CHAPTER
3.4 Size Exclusion Chromatography
3.4.3 Universal Calibration for Linear Homopolymers
We consider now how the elution volume axis of the raw chromatogram can be translated into a molecular weight scale. A series of commercially available anionically polymerized polystyrenes is particularly well suited for calibration of GPC columns. These polymers are available with a range of molecular weights and have relatively narrow molecular weight distributions (Section 2.5). When such a sample is injected into the GPC column set, the resulting chromatogram is narrower than that of a whole polymer, but it is not a simple spike because of the band broadening effects described earlier and because the polymer standard itself is not actually monodisperse. Since the distribution is so narrow, however, no serious error is committed by assigning the elution volume corresponding to the peak of the chromatogram to the molecular weight of the particular polystyrene.
(The molecular weight distributions of most of these samples are sharp enough that all experimental average molecular weights are essentially equivalent to within experimental error.) Thus, a series of narrow distribution polystyrene sam- ples yields a set of GPC chromatograms as shown inFig. 3.10. The peak elution volumes and corresponding sample molecular weights produce a calibration curve (seeFig. 3.12later) forpolystyrenein the particular GPC solvent and column set.
It turns out that combinations of packing pore sizes which are generally used result in more or less linear calibration curves when the logarithms of the polysty- rene molecular weights are plotted against the corresponding elution volumes.
Increasing molecular weight
Elution volume Amount of polymer in eluant
FIGURE 3.10
Gel permeation chromatography elution curves for anionic polystyrene standards used for calibration. The polystyrene standard samples were measured separately; use of a mixture of polymers may cause elution volumes of very high-molecular-weight standards to be erroneously low[18].
132 CHAPTER 3 Practical Aspects of Molecular Weight Measurements
It remains now to translate this polystyrene calibration curve to one that will be effective in the same apparatus and solvent for other linear polymers. (Branched polymers and copolymers present complications and are discussed separately later.) This technique is called auniversal calibration, although we shall see that it is actually not universally applicable.
Studies of GPC separations have shown that polymers appear in the eluate in inverse order of their hydrodynamic volumes in the particular solvent. This forms the basis of a universal calibration method sinceEq. (3-64)is equivalent to
lnð½ηMÞ5lnð2:5LÞ1ln lim
c-0V
(3-94) The product [η]M is a direct function of the hydrodynamic volume of the sol- ute at infinite dilution. Two different polymers that appear at the same elution volume in a given solvent and particular GPC column set therefore have the same hydrodynamic volumes and the same [η]Mcharacteristics.
The conversion of a calibration curve for one polymer (say, polystyrene, as in Fig. 3.11) to that for another polymer can be accomplished directly if the MarkHouwinkSakurada equations are known for both species in the GPC sol- vent. FromEq. (3-70), one can write
½ηiMi5KiMaii11 (3-95)
107
106
105
M
104
103
130 140
Elution volume (mL)
150 160
FIGURE 3.11
Polystyrene calibration curve for GPC, whereMis the molecular weight of the anionic polystyrene standard samples.
where the subscript refers to the polymer type. Thus, at equal elution volumes
½η1M15K1M1a1115½η2M25K2Ma2211 (3-96) and the molecular weight of polymer 2, which appears at the same elution volume as polymer 1 with molecular weightM1, is given by
lnM2511α1
11α2
lnM11 1 11a2
ln K1
K2
(3-97) The polystyrene calibration curve of Fig. 3.11 can be translated into that of any other polymer for which the MHS constants are known[19].
EXAMPLE 3-4
The MarkHouwinkSakurada constants for polystyrene in toluene at 25C are K54.1731023cm3/g and a50.65. And the corresponding constants for polypropylene in the same solvent are K521.831023cm3/g and a50.73. Using the following mono- disperse polystyrenes as calibration standards, construct a calibration curve for polypropylene in gel permeation chromatography. Also estimate the molecular weight of a fraction of an unknown polypropylene sample showing an elution volume of 150 cm3.
Molecular weight of the polystyrene standards Elution volume (cm3)
10,000 160
50,000 145
100,000 132
500,000 120
Solution
PS in toluene at 25C KPS54.1731023cm3/g
aPS50.65
PP in toluene at 25C KPP521.831023cm3/g
aPP50.73
lnMPP511aPS
11aPPlnMPS1 1 11aPPlnKPS
KPP
5110:65
110:73lnMPS1 1
110:73ln4:1731023 21:831023 50:954 lnMPS20:956
MPP5exp ln MPS0:95420:956 5exp ln MPS0:954
=expð0:956Þ 50:384 MPS0:954
134 CHAPTER 3 Practical Aspects of Molecular Weight Measurements
Molecular weight of the polystyrene standards Elution vol. (cm3) Mol. Wt. of PP
10,000 160 2,500
50,000 145 12,000
100,000 132 23,000
500,000 120 105,000
14.0 13.0 12.0 11.0 10.0 9.0 8.0 7.0
110 120 130 140
PP
PS
Elution volume (cm3)
ln M
150 160 170
The molecular weight of the fraction of the unknown PP sample showing an elution vol- ume of 150 cm3isB6000.
Note that the universal calibration relations apply to polymeric solutes in very dilute solutions. The component species of whole polymers do indeed elute effec- tively at zero concentration but sharp distribution fractions will be diluted much less as they move through the GPC columns. Hydrodynamic volumes of solvated polymers are inversely related to concentration and thus elution volumes may depend on the concentration as well as on the molecular weights of the calibration samples. To avoid this problem, the calibration curve can be set up in terms of hydrodynamic volumes rather than molecular weights. A general relation[20]is
V5 4π½ηM
μ14πLcð½η2½ηθÞ (3-98)
Here cis set equal to the concentration of the solution of a sharp distribution calibration standard used to establish the calibration curve, and [η]θ (Eq. 3-40)
can be calculateda priori[5]. For whole polymers, all species elute effectively at zerocand
V54π½ηM=μ (3-99)
The constant μ in Eqs. (3-98) and (3-99) must equal 10π L (instead of the numerical value given in [20]) in order to coincide with Eq. (3-64). This proce- dure is necessary for high-molecular-weight polymer standards in good solvents.
In other cases, the hydrodynamic volume calibration is equivalent to the infinite dilute [η]M method. With this modification, the calibration curve for narrow dis- tribution standards is converted to the form shown in Fig. 3.12, usingEq. (3-98) to translate M to hydrodynamic volumeV. The curve is then applied to analysis of whole polymers through the use ofEq. (3-99).
When MHS constants for a particular polymer are not known, they can be esti- mated from GPC chromatograms and other data on whole polymers of the partic- ular type [21]. It is not necessary to use fractionated samples in this method of determiningKanda.
A parameter J is defined as the product of intrinsic viscosity and molecular weight of a monodisperse speciesi. That is,
Ji ½ηiMi (3-100)
WithEq. (3-95),
½ηi5Jiaða11ÞK1=ða11Þ (3-101)
– 44
– 42
– 40
– 38
– 36
– 34
130 140
Elution volume (mL)
In V
150 160
FIGURE 3.12
Universal Calibration Curve in terms of Hydrodynamic Volume V and Elution Volume.
136 CHAPTER 3 Practical Aspects of Molecular Weight Measurements
and fromEq. (3-84),
½η5K1=ða11ÞX
wiJia=a11 (3-102) If two samples of the unknown polymer are available with different intrinsic viscosities, then
½η1
½η2
5X
i
ω1iJ1ia=a11 X
ω2iJ2ia=ða11Þ (3-103) Here the wi are available from the ordinates of the gel permeation chromato- gram and the Ji from the universal calibration curve of elution volume against hydrodynamic volume throughEq. (3-94) or (3-98). The intrinsic viscosities must be in the GPC solvent in this instance, of course. A simple computer calculation produces the best fita toEq. (3-103), and this value is inserted into Eq. (3-101) to calculateK. These MHS constants can be used withEq. (3-97)to translate the polystyrene calibration curve to that for the new polymer.
Note that this procedure need not be restricted to determination of MHS con- stants in the GPC solvent alone [22]. The ratio of intrinsic viscosities inEq. (3- 103)can be measured in any solvent of choice as long as thewiandJivalues for the two polymer samples of interest are available from GPC in a common, other solvent. The first step in the procedure is the calculation ofKand a in the GPC solvent as outlined in the preceding paragraph. The intrinsic viscosities of the same two polymers are also measured in a common other solvent. The data per- taining to this second solvent will be designated with prime superscripts to distin- guish them from values in the GPC solvent. In the second solvent,
Ji0 ½_ηiMi (3-100a)
For a species of given molecular weightMi,Eqs. (3-100a) and (3-100)yield Ji0 ½η0iJi=½ηi (3-104) WithEq. (3-70)
Ji05JiðK0=KÞMia02a (3-105) Then for the non-GPC solvent,Eq. (3-103)becomes
½η01
½η02
5X
ω1iJa0=ða11Þ=X
ω2iJa0=ða11Þ (3-106) As above, the wi and Ji values are available from the GPC experiment and intrinsic viscosities of the two polymer samples in the GPC solvent. The exponent acan be calculated as described in connection withEq. (3-103).
The computed best fit value of a inEq. (3-106)can be now used to calculate Kfrom:
K05½η0Ka0=ða11Þ=X
ωiJia0ða11Þ (3-107)
Here the MHS constantsKandafor the GPC solvent are used with the expo- nenta0 and the measured intrinsic viscosity [η0] of a single polymer sample in the non-GPC solvent.
This procedure is much less tedious than the method described inSection 3.3.2 for measurement of MHS constants. It may not necessarily produce the same K and a values as the standard fractionation method described earlier. This is because K and a are inversely correlated, as mentioned, and are also not entirely independent of the molecular weight range of the samples used.
Essentially the same K and a should be obtained if the two samples used with Eq. (3-103) or (3-106) and the fractions used with Eq. (3-71b) have similar molecular weights.
Because Eq. (3-80)defines Mw whena51, it is possible to estimate the Mw
of a sample by measuring the Mv for the polymer in two or three solvents with different values of the exponent a. A plot of Mv against a is linear and extrapo- lates to Mw at a51 [23]. This procedure is fairly rapid if single-point intrinsic viscosities (Section 3.3.4) are used. It can be employed as an alternative to light scattering although the latter technique is more reliable and gives other informa- tion in addition to the weight average molecular weight. The GPC method out- lined here is a convenient procedure to generate the MHS constants for this approximation ofMwfrom solution viscosity measurements.
It is possible in principle to deriveKandafrom a single whole polymer sam- ple for which [η] in the GPC solvent andMn are known[21]. This method is less reliable than the preceding procedure which involved intrinsic viscosities of two samples because the computations of Mn can be adversely affected by skewing and instrumental broadening of the GPC chromatogram.