving Intrinsic Viscosity
Say we are interested in a fast, approximate molecular weight of a polystyrene sample. We dissolve 0.10 g of the polymer in 100 ml of butanone and measure the flow times at 25°C in an Ubbelhode capillary viscometer. The results are
Pure butanone 110 s
0.10% Polystyrene solution 140 s
Starting with equation (3.84), and nothing that the flow time is proportional to the viscosity,
As an approximation, assume that the concentration is near zero, and the [h]
= 2.73 ¥ 102ml/g, equation (3.86). This obviates the extrapolation in Figure 3.14 that is required for more accurate results. Using the Mark–Houwink–
Sakurada relation, equation (3.97) and Table 3.11, we have
Note that the units of K are irregular, depending on the value of a. Usually the units of K are omitted from tables.
the molecule, defined by its hydrodynamic radius, can or cannot enter small pores in a bed of cross-linked polymer particles (76), the most common form of the stationary phase. The smaller molecules diffuse in and out of the pores via Brownian motion (see Figure 3.16) and are delayed. The larger molecules pass by and continue in the mobile phase.
The instrumentation most commonly used in GPC work is illustrated in Figure 3.17 (77). The stationary phase consists of small, porous particles. While
Figure 3.16 The size exclusion effect. The short chain can enter the pore, whereas the long chain will pass by.
Figure 3.17 An illustration of the modules that make up GPC instrumentation (77).
the mobile phase flows at a specified rate controlled by the solvent delivery system, the sample is injected into the mobile phase and enters the columns.
The length of time that a particular fraction remains in the columns is called the retention time (78). As the mobile phase passes the porous particles, the separation between the smaller and the larger molecules becomes greater (see Figure 3.18) (76).While separation of polymer chains according to size remains the most important experiment, there are many other aspects, as described below.
3.9.1 Theory of Gel Permeation Chromatography
With the exception of proteins and a very few other macromolecules, most polymers exhibit some form(s) of heterogeneity (79). The most important is the molecular weight distribution (MWD), sometimes called a molar mass distribution, or polydispersity index (PDI), equal to Mw/Mn. Another type of heterogeneity involves a distribution of chemical composition, including statistical, alternating, block, and graft copolymers, as described in Chapter 2.
Still another type of heterogeneity relates to functionality, particularly end groups. Macromolecules with terminal functional groups are usually called telechelics or macromonomers. Molecular architecture provides yet another
3.9 GEL PERMEATION CHROMATOGRAPHY 119
Figure 3.18 Illustration of the GPC experiment (81). The sample is injected into a solvent, which flows into a porous packed bed. The larger molecules flow straight through, whereas the small ones are temporarily held up.
type of heterogeneity, dictated by the shape of the chain. While most synthetic polymers form random coils, an increasing number of polymers are rod shaped, or form rings. Very many polymers are branched. Each of these types of heterogeneity must be taken into account when measuring molecular weights by relative methods.
There are two very popular relative methods of characterizing polymers with one or more of the above heterogeneities. Gel permeation chromatogra-phy (GPC), also known as size exclusion chromatograchromatogra-phy (SEC) or gel filtra-tion chromatography, is one of several chromatographic methods available for molecular weight (molar mass) and molecular weight distributions.While GPC has its greatest value for measuring the molecular weight and polydispersity of synthetic polymers, a closely related method, high-performance liquid chro-matography (HPLC) is more useful for separating and characterizing poly-mers containing functional groups, such as proteins and pharmaceutical polymers containing special active groups. Both of these methods depend on distribution coefficients.
3.9.2 Utilization of Distribution Coefficients in GPC and HPLC
Both GPC and HPLC processes relate to the selective distribution of an analyte between a mobile phase and a stationary phase. In general, the distri-bution coefficient, Kd, is given by (79)
(3.104) where VR represents the retention volume of the solute, Vi represents the interstitial volume of the column, and V is the volume of the stationary phase.
Expressed qualitatively, Kdis defined as the ratio of the analyte concentration in or attached to the stationary phase to that in the mobile phase, namely the partitioning between the mobile and stationary phases. The volume V, depend-ing on the system, can be comprised of the pore volume of porous particles, the active surface area, or the volume of chemically bonded stationary phase.
As with all such distribution coefficients, the quantity Kdis related to the Gibbs free energy, DG,
(3.105) After rearranging, we have
(3.106) Now the physical picture determines the sign and magnitude of the quantities above. The limited dimensions of the pores relative to the size of the polymer
K S
R H
d = Ê -RT
Ë
ˆ exp D D ¯ -RTlnKd =DH T S- D =DG
K V V
d V
R i
=
-chains causes DS of the polymer -chains to decrease. Interactions between the pore walls and the polymer chains are expressed in changes in DH, and are negative if the polymer and the wall are attracted to each other.
In the general case, Kdmay be expressed
(3.107) where the subscripts GPC and HPLC indicate quantities involving only entropic or enthalpic interactions, respectively. In the ideal GPC case, KHPLC
equals unity, and Kd= KGPC. Then,
(3.108)
Of course, for the ideal HPLC case, the reverse is true. Equation (3.106) expresses the general case, where both entropy and enthalpy are involved.
Usually one or the other must be reduced to substantially zero to use either GPC or HPLC.
3.9.3 Types of Chromatography
All types of chromatography utilize columns containing a finely divided sta-tionary phase and a solution of a mixture that passes through the columns, called the mobile phase. Analyte mixtures separate as they travel through the columns due to the differences in their partitioning between the mobile phase and the stationary phase.
The present interest primarily involves GPC, which uses porous particles to separate molecules of different sizes (80,81). Its most important use has been to determine molecular weights and molecular weight distributions of poly-mers. Polymer molecules that are smaller than the pore sizes in the particles can enter the pores, and therefore they have a longer path and longer transit time than larger molecules that cannot enter the pores. Motion in and out of the pores is governed by Brownian motion. Thus the larger molecules elute earlier in the chromatogram, while molecules that entered more and more pores elute later and later.
HPLC, by contrast, utilizes interactions between the polymers and the surface of the particles composing the stationary phase (82). Important HPLC methods include reverse-phase partition chromatography, normal-phase par-tition chromatography, adsorption chromatograpy, chiral stationary phases, partition chromatography, and ion chromatography. In reverse-phase chro-matography, the groups being analyzed are more polar than the stationary phase. In normal-phase chromatography, the groups are less polar. There are also various hybrid methods such as HPLC size exclusion chromatography, particularly for proteins and functional group macromolecules. The best
K S
d = Ê R
Ë ˆ exp D ¯ Kd =KGPCKHPLC
3.9 GEL PERMEATION CHROMATOGRAPHY 121
instruments today are sometimes called universal HPLCs, containing both GPC and HPLC columns and measurement capability; see Table 3.12 for specifications.
3.9.4 GPC Instrumentation
The most important parts of the instruments are the pumps for maintaining constant, pulseless rates of flow, the column types for the molecular weight range of analysis, and the detector system; see Table 3.12. Some of the popular types of solvent delivery pumps include syringe pumps, the plunger of which is actuated by a screw-fed stepper motor, and the dual piston reciprocating pumps, which may have the pistons in either series or parallel; see Figure 3.19 (79).
Table 3.12 GPC/HPLC universal type instruments available
Company Features
Waters Alliance Systems HPLC w/RI + UV detectors; HPLC oriented
Polymer Laboratories LC Accord w/UV-vis detectors, integrated HPLC/GPC RI, LS evaporation detectors added
Hewlett-Packard HP 1100 Quaternary pump w/degasser, thermostatted column compartment, w/fluorescence and refractive index detectors
Note: Each of these instruments is equipped with full sets of columns, a computer, and printer.
Abbreviations: GPC, gel permeation chromatography; HPLC, high-performance liquid chro-matography; RI, refractive index; LS, light-scattering; UV, ultraviolet, usually for measuring fluorescence; vis, visible light.
Figure 3.19 Dual piston pumps with parallel pistons deliver a smooth flow, while those in series are easier to maintain, since they have two check valves instead of four.
The type of column packing depends on whether the polymer is water soluble or organic soluble. For water soluble polymers, column packings consist of a range of materials, including silicas, hydroxyethyl methacrylate copoly-mers, chitosan, and highly cross-linked poly(vinyl alcohol). Organic soluble polymer-based columns most often contain porous, densely cross-linked poly-styrene, but porous silicas and highly cross-linked poly(vinyl alcohol) are also used. The size of the pores determines the size of the molecule that can diffuse in and out by Brownian motion. The larger molecules are restricted to enter-ing only the larger pores. Since the motion in and out of the pores is random, the residence time in the pores of the short chains is longer. Hence the larger, high molecular weight polymer chains elute first from the column. There are two major types of such columns. First, there are a series of columns each con-taining a specific pore size range. A series of three or even four such columns are required for the general case of any molecular weight between, say, 10,000 and 2,000,000 g/mol. The newer types are called mixed bed columns, because a range of pore sizes is included in each. Two such columns may be sufficient for molecular weight determination, speeding up the flow time. Usually one adds a guard column up front, which absorbs the gels and other unwanted material and thus prolongs the calibration and life of the columns.
There are several types of detectors; see Figure 3.17 (79). These are classi-fied as either concentration-sensitive detectors, or molar mass (molecular weight) sensitive detectors. The refractive index detector is most popular concentration-sensitive detector, measuring the change in refractive index as the concentration of polymer in the solution changes. These usually operate on some type of differential refractive index method or Fresnel refraction. Of course, most of the time substantially pure solvent flows. When the polymer chains arrive at the detector, then the refractive index of the solution changes, providing a measure of the polymer concentration. While most polymers have a different refractive index than the solvent (usually higher), if the refractive indices of both the polymer and solvent are substantially the same, the method cannot be used.
Another detector group of methods involves the input of ultraviolet light, with the output being fluorescence or absorption by the polymer. There are two different methods of detecting the ultraviolet light interaction with the polymer. If the polymer fluoresces, then the detector can be placed at an angle to the light beam path and the fluorescence intensity level measured, usually in the visible range. Another method measures the attenuation of the beam directly, from whatever cause. Polymers that neither absorb nor fluoresce at the incoming wavelength cannot be detected by this method. However, instru-ments that measure the attenuation of the beam directly are more popular than the fluorescence units in practice, apparently because they are more useful. Some models come with diode array detectors to give greater control over the wavelengths of light being utilized. Other types have variable wave-length inputs, or a dual wavewave-length detector system, such that two ultraviolet wavelengths can be utilized simultaneously, enhancing copolymer analysis, for
3.9 GEL PERMEATION CHROMATOGRAPHY 123
example. For aromatic polymers such as polystyrene, this provides an alter-nate and very powerful detector system, since it absorbs strongly in the ultraviolet.
Other methods include a density detector utilizing a mechanical oscillator, and also an evaporative light-scattering method. In this latter method the sample is nebulized (evaporated). Each droplet that contained nonvolatile material will form a particle. This aerosol causes light to be scattered, result-ing in a method to determine the concentration of solute.
Molecular-weight-sensitive methods include light-scattering, viscometry, and the like. Since light-scattering and viscometry measure different averages (see Section 3.6 and Section 3.8, respectively), the results will be somewhat different. If two angles are used in the light-scattering detectors, the radius of gyration of the polymer chains can be determined. Thus not only can the molecular weight distribution be determined, but also the radii of gyration distribution.
Analysis of the literature shows a surprising tendency to “mix and match”
parts of instrumentation from different supply sources, including software (83–85).
3.9.5 Calibration
Noting GPC is a relative molecular weight method, such instrumentation needs to be calibrated. Narrow molecular weight distribution, anionically syn-thesized polystyrenes are used most often for the purpose. Other polymers used for calibration include poly(methyl methacrylate), polyisoprene, polybu-tadiene, poly(ethylene oxide), and the sodium salt of poly(methacrylic acid).
Molecular weight ranges available start at low oligomers of only a few hundred g/mol, up to 20,000,000 g/mol. In all cases, use of narrow molecular weight dis-tribution standards is preferred.
Since the polymer chains are separated on the basis of their size, rather than their molecular weight per se, calibration via polymers other than the one of actual interest carries an absolute error. Clearly, selecting a polymer with an Rg2/M ratio similar to that of interest is preferred. Today, except for special pur-poses, most polymer scientists are willing to accept the absolute error to be able to determine the approximated molecular weight and the molecular weight distribution of a polymer rapidly, usually in about half an hour. For most random coil synthetic polymers, the error is less than about 30%. If the actual molecular weight needed is important, then the instrument must be cal-ibrated with the polymer in question. Section 3.9.7 describes a universal method to obtain more realistic results in the general case.
Calibration usually involves the determination of the elution volume for a series of narrow molecular weight standards. If the molecular weight is very low, as in oligomers, peaks from individual species may be used. For higher molecular weights, the peak average is used. Usually an assumption is made of a linear relation between log M and a function of the elution volume, such
as a polynomial. As with all such fits, care must be exercised to use standards that cover the entire molecular weight range to be studied, and perhaps a bit more. For completion, other types of chromatography include thin layer (86), gas (87), and supercritical chromatography (88), among others.
Today, data acquisition and processing are usually computer controlled.
There are four transformations required of the raw chromatographic data to provide results as usually reported; see Figure 3.20 (79): (a) conversion of elution time to elution volume, (b) conversion of elution volume to molecu-lar weight, (c) conversion of detector response to polymer concentration, and (d) conversion of polymer concentration to weight fraction. Quantification of plate count and resolution, as well as calibration are discussed further in ASTM D5296-97 (89).
3.9.6 Selected Current Research Problems
Pasch and Trathnigg (79) describe a basic determination of the molecular weight distribution of a suspension polymerized polystyrene, using dibenzoyl peroxide as an initiator. Calibration of the GPC instrument utilized anioni-cally polymerized polystyrene standards. Five mm cross-linked polystyrene particles were used in the columns, the mobile phase solvent was chloroform, and the detectors were refractive index and density based. Ethanol was added as an internal standard for flow rate correction. The results for the refractive index detector are shown in Figure 3.21 (79). The weight-average molecular weight of this sample is approzimately 63,000 g/mol.
3.9 GEL PERMEATION CHROMATOGRAPHY 125
Figure 3.20 The molecular weight and molecular weight distribution are determined with stan-dards precalibrated via an absolute method such as light-scattering.
In the case of copolymers, any single detection method will have variable sensitivity for each type of mer. If the copolymer composition is itself a vari-able, then the use of dual or even multiple detectors will be required for accurate results. Calibrations for both homopolymers should be followed, if possible, by an analysis of homopolymers of similar molecular weights, before attempting an analysis of the copolymers themselves.
The molecular weight of each component in a polymer blend may also be determined. In a model experiment, poly(methyl methacrylate), PMMA, mol-ecular weights were estimated in the presence of polystyrene, PS (90). Anion-ically polymerized polystyrene and free radAnion-ically polymerized poly(methyl methacrylate) were dissolved in tetrahydrofuran in a 50/50 w/w mix. A dual detector GPC was used, equipped with refractive index (RI) and ultraviolet (UV) detectors.
The refractive index detector measures both polymers simultaneously; see Figure 3.22 (90). However, the sensitivity to each polymer differs, being a func-tion of the difference between each polymer refractive index and the solvent.
The values are summarized in Table 3.13 (90). This particular system is nearly three times more sensitive to the polystyrene.
The ultraviolet detector only observes the phenyl groups on the poly-styrene, which fluoresce. After subtracting out the polystyrene component, the poly(methyl methacrylate) component remains (bottom curve, Figure 3.22).
The results in Table 3.13 show that the result is approximately 30% in error, but not so bad considering that poly(methyl methacrylate) is the less sensitive polymer in the measurements.
While much of the work done today involves more or less straightforward characterization of molecular weights and molecular weight distributions of
Figure 3.21 The molecular weight distribution of polystyrene PS 50000 with stationary phase Phenogel M, mobile phase, butanone.
polymers, a great deal of research involving much greater complexity is in progress. Current research problems include chain geometry and solution aggregation (83), ABC triblock copolymers as topological isomers (84), and hyperbranched polymers (85); see Section 14.5.
3.9.7 The Universal Calibration
Beginning with the Mark–Houwink–Sakurada relationship, equation (3.97), it is easy to show that the average molecular size is given by
(3.109) where represents the root-mean-square end-to-end distance of the polymer chain.
r02
h a
[ ]M=F
( )
r02 3 2 33.9 GEL PERMEATION CHROMATOGRAPHY 127
Figure 3.22 Analysis of the molecular weights of the polymers in a polymer blend of poly-styrene and poly(methyl methacrylate). This method requires a dual detector GPC system.
Table 3.13 Molecular weight determination of poly(methyl methacrylate) in blend Theoretical
Refractive Area under Molecular Weight, Molecular Weight, Component Index Curve, % nominal, g/mol determined, g/mol
Polystyrene 1.592 69.4 Mn= 2.66 ¥ 105 —
Mw= 2.90 ¥ 105
Poly(methyl 1.4893 30.6 Mn= 4.01 ¥ 105 Mn= 6.3 ¥ 105
methacrylate) Mw= 8.82 ¥ 105 Mw= 1.06 ¥ 106
Tetrahydrofuran 1.408 — — —
The right-hand side is proportional to the polymer’s hydrodynamic volume (91). A new aspect of GPC calibration arises from the recognition that a polymer’s hydrodynamic volume might form the basis for molecular weight determination. Since GPC depends on the hydrodynamic volume per se rather than its molecular weight per se, a new calibration method is suggested. This is the “universal calibration,” which calls for a plot of [h]M versus elution volume.
Figure 3.23 (92) illustrates the universal calibration procedure for poly(vinyl acetate) and polystyrene. Note that the two sets of data lie on the same straight line. The universal calibration is valid for a range of topologies and chemical compositions. However, it cannot be used for highly branched materials or polyelectrolytes, which have different or varying hydrodynamic volume relationships. The universal calibration procedure is especially useful for estimating the molecular weight of new polymers, since the intrinsic viscosity is usually easy to obtain. The procedure also tends to correct for differences in the hydrodynamic relationships when several polymers are compared, and only one of them (e.g., polystyrene) is used as the calibration material.
3.9.8 Properties of Cyclic Polymers
In the earlier sections, the polymer chains were assumed to be linear. How do the basic solution properties differ if the chain is cyclic in nature, resembling
Figure 3.23 The universal calibration curves for polystyrene and poly(vinyl acetate) (94). The number 5 in the x-axis units means that the scale is in siphon “counts” of 5 cm3, so that the x-ordinate 30 corresponds to an elution volume of 150 cm3. (R. Dietz, private communication, November 1984.) Mˆris the “peak” GPC molecular weight, usually the unknown, Mˆrvalues are close to the geometric mean of Mnand Mw.
a pearl necklace? Bielawski, et al. (92) synthesized a series of cyclic polyocte-namers of high molecular weight, starting from cis-cyclooctene monomer.
These polymers had the structure,
(3.110) Molecular weights ranged up to 1.2 ¥ 106g/mol. (It should be remarked that while low molecular weight cyclic polymers are relatively easy to synthesize starting with reactive groups and dilute solutions where the ends of different chains have only a low probability of finding one another, controlled synthe-sis of a high molecular weight cyclic polymer is difficult.)
The properties of the linear and cyclic polymers are compared in Figure 3.24 (92). As one might expect, the more compact cyclic polymers possessed smaller hydrodynamic volumes (i.e., they eluted later via GPC in Figure 3.24A). They had lower intrinsic viscosities than their linear analogs, with [h]cyclic/[h]linear= 0.4 (Figure 3.24B). The Mark–Houwink–Sakurada parameter a was 0.7 in both cases.
CH2 CH2 CH2 CH2 CH2 CH2 CH CH n
CH CH
3.9 GEL PERMEATION CHROMATOGRAPHY 129
Figure 3.24 A comparison of the physical properties of cyclic and linear polyoctenamers.
(A) Plot of GPC elution volume vs. molecular weight. The molecular weights were determined by light-scattering and considered absolute. (B) A traditional plot of log intrinsic viscosity vs. molecular weight. (C) Plot of the mean square radius of gyration vs. molecular weight.
The root-mean-square radius of gyration, Rg, was measured using GPC coupled to a multiangle light-scattering detector. The corresponding ratio Rg2(cyclic)/Rg2(linear) was found to be approximately 0.5 over a wide range of molecular weights, Figure 3.24C, as predicted long ago by Zimm and Stockmayer (93). Studies with a MALDI-TOF mass spectrometer (Section 3.10) revealed that the molecular weights were multiples of 110.2 g/mol (C8H12), the monomer employed.
The important conclusion here is that cyclic polymers have more compact solution structures. Some cyclic polymers occur in nature. As model materials, they make one-ring networks.