The Instrument Spreading Correction in GPC. II. The General Shape Function Using the Fourier Transform Method with a Nonlinear Calibration Curve

Abstract

The instrument spreading function suggested in Part I of this series is investigated for use with the Fourier transform method for generating corrected elution volume chromatograms. The instrument spreading parameters are obtained using linear theory on narrow molecular weight distribution standards, as indicated in Part I. The corrected chromatogram is then combined with a nonlinear molecular weight calibration curve which was fit with a function suggested by Yau and Malone to generate true values of the number- and weight-average molecular weights.

The instrument spreading function is shown to qualitatively and quantitatively describe the dispersion, skewing, and flattening effects ordinarily found in GPC chromatograms due to imperfect resolution by the GPC columns. The Yau-Malone function is shown to be a very useful function for fitting nonlinear molecular weight vs elution volume calibration data. Although the Fourier transform method is shown to work well with analytically generated data, it is shown that a number of numerical problems must be overcome before it can quantitatively produce corrected elution volume chromatograms. Some of these numerical problems are discussed.     

The Instrument Spreading Correction in GPC. I. The General Shape Function Using a Linear Calibration Curve

Abstract

A general shape function is proposed for describing the instrumental spreading behavior in gel permeation chromatography (GPC) columns due to axial dispersion and skewing effects. The general shape function contains statistical coefficients which describe the axial dispersion, skewing, and flattening of ideal monodisperse standards. A method denoted as the “method of molecular weight averages” is used to derive equations to correct GPC number- and weight-average molecular weights and intrinsic viscosities calculated from linear molecular weight calibration curves. The validity of these equations is experimentally verified with data for polystyrene, polybutadiene, and polyvinyl chloride polymers in tetrahydrofuran. The physical significance of the correction equations and their statistical coefficients is discussed in relation to the observed GPC chromatograms. Application of this shape function to the numerical Fourier analysis method for correcting differential molecular weight distribution (DMWD) curves is outlined. Also, a method is presented for obtaining corrected DMWD curves from a fitted molecular weight calibration curve corrected for instrument spreading by use of the hydrodynamic volume concept.     

A generalized method for correcting instrument spreading in gel permeation chromatography

A general method of chromatogram correction for skewed instrument spreading in gel permeation chromatography is presented. The correction method is so general that there is no restriction on the shape of the spreading function. It admits nonsymmetric, non-Gaussian as well as nonconvolution type. Aspects of solution techniques are discussed and an illustrative example is given to elucidate the method.     

Polymer reactors and molecular weight distribution. VIII. A method of interpreting skewed GPC chromatograms

A new-method of interpreting GPC chromatograms which accounts for skewing and symmetrical axial dispersion has been developed. General relationships for a symmetrical axial dispersion correction and for a skewing correction are derived. The method has been verified experimentally for unimodal chromatograms and linear calibration curves over a wide range of GPC operating conditions, polymer molecular weights and polydispersities. Measurements of h and skewing factors were obtained by a once-through technique. The need for performing reverse flow experiments has been eliminated. Artificial oscillations in the corrected chromatogram due to step size (Method of Pierce-Armonas), and to number of terms in a polynomial expansion (Method of Tung and Method of Smith) are eliminated. The method has yet to be evaluated for nonlinear calibration curves and multi-modal distributions. However, suggestions for its application in these circumstances are presented.     

The determination of particle size distribution by hydrodynamic chromatography. An analysis of dispersion and methods for improved signal resolution

Various methods are described and compared for the determination of particle size distributions (PSD) in the submicron range by a technique known as hydrodynamic chromatography (HDC). Data are presented for a series of monodisperse latexes to establish the validity of the Mie theory of light scattering in describing the detector optical density signal. Analyses for the PSD involve corrections to the experimental HDC chromatograms for the effects of dispersion and are broadly classified as integral and numerical methods. Comparisons of calculations are made to chromatograms for polydisperse latexes as well as synthetic, discontinuous distributions and show the critical role of the optical density–particle size relationship in determining resolution and calculation stability. An integral method involving a non-Gaussian form for the dispersion function and a polynomial expansion for the chromatogram and an iterative numerical method involving modifications of a previously published technique are shown to give the best results for the PSD. The discussion includes an analysis of the possibility of improved signal resolution using turbidity in the absorption wavelength region and refractive index measurements. The conclusion is reached that increased resolution with turbidity is preferable to refractive index measurement since lower particle concentrations can be used.     

SEC–viscometer detector systems. II. Resolution correction and determination of interdetector volume

The closely related topics of resolution correction and the determination of interdetector volume were examined. In determining the interdetector volume (δ) by searching to superimpose two types of intrinsic viscosity calibration curve (one from narrow standards and one from a broad standard), the underlying equation based upon symmetrical axial dispersion theory was derived. This equation combined with experimental results showed that the local intrinsic viscosity value is very similarly affected by interdetector volume and by band spreading. The result supported the idea of using an effective δ to effect an axial dispersion correction to local intrinsic viscosity data. However, it also increased the difficulty of finding both δ and standard deviation (δ) simultaneously by numerical search. Furthermore, attempts to apply the method to the chromatograms of narrow standards showed inadequate superposition. Following calculation of skewing factors, the superposition problems were attributed to skewing of the chromatograms of the monodisperse polymer standards used. © 1993 John Wiley & Sons, Inc.     

The instrument spreading correction in GPC. III. The general shape function using singular value decomposition with a nonlinear calibration curve

The Gram-Charlier series was suggested as an empirical instrument spreading function in the first paper (part I) of this series. In the second paper (part II) of this series, the Fourier transform method was used together with the suggested series to solve Tung's integral equation. In this paper, an alternate method for solving Tung's equation is proposed which eliminates some of the limitations of the Fourier transform method. In the approach used in this study, Tung's integral equation is approximated by a set of linear equations. Since no unique least-squares solution can be computed, a closely related problem whose solution closely approximates the original problem is formulated and solved using singular value decomposition. By avoiding the use of the smallest singular values and forcing the equality of the areas of the corrected and the uncorrected chromatograms, an approximate solution to the original problem is obtained in which the oscillations inherently present due to the ill-posed nature of the problem are filtered out. The performance of the method with the experimental data given in Part II is indicated.     

基于工业色谱仪的上位机色谱图处理

工业色谱仪是一种精密的过程分析仪器,但其现场运行环境一般比较恶劣。为此采用LabWindows/CVI开发了工业色谱仪的仿真系统用于监控仪器的运行状态,主要研究对采集到的谱图数据进行谱峰参数的识别与显示,最终实现色谱峰的识别,并可以通过用户界面查看历史、实时谱图信息,对分析结果标定显示。结果表明:该方法可以对谱图数据进行有效的分析与显示。     

A new method for calculating and correcting molecular weight distributions from gel permeation chromatography

A new method for calculating and correcting molecular weight distributions of polymer samples from GPC chromatograms is presented. The integral equation which relates the true molecular weight distribution of polymer sample to the chromatogram is reformulated into an equivalent variational problem of quadratic functional. The method of steepest descent in the function space is then applied to the minimization problem to obtain the true molecular weight distribution. This method is efficient and reduces some of the oscillation problems encountered in the previous methods. Examples are given.     

Calibration of instrumental spreading for GPC

The proper use of the method for correcting instrumental spreading in GPC requires a precise calibration of the spreading characteristics of the instrument. Heretofore, such a calibration could be obtained only through the tedious reverse-flow experiments. A more rapid method of calibrating instrumental spreading is presented in this work. This method uses the leading halves of the chromatograms of several standard polystyrene samples. These chromatograms are normally used in the calibration of molecular weight; additional experimental steps are therefore not required. The calculation of the instrumental spreading characteristics from these chromatograms is also relatively simple. The instrumental spreading characteristics were found to depend on the elution volume but not on the nature of the polymer. Thus, calibration results from using polystyrene standards can be used to treat chromatograms for other polymers. For the present GPC instrument, the spreading was found to reach a maximum at an elution volume near 400,000 in polystyrene molecular weight. The existence of this maximum is in agreement with observations made by other investigators and is and indication that diffusion in the mobile phase is not an important contribution to instrumental spreading. The problem of skewing or tailing is discussed. Indication of skewing was observed for one of the higher molecular weight polystyrene samples but the extent of skewing was not severe at the present flow rate of 2 ml/min.     

Graphic aid for interpreting gas chromatograms

A simple semilogarithm plot of elution volumes can be constructed as an aid to identify components that are members of the same homologous series. This technique is especially useful in organoleptic and flavor studies for identifying small peaks in any gas chromatogram. The method is applicable to chromatograms containing 20 or more peaks, which do not have to be completely resolved. Any member of the homologous series within 4 or 5 carbons of the unknown can be used as a standard for identification. The tentative plots also help in examining chromatograms obtained with hydrogen flame or with β ray-type detector equipment where samples are too small for collection and identification by chemical and physical tests. Presented at fall meeting, American Oil Chemists' Society, 1961. This is a laboratory of the Northern Utilization Research and Development Division, Agricultural Research Service, U.S.D.A.     

Statistical moments of elution bands and their application in gel permeation chromatography of polydisperse macromolecules

Equations have been derived that relate the statistical moments of uncorrected and spreading-corrected chromatograms for a general form of the spreading function in gel permeation chromatography of polydisperse macromolecules. The first moment (centroid) of the chromatogram is shown to be directly given by the centroid, In M^*, of a suitably defined molecular weight distribution function of the polydisperse sample, regardless of the position of the calibration dependence, provided it is linear. Both the molecular weight M^* associated with the centroid of the chromatogram and its second central moment (variance) are but little sensitive to the shape of sample molecular weight distribution and can be easily calculated from the polydispersity index M_w/M_n, at least for polymers of a not excessively broad distribution. The derived relations are shown to find application in the calibration of GPC columns by means of characterized, polydisperse standards and in the separation of independent contributions to peak width which originate in sample polydispersity and in band broadening processes in the column. Improved column- and packing performance criteria are also proposed.     

A valid method of calibration which includes correction for dispersion in gel permeation chromatography

Two independently derived distribution function methods validate both the calibration curve and the dispersion correction of the “effective linear calibration” method used in gel permeation chromatography (GPC). Experimental conditions are specified for making the method more useful by permitting linear extrapolation of the calibration line, and for using a minimum number of standards. The independent methods quantitatively relate known differential of integral distribution functions for standard samples to their respective chromatograms. As such, they are useful calibration methods also, but are limited in scope and range.     

Profiles of Spreading Sessile Drops of Viscous Polymer Melts

The profile or shape of spreading drops of viscous polymer melts has always raised some questions regarding the basic forces inherent in and controlling the wetting phenomenon. The occurrence and nonoccurrence of “projecting feet” for spreading sessile drops has puzzled experimentalists for some time. Recent work for a homologous polymer series, differing only in molecular weight and molecular weight distribution and examined over a wide temperature range, has emphasized that an advancing projecting foot does occur and is dependent upon the molecular weight of the polymer material. The projecting feet can be interpreted as an effect resulting from the viscoelastic response of the material which can occur when the molecular weight of the polymer is higher than the characteristic critical molecular weight of entanglement of the material. For low molecular weight polymers the strong interfacial forces are not impeded by the bulk viscous response of the material whereas the present evidence indicates that, for high molecular weight material, the elastic or entangled component of the polymer may retard complete bulk flow or the redistribution of the polymer in harmony with the advancing interface. Previous inconsistencies in the literature regarding the observations of projecting feet may be explained by this model.     

Chromatogram broadening of proteins and dextrans in size exclusion chromatography

The mechanisms governing the broadening of experimental chromatograms for proteins and paucidisperse dextrans were studied on TSK-G2000SW and TSK-G3000SW columns. Within the conditions studied, the chromatogram variance for all solutes increased linearly with increasing effluent flow rate. As predicted by current theories of the kinetics of size exclusion chromatography, this flow rate dependence is caused mainly by slow mass transport of the solute within the stationary phase of the column. Restricted diffusion within the stationary phase was dependent upon the ratio of solute molecular size to column pore radius and was similar for both proteins and dextrans. In comparison with results for monodisperse proteins, the broader chromatograms produced by dextrans were due to sample polydispersity and not to differences in solute column spreading. Corrections for column spreading on these columns are small for the determination of integral properties of polymers but may be significant when molecular weight distributions are of interest.     

On the selectivity of adsorption chromatography

In this paper, the shape of chromatograms obtained by elution of copolymers with a mixture of solvent-nonsolvent of variable composition has been studied from a theoretical point of view. It is assumed that the thermodynamic quality of the mixture at which the copolymer is eluted depends only on its composition and not on its mass and structure. If one characterizes the polydispersity in composition, assuming that each constituent obeys a molar mass distribution of the Zimm–Schulz type, one can draw the following curves: concentration of the copolymer versus eluant composition. They depend on three parameters, including composition of the copolymer and the polydispersity of each constituent. The most striking result is that the shape of the chromatogram changes dramatically when one increases the polydispersity of either constituents. For low polydispersities, one obtains a classical peak. For large polydispersities, the chromatogram has a minimum for the intermediate values of composition and presents peaks in the vicinity of the pure homopolymers. This can be explained easily by a qualitative argument. © 1996 John Wiley & Sons, Inc.     

Correction of instrument spreading in gel-permeation chromatography

Present methods of correcting instrument spreading (resolution correction) in GPC are either too cumbersome to use or inaccurate when the correction is large. Two new methods which are both accurate and simple to use are presented in this work. The first method using the technique of Fourier analysis is more general and can be used to correct non-Gaussian instrument spreading. The second method using a fourth-degree polynomial requires a Gaussian instrument spreading function. The instrument spreading function may vary with respect to the elution volume in both methods.     

Solution of Tung's axial dispersion equation by numerical techniques

Numerical methods for the solution of Tung's axial dispersion equation have been developed and comprehensively evaluated. These methods are general and can be applied where the instrumental spreading function is unsymmetrical and nonuniform. Computation times required are comparable to those of the method of Chang and Huang being about 10 sec per case on the CDC6400 computer. Memory requirements are minimal and this should permit their use with minicomputers for data acquisition and processing.     

Detection of pressed hazelnut oil in admixtures with virgin olive oil by analysis of polar components

Analysis of the polar fraction from virgin olive oil and pressed hazelnut oil by high-performance liquid chromatography showed marked differences in the chromatograms of the polar components in the two oils. Six commercial samples of pressed hazelnut oil and 12 samples of virgin olive oil (or blended olive oil including virgin olive oil) were analyzed. The phenolic content of the pressed hazelnut oil samples was 161±6 mg·kg⁻¹. Inspection of the chromatograms showed that the pressed hazelnut oil extracts contained a component that eluted in a region of the chromatogram that was clear in the olive oil samples, and consequently this component could be used to detect adulteration of virgin olive oil by pressed hazelnut oil. The component had a relative retention time of 0.9 relative to 4-hydroxybenzoic acid added to the oil as an internal standard. The ultraviolet spectrum of the component showed a maximum at 293.8 nm, but the component could not be identified. Analysis of blends of oils showed that adulteration of virgin olive oil by commercial pressed hazelnut oil could be detected at a level of about 2.5%.     

Quantitative size exclusion chromatography of polypropylene I: Method development

Polypropylene was analyzed by size exclusion chromatography (SEC) at 145°C using a single-differential refractometer detector. The objective was to provide data for characterization of polypropylene degradation during a reactive extrusion process. Two antioxidants [tetrakis(methylene(3,5-di-tert-butyl-4-hydroxyhydrocinnamate)) methane (I) and octadecyl 3,5-di-tert-butyl-4-hydroxyhydro cinnamate (II)] were tested for their ability to prevent thermal degradation of the polypropylene during sample preparation. The use of 0.20 wt% of (I) was effective during the 36–48 h required to completely dissolve the samples in trichlorobenzene for SEC analysis. “Reshaping” of the chromatograms by resolution correction demonstrated that, while the molecular weight averages were changed by 8% because of axial dispersion, most of the individual heights of the distributions were changed by less then 2%. Tail heights of the distributions were more affected but were also shown to be highly imprecise. Selecting individual heights of the distributions rather than molecular weight averages therefore minimized axial dispersion error and also circumvented errors in molecular weight averages originating from dilution of distribution tails below detector sensitivity limits. Various forms of distributions were examined and the equations linking the chain length distribution predicted by polymerization kinetic models to the SEC chromatogram are presented. The analytical method developed provided precise data for kinetic modeling. However, absolute accuracy requires further assessment.