JP4439211B2 - Method for determining weight average molar mass, mean square radius, and second virial coefficient of unfractionated sample - Google Patents

Description

Background Molecules in solution are generally characterized by their weight average molar mass M _w , their mean square radius <r _g ² >, and a second virial coefficient A ₂ . The latter is an indicator of the interaction between the molecule and the solvent. For unfractionated solutions, these properties show how they scatter light, as described in an original paper published by Bruno Zimm in 1948 in Non-Patent Document 1. It may be determined by using and measuring. Light scattered from a small amount of solution is measured over a range of angles and concentrations. Properties derived from light scattering measurements are related via the following equation developed by Jim:

_Where R (θ) is the excess Rayleigh ratio measured in the direction θ per unit solid angle, defined as R (θ) = [I _s (θ) −I _solv (θ)] r ² / I ₀ V _Where I _s (θ) is the intensity of light scattered by the solution as a function of angle, I _solv (θ) is the intensity of light scattered from the solvent as a function of angle, and I ₀ Is the incident intensity, r is the distance from the scattering volume to the detector, V is the illuminated volume seen by the detector, P (θ) is the shape factor of the molecule being scattered,

K ^* = 4π ² (dn / dc) ² n ₀ ² / (N _a λ ₀ ⁴ ), N _a is the Avogadro number, dn / dc is the refractive index increment, and n ₀ is Solvent refractive index, λ ₀ is the wavelength of incident light in vacuum. The form factor is related to the mean square radius by the following equation:

The collection of light scattering data over a range of scattering angles is more commonly referred to as multiangle light scattering (MALS). The data is then fitted to equation (1) and M _w , <r _g ² >, and A ₂ are extracted. For this purpose, the inverse of equation (1) is more commonly used and it may be written as:

  There are several ways to fit the data to equation (3). The most common method in history is to mimic the graphical method presented by Jim with software. This is often referred to as the Jim plot method. Alternatively, a comprehensive nonlinear least squares fit method described below may be used.

One powerful method for characterizing molecular solutions is to fractionate the sample first by chromatographic means such as size exclusion chromatography (SEC) and then measure the scattered light and concentration as a function of the elution volume v. It is. If the fractions are well separated, each volume sample can be considered monodisperse in nature. If A ₂ is known from previous experiments, or if the concentration is low enough that the effect of A _{2 on} the scattered light is negligible, the data is fitted to equation (1) or (3) and the distribution M (v ) And r _g ² (v) may be extracted. This is routinely performed by commercially available software such as the ASTRA® program developed by Wyatt Technology Corporation of Santa Barbara, California.

A figure of merit (FOM) may be defined that characterizes when the A ₂ term may be ignored. Equation (1) shows that A ₂ is a prefactor for the c ² term. By comparing the size of the terms in parentheses in equation (1), the FOM may be defined as follows:

As assumed in the derivative of the Jim equation, the second virial coefficient has only a slight effect on the light scattering signal when FOM << 1. If you want to measure the second virial coefficient by light scattering, the FOM must be large enough that the effect can be measured with good accuracy, but requires a higher-order concentration term in equation (1) It must not be so large. Since SEC columns dilute samples about an order of magnitude per column, the concentration resulting from chromatographic separation is typically low enough that A ₂ can usually be ignored. Details of chromatographic separation methods, definitions and calculations of mass and size moments, and explanations of terms used to describe the associated distributions can be found in the 1993 general review by Wyatt in Non-Patent Document 2.

In summary, there are two modes of light scattering measurement. In batch mode, a series of light scattering measurements on a single sample are made at different concentrations. M _w , <r _g ² >, and A ₂ are determined by the concentration and angular dependence of the scattered signal. In chromatographic mode, the composition of the sample changes as the sample elutes, so a priori knowledge of A ₂ is required, but the distributions M (v) and r _g ² (v) are measurable. From these distributions, mean values M _w and <r _g ² > may be calculated. Note that the value calculated from the fractionated sample measurement should be the same as the value measured from the batch sample. The difference arises from the assumption that A ₂ = 0 and distortion due to band expansion between detectors.

A ₂ can be determined from batch measurements, but the question remains whether the second virial coefficient can be accurately measured if the sample concentration varies continuously with time as in chromatographic elution. US Pat. No. 5,129,723 by Howie, Jackson, and Wyatt describes a method for injecting an unfractionated sample into a MALS detector following elution and thorough mixing. It was done. This procedure created a sample peak that passed through the light scattering detector and its profile was assumed to be proportional to the concentration profile of the sample that was eluted but not yet fractionated. Since the mass distribution in each flake is the same, each point of the profile is proportional to the sample concentration × weight average molar mass at that point, with reference to equation (3) and A ₂ = 0 Assumed by setting. Based on this, a set of these points could be used to generate a Jim plot, and the associated weight average molar mass, mean square radius, and second virial coefficient were derived. Since the knowledge of the total mass injected was sufficient to convert the sample peak curve to a concentration profile, a concentration detector was considered unnecessary. However, this method was flawed. Because A ₂
This is because the assumption that is 0 is inconsistent with the derived result that it is not.

  The second method is to use a chromatographic configuration, in which case both light scattering and concentration detectors are used. When injecting monodisperse samples or developing chromatographic methods to fractionate monomer peaks, the weight average molar mass in each elution fraction must be constant throughout each peak. From the MALS and concentration data, a Jim plot analysis may be performed from values at several different slices or sets of slices of the elution profile. The weight average molar mass, the mean square radius, and the second virial coefficient may then be derived. However, for most proteins, the mean square radius is too small to be measured accurately.

While this method may work in principle, there are practical difficulties that prevent it from becoming generally applicable. The experimental equipment described above requires two detectors. Since fluid must pass through capillaries and fittings as it moves between detectors, mixing and diffusion results in “band expansion between detectors”. The downstream peak is “broader” than the upstream peak. This means that a monodisperse sample produces a measured mass distribution that appears polydisperse unless the effects of expansion are taken into account. This effect is particularly problematic for proteins. This is because their FOM is usually much smaller than 1 and the second virial coefficient contributes a little to the scattered light. Errors associated with band expansion usually dominate and the derived second virial coefficient is significantly distorted. Various analytical corrections of band expansion have been developed over the years, but the contribution of A ₂ to the light scattering signal is so small that the resulting value of A ₂ is just that of the band extension used. Depends on the model sensitively. Due to this sensitive dependence on the model, this method was unreliable.

Since the molar mass of protein for a monodisperse sample in a suitable buffer solvent is often easily measured by MALS, mass spectroscopy, or direct sequencing, US Pat. The “Method for measuring the second virial coefficient” emphasized the usefulness of avoiding the generation of a large number of concentrations necessary to produce an entire Jim plot. However, when the method described therein is extended to a more general number of concentration aliquots, it will be shown that the method no longer requires a priori knowledge of molar mass. In addition, the accuracy is improved due to the averaging achieved by the extended method.

Many scientists use on-line methods to create Jim plots, and the aforementioned ASTRA® program makes decisions much more easily than earlier manual methods. Currently, the best means for measuring A ₂ of unfractionated samples is to prepare aliquots at different concentrations, and use a syringe or syringe pump directly to the light scattering cell of the MALS instrument. And then inject. As the infusate fills the cell, each aliquot creates a gradient up to a plateau for pure solvent baseline scattering. From a selected range of points, collectively referred to as a peak, on each of the plateaus, the software accepts the corresponding input of the prepared concentration and creates a Jim plot. This approach has several drawbacks, the most important of which is the need to prepare and use a relatively large amount of sample. When injecting into a flow cell, the need to produce a flat plateau means that the cell must be overfilled several times. For a flow cell with an internal volume of 80 μl, more than 500 μl of sample is required for each aliquot. For current pharmaceutical research, this requirement makes the current approach infeasible. Often, this amount of sample cannot be synthesized in practice. In addition, for proteins and similar biopolymers, each injected sample must be dialyzed prior to injection. This significantly increases the preparation time and effort required to make the measurement.

Finally, in the present invention, it is shown that the integration between peaks eliminates the dependence on the peak shape and the correction of band expansion is reduced to a single parameter. Sensitive dependencies on the extension model are removed, making this method practical and reliable. A ₂ can be measured very easily not only in the presence of band broadening for monodisperse samples such as proteins, but also when unfractionated samples are injected directly into the light scattering measurement cell.
US Pat. No. 6,411,383 Bruno Zimm, “Journal of Chemical Physics”, 1948, 16, 1093-1099 Wyatt, “Analytica Chimica Acta”, 1993, 272, 1-40.

SUMMARY OF THE INVENTION The main object of this invention is to extend the method described in the parent invention to a method for directly extracting M _w , <r _g ² >, and A ₂ from a series of infusions having different concentrations. To eliminate the need to know M _w a priori. Also disclosed are two fitting techniques for extracting these parameters. One method is based on comprehensive non-linear least squares fit data. The second analysis technique is modeled after Jim's graphical method. The resulting plot is referred to as the Trainoff-Wyatt (TW) plot to clearly distinguish it from the Jim plot. The object of the present invention is to automate the process and eliminate manual sample preparation as much as possible. Another object of the present invention is to make more accurate measurements by taking advantage of the implied averaging in the multiple injection technique. Finally, an object of the present invention is to make such a determination using a minimum amount of sample.

The aforementioned US Pat. No. 6,411,383 by Wyatt makes it possible to determine the second virial coefficient of a sample composed of a monodispersed molar mass distribution in a solvent directly from a single injection. About. The same method can be applied to a class of unfractionated samples. The present invention is for a more general application using a series of concentrated infusion solutions. The Wyatt patent requires a priori knowledge of the weight average molar mass of the sample. The invention determines M _w , <r _g ² >, and A ₂ using analytical techniques similar to those presented by Jim.

  In addition, the Wyatt patent assumes that the band extension between detectors is negligible. It is often desirable to minimize the amount of sample used for the measurement. Therefore, a small amount of infusate is used, resulting in a narrow peak. In this case, the extension of the band between detectors cannot be ignored. The present invention presents several methods that take this effect into account.

  This method begins with preparing a set of j dilutions of a sample for injection into a SEC column set. If an autosampler is available, a single sample may be provided for subsequent automatic dilution to a defined concentration. There are two ways to use an autosampler to prepare the dilution. First, it can inject smaller quantities of sample into the injection loop without filling it up. Alternatively, the autosampler can pre-dilut the sample before filling the injection loop and then fill it completely. The latter method is preferred. This is because it produces a consistent concentration profile with different amplitudes for each infusate.

  The analysis requires a set of decreasing concentrations of sample to be measured. For proteins, each sample aliquot must be dialyzed. Introducing a set of SEC columns through which each sample flows eliminates this requirement. This is because dialysis occurs while the sample passes through the column. Following the column, a MALS detector and a concentration detector are connected in series. For small molecules whose mean square radius is too small to be measured, measurements at a single scattering angle such as 90 ° are sufficient, but the accuracy of the determination will be reduced. These are the conventional elements of standard separation by SEC and provide an absolute determination of the eluted molar mass present in the sample. If neither dialysis nor fractionation is required, the separation column can be eliminated and the sample may be injected directly into the flow cell of the light scattering instrument. Alternatively, dialysis may be achieved without separation using a protective column.

Detailed Description of the Invention Theoretical Description The analytical method is now addressed. Consider a set of injections _j , each corresponding to a total injection mass m _j . Returning to equation (1), for each successive peak j and scattering angle θ _k , the excess Rayleigh ratio and the concentration peak are added.

  Each sum is determined over all i in peak j. Note that each sum may have a different number of terms. This is because each peak may have a different width and may be spread by the effects of band expansion between detectors.

The coefficient Δv multiplied by the left side of the equation (5) is just the area under the excess Rayleigh ratio peak .

I will call it . It may be calculated directly . Therefore, equation (5) may be rewritten as follows .

Where

Is proportional to the total mass eluted,

It is .

Since the concentration detector produces c _ij at each elution volume, the sums D _j and C _j of equation (6) can be easily calculated over the entire extended range of concentration peaks. The excess Rayleigh ratio R _ij (θ _k ) is first calculated by subtracting the pure solvent light scattering value from the measured solution value as defined immediately after equation (1) above. Details are given in the previously cited paper by Wyatt in Analytica Chimica Actor. Here, the reciprocal of equation (6) is taken to obtain the following equation.

  For this reason,

Where κ _j = D _j / C _j . Equation (8) is called the Trenoff-Wyatt (TW) equation.

Equation (8) presents the essence of the present invention. Of particular importance is the measurement and calculation of C _j and D _j over a range of elution slices sufficient to contain all the elements of the elution sample. The number of flakes included in these sums is generally due to the effects of band expansion as described above.

More than the number of flakes used in the calculation.
So far, the effect of band expansion between detectors on downstream detectors has been ignored. The effect that it has on equation (8) is examined. Assume that the concentration detector is downstream of the light scattering detector. Further assume that the effects of mixing and diffusion are linear and that we can write:

Where c ^m (t) is the measured concentration signal, B (τ) is the standardized expansion kernel, and c (t) is the concentration measured when there is no extension between detectors. . To simplify the explanation, the notation was switched to continuous display. Consider the effect this has on C and κ in equation (8). When C is calculated from the measured signal, the following equation is obtained.

This is natural because B (τ) is standardized. This is expected because C is proportional to the injected mass and the expansion does not affect the injected mass. However, it can be seen from the following equation that the expansion affects κ through the measured quantity D ^m .

  Consider the third integration on the right. This is the form of the superposition integral of the extension function with itself. It is simply a function of the parameter τ−τ ′. Rewriting the third integral of equation (14) using the superposition integral theorem yields:

Where

Is the Fourier transform of the extended kernel and is defined by the following equation:

It is further assumed here that the characteristic time scale of the expansion function is much smaller than that of the concentration peak. Each of the two time scale is referred to as tau _b and tau _p. In that case,

Is a function with a broad peak with a characteristic width π / τ _b . Equations (15) and (14) denote it as τ _b → 0,

, And the limitation of D ^m → D, as expected .

The conclusion is that the extension between detectors distorts only the κ term of the TW formula. From equation (14) it can be seen that the effect of expansion is to suppress the peak maximum and to increase the value in “wing”. This means that the measured value D ^m ≦ D. This in turn implies that the computer-calculated value of A ₂ is overestimated. There are several approaches to correcting for expansion data. Conceptually simplest and least useful is deconvolution of equation (9) to obtain the following equation, which is an unexpanded concentration profile.

It can then be used to directly compute D. There are several problems with this approach . One is that there is a need to know the complete form of the extension function . Another problem is that the deconvolution integration process is numerically unstable . Ratio for large values of ω

Please consider .

and

Since both are derived from noisy physical measurements and both tend to zero for large values of ω, this ratio will have a large variation . If an inverse Fourier transform is performed, the result will therefore have increased high frequency noise . This can still be filtered, but this procedure cannot be justified from the original principle . To make matters worse, the inverse Fourier transform can produce physically impossible results such as negative values of concentration or non-causal ringing . In fact, these problems make this procedure impractical . Finally, if these problems can be overcome, the process of making a series of separate injections can be eliminated, and instead, continuous concentration fluctuations along one injection peak can be used as individual parameters in equation (3). Can be used to fit .

  For a more practical method, consider the special case where both c (t) and B (τ) are Gaussian. In that case, equation (14) can be explicitly computed by the following definition.

Where c is proportional to the total injected mass and is different for each injected solution. Inserting equation (18) into equation (9) yields another Gaussian distribution with width τ _cm ² = τ _c ² + τ _b ² . Also, the following formula is found:

However, recall that τ _c is the Gaussian width of the unexpanded peak and is generally unknown. However, in the small extension (τ _b << τ _c ) limit, it can be approximated by the width of the light scattering peak. Thus we can write

Both of these quantities can be measured directly . Of course, if the peak and extension function deviate from the glass profile, it is expected that equation (20) will begin to decompose . However, it is reasonable to assume that equation (20) is valid for τ _b << τ _c , which is

Imply that .
Let's go back to the general problem. When producing a dilution using an autosampler, the infusion has, as a sufficient approximation, the same shape but different amplitude.

  Assuming they have the same shape, the jth concentration profile can be written as:

Where m _j is also the injected mass of the j th injection. Inserting this into equation (11) yields the following equation:

Next, the ratio D ^m ₀ / D ₀ can be approximated using equation (20). Alternatively, this ratio can be measured by running a conventional plateau method for measuring A ₂ and a TW plot for a reference sample and then adjusting the ratio until the two methods match. It is. This ratio is effectively a measure of the extension between detectors and should be independent of the sample used. Thus, once the system is characterized by the reference sample, the measured ratio can be used for analysis of the next unknown sample.

Collection of experimental data To determine the weight average molar mass M _{w of} the molecular solution, the mean square radius <r _g ² >, and the second virial coefficient A ₂ , a set of j samples of different concentrations are sequentially shown in FIG. It is injected into a chromatography system as shown. The solvent is drawn by the pump means 1 from the solvent tank 2 through the degasser 3 and then pumped through the filter means 4. The degasser 3 is generally used to remove dissolved gas from the solvent. This is because such gases can then produce small bubbles in the solution, which can harm the desired measurement from the solution itself. Filter means 4 is generally incorporated as shown to remove residual particulate matter from the solvent that may impair the desired measurement. An aliquot of sample 5 from which the weight average molar mass, mean square radius, and second virial coefficient are to be derived by the method of the present invention is injected by the injection means 6 and passes directly through the light scattering MALS detector 8. Diluents can be prepared in advance and injected manually.

Alternatively, they can be generated by programming the autosampler to reduce the volume of the undiluted sample, or preferably to dilute the sample and inject a fixed volume. . Usually, the diluent ranges over an order of magnitude or greater. If the sample requires prior dialysis and / or fractionation, the selected separation column set 7 may be placed in front of the MALS detector 8. In the absence of aggregates, dialysis may often be accomplished by the use of a protective column only, or may be omitted if dialysis is not required. Each sample 5 consecutive passes through the MALS detector 8, flows through the concentration detector 9, shown as a differential refractive index detector DRI, whereby the sample concentration is measured at each slice interval Delta] v _i. If the collection interval Δv _i is equidistant, Δv _i = Δv is a constant. The resulting light scattering and concentration signals are then stored and processed by computer means 10 and for each injected aliquot j, the excess Rayleigh ratio R _ij (θ for each slice i at each measured scattering angle θ _{k. k} ) is calculated. The computer means 10 also computes the molecular features including mass and size and their distribution. The sample concentration detector 9 is generally a DRI detector, but may be replaced with an ultraviolet absorption detector. An evaporative light scattering detector may also be used to monitor the concentration of each eluted sample, but such devices may require special calibration because their response is generally non-linear.

Fitting Data and Extracting Molecular Parameters There are two basic ways to fit light scattering and concentration data to extract M _w , <r _g ² >, and A ₂ from equation (8). The first method is based on Jim's graphical method and the functional form of Equations (3) and (8). It consists of fitting a subset of the data for substitution into the zero scattering angle and zero concentration. First, each angle data is fitted to the following equation as a function of κ _j .

In the formula, a _{l, k} is a fit parameter, and N _l is the order of density fit. When N _l = 1, a linear fit is performed, and when N _l = 2, a quadratic fit is performed. Note that a _{0, k} is the substitution of the kth detector angle into the zero concentration. Next, the data of each injection solution is fit to the following formula.

In the equation, b _{j, m} is a fitting parameter, and N _m is the degree of angle fitting. Note that b _{j, 0} is substitution of data into the zero scattering angle. Finally, according to Jim, the fit factor is then fitted to the following formula:

In the equation, u _m and v _l are fit coefficients. They are related to molecular properties by the following formula:

  Fitting the measured data into the form of equations (24), (25) and (26) may be done in a statistical sense, so that the data used to perform these fits are It should be clear that it may be measured by the measured standard deviation. These are standard techniques and do not require further explanation.

Equation (27) implies that M _w can be computed from both angular and concentration fits. The two methods should match, but in the presence of experimental noise they are often slightly different. This can be used as a consistency check to determine the accuracy of the method. The average amount can be defined as follows:

Furthermore, the fit described in equations (25) and (26) can be better visualized by a plot similar to the one presented by Jim. These are referred to as Trenoff-Wyatt plots to distinguish them clearly from Jim plots. To create a TW plot, the data and fits in equations (25) and (26) are plotted on the abscissa and sin ² (θ _k / 2) + kκ _j on the ordinate as shown in FIG.

Graph by plotting. The value k is called “elongation factor” and is selected so that data from different infusates do not partially overlap. It only affects the scale of the plot, not the parameters determined by the fit.

  The second method is to perform a comprehensive fit to the entire data set. This is different from the Jim method, which performs a series of fits on a subset of the data. Jim's method was developed in the 1940s without the use of numerical fitting software. A comprehensive method is to define a fit function that models the data and perform a non-linear least squares fit of the data to the fit function using a standard algorithm such as the Marquardt method. The fit model is shown below.

_Where a _lm is the fit parameter and N _l and N _m are the fit orders of the angle fit and κ fit, respectively. The fit parameter is related to the physical quantity by the following equation.

  A TW plot may also be created from the results of the global fitting method. The only difference is that in this case the fit from equation (31) is used. Note that, unlike the above-described fitting method, the molar mass is clearly calculated in the equation (32).

Method Example To demonstrate the usefulness of this method, a measurement of molecular parameters of standard polystyrene dissolved in toluene is presented. Samples from Pressure Chemical Corporation have a molecular weight of about 200 kD and are known to have a linear random coil structure. Furthermore, this sample is very nearly monodispersed. It has a polydispersity of less than 1.01, indicating that there is no substantial amount of aggregate.

Two samples were used to characterize this sample. The first method is the traditional gym method. A parent sample was prepared at a concentration of 1.808 mg / ml. A series of known dilutions were prepared by diluting the parent sample. The resulting concentrations were 1.808 mg / ml, 1.4582 mg / ml, 1.0703 mg / ml, 0.7342 mg / ml, 0.3745 mg / ml, and 0.1879 mg / ml. These were injected using a 500 μl injection loop such that the MALS instrument flow cell was overfilled. A protective column was used to remove dust and separate dissolved gas from the sample. The raw signal 11 from the 90 ° light scattering detector is shown in FIG. The plateau was clearly visible and a narrow range of data on the plateau for each peak was averaged and used to create the Jim plot shown in FIG. The measured amount is M _w = 2.32 × 10 ⁵ g / mol,

And A ₂ = 4.84 × 10 ⁻⁴ molml / g ² . The data is high quality but requires a large amount of sample.

The second method is the subject of this invention . The same sample solution was injected, but with a 107 μl injection loop, the flow cell was not completely filled and the plateau was not achieved
I made it . Data from one injection is shown in FIG . It indicates that the 90 ° light scatter signal 11 is superimposed on the DRI signal 12 . The DRI signal is timed to compensate for the delay introduced by the inter-detector volume . Both are baseline subtracted . In addition, for clarity of explanation, the DRI peak has been scaled to 90% of the light scattering peak . From each infusion solution

, C _j , and κ _j are computed without correcting for band broadening between detectors . The resulting TW plot using the Jim Fit method is shown in FIG . Line 13 is a fit defined by equation (26), and line 14 is a fit defined by equation (25) . The results obtained are: M _w = 2.10 × 10 ⁵ g / mol,

, And was _{^{A 2 = 5.13 × 10 -4 molml}} / g 2.
Next, fitting the same data using the global fit method as shown in FIG. 6 produced the same results for the molecular parameters . Finally, the band extension of κ _j is corrected using the Gaussian approximation of equation (20) . The resulting TW plot is substantially the same as FIG . The derived molecular parameters are M _w = 2.10 × 10 ⁵ g / mol,

, And _{^{A 2 = 5.13 × 10 -4 molml}} / g 2. Band extension correction is M _w and

Has a negligible effect on it, but it changes A ₂ by about 10% . The Gaussian approximation result is within 3% of the traditional Jim plot method .

Finally, the result of the Gaussian approximation to the band extension correction between the detectors is compared with the result of the calibration method. In the above analysis, the Gaussian approximation was computed separately for each peak. Assuming that the extension is one system parameter and is the same for each peak, the average value D ^m /D=1.0099 can be computed. From the calibration method, it can be seen that D ^m /D=1.060. The two methods are therefore consistent within 4%, which is within experimental error.

  As will be apparent to those skilled in the art of light scattering, there are many obvious variations of the method that has been invented and described that do not depart from the fundamental elements listed for practice. All such variations are quite obvious implementations of the present invention described so far and are included by reference to the claims.

FIG. 3 is a diagram of the main chromatographic element connections of a preferred embodiment of the present invention. FIG. 4 is a diagram of a 90 ° light scattering signal for a series of infusates used to construct a Jim plot of 200 kD polystyrene in toluene shown in FIG. This shows the plateau achieved with each infusate and the range of samples averaged together. The vertical axis is the raw detector voltage minus the solvent baseline voltage. FIG. 2 is a traditional Jim plot of a narrow 200 kD standard polystyrene dissolved in toluene. Samples were made by serial dilution of the parent solution. FIG. 6 is a diagram of a single infusion used in the Trenoff-Wyatt plot shown in FIG. Trace 11 is a raw 90 ° light scatter signal. Trace 12 is a DRI signal. It has been scaled so that its peak is 90% of the light scattering peak, and timed to correct the interdetector delay volume. The sagging feature in the DRI signal is due to the dissolved gas in the sample, which is separated by a protective column. Some differences in peak shape are due to band expansion between detectors. Note that it does not stay constant. This is one of the six infusions used to create FIG. It is a figure of the Trenoff-Wyatt plot using the Jim fit method. Line 13 on the left of the plot is a projection of the data for each infusate to the zero scattering angle. The slope of this projection determines the A _2. The bottom line 14 of the plot is a projection of the data for each detector to zero κ. The slope of this projection determines <r _g ² >. The intersection of these two lines is 1 / _Mw . FIG. 4 is a diagram of a Trenoff-Wyat plot using a global fit method. The same data used in FIG. 5 was used.

Claims (9)

A method for determining a weight average molar mass M _w , a mean square radius <r _g ² >, and a second virial coefficient A ₂ of an unfractionated sample, comprising:
A) preparing a series of n dilutions of the sample in a suitable solvent means, the dilutions spanning a range of concentrations of about 10 times each other , the method further comprising:
B) providing a solvent means tank;
C) providing pump means for causing the solvent means to flow sequentially through the multi-angle light scattering detection means and the concentration detection means;
D) sequentially injecting an aliquot of each of the n sample diluents;
E) During each elution of the aliquot , as it passes through the multi-angle light scatter and concentration detector means, the multi-angle light scatter and Collecting and storing concentration data in a computer means;
F) From the collected concentration data value c _ij over the total concentration elution element ij of each of the sample dilutions j, n sums corresponding to the n sample injections are obtained by the computer means.

(Where j = 1 to n);
G) From the collected concentration data value c _ij over the total concentration elution element ij of each of the sample diluents j, by the computer means, n sums corresponding to the n sample injection solutions

(Where j = 1 to n);
H) calculating an excess Rayleigh ratio R _ij (θ _k ) from the collected light scattering data at each scattering angle θ _k for each sample diluent j and each elution i;
I) n total sums for each scattering angle θ _k and sample diluent j over the total light scattering elution element i of the sample diluent j by the computer means

Forming (j = 1-n);
J) forming the ratio D _j / C _j = κ _j ;
K) The ratio κ _j and the sum for all angles k and n sample dilutions

Deriving the weight average molar mass, the mean square radius, and the second virial coefficient of the unfractionated sample. The weight average molar mass, the mean square radius, and the second virial coefficient of the unfractionated sample are:
a)

The function by statistical means

Fitting , where N _θ is the maximum degree of angular variation fit and N _κ is the maximum degree of κ variation to be fitted;
b) then determining the coefficients a ₀₀ and a ₀₁ ;
c) The weight average molar mass

, The mean square radius

And the second virial coefficient

The method of claim 1, wherein the method is derived by the step of: The weight average molar mass of the sample that is not the fraction the mean square radius, and the virial coefficients are derived from the plot,
a) The data for each angle is a function of κ _j

( _Where a _{l, k} is the fit parameter and N _l is the order of the density fit)
b) Data for each infusion is

( _Where b _{j, m} is the fit parameter and N _m is the degree of the angle fit)
c) The fit factor is

(Where u _m and v _l are fit coefficients)
d) The weight average molar mass is determined from M _w = [1 / u ₀ + 1 / v ₀ ],
e) the second virial coefficient A ₂ = v _1/2 ;
f) The mean square radius

The method of claim 1, wherein The method according to claim 1, wherein the factor D _j of the ratio D _j / C _j = κ _j is corrected for band extension. The multi-angle light scatter detector means is replaced by a single angle light scatter detector means at a selected angle θ and the sum

The method of claim 1, wherein is formed for a single angle accordingly.   6. A method according to claim 5, wherein the single angle light scatter detector means has an angle [theta] = 90 [deg.].   The method of claim 1, wherein the concentration detector means is a differential refractive index (DRI) detector.   The method of claim 1, wherein the concentration detector means is an ultraviolet (UV) absorption detector.   The method of claim 1, wherein the concentration detector means is an evaporative light scattering detector.

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