Axial Dispersion Correction for the Goldwasser Method of Absolute

Chapter 4

Axial Dispersion Correction for the Goldwasser Method of Absolute Polymer M Determination Using SEC-Viscometry n

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

Copolymers and polymer blends, especially where there is compositional drift across the sample molecular weight distribution (MWD), present serious problem to the absolute MW determination by conventional size exclusion chromatography (SEC or GPC for gel permeation chromatography) using only a single concentration detector, which is generally affected by the sample chemical compositional differences. Since sample specific refractive index increment (dn/dc) changes with chemical composition, MW determination of these kinds of samples also presents difficulty for SEC using an on-line light scattering detector. A complete MWD of polymer samples with compositional variances is difficult. However, the introduction of a SEC-viscometry method, known as the Goldwasser Method (1) makes it possible to determine a number-average molecular weight value (Mn) of polymer samples with complex compositional heterogeneity. The Goldwasser method is based on the SEC universal calibration principle. This unique SEC method uses only the SEC data collected on an on-line viscosity detector to determine the Mn of polymer, regardless of its chemical structural differences. The method requires no concentration detector. In this work, an axial dispersion correction factor for the Goldwasser Mn value is developed. The theoretical basis of this correction factor is derived from the use of a Gaussian instrument band-broadening function as the axial dispersion peak shape model. Substantial improvement in accuracy of the Goldwasser Mn value is expected with this correction factor. It is hoped that this improvement will help to make this important method more widely useful in the polymer characterization field. BACKGROUND (1) Universal Calibration (UC) Principle (assuming no axial dispersion) At any GPC elution volume (e.g. the j-th retention volume), all the i-th polymer molecules will co-elute at this j-th retention volume if,

44

© 1999 American Chemical Society Provder; Chromatography of Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1999.

45 Μ\ηΙ=Μ [η] 1

=

1

.^Μ{η\=Η

ί

(Ο

where M, [??], and H, are the polymer molecular weight (MW), intrinsic viscosity (TV), and hydrodynamic volume (HV), respectively. Or,

M,=^,

Or,

(2)

The observed intrinsic viscosity at the j-th GPC retention volume is the weighted-average of the intrinsic viscosity values of all the i-th molecules co-eluting at that retention volume: CΗ 1 J

Or,

*

Σ,ζ " Σ,ζ

Σ^,/Σ,^/Μ) ^

(A/.-[7D,=tf,

where H =H X

2

w (4)

= ...=#,=#,

(5)

What is important in equation 3 is that the proper molecular weight average to use with universal calibration methodology is the number-average molecular weight. Equation 4 is an important theoretical result developed by Hamielec and Ouano (2). Equation 4 and 5 shows the proper statistical averages of MW and IV that are required to treat the universal calibration result of heterogeneous polymer samples at every GPC retention volume, assuming no GPC axial dispersion. With axial dispersion, the assumption stated in Equation 5 can no longer to be true (see equation 8 in the later theoretical results section). Current Goldwasser Method (1)

M n.BulkSampU =

C

C

where, η$ρ\& the sample specific viscosity.

Provder; Chromatography of Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1999.

46 Equation 6 gives the Mn calculation in the conventional SEC-viscometry method with universal calibration, i.e., where the intrinsic viscosity is measured at each elution slice. The numerator in equation 6 is proportional to the sample weight injected and the *lsp value in the denominator is proportional to the excess pressure-drop of an online viscometer signal. Based on these observations, Goldwasser (1) derived equation 7 below to show that the bulk sample Mn value can be obtained by using only the viscosity detector signal, without the use of the concentration detector signal.

— SampleWeightlnjected M n.BulkSampU = ~ ; ~ — ViscosiiyDetectorSignal h Η

— : = M n.Goldwasstr

r

Under the condition of accurate sample weight injected and total sample recovery, a same Mn value is obtained either by the conventional SEC-viscometry of equation 6, or by the Goldwasser method of equation 7.

Comments: (1) Positive Features: this Mn determination is generally applicable to all polymer structural heterogeneity, that includes polymer branching, chain rigidity, polymer blends and copolymers with compositional and chemical structural differences, as long as that the GPC separation behaves normally and obeys the universal calibration principle. (2) The current Goldwasser method does not account for the reality of the GPC axial dispersion problem. The problem is that, due to mixing effects, not all molecules eluted at the same retention volume have exactly the same hydrodynamic volume. Equation 5 is valid only if there is no axial dispersion and non-size-exclusion effects.

THEORETICAL RESULTS Under axial dispersion circumstances, the hydrodynamic volume values of different molecules co-elute at the same SEC retention volume are no longer all of the same value. With axial dispersion, there will be local polydispersity in the hydrodynamic volume values within each individual SEC retention volume. The challenge is to question what hydrodynamic volume average should be used to account for the axial dispersion effects. The presence of axial dispersion affects the hydrodynamic volume. In order to understand how axial dispersion affects the Mn value calculatedfromthe SEC-viscometry data, it is important to understand first how axial dispersion affects the relationship of SEC hydrodynamic volume versus the retention volume. The mathematical derivations of these concepts are presented below in the four inter-connected sections.


(7)

47 (1) Generalization of UC Equation to Include Axial Dispersion and Non-SEC Perturbations

" If, ~ If, M,

H.

(8)

r

Σ(—) Or,

(Λ/„·[7]„), = ( # „ ) ,

(9)

where Ha which equals to the first part of equation 8 is defined as the number-average hydrodynamic volume of the polymer molecule. Unlike equation 4, equation 9 is more generally applicable for all cases where not all the i-th molecules eluting at the j-th retention volume have the same hydrodynamic volume, i.e., in equation 8, not all Hi = Hj. (2) Experimental Evaluation of

' (

c

H (v) n

11 ' (10)

.), = Η

Equation (10) provides a means to experimentally evaluate H (y) as a function of retention volume ν , through the use of the viscosity detector signal alone, without the use of the RI or other concentration detector signal. n

(3) Effect of Gaussian Axial Dispersion on

M (y) n

We let JF(v)and J^(v)to represent the true concentration and viscosity detector elution profile respectively, that has not been affected by axial dispersion. And, we let F(v) and F (v)to represent the experimental concentration and viscosity detector elution curve n

respectively, that has been affected by axial dispersion. And, we let G(v-y) to represent a Gaussian axial dispersion function having a standard deviation of σ for the axial dispersion sigma value, expressed in retention volume units. The effect of axial dispersion on the local Mn values at every retention volumes can be formulated as the following.


48 n

n

M (y)=the M value of polymer observedat the retentionvolume" ν" n

n

rW(y)G(y-y)dy

Cw(y)G(y-y)dy

J— 00

J— 00

fw(y).G(y-y)IM (y)dy

r^(y)G -

n

(v

y)

fy(y)G(v-y)dy r-°W(yy\nly)\iJiy-y) J-« H,(y)

£[w(y) bh)l G(v - y)dy

iy{y)G(y-y)dy

f > ) ' [ è ) ] G ( v - # ' ^ M H ^ [y,(y)G(v-y)dy

Η {ν) ηη

M »

>W,(y)-G(y-y) dy

(ii)

M »

The quantity H (v) is defined as the argument in the square brackets in equation 11. B y nTJ

applying the mathematical analogy of equation A-9 given in a previous paper (3), we have:

e

2

(12)

H (v) t

Therefore, H (v) nn

W.(v) W.(v)

^ (ν η

+

-e Ησ) 2

W)0

H,(v)

(13)

2

where H (v) represents the true linear U C calibration line not affected by axial dispersion, t

H (v)=H e- * t

x

H

v

(14)

(4) Effect of Gaussian Axial Dispersion on The Mn Value of The Bulk Sample The substitution of the Mn(v) expression of equation 13 into the definition of the bulk sample M n calculation, we have:


49

Μη.

J—00

Sample — "F(v)' I ^ A/„(v) ν)

>

-

i(W,Y 2

*>)

J

2

e

M

2

Λ

2

n.sample = £

- ^ < χ ) 2 2 2

2

=e

'

1

—β

N

dv

2

· Mn.Goldwasser

rf^-dv ]

~Η,{ν)

.


(15)

50

It is noted that the expression in the square bracket in equation 15 is the same expression (except that it is written in the integral notation here) for the M n value of the conventional SEC-viscometry, and the M n value of the Goldwasser method described previously in equation 6 and 7.

Or,

Mnjru.

= β

2

^

with, the proper correction factor = Ζ

-M .Gc4a^sr

(16)

.

(17)

n

2

+

-(Η σ) 2

2

Thus, 6 is the proper factor to correct the apparent or the Goldwasser numberaverage molecular weight to an axial-dispersion corrected or true number-average molecular weight obtainable from SEC-viscometry and universal calibration methodology.

DISCUSSION The result of this work (equation 16) indicates that, without axial dispersion correction, the current Goldwasser method of sample M n determination using GPC-viscometry would underestimate the true M n value for the sample. This same correction factor is also applicable to the M n value calculated directly in the conventional SEC-viscometry approach using universal calibration where intrinsic viscosity is measured at each elution slice. 2

+

-(Η σ) 2

2

It is interesting that the M n correction factor β is not a function of the sample elution curve shape. That means, the percentage error caused by symmetrical axial dispersion as approximated here by a Gaussian model will be the same for all samples, regardless of their M W D curve shape or their chemical or structural heterogeneity. The correction factor here is developed on the model of symmetrical Gaussian axial dispersion function and linear universal calibration curve. This correction factor is expected to work well in GPC experiments using linear columns with good column efficiency. It is possible to generalize the correction factor for non-linear U C calibration and skewed axial dispersion function if the need exists. To obtain proper correction of the M n value, it is important that the viscosity detector volume-offset should have been properly aligned with the concentration detector signal in the software, and thus, aligned with the position of the universal calibration curve. Improper volume-offset alignment in the software can affect the Goldwasser M n values. In fact, a small intentional shift of the viscosity GPC tracing to a shorter retention time will tend to adjust the Goldwasser M n values upward, closer to the true M n value. It has been shown that an intentional shift in viscosity detector signal to shorter retention is equivalent to the effect of correcting for symmetrical axial dispersion in GPC-viscometry (4).


51 ACKNOWLEDGMENT The author is deeply indebted to Professor S. T. Balke and Dr. T. H . Mourey who have kindly pointed out that they have also obtained the same Goldwasser M n correction factor by using a very different mathematical approach (5). It is the hope of this author that readers may still find the insights behind the mathematical derivations presented in this manuscript somewhat interesting and contributing to the better understanding of the problem. REFERENCES 1. Goldwasser, J. M., in Chromatography ofPolymers: CharacterizationbySEC and FFF; Provder, T., Ed.; ACS Symposium Series 521; American Chemical Society: Washington, D.C.; 1993, p.243. 2. Hamielec, A. E., and Ouano, A. C., J. Liq. Chromatogr., 1978, 1, p.111. 3. Yau, W. W., Stoklosa, H. J., and Bly, D. D., J. Appl. Polym. Sci., 1977, 21, p.1911, equations A7-A9. 4. Yau, W. W., in Proceedings of the International GPC Symposium '87, published by Waters Corporation, Milford, MA, 1987, p. 148. 5. Balke, S. T., Mourey, T. H., and Harrison, C. Α., J. Appl. Polym. Sci., 1994, Vol. 51, pp. 2087-2102.


Axial Dispersion Correction for the Goldwasser Method of Absolute

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