The Chevron Approach to GPC Axial Dispersion Correction

Chapter 3

The Chevron Approach to GPC Axial Dispersion Correction Wallace W. Yau

Chromatography of Polymers Downloaded from pubs.acs.org by UNIV LAVAL on 07/12/16. For personal use only.

Chevron Chemical Company L L C , Kingwood Technical Center, 1862 Kingwood Drive, Kingwood, T X 77339-3097

Proper correction for axial dispersion in gel permeation chromatography (GPC, or SEC for size exclusion chromatography) has not been adequately addressed in the available commercial softwares today. In this paper, we will propose a new computer algorithm to address this problem. Our approach is built on the theoretical foundations previously described in the GPCV2 paper published by Yau, et. al (1), where GPCV2 stands for an improved version, or a 2nd version of the well-known Hamielec's broad -standard GPC calibration method (2). Most commercial GPC softwares provide the dispersion-correction capability of the Hamielec method. Very few of them have adopted the GPCV2 improvement. While the GPCV2 method is better, it still has its problems. GPCV2 corrects for the axial dispersion effect in the calculated molecular weight values of polymer samples, but it does not remove the axial dispersion effect in the reported molecular weight distribution (MWD) curve of the sample. Our computer algorithm at Chevron removes this discrepancy. In this paper, we will present the theoretical basis for two Chevron methods of GPC calibration and data processing.

With the 5-μπι and 10-μιη high-resolution GPC packings that are widely available nowadays, there has been a common tendency among GPC practitioners to disregard the effect of column axial dispersion or instrumental band-broadening problems in conventional GPC experiments. In many applications, this neglect of axial dispersion correction is quite justifiable. The reasons for the present author to revisit the subject of axial-dispersion correction are the following: (1) For the polyolefln work in the author's laboratory, samples with the high-MW-end of the MWD curves extending up to 10 million MW are not uncommon. In this case, in order to avoid shear degradation, there is little choice but to use larger particle size (20-μπι) columns and accept the low resolution condition of the GPC experiment. (2) There is a need for GPC software that corrects the sample MWD curve for the axial-dispersion effect. In industry, the overlay of MWD curves often is more useful for solving polymer problems than the tabulation of average-

© 1999 American Chemical Society

35

36 MW values. But, without a band-broadening correction, the MWD-overlay approach is affected by the column resolution changes that occur when columns deteriorate with time, or whenever an old column set is replaced. (3) Broad-standard calibration, when used improperly, can lead to unacceptably large MW errors even when high-resolution columns are used. Unexpectedly large errors in the MWD curves can also occur.


Figure 1 is a quick overview of the choice of GPC calibration algorithms commonly available in the commercial software today. These algorithms are briefly discussed below. In the discussions that follow, we use the word "standard" to represent the calibration standard of know molecular weight, and the word "sample" to represent a polymer sample with the molecular weight of which is being determined. If several narrow standards of known MW of the same chemical structure as the sample are available, the true calibration line for a GPC system [i.e., Μχ(ν) with the true calibration constants of Ό\ and D2] can be obtained experimentally by using the GPC peak positions of the narrow standards. The sample MW values calculated using this narrow-standard calibration are not corrected for the axial dispersion effect. But, these MW errors are often very small for high-resolution columns. These errors are also highly predictable and symmetrical, i.e., the Mw value will be slightly too high, and the Mn value will be slightly too low by roughly the same percentage. Although this method does not correct the sample MWD curve for axial dispersion, it would not cause gross errors of shifting the MWD curves to wrong MW regions, as it could in the case of the Hamielec method described below. The problem of the narrow-standard method is its limited applicability to just a handful of polymer types where there are narrow standards commercially available. The Hamielec broad-standard method uses the known Mw and Mn values of a broad standard and has the computer search for an effective calibration line [i.e., MJI(V) with the effective calibration constants of D f and D2'] from the experimental GPC elution curve F(v) of the broad standard. Since the experimental elution curve of the standard has been affected by axial dispersion, the Hamielec method partially compensates for the axial dispersion error if the sample is of the similar MW and MWD features as the standard. However, this method can lead to gross inaccuracy of sample MW values if the elution volume of the sample deviates substantiallyfromthat of the broad standard. The axial dispersion in the broad standard causes the Hamielec Μβ(ν) to rotate counter clockwisefromthe true calibration curve Μχ(ν). The pivot point is near the center of the broad-standard elution curve. To the left of this point, MJJ(V) is lower than Μχ(ν) and therefore will underestimate the sample MW. Conversely, for samples that elute later than the broad standard, the Hamielec method can grossly overestimate the sample MW. For the same reasons, this rotation of MJJ(V) can also cause erroneous shifts of the sample MWD curves to the wrong MW regions. Therefore, the MW errors caused by axial dispersion becomes a much more complex problem and a much less predictable challenge in the Hamielec method as compared to the narrow-standards method. Because of this reason, the axial dispersion problem of the Hamielec method is highly dependent on the

37 polydispersity or the Pd value of the broad standard [Pd is defined as the ratio of Mw/Mn]. The broader the standard the better, with less rotation of the MJJ(V) line. For


polyolefins, the use of broad standards having polydispersity value of 10 or more is

Calibration Methods

Mw, M n , Pd C^rre^ons

MWDoverlay JPjPeçisiQO

No

Poor

Limited to polymer samples of same structures as the std.

• Broad Standard -Hamielec method Mw=DiZF(v)e" M =Di'/ZF(v)e

Partial

Very Poor

Correction limited to samples of similar MWD as the standard

• Broad Standard - duPont GPCV2 Mw=D,e^IF(v)c-

Yes

Poor

• Narrow Standards - peak position calibration curve

D2,V

+D2,V

N

D 2 V

Remarks

Corrections apply to samples of any MWD

where, F(k)^,, = J l F ^ - G ( V -y)-dy and,

σ rtirc^ Figure 1.

Axial Dispersion Problem in Existing GPC Softwares.

38 possible. The situation is more difficult for analyzing condensation polymers like Nylon, Dacron, and so on, where the polymer polydispersity values are quite low, staying around 2 or so. The errors of the Hamielec method using a broad standard of polydispersity of 2 can be extremely high andrisky,unless all samples are of similar MW and MWD to that of the standard. For such cases, one should consider creating a broad standard of much higher polydispersity by blending together several standards of very different MW values.


The GPCV2 method uses mathematical formulations (see Figure 1) of Mw and Mn that have already accounted for the column axial dispersion effect and therefore allows the search for the true calibration curve Mj(v) and the true calibration constants of Ό\ and D 2 parameters, where the effect of axial dispersion is measured by the standard deviation σ of a narrow standard GPC elution peak. This GPCV2 method represents an attempt to minimize the counter-clockwise rotation of the broad-standard calibration curve, and thus to nurrimize the MW accuracy problems created by the calibration line rotation. The method provides the correction of the sample Mw, Mn and Pd values, but still provides no correction to the MWD curve. A glaring discrepancy exists in GPCV2 where the calculated sample MW values are now closer to the true values, but the calculated MWD display remains too broad as compared to the true MWD of the sample. This discrepancy is removed in the Chevron methods reported below. CHEVRON METHOD-1 Chevron Method-1 is a direct extension of the GPCV2 method by introducing one additional step of an "effective sample calibration search" for the M(v) line of every sample with its own calibration constants of the Di" and D 2 " parameters. The method is outlined in Figure 2. The thought behind Chevron-1 is as follows, referring to Figure 2. One can expect that, much like the behavior of the broad standard in the Hamielec method, every polymer sample would prefer to have its own effective MW-calibration curve that would be consistent with the sample's true MW values under the existing axial dispersion conditions. Therefore, for every sample, there will be a unique M(v) curve that will have its rotation and pivot point located justrightto produce the correct Mw and Mn values for the sample. The effect of applying this M(v) calibration line [which is less steep than the true calibration line Mj(v)] to the experimental F(v) elution curve of the sample would then act as if it is compressing the peak width of the F(v) curve into a narrower MW window, and therefore result in a sharper and narrower sample MWD curve. The result is that the axial dispersion effect on the sample F(v) curve is now removed from the calculated sample MWD curve. This correction of axial dispersion in sample MWD helps to make it more meaningful to compare sample MWD-overlay curvesfromGPC results obtained under different column resolution conditions. The first 3 steps in the Chevron-1 algorithm outlined in Figure 2 are the same steps that exist in GPCV2. Steps 4 and 5 are added here to achieve the desired effect of removing axial dispersionfromthe sample MWD curve.


39

Ο

20

40

80

100

80

120

Retention Volume (V) Calibration Method

Mw,M ,Pd Corrections

MWD-overlay Precision

Chevron-1

Yes

Better

N

Remarks Corrections apply to bell-shaped MWD and linear columns

Chevron-1 Algorithms: 1. Use narrow-standard calibration if available, or, obtain GPCV2 calibration linefrombroad standard. 2. Calculate Mv, expt-i and Mn, «ρηfromF(\%m ie. 3. Calculate Mv. true and Mn, true of the sample from: Mw.tmcF β ^ Mw.expt'l, and, Μν,ιπκΡ é ^ M .expfl . P

7

2

a 9 / 2

N

D

v

4. Search the sample M(v) function [every sample has its own unique M(v) =iEfc"* ] : MwHDrZF(v)e 2\ano\ MN=Dr/ZF(v)e 2" . 5. Calculate sample MWD curve from F(v) and M(v) of the sample. D

Figure 2.

+D

v

Chevron Method-1 for GPC Axial Dispersion Correction.

40


One may notice that the assumption of linear calibration is implicit in the Hamielec method, GPCV2, and also in Chevron-1. The linear calibration assumption is made to improve the precision of these methods, although, at the expense of the versatility of the methods. For instance, the Chevron-1 method described here is expected to work well with the usual bell-shaped GPC elution curves and linear columns. But, for samples with elution curves very different from the usual bell-shaped GPC curve, one may need to consider using the Chevron-2 method described below. The Chevron-2 method is more easily applicable to situations of non-linear calibration curve as well. However, it is all a matter of compromise: Chevron-2 is more versatile, but Chevron-1 gives better precision and is less affected by detector noises. CHEVRON METHOD-2

The linear calibration approximation used in Chevron-1 is not expected to work well, for example, with a bimodal MWD sample as illustrated in Figure 3. Each component in this bimodal elution curve would prefer to have its own "effective" calibration line. Since the pivot points of the two components are separated, there will be a discontinuous break of the M(v) curve in-between the two component peaks. Therefore, the shape of the overall "effective" MW-calibration curve can be quite complicated. Fortunately, the formulations developed in the GPCV2 paper (1) can be used to estimate the shape of these "effective" MW-calibration curves for samples of any MWD and GPC elution curve shapes. These formulations are shown in Figure 3, step 2 of the Chevron-2 algorithm, where Mw(v) and Mn(v) stand for the local weight-average and number-average MW values at any retention volume ν along the sample GPC elution curve. And, F(v) stands for the experimental GPC elution curve at the retention volume v, while F(v-D2a ) stands for the experimental GPC elution curve at a retention volume of (v-D2a ). 2

2

Same as the Chevron-1 method, Chevron-2 also has the effect of removing axial dispersionfromMWD by compressing the F(v) curve into a narrower MW region by way of the counter-clockwise rotation of the "effective" MW-calibration curve of the sample. However, in addition to narrowing down the overall MWD breadth of the MWD window, Chevron-2 also improves the resolution of the MWD curve. This resolution enhancement effect can be explained by a close examination of the bimodal example in Figure 3. The M(v) curve of this bimodal sample can be approximated by the MQ(V) curve calculated in step 3 of Chevron-2 algorithm. The geometric-average MQ(V) curve is situated inbetween the Mw(v) and Mn(v) curves shown in the graph. One sees that, this MQ(V) curve is flat on the two sides, but has a steep slope in the transition region between the two pivot points. We knowfromthe previous explanation of the Chevron-1 method that theflatslope at the two ends would have the effect of causing the narrowing of the overall MWD curve. The opposite is true in the middle overlapping part of the bimodal peaks where the MQ(V) slope is steep. In this middle portion, the spacing between the F(v) data points will be more stretched out to cause a valley to be developed in the middle of the sample MWD curve. All these manipulations are automatically taken care of by digital


41

Calibration Method

M , M , Pd Corrections

MWD-overlay Precision

Chevron-2

Yes

Best?

w

N

Remarks Corrections apply to samples of any MWD, possible with non-linear columns, has resolution-enhancement feature, but more sensitive to baseline noise than Chevron-1.

Chevron-2 Algorithms: 1. Use narrow-standard calibration(linear or non-linear) if available, or, obtain GPCV2 calibration line from broad standard. 2. For every sample. calculate: F(v) _F(v-zyn e -M (V) A/ (v) = β -Af (v) A/„(v) = r

W

2

r

F(v)

=Z W ^ r M

'

Λ

" '

2

F(y+D a )

T

2

=1+ £ [ P ( v ) / M (v)]} N

3. Calculate the geometric average molecular weight:

M {v)=QM (yYM (y)) G

w

N

4. Calculate sample MWD curve from F(v) and Mo(v) of the sample.

Figure 3.

Chevron Method-2 for GPC Axial Dispersion Correction.

42 computation following the step 4 of the algorithm using the equation (1) described below. Since we are no longer dealing with linear approximation, the local slope of the MQ(V) curve will have to be accounted for in step 4 to calculate the sample MWD curve. The desired y-axis value of a MWD curve would have to be calculated rigorously using the following equation:


logA* dV

F(y) S(v)

(i)

where S(v) is the slope of the log MQ(V) calibration curve at the retention volume v. Since S(v) is sensitive to the signal-to-noise level of the GPC detector, the log MQ(V) calibration curve needs be curve-fitted segmentary to improve the precision of the S(v) calculation under noisy detector conditions. DISCUSSION

In summary, the key issues this manuscript tries to address are the following: (1) Highprecision GPC-MWD overlay curves are more useful than the tabulated Mw, Mn, and polydispersity values for solving industrial polymer problems. (2) GPC column efficiency deteriorates, as the column axial dispersion effect becomes worse with time. Currently, it is very difficult to obtain good MWD overlay curves for samples that are GPC-analyzed on different days, in different weeks, or in different months. (3) Chevron-1 or -2 has the axial dispersion correction feature for the sample MWD curves to allow more meaningful comparison of MWD-overlay curves obtained at different times. Although Chevron-2 is more accurate than Chevron-1 and also adds some "resolutionenhancement" or "deconvolution effect" to thefinalMWD, it has its drawback of being more sensitive to detector noise than Chevron-1. Therefore, the choice of which method to use depends largely on the nature of the sample elution curve shape and the GPC detector signal-to-noise conditions. In theory, Chevron-2 is also applicable for nonlinear calibration curve by using the local calibration slope of D (v) at every retention volume. Such practice however will add more detector noise effects and more precision problems to the calculations when dealing with real experimental data. 2

What are presented in this paper are two theoretical models of GPC axial dispersion correction. The current models were developed based on a symmetrical Gaussian axial dispersion function of a constant σ-value for the axial dispersion standard deviation. The application of the methods to real data to study the effect of detector noise on the two methods will be included in a future report. Further complications may occur in real data applications, the σ-value can change with elution volume, especially at high molecular weights (and with low resolution columns!). Also, the axial dispersion function may sometimes deviate from Gaussian to have a skewed component. These problems are real

43 and they are very important issues that need be addressed for future improvements over the two current methods described in this paper.

REFERENCES 1. Yau, W. W., Stoklosa, H. J., and Bly, D. D., J. Polym. Sci., 1977, 21, p.1911.


2. Balke, S. T., A. E. Hamielec, A. E., LeClair, B. P., and Pearce, S. L., Ind. Eng. Chem., Prod. Res. Develop. 1969, 8, p.54.

The Chevron Approach to GPC Axial Dispersion Correction

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