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TitleMolecular Weight of Polymers
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Table of Contents
                            Polymer Chemistry
	Table of Contents
	Chapter 3. Molecular Weight of Polymers
		INTRODUCTION
		SOLUBILITY
		AVERAGE MOLECULAR WEIGHT VALUES
		FRACTIONATION OF POLYDISPERSE SYSTEMS
		CHROMATOGRAPHY
		OSMOMETRY
		END-GROUP ANALYSIS
		EBULLIOMETRY AND CRYOMETRY
		REFRACTOMETRY
		LIGHT SCATTERING MEASUREMENTS
		ULTRACENTRIFUGATION
		SMALL-ANGLE X-RAY SCATTERING
		MASS SPECTROMETRY
		VISCOMETRY
		SUMMARY
		GLOSSARY
		EXERCISES
		BIBLIOGRAPHY
                        
Document Text Contents
Page 1

3
Molecular Weight of Polymers

3.1 INTRODUCTION

It is the size of macromolecules that gives them their unique and useful properties. Size
allows polymer chains to act as a group so that when one part of the chain moves the
other parts are affected, and so that when one polymer chain moves, surrounding chains
are affected by that movement. Size allows memory to be imparted, retained, and used.
Size allows cumulative effects of secondary bonding to become dominant factors in some
behavior. Thus, the determination of a polymer’s size adds an important factor in under-
standing its behavior.

Generally, the higher the molecular weight, the larger the polymer. The average
molecule weight (M) of a polymer is the product of the average number of repeat units
or mers expressed as n̄ or DP times the molecular weight of these repeating units. M for
a group of chains of average formula (CH2 CH2)1000 is 1000(28) � 28,000.

Polymerization reactions, both synthetic and natural, lead to polymers with heteroge-
neous molecular weights, i.e., polymer chains with a different number of units. Molecular
weight distributions may be relatively broad (Fig. 3.1), as is the case for most synthetic
polymers and many naturally occurring polymers. It may be relatively narrow for certain
natural polymers (because of the imposed steric and electronic constraints), or may be
mono-, bi-, tri-, or polymodal. A bimodal curve is often characteristic of a polymerization
occurring under two distinct pathways or environments. Most synthetic polymers and
many naturally occurring polymers consist of molecules with different molecular weights
and are said to be polydisperse. In contrast, specific proteins and nucleic acids, like typical
small molecules, consist of molecules with a specific molecular weight (M) and are said
to be monodisperse.

Since typical small molecules and large molecules with molecular weights less than
a critical value (Z) required for chain entanglement are weak and are readily attacked by
appropriate reactants, it is apparent that the following properties are related to molecular

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Page 2

Figure 3.1 Representative differential weight distribution curves: ( |�|�|�|�|�|�) relatively broad
distribution curve; (—o —o —o —o ) relatively narrow distribution curve; (————) bimodal distribution
curve.

weight. Thus, melt viscosity, tensile strength, modulus, impact strength or toughness, and
resistance to heat and corrosives are dependent on the molecular weight of amorphous
polymers and their molecular weight distribution (MWD). In contrast, density, specific
heat capacity, and refractive index are essentially independent of the molecular weight at
molecular weight values above the critical molecular weight.

The melt viscosity is usually proportional to the 3.4 power of the average molecular
weight at values above the critical molecular weight required for chain entanglement, i.e.,
� � M3,4. Thus, the melt viscosity increases rapidly as the molecular weight increases and
more energy is required for the processing and fabrication of these large molecules. How-
ever, as shown in Fig. 3.2, the strength of polymers increases as the molecular weight
increases and then tends to level off.

Thus, while a value above the threshold molecular weight value (TMWV; lowest
molecular weight where the desired property value is achieved) is essential for most practi-
cal applications, the additional cost of energy required for processing extremely high
molecular weight polymers is seldom justified. Accordingly, it is customary to establish
a commercial polymer range above the TMWV but below the extremely high molecular
weight range. However, it should be noted that since toughness increases with molecular
weight, extremely high molecular weight polymers, such as ultrahigh molecular weight
polyethylene (UHMPE), are used for the production of tough articles such as trash barrels.

Oligomers and other low molecular weight polymers are not useful for applications
where high strength is required. The word oligomer is derived from the Greek word oligos,
meaning “a few.” The value for TMWV will be dependent on Tg, the cohesive energy
density (CED) of amorphous polymers (Sec. 3.2), the extent of crystallinity in crystalline
polymers, and the effect of reinforcements in polymeric composites. Thus, while a low
molecular weight amorphous polymer may be satisfactory for use as a coating or adhesive,

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Page 24

For polymer solutions containing polymers of moderate to low molecular weight, P� is 1
and Eq. (3.19) reduces to

Hc



1

Mw
(1 � 2Bc � Cc2 � . . .) (3.20)

Several expressions are generally used in describing the relationship between values
measured by light scattering photometry and molecular weight. One is given in Eq. (3.20)
and the others, such as Eq. (3.21), are exactly analogous except that constants have been
rearranged.

Kc/R � 1/Mw (1 � 2Bc � Cc
2 � . . . (3.21)

At low concentrations of polymer in solution, Eq. (3.21) reduces to an equation of
a straight line (y � b � mx):

Hc



1

Mw


2Bc

Mw
(3.22)

When the ratio of the concentration c to the turbidity � (related to the intensity of scattering
at 0 and 90�) multiplied by the constant H is plotted against concentration, the intercept
of the extrapolated curve, is the reciprocal of Mw and the slope contains the virial constant
B, as shown in Fig. 3.15. Z0 is directly related to P�, and both are related to both the size
and shape of the scattering particle. As the size of the polymer chain approaches about
one-twentieth the wavelength of incident light, scattering interference occurs giving a

Figure 3.15 Typical plot used to determine M�1w from light scattering data.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

Page 25

scattering envelope that is no longer symmetrical. Here the scattering dependency on
molecular weight reverts back to the relationship given in Eq. (3.19), thus, a plot of Hc/
� vs. c extrapolated to zero polymer concentration gives as the intercept 1/MwP�, not 1/Mw.
The molecular weight for such situations is typically found using one of two techniques.

Problem

Determine the apparent weight-average molecular weight for a polymer sample where the
intensity of scattering at 0� is 1000 and the intensity of scattering at 90� is 10 for a polymer
(0.14 g) dissolved in DMSO (100 mL) that had a dn/dc of 1.0.

Most of the terms employed to describe H and � are equipment constants, and
their values are typically supplied with the light-scattering photometer and redetermined
periodically. For the sake of calculation we will use the following constant values: K′ �
0.100 and � � 546 nm. DMSO has a measured refractive index of 1.475 at 21�C.

� � K′ n2
i90
i0

� (0.100) (1.475)2 � 101000� � 2.18 � 10�3
At 546 nm H � 6.18 � 10�5n0

2 (dn/dc)2 � 6.18 � 10�5 (1.475)2(1.0)2 � 1.34 �
10�4.

Typically, concentration units of g/mL or g/cc are employed for light-scattering
photometry. For the present solution the concentration is 0.14 g/100 mL � 0.0014 g/mL.

Hc



1.34 � 10�4 � 1.4 � 10�3

2.18 � 10�3
� 8.6 � 10�5

The apparent molecular weight is then the inverse of 8.6 � 10�5 or 1.2 � 104. This is
called “apparent” since it is for a single point and not extrapolated to zero.

The first of the techniques is called the dissymmetrical method or approach because
it utilizes the determination of Z0 vs. P� as a function of polymer shape. Mw is determined
from the intercept 1/MwP� through substitution of the determined P�. The weakness in
this approach is the necessity of having to assume a shape for the polymer in a particular
solution. For small Zo values, choosing an incorrect polymer shape results in a small error,
but for larger Z0 values, the error may become significant, i.e., greater than 30%.

The second approach uses a double extrapolation to zero concentration and zero angle
with the data forming what is called a Zimm plot (Figs. 3.15 and 3.16). The extrapolation to
zero angle corrects for finite particle size effects. The radius of gyration, related to polymer
shape and size, can also be determined from this plot.

The second extrapolation to zero concentration corrects for concentration factors.
The intercepts of both plots is equal to 1/Mw.

The Zimm plot approach does not require knowing or having to assume a particular
shape for the polymer in solution.

Related to the Zimm plot is the Debye plot. In the Zimm approach, different concen-
trations of the polymer solution are used. In the Debye, one low concentration sample is
used with 1/Mw plotted against sin

2 (�/2), essentially one-half of the Zimm plot.
Low-angle laser light-scattering photometry (LALLS) and multiangle low-angle

laser light scattering photometry (MALS) take advantage of the fact that at low or small
angles the form factor, P�, becomes 1, reducing Equation 3.19 to 3.20 and at low concentra-
tions to 3.22.

A number of automated systems exist with varying capabilities. Some internally
carry out dilutions and refractive index measurements, allowing molecular weight to be

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Page 47

32. For solution to occur G must be. (a) 0, (b) �0, or (c) �0.
33. Will a polymer swollen by a solvent have higher or lower entropy than the solid

polymer?
34. Define the change in entropy in the Gibbs free energy equation.
35. Is a liquid that has a value of 0.3 for its interaction parameter (
i) a good or a poor

solvent?
36. What is the value of G at the � temperature?
37. What term is used to describe the temperature at which a polymer of infinite molecu-

lar weight precipitates from a dilute solution?
38. At which temperature will the polymer coil be larger in a poor solvent: (a) at the �

temperature, (b) above the � temperature, or (c) below the � temperature?
39. If � for water is equal to 23.4 H, what is the CED for water?
40. What is the heat of mixing of two solvents having identical � values?
41. If the density (D) is 0.85 g/cc and the molar volume (V) is 1,176,470 cc, what is

the molecular weight?
42. Use Small’s molar attraction constants to calculate � for polystyrene.

H H
� �

�(�C�C�)�
� �

H C6H5

43. Calculate M for a polymer having a � value of 10 H and a Tg value of 325K (M �
chain stiffness).

44. Why do � values decrease as the molecular weight increases in a homologous series
of aliphatic polar solvents?

45. Which would be the better solvent for polystyrene: (a) n-pentane, (b) benzene, or
(c) acetonitrile?

46. Which will have the higher slope when its reduced viscosity or viscosity number is
plotted against concentration: a solution of polystyrene (a) in benzene or (b) in
noctane?

47. What is the value of the virial constant B in Eq. (3.32) at the � temperature?
48. When is the Flory equation [Eq. (3.31)] similar to the Mark-Houwink equation?
49. What is the term used for the cube root of the hydrodynamic volume?
50. Explain why the viscosity of a polymer solution decreases as the temperature in-

creases.
51. Which sample of LDPE has the higher average molecular weight: (a) one with a

melt index of 10 or (b) one with a melt index of 8?

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