Mol Wgt
• Recognize the influence of molar mass (molecular
weight) on polymer properties.
• Understand the relationship between molecular weight and degree of polymerization .
• Understand the significance of number-average (M n and weight-average (M w
) molecular weights.
)
• Understand the nature of polydispersity .
• Recognize that the molecular weight of a polymer influences its physical properties.
X
• The molecular weight (M) of a homopolymer is the
sum of the masses of the repeat units (M polymer chain:
0
) in the
M = XM
0 where X is the degree of polymerization , the number of repeat units in the chain.
• The molecular weight of a copolymer is based on a weighted average of the masses of all repeat units
(M i0
, M j0
M k0
, etc.) in the polymer chain:
M
0 copoly = χ i
M i0
+ χ j
M j0
+ χ k
M k0
+ … where χ i
, χ j
, χ k
, etc. are the mole fractions of the repeat units.
2-1

Mol Wgt
• Most polymers are polydisperse — they contain more than one chain length.
• The average distribution of chain masses can be described in more than one way:
– M n
, the number-average molecular weight
– M w
, the weight-average molecular weight
– M z
, the z-average molecular weight
– M v
, the viscosity-average molecular weight
• M z
≥ M w
≥ M v
> M n
• Each value is determined by an aspect of polymer structure.
n = 110
Signal
Intensity n = 100 n = 120
H
C
H
C
H n n = 130 n = 140
9000 11000
M
13000
Adapted from K. Rollins et al., 1990 Rapid Commun. Mass Spectrom., 4, 355-359
15000
2-2

Mol Wgt
M n
• The number-average molecular weight (molar mass) of a polymer containing N i molecules of mass M i the arithmetic mean of the molar mass distribution: is
M n
Σ
N i
M i
= ————
Σ
N i
• M n determines the polymer’s colligative properties and tensile strength (= C
1
– C
2
/M n
).
• M n may be determined directly by end-group analysis , osmometry , ebullioscopy (bp elevation), and cryoscopy (fp depression).
M w
• The weight-average molecular weight (molar mass) is the sum of the products of the molar mass of each
fraction multiplied by its weight fraction (w i
).
In terms of w i
M w or numbers of molecules, M w is
=
Σ w i
M i
M w
Σ
N i
M i
2
= ————
Σ
N i
M i
• M w accounts for the distribution of molar mass in the polymer.
• M w may be determined directly by light scattering .
2-3

Mol Wgt
• The molecular weight distribution, or polydispersity index , is the ratio of the weight-average molecular weight to the number-average molecular weight:
M
PDI = ——
M w n
• The polydispersity index of a monodisperse polymer is 1.00.
• The polydispersity index increases as the polymer distribution broadens.
You have a polymer sample that contains the following molecules:
M, Da*
1,000,000
700,000
400,000
100,000
50,000
Total:
*Da = dalton, g/mol
N
2
5
10
4
2
23
What are M n
, M w
, and the polydispersity index?
2-4

Mol Wgt
M n
M n
Σ
N i
M i
= ————
Σ
N i
10,000,000 Da
= —————————
23
M n
= 435,000 Da
M, Da
1,000,000
700,000
400,000
N
2
5
10
100,000
50,000
4
2
Totals: 23
M w
=
Σ w i
M i w i
=
N i
M
————
Σ
N i
M i
M w
= 609,500 Da
N·M, Da
2,000,000
3,500,000
4,000,000
400,000
100,000
10,000,000 w
0.20
0.35
0.40
0.04
0.01
1.00
w·M, Da
200,000
245,000
160,000
4,000
500
609,500
M w
M w
Σ
N i
M i
2
= ————
Σ
N i
M i
6.095
× 10 12 Da 2
= ————————
1.000
× 10 7 Da
M w
= 609,500 Da
M, Da N
1,000,000
700,000
400,000
2
5
10
100,000
50,000
4
2
Totals: 23
M
PDI = ——
M w n
609,500 Da
PDI = ———————
435,000 Da
PDI = 1.40
N·M, Da
2,000,000
3,500,000
4,000,000
400,000
100,000
10,000,000
N·M 2 , Da 2
2.000
× 10 12
2.450
× 10 12
1.600
× 10 12
0.040
× 10 12
0.005
× 10 12
6.095
× 10 12
2-5

Mol Wgt
M z
• The z-average molecular weight (molar mass) is
M z
Σ
N i
M i
= ————
3
Σ
N i
M i
2
• M z is especially sensitive to the presence of high-MW chains.
• M z may be determined directly by sedimentation equilibrium (ultracentrifugation) and light scattering .
M v
• The viscosity-average molecular weight (molar mass) is :
Σ
N
M v i
M ( 1+a ) 1/ a i
= ——————
Σ
N i
M i where the exponent a (0.5 ≤ a ≤ 2.0) is determined by the polymer, solvent, and temperature.
• For typical polymers, M w
> M v
> M n
.
• M w
= M v when a = 1.
• M v may be determined indirectly by dilute solution viscometry .
2-6

Mol Wgt
RI
M n
= 195,322
M w
M z
= 220,715
= 259,299
PDI = 1.13
M n
= 181,986
M w
M z
= 260,091
= 417,392
PDI = 1.43
5.0
5.5
log M
6.0
6.5
Method
Static light scattering
Type*
A
M i w
Range, g/mol
> 100
Dilute solution viscometry
Small angle X-ray scattering
Size-exclusion chromatography
Membrane osmometry
Ebullioscopy, cryoscopy
End group analysis (titration)
Vapor phase osmometry
R
A
R
A
A
E
A v w n,w,z n n n n
> 200
> 500
> 1,000
> 5,000
< 20,000
< 40,000
< 50,000
Sedimentation equilibrium
Mass spectrometry
Dynamic light scattering
A
A
R w,z n,w,z z
< 1,000,000
< 1,500,000
< 10,000,000
*A = mass calculation requires no assumptions about polymer structure, E = mass calculation requires information about polymer structure, R = mass calculation requires information about polymer structure and polymer-solvent interactions.
Adapted from H.-G. Elias, “An Introduction to Polymer Science”, 2 nd Ed, 1997, p 31
2-7

Mol Wgt
Signal
Intensity
M w
= 11,980 (115n)
M n
= 11,980 (114n)
PDI = 1.01
M z
= 12,060 (116n)
Polystyrene
Oligomer
H
C
H
C
H n
9000 11000
M
13000
Adapted from K. Rollins et al., 1990 Rapid Commun. Mass Spectrom., 4, 355-359
15000
O
( CH
2
)
4
O − C
O
( CH
2
)
4
C − O m
( CH
2
)
4
O −
O
C
O
( CH
2
)
8
C − O n
Copolyester
Signal
Intensity
M n
= 3,940
M w
= 4,320
PDI = 1.10
M z
= 4,780
3000 4000 5000
M
6000 7000 8000
Adapted from M.S. Montaudo et al, 1998 Rapid Commun. Mass Spectrom., 12, 519-528
2-8

• M n
: Brittleness (resin versus elastomer), stress- strain properties
• M w
: Tensile strength, hardness
• M z
: Flex life, stiffness, melt viscosity
• M v
: Solution viscosity, extrudablility, molding properties
Mol Wgt 2-9