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Chapter 18

Using Multiple Detectors to Study Band Broadening in Size-Exclusion Chromatography

Downloaded by PENNSYLVANIA STATE UNIV on September 7, 2012 | http://pubs.acs.org Publication Date: November 4, 2004 | doi: 10.1021/bk-2005-0893.ch018

MilošNetopilík Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, 162 06 Prague 6, Czech Republic

This chapter reviews the sources of band broadening, including the separation process itself, restricted to the case of a chromatographically-simple polymer and Gaussian bandbroadening function. Methods of determining band broadening using multiple detection are also reviewed, as based on the latest contributions of the author to the problem.

Size-exclusion chromatography (SEC) is a method of separation of polymers according to molecular weight where, in the case of chromatographysimple polymers the latter is related to their hydrodynamic volume (1). Soon after invention of the method (2) mass detection of the polymer, frequently with a refractive index (RI) detector, was supplemented by viscometric (3-5) and light-scattering (6-8) detection. Chromatographic separation of polymers is possible in the adsorption mode, where the molecules of a polymer (analyte) in the mobile phase (MP) are temporarily adsorbed on (or react with) the solid phase (SP) by van der Waals forces. In SEC mode, due to their thermal motion the molecules of the analyte (analyzed polymer) are temporarily captured in the pores of the SP with the capture probability determined by the size of the analyte and by the accessible portion of the inner pore volume, or in the combination of the two (adsorption and size-exclusion) modes (9). In the SEC mode, the flux of the analyte between the phases is caused by the difference, produced by flow of the MP, in the entropie part of the chemical potentials between the phases (10). In both modes,

302

© 2005 American Chemical Society

In Multiple Detection in Size-Exclusion Chromatography; Striegel, A.; ACS Symposium Series; American Chemical Society: Washington, DC, 2004.


303 the fractionation is accomplished by the transport of the molecules in the MP and by the delay of molecules in/on the SP. The mechanism of chromatographic separation was described using a kinetic formalism based on the probabilities of adsorption and desorption (77). This description has been developed and is suitable for the adsorption mode. The SEC mode was described by postulating an equilibrium of the analyte in the MP and in the pores of the SP in a close region (72), combined with displacement. Both descriptions were proven to be equivalent (13). From the theoretical analysis it follows that the separation process is one of the sources of band broadening (77,75), a process which decreases the resolution power of the separation system (14). However, in the case of an effective separation system and an analyte with medium-broad molecular weight distribution (MWD), the experimental error caused by band broadening is comparable with errors caused by other sources of error (75) and in practice is frequently neglected. This review, based mainly on the author's contributions, presents the separation and detection processes from the viewpoint band-broadening (axial dispersion) of chromatographically-simple analytes. The interactions of a molecule of such analyte with the SP can be described (72) by the mean fraction of analyte in the MP in a restricted region (plate) forming a part of the separation system (column) from the total amount in the MP and the SP. Equivalently (75), the interactions can also be described by the probabilities per unit time of adsorption and desorption on/from the SP. Both approaches lead to a bandbroadening function which is principally skewed and tends, with increasing number of interactions of the analyte molecules with the SP, to the symmetrical Gaussian distribution and that describes basic features of band broadening. This agrees qualitatively both with experiment (16) and with the approach based on the distribution of times spent by molecules in pores of the SP (16). If only one detector is employed, band broadening can be estimated only indirectly. An elution curve identical to the spreading function, the broadness of which is a measure of band broadening, is obtained only for an analyte of uniform in molecular weight, M, which is usually a low-molecular-weight (M) substance (e.g., toluene) in organic-phase separation systems or proteins (77) in aqueous MP, and its characteristics can be determined directly. In organic MP, however, only reference standards with MWD, but not uniform in M, are available. To characterize their MWD the extent of band broadening must be known precisely. Absolute methods, such as sedimentation equilibria, used for the determination of the weight-to-number-average molecular weight ratio, Mw/A/n, which is used for checking the SEC results (2) are now obsolete and there is a lack of methods for M and M / A / determination. Therefore, the potential of multidetector SEC is to be used to get all information from the SEC data. This is possible preferably using multiple detection (18-22). n

w

n


304 The analysis of a system with a dual detection can yield, in principle, two values of the M /A/ ratio. Their difference is a measure of the band broadening of a particular system. The band broadening obtained using the calibration dependence, i.e., the logarithmic dependence of Mon elution volume (F), in a wide range of V (and M) is overestimated. On the contrary, the band broadening obtained from a local calibration dependence, i.e., obtained from the dual detection of M by the combination a light scattering (LS) with a concentrationsensitive detector (differential refractometer (8)) is underestimated. The correct Λ / w / M i ratio lies between the two values and can be assessed by a correction procedure. The correction method can be based (23) either on any numerical (point by point) deconvolution procedure of the elution curves (EC) (14,24-26) or on the correction of the molecular weight averages calculated from the concentration EC and a broad-range calibration dependence (30) and from the dual detector record (22). As the latter method is based on the approximation of the sample MWD by the log-normal function, it is used preferably on analyses of narrow-MWD samples where the point-by-point methods tend to instability (25) and the correction of the dual detector record is prone to fail (27). This makes SEC with dual or multiple detection an excellent tool for studying band broadening and characterization of the MWD of such samples (22). In the following sections band broadening will be discussed from the viewpoint of multiple detection.


w

n

Formulation of Band Broadening Band broadening is described by an equation proposed by Giddings and Eyring (//) and frequently referred to as the Tung (14) equation (i) relating experimental and theoretical elution curves, F(V) and W(y), respectively, where G(V,y) is the band-broadening (spreading) function, which is the (hypothetical) elution curve of the individual species uniform in M (it is identical with the elution curve only for analytes uniform in M). This function of two variables for elution volume, V and v, gives the contribution to the experimental elution curve F(V) at elution volume Κ of a fraction (band) of polymer W(y)dy with molecular weight M related to elution volume by an equation called 'calibration dependence', in the first approximation linear,

\nM=A+By


(2)

305 where A and Β are constants and y is the elution volume of the maximum of the spreading function (12)

y=v /p

(3)

Q

where V is the excluded volume of the separation system and ρ the mean fraction of the analyte in the MP from the total amount in the MP and in/on the SP. Solving equation (1) means finding W(y) from the known F(V). It is called the inverse problem and several mathematical methods were developed for this purpose (14,24-26). Equation (1) can be solved analytically under the condition that the spreading function is approximated by the (Gaussian) normal distribution (28,29)


0

1 G

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