Classifications

• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
  • G01N15/00—Investigating characteristics of particles; Investigating permeability, pore-volume, or surface-area of porous materials
  • G01N15/02—Investigating particle size or size distribution
  • G01N15/0205—Investigating particle size or size distribution by optical means, e.g. by light scattering, diffraction, holography or imaging
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
  • G01N30/00—Investigating or analysing materials by separation into components using adsorption, absorption or similar phenomena or using ion-exchange, e.g. chromatography or field flow fractionation
  • G01N30/02—Column chromatography
  • G01N30/86—Signal analysis
  • G01N30/8693—Models, e.g. prediction of retention times, method development and validation
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
  • G01N30/00—Investigating or analysing materials by separation into components using adsorption, absorption or similar phenomena or using ion-exchange, e.g. chromatography or field flow fractionation
  • G01N30/02—Column chromatography
  • G01N30/88—Integrated analysis systems specially adapted therefor, not covered by a single one of the groups G01N30/04 - G01N30/86
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
  • G01N11/00—Investigating flow properties of materials, e.g. viscosity, plasticity; Analysing materials by determining flow properties
  • G01N2011/0026—Investigating specific flow properties of non-Newtonian fluids
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
  • G01N30/00—Investigating or analysing materials by separation into components using adsorption, absorption or similar phenomena or using ion-exchange, e.g. chromatography or field flow fractionation
  • G01N30/02—Column chromatography
  • G01N30/88—Integrated analysis systems specially adapted therefor, not covered by a single one of the groups G01N30/04 - G01N30/86
  • G01N2030/8809—Integrated analysis systems specially adapted therefor, not covered by a single one of the groups G01N30/04 - G01N30/86 analysis specially adapted for the sample
  • G01N2030/884—Integrated analysis systems specially adapted therefor, not covered by a single one of the groups G01N30/04 - G01N30/86 analysis specially adapted for the sample organic compounds
  • G01N2030/885—Integrated analysis systems specially adapted therefor, not covered by a single one of the groups G01N30/04 - G01N30/86 analysis specially adapted for the sample organic compounds involving polymers
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
  • G01N30/00—Investigating or analysing materials by separation into components using adsorption, absorption or similar phenomena or using ion-exchange, e.g. chromatography or field flow fractionation
  • G01N30/02—Column chromatography
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
  • G01N30/00—Investigating or analysing materials by separation into components using adsorption, absorption or similar phenomena or using ion-exchange, e.g. chromatography or field flow fractionation
  • G01N30/02—Column chromatography
  • G01N30/62—Detectors specially adapted therefor
  • G01N30/78—Detectors specially adapted therefor using more than one detector
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
  • G01N30/00—Investigating or analysing materials by separation into components using adsorption, absorption or similar phenomena or using ion-exchange, e.g. chromatography or field flow fractionation
  • G01N30/02—Column chromatography
  • G01N30/86—Signal analysis
  • G01N30/8651—Recording, data aquisition, archiving and storage
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
  • G01N33/00—Investigating or analysing materials by specific methods not covered by groups G01N1/00 - G01N31/00
  • G01N33/44—Resins; rubber; leather
  • G01N33/442—Resins, plastics

Definitions

• the present invention relates to analysis of data from gel permeation chromatography (GPC) instrumentation, and more particularly to a computer system and method for analyzing data to determine radius-of-gyration, as well as absolute molecular weight and intrinsic viscosity of a polymeric distribution.
• GPC gel permeation chromatography
• the molecular structure of a synthetic polymer determines its end-use and its process characteristics, such as hardness, tensile strength, drawability, elastic modulus, and melt viscosity.
• Important determinants that define the molecular structure of synthetic polymers include the chemical nature of their repeating units, their molecular weight distribution, their molecular topology (branching), and their conformation law.
• a molecule's effective size determines its elution volume.
• two polymers may have the same molecular weight, but may differ in composition or branch topology. Such differences can lead to different elution volumes.
• Elution volume by itself determines a molecular weight distribution in a relative, not an absolute sense.
• GPC gel permeation chromatography
• SEC size exclusion chromatography
• the conformation law (CL) of a polymeric sample is the molecules' radius of gyration (r g ) distribution versus molecular weight. Both the INL and CL give important information relating to branching properties and the conformation of the molecules. The accuracy and the precision of the estimation of the MWD, the WL, and the CL depend not only on the quality of the data obtained from the chromato graphic system, but also on the details of the data analysis methodology. Thus, data analysis methods become an essential element in the GPC analysis of polymers.
• each detector At evenly spaced time intervals, each detector records a measurement of the properties of the separated sample as it elutes from the column and passes through a detector's flow cell. Each measurement , averaged over a narrow time range, corresponds to a narrow range in the sample's molecular weight distribution.
• the molecular weight range corresponding to a measurement recorded at a single time interval is referred to as a "slice".
• a slice therefore refers to measurements that are obtained by sampling the detector responses at elution time t,-.
• a slice can be referenced by its slice number i, its elution time ti, or its elution volume V ⁇ .
• elution volume is obtained by multiplying the elution time of a slice by the nominal flow rate of the pump.
• the radius of gyration r g of a molecule is the root-mean-squared radius averaged over all conformations.
• the quantity r g (V ; ) is the radius of gyration of a molecule at slice i.
• the quantity r g (V j ) needs to be determined for each slice over the whole peak region in order to determine the conformation law of a polymeric sample.
• the size of the molecule affects how the intensity of scattered light changes as a function of the scattering angle at which the scattered intensity is measured.
• r g (F ⁇ ) is obtained from analysis of the data obtained from a light-scattering (LS) detector.
• the prior art analysis first obtains the radius of gyration r g i ⁇ r g (f.) independently from each slice. Model curves can then be fit to these values r . in order to obtain additional results.
• the value for the radius of gyration is adjusted for each slice to obtain the fitted value r . for each slice.
• the radius of gyration needs to be determined for each slice over the whole peak region. Detector noise present in the measured values of the Rayleigh ratios AR t j introduces error in all quantities derived from them. Of particular concern in the estimation of the radius of gyration. It is important to obtain a smooth representation of the radius of gyration as a function of volume, since result is used in subsequent, noise-sensitive calculations.
• a major problem in the analysis of data is that the noise present in the detector responses introduces errors in the quantities computed from slice measurements.
• Each of the known detectors employed for GPC contains non-idealities in their responses. Typically, these non-idealities fall into two categories, baseline drift and stochastic detector noise. Detector noise can also be referred to as system noise. Baseline drift in a thermally stabilized chromatograph is accurately compensated for by baseline correction procedures.
• Detector noise is an irreducible component of the measurement process.
• the origin of this noise seen as fluctuations in the baseline, is the result of several fundamental phenomena.
• One is the shot noise of the light sources such as in RI and LS detectors.
• Other origins are thermal noises associated with amplifiers in all detectors; fluctuations in the pump flow rate; and thermal variations. Particulate, contaminants, and bubbles can also add additional noise components to the signal.
• the effect of the detector-noise-induced error in the slice-measurements is to introduce error in the quantities log (R, / c, ) and log ( ⁇ sp , jc t ) Because c. is in the denominator, the noise in these quantities increases as the response in the concentration profile decreases. Because of the logarithm, the noise in these quantities also increases as the response of the molecular- weight-sensitive detector decreases. Thus, the noise in
• the inventive method employs a GPC chromatograph system followed by one or more detectors.
• the detectors contemplated within the scope of the invention are a multi- angle laser light-scattering (MALLS) detector, a refractive index (RI) detector, and a viscometer (V) detector.
• MALLS multi- angle laser light-scattering
• RI refractive index
• V viscometer
• the RI, MALLS, and V detectors are used to measure, respectively, the concentration c t , the Rayleigh ratio ⁇ R. . and the specific viscosity ⁇ sp>i , for each slice i.
• a MALLS detector refers to a multi-angle light scattering detector.
• An example of such a system is an integrated Alliance GPCN 2000 system by Waters Corporation, Milford, MA that incorporates an on-line differential refractometer (RI) and a differential capillary viscometer (V) detector. Included in the system is a dual- angle static light scattering (MALLS) detector PD 2040 by Precision Detectors, Franklin, MA.
• MALLS dual- angle static light scattering
• a set of three 10 ⁇ m Waters Styragel columns (7.8 mm ID. x 300 mm) are an example of columns that can be used for size-exclusion separation other columns known in the art are contemplated within the scope of the invention.
• the present invention provides several data analysis methods that can be applied to the data acquired from a chromatographic system that includes a multi-angle laser light- scattering (LS) detector by itself, or in combination with either or both an RI or N detector.
• LS multi-angle laser light- scattering
• the detector configurations considered within the scope of the invention are four configurations: LS, LS-RI, LS-V, and LS-RI-N.
• the inventive methods contemplate one or more novel algorithms specific to each detector configuration.
• the radius of gyration, r g of a molecule is its root-mean-squared radius averaged over all conformations. It is well know in the art that the size of the molecule, as measured by its radius of gyration, affects how the intensity of scattered light changes as a function of the scattering angle at which the intensity is measured. Thus measuring the intensity of scattered light at different angles with respect to the incident beam is a method by which the radius of gyration of a molecule can be measured is well described in the text of (Mendichi, R. Radius of Gyration Measurements by GPC-SEC in Encyclopedia of
• FIG. 1. is an overview of a Gel Permeation Chromatography system which provides data to be analyzed by a system including the present invention
• FIG. 2 is an illustrative computer system for analyzing data according to the present invention
• FIG. 3 A shows the form factor of the line model
• FIG. 3B shows the form factor curves sample at seven angles
• FIG. 3C illustrates the effects of noise on measurements of the form factor
• FIG. 4A shows the difference in Rayleigh ratios from different scattering functions as compared to linear model
• FIG. 4B is a graph showing simulation of the MWD of a sample.
• the X-axis is log MW, the Y-axis is proportional to dW/dlogMW;
• FIG. 5 depicts three intrinsic viscosity laws are plotted, which are the linear Mark-Houwink law (1), the Zimm-Stockmayer IN law (2), and a quadratic polynomial law (3)
• the x-axis is log MW and the y-axis is log[ ⁇ j;
• FIG. 6 A is a graph showing a molecular weight calibration curve
• FIG. 6B is a graph showing concentration profile.
• the x-axis is elution volume, and the y- axis is log MW;
• FIG. 7 graphically depicts the three detector responses versus elution volume.
• the three detector responses from the RI, LS, and N detectors are refractive index, Rayleigh ratio and specific viscosity of the sample.
• the x-axis is elution volume (proportional to slice number).
• the y-axis is arbitrary; and
• FIG. 8 depicts the detector responses with the inclusion of baseline noise.
• FIG. 1 An illustrative gel permeation chromatography (GPC) system for determining radius of gyration, molecular weight and intrinsic viscosity of a polymeric distribution is illustrated in Fig. 1.
• a solvent reservoir 10 provides solvent that is pumped by a solvent pump 12.
• the polymeric sample 14 is introduced into the solvent flow.
• the solvent along with the polymeric sample 14 then passes through a set of columns 16 which contain the chromatographic bed.
• the solvent then passes through a detector 18, where one or several detectors analyze the solution and determine the presence and properties of polymer chains. This information is supplied to a data processor 20, as shown by arrow 22.
• the waste solution then collects in a container 24.
• the GPC separation is effected by a pump that forces the solvent through the set of columns 16 containing packed beads (the cliromatographic bed) that possess a distribution of pore sizes.
• a chromatograph employing an Alliance 2690 Solvent Delivery System, Waters Corporation, Milford, MA, and Styragel Columns, Waters Corporation, Milford, MA, is used.
• a set of three 10 ⁇ m Waters Styragel columns (7.8 mm I.D. x 300 mm) are an example of columns that can be used for size-exclusion separation. These are two HT 6E mixed bed linear columns with effective MW range 5xl0 3 - 10 7 g/mol, and one HT 2 with effective range 10 2 -10 4 g/mol.
• An injector mechanism introduces the polymer sample into the flowing solvent stream.
• the Alliance 2690 contains such an injection mechanism. Injection of the sample 14 into the flowing stream forces it through the column set 16. The separation is effected by a sieving mechanism. The pores exclude the larger molecules and retain the smaller molecules. Larger molecules see a column set having a relatively small effective volume. Smaller molecules see a column set having a relatively large effective volume. Consequently, larger molecules elute first, and the smaller molecules elute later.
• the detectors 18 follow the column set 16 and measure the physical properties of the eluent.
• the detectors follow the column set and measure the physical properties of the eluent as a function of time after injection of the sample.
• Typical detectors used in GPC measure refractive index (RI), viscosity (N), or the light-scattering (LS) of the eluent.
• the typical signal output of detectors is a fluctuating analog voltage.
• the analog signal is samples at a regular period, typically once per second.
• the sampled voltages are converted to digital form by an A/D converter.
• a computer is part of the GPC cliromatographic system.
• the computer controls the cliromatographic pump, injectors and detectors, as well as recording the digitized data from the detectors.
• the digitized measurements and the times of the sampling are stored in a computer memory, such as a disk drive.
• the software that displays and processes the data resides within the computer.
• Our algorithm resides in the computer. After the separation is complete, the algorithm is used to process the data.
• the data processor 20, as depicted in FIG. 2, includes a database 26 and a processor 28.
• the database 26 includes long-term storage such as magnetic disks, tapes and opto- magnetic discs. It is contemplated within the scope of the invention that other long term storage known in the art may be used.
• the database 26 in this illustrative embodiment is a relational database.
• the processor 28 accesses the data stored in the database 26 for processing and producing information that may be stored back in the database 26, or displayed on a workstation 30, or printed out (not shown).
• the data processor 20 can also include instrumentation to process and convert the data 22, including A/D (analog to digital) conversion circuits, sample and holds, and other devices (not shown).
• the data processor is a general purpose computer, such as an Intel Pentium® based personal computer running Microsoft Windows® XP or Windows NT. It is contemplated within the scope of the invention that other processors known in the art may be used.
• the processing application software is Millennium 32 ® as produced by Waters Corporation of Milford, Mass. Millennium 32 ® is a chromatography information and systems manager that allows a chromatographer to control instrumentation, and to acquire, store and retrieve data obtained from detectors. It also allows the chromatographer to process that data and store, retrieve display, and print the processed results in graphical and tabular form.
• Another component is the application software that processes and extracts information from the data and presents it in reports.
• a report generator allows the user to create, manage and print reports.
• a graphical user interface allows the user to interact with all these subsystems and components from the monitor.
• the illustrative embodiments of the present invention is implemented within the data processing component ofMillennium 32 .
• the choice of detectors determines what we can learn about a polymeric distribution, h a first illustrative embodiment, the data from the LS detector that determines the radius-of-gyrationr . If only a LS detector is used then all we will learn is the value of r for each slice r g ,. ⁇ r (V ⁇ ) .
• the first inventive method implements a more accurate computation of R . from the LS data.
• a LS-RI configuration is employed, the method disclosed for this dual-detector configuration allows a more accurate computation of R g ,. the conformation law, (CL), R g ( ) , and the MWD for the polymeric sample.
• the CL is an important intrinsic property of polymer samples.
• the conformation law is that of a power law with respect to molecular weight, where r g ⁇ M a .
• the parameterized form of the conformation law is commonly written as
• the column calibration H(l ⁇ ) is known, this method is presented to obtain the sample's IN law from this data and the column calibration.
• the column calibration H( ⁇ ) is the column's hydrodynamic volume versus elution volume.
• H (V t ) can be obtained from the data obtained from GPC analysis of narrow, mono-dispersed standards.
• the triple detector configuration of LS-RI- V includes a method that determines R g (V t ) , CL, MWD, and IN, and H(V) .
• the column calibration H (V) is known from independent measurement, then we can compare H(V) measured two ways (once from the triple detection data and once from narrow standards) to obtain a measurement of local polydispersity of the sample.
• the RI detector When included in a system, the RI detector is used to measure a peak's concentration profile. In place of an RI detector, it is contemplated within the scope of the mvention that a suitably calibrated UV/Nis absorbance detector, evaporative light scattering detector (EVS), or infrared (IR) detector can be used to measure a peak's concentration profile.
• a suitably calibrated UV/Nis absorbance detector, evaporative light scattering detector (EVS), or infrared (IR) detector can be used to measure a peak's concentration profile.
• a polymeric sample contains a distribution of chain lengths.
• the sample's MWD can be described by the relative mass (or concentration) per unit interval of molecular weight. Because of the kinetics governing the synthesis of polymers, these distributions are conventionally described by the relative mass per unit of the logarithm of MW, as shown in FIG. 4B.
• the horizontal axis is log MW and the vertical axis is proportional to dW/dlogMW.
• Each polymeric sample possesses a refractive index increment described by dn/dc, the change in the solution's refractive index per unit change of sample concentration.
• Rayleigh light scattering describes the scattering of light by a polymer chain, when the wavelength ⁇ of the incident light is comparable to or larger than the size of the chain.
• an incident beam of un-polarized light of intensity I 0 low sample concentration c, low scattering angle ⁇ , and small particle size.
• the sample in a unit volume of solution scatters the light into direction ⁇ with respect to the beam. That part of the intensity of the light that scattered by the sample I( ⁇ ) and received at distance r is given by
• K * is the optical constant expressed in terms of numerical and physical constants, as
• N A is Avogadro's number
• n 0 is the refractive index of the solvent
• Rayleigh scattering occurs when the intensity of scattered light is proportional to the product of sample concentration and molecular weight. The scattering will, in general, also depend on the scattering angle. Light scattered near zero angle deflection, (also called low angle light scattering) is almost exactly proportional to the product of concentration times mass and is nearly independent of the size (radius of gyration) of the molecule.
• Scattering at large angles will generally have reduced scattering intensity as compared to 0 degrees.
• angles larger than zero degrees the ratio of size of the molecule (as measured by its radius of gyration) to the wavelength of the beam determines the rate of reduction of intensity with angle. The larger the size, the more rapid the reduction.
• Modern light-scattering detectors designed for on-line chromatographic separations, illuminate the sample in a flow cell with a polarized laser, and the intensity of the scattered light is measures simultaneously (or nearly simultaneously) at two or more angles.
• a Precision Detector LS detector measures scattering at 15 and 90 degrees.
• a Wyatt mini-Dawn measures scattering at three angle, 45, 90, and 135 degrees.
• the LS detector measures the Rayleigh ratio, ⁇ R,. . is subscripted by andy.
• the index i refers to the slice number and the index j refers to the scattering angle 0..
• AR ⁇ ⁇ ARi V ⁇ j gives the Rayleigh ratio at elution volume V ⁇ and angle ⁇ ⁇ .
• the inventive method addresses the subtle interactions that a more complex expression of the excess Rayleigh ratio takes into account. These subtle interactions involve the size of the scattering particle (as particle sizes become comparable to ⁇ ), the concentration of the solution (as chain-chain interactions described by the second virial coefficient become important), and the observation of scattering at non-zero angles.
• the scattering form factor is introduced to take into account the cancellation (interference) effects of lights scattered from different parts of a molecule when its dimension is close or exceed 1/20 of the wavelength in the solvent.
• the definition is:
• the scattering function depends on the size and shape of the molecule, as indicated by the three models (sphere, coil and rod) below.
• the "linear” model represents a straight line in Zimm plot ( ⁇ Re "1 ⁇ sin 2 ( ⁇ /2)). The result ⁇ 2 of the fit for a given model should indicate the goodness of modeling.
• FIG. 3 A shows the form factor of the line model, P Vme (u) l + ⁇ /3 '
• each curve represents the ratio of a molecule's rtile to the laser wavelength normalized to the solvent index of refraction ⁇ n ⁇ ⁇ ln .
• a molecule whose size is small compared to the ⁇ n exhibits constant form factor at each angle.
• the scattered intensity at zero angles is unchanged, but the rate of fall off in scattered intensity becomes larger. It is the measurements of form factor over a range of angles that allow the measurement of r g of the molecule from LS detection.
• FIG. 3B shows the same curves, but sampled only at seven angles, 45, 60, 75, 90,105, 120, and 135 degrees. Obviously LS detectors that collect measurements at fewer angles provide less information with which to obtain accurate estimates of r g .
• FIG. 3C illustrates the effects of noise on measurements of the form factor.
• Figure 4A shows how significantly the different models affect the processing results, depending on the gyration radius (difference in Rayleigh ratios as compared to linear model):
• the inventive method makes better use of the data in the tails of the distribution than do methods of the prior art.
• the inventive method replaces the slice-by-slice computations of critical quantities with parameterized models that are fit simultaneously to all the data collected at each slice. It is this simultaneous use of LS data obtained at all slices and at all scattering angles that improves the estimate of the radius of gyration of the molecule as a function of elution volume or molecular weight.
• the methods of the prior art determine r . for each slice as a first step
• the inventive method gives better results because it is significantly less sensitive to the effects of detector noise than are the methods of the prior art.
• the extreme sensitivity to detector noise occurs when the independent fits to each slice are performed; the disclosed method, by bypassing this step, avoids introducing excess noise into the determination of r s,r
• model parameters can be properties intrinsic to the polymer distribution, such as branching parameters.
• P m is the array of polynomial coefficients, where this model is fit directly to the light scattering data. This fit is carried out by rearranging the terms in the ⁇ 2 fit so that that R g ( ⁇ ) is determined only by the ratio of the measured intensities of the light-scattering channels.
• R g ( ⁇ ) is determined only by the ratio of the measured intensities of the light-scattering channels.
• the subscript 0 refers to the scattered intensity measured at one of the channels, e.g.,
• the inventive method is that the Rayleigh ratio at zero angle, ⁇ R, 0 , is not a fitting parameter in this formulation. This advantage becomes crucial both in the tails of the chromatogram and in the low molecular mass (late eluting) region of the GPC chromatogram.
• the fitting parameters ⁇ R, 0 can become highly correlated with the parameters that describe r .
• the inventive method produces stables estimates of r g even in the tail and low mass regions.
• uy is calculated from model r at i-th slice:
• Vj is the elution volume at i-th slice
• This formulation of ⁇ 2 takes advantage of the fact that the model of r g is fit to all the light-scattering data directly. The least-squares fit is carried out over the whole of the peak region.
• This model can be generalized to weight each channel separately, as different channels have different noise.
• the weight function for each channel is determined by the detector noise level in that channel.
• the least-squares fit can still be carried out over the whole of the peak region taking into account weighting as follows.
• the denominator is a model of the noise present in the numerator.
• the slices having the highest signal to noise (typically found in the heart-of-the peak) automatically determine the properties of the r g (V) curve.
• the slices having the lowest signal to noise have little effect on the determination, of the curve.
• the inventive method allows all data to be used, including data in the tails of the distribution. Thus it is not necessary to restrict the fit to a region of high signal to noise. This is, unlike methods of the prior art that are typically affected by the noise in the tails of the peak and therefore require the user to manually restrict the fit to a good data region, typically the heart of the peak. Unlike the inventive methods, the methods of the prior art require the choice of a good data region.
• the Rayleigh ratio at zero angles for all slices is determined according to the invention only by the calibration of the LS detector.
• LS calibration is well known in the prior art, and preferably consists of performing a GPC separation of a single chemical component with a known molecular weight. The calibration is obtained for the reference channel.
• the zero-order polynomial fit is used in narrow LC-LS peak processing to calculate the r g (V), while the 1 st order r g (N) fit is used in narrow
• RI or UN detector
• the first is, again, the fit of the model of r (P) to the LS data alone, exactly as described above.
• the second inventive step is the fit of a polynomial model to both the RI and LS data.
• the LS data is not 0 (V, ) (as in Gorenstein et al.), but rather AR ; j , the data obtained at each slice and at each scattering angle
• the polynomial model is of the molecular weight as a function of elution volume (as in Gorenstein et al.):
• Q m is the array of polynomial coefficients.
• the slice determine values for r g i are calculated from the normalized LS data as described above. From these the scattering form factor is obtained for each slice at each angle, , P tJ ⁇ P(k 2 r X j ) .
• the method fits the model M(V;Q m ) by optimizing the following fit:
• the fit is to the Rayleigh ratio data obtained at all slices and at all angles.
• a weight factor can be included to take into account differing noise levels between the LS channels. This fit can be applied to the complete distribution without the need to select a good data region, or otherwise restrict the fit to the heart of the peak.
• the zero-order polynomial fit is used in narrow LC-LS peak processing to calculate the M(N), while the 1 st order M(V) fit is used in narrow GPC/V-LS peak processing.
• the calibration model is based on a model of molecular weight versus elution volume
• the structural model replaces M(V;Q m ) with a model of r g as a function of molecular weight.
• r g (M;P m ) The preferred relationship between r g and M can be a polynomial function, expressed as
• the structural fit is carried out in two steps.
• the first step is still the determination of r . obtained solely from the LS detector, as described above. Again, this fit provides the values P. . ⁇ P(k 2 R g 2 i X j ) of the scattering function for each slice and scattering angle.
• the model r g (M;Q m ) is calculated by the following fit:
• this inventive method produces a smooth model for conformation law r (M ;K a ⁇ ) or r g (M ; Q m ) that fits directly to all of the data for both detectors.
• an intermediate model M(V) is not needed, and no noise- sensitive slice value are computed.
• Any desired slice value such as the column calibration M(Vi) can be computed from the model fits obtained by the method disclosed herein. In Gorenstein et al., these same structural models were fit to RI-LS data, but the fit to the LS data only involved the Rayleigh ratio at zero angles. The inventive method fits these structural modes to LS data obtained at each slice and at each angle.
• This calibration curve H,- can be obtained using standard techniques.
• the calibration curve can be obtained using the injection of narrow (mono-dispersed) polymer standards. If the standards have known molecular weight, then RI-N detection provides the hydrodynamic volume at the retention volumes at which the standards elute. If the molecular weight of the standards is not known, then RI-LS detection can be used to determine each standard' s molecular weight.
• log[ ⁇ ] is modeled as a polynomial of elution volume.
• r g is first modeled as a polynomial of elution volume r g (N) and is calculated first from the LS data. Then the following fit is performed, which determines the model for log [ ⁇ ] :
• ⁇ is the hydrodynamic volume read at i-th slice read from the external universal calibration curve
• [ ⁇ ] are the modeled values at the i-th slice
• the P tJ are obtained from the LS data as described above. Again, the concentration and light-scattering data are used to constrain the model.
• the intrinsic viscosity [ ⁇ ] is modeled as a function of molecular weight M as following:
• the next step in the analysis is to then model r as a polynomial of elution volume r g (N) using the LS data so as to obtain r . and P u , as has been described above. Finally, the following fit is performed, which determines the parameters that specify the model parameters S m .
• RI-V data is processed according to the method described in Gorenstein et al., to obtain the IVL.
• a model of the sample's intrinsic viscosity is determined using either a calibration or structural fit.
• the last step in this inventive method is to combine the intrinsic viscosity model and the MWD determination to obtain H t , the universal calibration curve.
• Table 1 summarizes the first approach. The first row shows that first step involves only the LS detector and obtains r g (V)
• the second row shows that with the inclusions of RI (or UV) data that gives a, we can perform a calibration fit or structural fit that gives the MWD and the CL.
• the third row describes the possible ways of combining the RI/UV and V data to obtain information related to the intrinsic viscosity of the sample.
• a calibration model or structural model (IVL) can be used.
• the fourth row shows that we can combine the intrinsic viscosity data and MWD to obtain the column calibration, or hydrodynamic law H(Vj).
• S m is the array of polynomial coefficients to be determined by a fitting procedure.
• This model for H is calculated first from using data from all three detectors. The fit is performed by minimizing the following
• the log[ ⁇ ] is modeled as a polynomial of elution volume.
• T m is the array of the log[ ⁇ ] polynomial coefficients.
• the non-linear fit optimizes the ffoolllloowwiinngg ⁇ ,, uussiinngg data from two detectors LS and RI , and the previously determined values of H and P tj
• Another possibility is to optimizes the following ⁇ 2 , using data from two detectors LS and V , and the previously determined values of H, and P
• This formulation may be advantageous for polymers that have little RI response due to a preponderance of molecules having high molecular weight.
• the light scattering signal, the Rayleigh ratio, is most commonly expressed in the following way in terms of concentration c, molecular weight M:
• Equation 3.134 of "Introduction to Polymers ' " 2 nd Edition by R. J. Young and P. A. Lovell, P.186:
• a 2 is the second viral constant.
• the above formulas apply to the condition that 2 A 2 c « M.
• the constant is default to 0, and the units is ml.mol/g 2 .
• the equation is as simple as for zero A 2 . Therefore, calculation is performed with zero A 2 . After the calculation is done, the result M' is corrected to converted it back to M.
• the pressure drop across a capillary tube P is proportional to the viscosity of the liquid flowing through the tube. If P 0 is the pressure drop due to the solvent alone, then the specific viscosity ⁇ sp of a slice containing a polymer in solution is defined as
• the intrinsic viscosity (IV) is defined as ratio of the slice's specific viscosity ⁇ sp divided by its concentration, c, in the limit of low concentration.
• the intrinsic viscosity of a polymer varies with its molecular weight.
• the intrinsic viscosity law describes the dependence of logarithm of the sample's intrinsic viscosity on the logarithm of its molecular weight.
• the logf ⁇ ] is in general is proportional to the logarithm of the chain's molecular weight, logM,.
• the empirical Mark-Houwink intrinsic viscosity law expresses this linear relationship as
• Polymers can be branched.
• Zimm and Stockmeyer developed a physical model of long-chain branched polymers, described in Zimm, B. and Stockmayer, W., 1949, J. Chem. Phys. 17, 1301-1314, which is incorporated by reference in its entirety. Based on this work, a model is developed describing the intrinsic viscosity for polymers with long-chain branching, which is called the Zimm-Stockmeyer (ZS) law.
• ZS Zimm-Stockmeyer
• the ZS law assumes that at low molecular weight region the polymer is essentially un-branched. In this region, the intrinsic viscosity law is asymptotically linear. The asymptotic slope of the intrinsic viscosity law at low molecular weight is described by the Mark-Houwink constants, K and ⁇ .
• the molecular weight at which a polymer branches is a stochastic process described by a branching probability.
• the value ⁇ is defined as the branching probability per Dalton.
• a typical values for ⁇ is 0.00001 per Dalton.
• the ZS intrinsic viscosity for each slice i is a function of the molecular weight of that slice, Mi and depends on the parameters K, ⁇ , ⁇ and ⁇ , as follows:
• the law can be regarded as a generalization of the Mark-Howink law and can describe two possible types of branched polymers, a three-branch point and a four-branch point.
• a third intrinsic viscosity law is simply an empirical description of a sample's intrinsic viscosity based on a polynomial expansion. This "law" is also formulated as an extension of the Mark-Houwink law, so that
• lo l l°g- ⁇ + oi ⁇ log ,. + a 2 log 2 M, +-- - + a N log w M,. Eq. (64)
• N is the order of polynomial
• K and ⁇ , ⁇ are the Mark-Howink constants.
• FIG. 5 shows all three intrinsic viscosity laws supper imposed, the linear Mark- Houwink law (36), the Zimm-Stockmayer JN law (37), and a quadratic polynomial law (38).
• this curve would universal, in the sense it would apply to all samples.
• FIG. 6 A plots the log of the MW versus elution volume for the sample described in FIG. 4B.
• FIG. 6B plots the concentration profile of the sample, c,-, versus slice i or elution volume. This concentration profile that results from the samples molecular weight distribution as shown in FIG. 4B is determined by this molecular weight calibration curve and the sample's refractive index increment dn/dc.
• the MW calibration curve must be determined by a calibration procedure employing the collected data.
• hydrodynamic volume measures the viscosity per chain, in contrast to intrinsic viscosity, which measures viscosity per concentration.
• use narrow standards of any suitable material can be used to construct a hydrodynamic volume calibration curve.
• RI-V detection determines the hydrodynamic volume for each standard.
• the standard's hydrodynamic volumes is plotted versus elution volumes, and polynomial curve is fitted to these data to determine the hydrodynamic volume calibration curve. This curve gives the hydrodynamic volume of the column for the elution volume spanned by the narrow standards.
• This curve is referred to as either as the universal calibration of the column set, or as the hydrodynamic volume calibration curve. For such a curve to be useful, it must span an elution volume that encompasses the sample's mass distribution.
• a properly calibrated RI detector responds to the solution's refractive index at each slice. Given the output of this detector, subtracting the detector's baseline response and dividing by dn/dc for the sample gives the concentration profile of the sample, c, for the zth slice.
• the concentration profile c,- can also be obtained by knowing the mass of material injected into the chromatograph with the assumption that the RI detector responds linearly to sample concentration.
• the LS detector responds to the sample by scattering light. Subtracting the detector's baseline response and applying a detector calibration procedure gives the excess Rayleigh ratio due to the sample, R,-, for each slice.
• the viscometer detector responds to the solution viscosity. Subtracting the detector's baseline response, and dividing by the viscosity of the baseline gives the specific viscosity of the sample, ⁇ spi , for each slice.
• IVL intrinsic viscosity law
• the viscometer measure's a sample's specific viscosity, which is the fractional increase in the viscosity of a solution due the presence of a sample.
• the sample's intrinsic viscosity is the ratio of the sample's specific viscosity to its concentration.
• the TVL of a sample is its intrinsic viscosity as a function of its molecular weight. RI-V detection of the sample, together with molecular weight calibration provided by the narrow standards provides a means to measure the sample's IVL.
• the specific viscosity ⁇ sp is the fractional increase in viscosity due to the presence of a sample.
• the pressure drop across a capillary tube P is proportional to the viscosity of the liquid flowing through the tube. If P 0 is the pressure drop due to the solvent alone, then the specific viscosity ⁇ sp of a slice containing a polymer in solution is defined as
• the intrinsic viscosity (IV) is defined as ratio of the slice's specific viscosity ⁇ sp divided by its concentration, c, in the limit of low concentration.
• the baseline-correct and calibrated detector responses give three signal profiles for each slice measurement of the sample: the concentration c,-, Rayleigh ratio R,-, and specific viscosity ⁇ sp ⁇ for slice i.
• Each of these detectors contains non-idealities in their responses. Typically, these non-idealities fall into two categories, baseline drift and stochastic baseline noise.
• Baseline noise is an irreducible component of the measurement process.
• Each of these detectors contains baseline noise at some level.
• the origin of this noise, seen as fluctuations in the baseline, is the result of several fundamental phenomena.
• One is the shot noise of the light sources in the RI and LS detectors.
• the other are the thermal noises associated with amplifiers in all detectors, and the third are fluctuations in the pump flow rate and thermal variations. Particulates, contaminants, and bubbles can also add additional noise components to the signal.
• the net result of these effects is manifested as an addition of stochastic noise to the output of each detector.
• the noises can be described as Gaussian deviates having zero mean and a well-defined standard deviation.
• the standard deviation of baseline noise will in general be different for the different detectors, but each detector's noise is constant throughout the separation.
• FIG. 4B shows the simulated MWD that is used.
• the column calibration in FIG. 6A applied to the MWD gives the concentration versus elution volume as it elutes from the column set 39, or c(V) also plotted in FIG. 6B.
• the excess Rayleigh ratio describes the fraction of incident light scattered by a compound.
• the excess Rayleigh ratio is proportional to the product of the concentration c, and molecular weight M, and is given by
• the excess Rayleigh ratio versus elution volume 42 is also shown in FIG. 7.
• the light-scattering profile which is the excess Rayleigh ratio for each slice R, is obtained by multiplying each slice in the concentration profile in FIG. 6B by MtK .
• the specific viscosity r ⁇ sp is the fractional increase in viscosity due to the presence of the sample.
• the specific viscosity is simulated by multiplying, slice-by-slice, the concentration times the intrinsic viscosity [ ⁇ ]i for that slice.
• the slice's intrinsic viscosity is obtained from the Zimm-Stockmayer intrinsic viscosity law 37, plotted in FIG. 5.
• the SN versus elution volume obtained from the viscometer detector is shown in FIG. 7 as curve 44.
• the hydrodynamic calibration curve is obtained from narrow molecular weight standards.
• simulated column sets universal calibration are obtained for each slice by multiplying each slice's the sample's molecular weight times its intrinsic viscosity.
• the inputs to this calculation are the column's molecular weight calibration curve FIG. 6A and the sample's intrinsic viscosity law (Fig. 5, curve 37).
• Fig. 7 shows the detector profiles based on the instrumental responses to the concentration profile described in FIG. 6A.
• the appropriate level of detector noise to is added to each of the simulated signals. Detector noise is simulated by adding to each point in each profile zero-mean random Gaussian deviates with standard deviations chosen to approximate typical detector performance.
• FIG. 8 shows the profiles after the addition of the baseline noise to each of the detector responses. Shown in FIG. 8 are the RI 40, LS 42 and V44 detector responses.
• minimization procedures described above require that initial parameters values be found and that an initial value for be computed. These parameters are then iteratively adjusted until the minimum of is found.
• the determination of initial parameter values can be accomplished by a variety of standard methods. Such methods include the manual estimation of values that are known to approximate the expected final parameter values; the adoption of typical values of parameters; or the implementation of a separate algorithm that determines initial parameters values from a subset of the data to be analyzed.
• the subsequent iterative adjustment of the parameters can be accomplished by a variety of standard methods. Methods that find the minimum of a function of N-variables include ⁇ ewton-Raphson, Levenberg-Marquadt, simplex, gradient search, and brute force search. Such iterative adjustment procedures are described in the Numerical Recipes in C, The Art of Scientific Computing, Second Edition, (1992) W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Univ. of Cambridge, pages 394-455.
• RI, V and LS detectors are described in the embodiments, herein, it will be appreciated that other detectors, such as other models and/or from other manufactures, can be implemented.

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