Molecular Weight Averages from Gel Permeation ... - ACS Publications

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Downloaded by UNIV OF MONTANA on January 25, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch013

Molecular Weight Averages from Gel Permeation Chromatography Using the Universal Calibration Method EDGAR NICHOLS

1

Gulf Oil of Canada Ltd., Montreal, Quebec, Canada

In GPC, the product [η]M, (or the hydrodynamic radius R ) has been widely accepted as a universal calibration parame ter. In the Ptitsyn-Eizner modification of the Flory-Fox equation the quantity Φ, which relates the dimensional pa rameters to the above product, is taken as a variable. The value of Φ depends upon molecular expansion in solution as represented by a function f(e). Because of this dependence polymeric species having the same [η]M value cannot have the same statistical dimensions (radius of gyration or end -to-end distance) unless they have the same e value. Thus, if [η]Μ is a universal calibration parameter, the statistical parameters cannot be used as such. A method is presented for obtaining the M /M ratio from GPC data even though universal calibration is used. e

w

n

T n G P C the product [ 7 7 ] M has been widely accepted as a universal * calibration parameter, where [ 7 7 ] is the intrinsic viscosity and M is the molecular weight. This product is defined by the Einstein-Simha viscosity expression (1) as J

[η]Μ

=

Φ β 0

β

(Ι)

3

where R is the radius of a hydrodynamic sphere and Φ is a constant having the value 6.308 Χ 10 (cgs units). Flexible polymers in dilute e

0

24

1 Present address: NL Industries, W . Caldwell, N. J. 07006

148 In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

13.

NICHOLS

149

Universal Calibration Method

solution behave as spheres of radius R = Ré, where R is the root-meansquare radius of gyration. The [η]Μ product for such polymers, then, is given by e

M M

= Φ ξ * R*

(2)

0

and for the end-to-end distance h, [η]Μ = Φ ξ 1 / ( 6 )


0

3

3

(3)

3 / 2

The value of ξ can be expressed by the Ptitsyn-Eizner relation (2) as ξ = [ξθ /(β)] 3

(4)

1/3

where ξβ refers to an ideal or theta solvent and f(e)

= 1-2.636 + 2.86e

(5)

2

The parameter ( e ) is the exponential dependence of a on M where a is the molecular-expansion coefficient. From the value of 2.86 Χ 10 found by Ptitsyn and Eizner for ΦΘ, it follows that ξβ = 0.875. The ratio (R /R)* = ( £ ) is thus implicit i n the value of Φ i n the Flory-Fox equation and has a value of 0.49, corresponding to the FloryFox Φ value of 2.1 χ 10 . It is clear from Equations 1, 2, and 3 that [ 7 7 ] M cannot be related to the statistical polymer dimensions h and R without a knowledge of ξ, i.e., Φ, which varies with solvent for a given polymer. It follows, that if all species having the same [ 7 7 ] M elute to gether from the G P C columns, then only R can be the universal_parameter, since ξ w i l l not be the same for all solute-solvent pairs and h and R w i l l not be equally correct for universal calibration. 2

23

3

e

23

e

Polydispersity It can be shown following the arguments of Newman et al. (3), that only the number-average molecular weight and hydrodynamic radius are valid when the Einstein-Simha viscosity expression is applied to whole polymers. Higher moments require a polydispersity factor. Thus,

(6)

(M)

n

(Re)*

(M)

u

In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

(7)

150

P O L Y M E R

M O L E C U L A R

W E I G H T

M E T H O D S

where q is a heterogeneity factor having the definition i n this case of {M)

L-J

w

m

Values of R can be calculated from Equation 6 for narrow fractions of known molecular weights and plotted as a function of V , the retention volume. and can then be calculated from the G P C molecular-size distribution curve for unknown whole polymers. T o obtain < M > / < M > from Equation 8 requires a value for q. B y use of both the appropriate value for ξ i n Equation 2 and the Mark-Houwink viscosity expression, one may write e

r

e

n

e


W

w

W

Φ (R)\

(9)

= Q K (M) «+i n

v

n

where Φ is equivalent to Φ ξ , Q is a heterogeneity factor, and K is a constant for a given polymer-solvent system and a = (3e + l ) / 2 . Thus one may write 3

0

/tf\3 W

n

V

=

QnKv(M)

1+a

n

(ξ) 6.308 Χ 10 3

24

v

;

F r o m solution theory it can be shown that K can be written as V

„ K

4.291 Χ 10**ξ*α*(Β/Μ ν*)*

,

ο

«

=

M^h

i

n

^1)

where (B/M ) is an intrinsic constant which w i l l be represented by A . Since a ~> M , equations 10 and 11 can be combined as m

0

e/2

(R)\

= Qn [ ^ ]

(12)

(M) *» N

where C is a temperature-dependent constant. It follows then that T

„ Γ»Ί - Γ^Ί Γ< >»Ί r< >»T iWnl ~ lo:l imvn] imi β

3

Μ

3/2

M

e/2

9

* n

(15)

and therefore, from Equation 13, q = D*

(16)


e/2

This means that the only effect of the q factor on the [77] - M relationship is that of the excluded volume due to coil expansion in a good solvent. It follows from Equations 8 and 16 that (17) It can be seen then that one need not use Equation 7 at all, since can be obtained from Equation 6 and < M > ^ / < M > from Equation 17, thus permitting < M > ™ to be calculated.. If one is dealing with an unknown whole polymer, Equation 6 permits the determination of < M > with no knowledge of solute-solvent interaction. To find < M > / < M > and hence requires a knowledge of (e) for the application of Equation 17. This is easily obtained from G P C provided two or more samples of a different molecular weight can be found. Equation 6 permits the determination of < M > for such samples from which the Mark-Houwink exponent (a) can be determined from the relationship in Equation 9, i.e., W

W

n

W

W

W

W

log (R)

n

=

(1 + «) log (M) ( 3 )

n

(18)

+ constant

Experimental The calibration was established with polystyrene standards sup plied by Waters Associates. The broad-distribution poly (vinyl chloride) standards were obtained from Arro Laboratories, Joliet, 111. Other P V C samples studied were obtained from Shawinigan Chemicals Division, Gulf O i l of Canada L t d . For all G P C analyses, small sample loads (ca 4 mg) and slow rates ca 0.76 m l / m i n ) were used to maximize resolution. Results and

Discussion

Using the method outlined above, values of (a), (e), and [èe f(e)] = ξ have been calculated for poly ( vinyl chloride ), poly ( vinyl acetate ), and polystyrene in tetrahydrofuran ( T H F ) at 25°C, and are shown in Table I. s


1/3

152

P O L Y M E R M O L E C U L A R WEIGHT METHODS Table I.

Viscosity P a r a m e t e r s — T H F , 2 5 ° C

Polymer PVC PS PVAc Table II.

a

e

ξ

0.76 0.70 0.64

0.179 0.134 0.090

0.748 0.778 0.805

A Comparison of Data Obtained for


From GPC and Equations 6 and 17 PVC Resin Code No.

(Re)n,

400-2 400-3 400-4

65.4 88.8 100.5

Ang.

(Re)n

83.2 113.0 126.5

1.27 1.27 1.26

Table III.

PVC

Resin A Β C D

w

(M)

n

25,100 41,000 53,600

.

(M)

n

2.6 2.6 2.45

65,100 106,500 131,500

Comparison of D a t a Obtained for Some from a M a r k - H o u w i n k

(R ) Ang. e

(M)

(Re)w

(Re)w)

Ang.

ny

67.5 73.60 86.0 90.0

(R,)., Ang.

(Β*)» obtained from the universal calibration curve and the G P C trace, the dispersity index D can be easily determined. W

n


13.

NICHOLS

Universal Calibration Method

153

Table I compares results obtained i n this manner with those reported by Arro Laboratories for 3 broad-distribution P V C standards. Table II does the same for several P V C commercial suspension resins from Shawinigan, using a Mark-Houwink expression as a basis for comparison of < M > values. W

P V C Standards by G P C and by Absolute Methods


By Light Scattering (L) or Osmometry (0)

(M) (0)

(M) (L)

25,500 41,000 54,000

68,600 118,000 132,000

n

M 25°C-THF,

ml/gram

W

70 113 125

P V C Suspension Resins by G P C and by Calculation Viscosity Expression From GPC and Equations 6 and 17

.

Molecular Weight Averages from Gel Permeation ... - ACS Publications

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