Lecture 2
Molecular Weight (MW) and MW
Distribution
Chapter 5
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Objectives
• Molecular weight
• Number average
• Weight average
• Polydispersity Index (PDI)
• Properties that are molecular weight dependent
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Molecular Weight
• i: degree of polymerization (# of monomer units)
Mo: molecular weight of
Mi: molar mass of
• Typically all chains are not equally long but display a variation
• Monodisperse: equal chain lengths, specific to proteins
• Polydisperse: unequal length, specific to most synthetic molecules
• Therefore we need to define an “average” molecular weight
• number average,
• weight average,
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Molecular Weight Distribution
• Number average molecular weight:
Mn
Ni: the number of molecules of length (degree of polymerization) i.
Mi: the molecular weight of a polymer chain corresponding to a
;
and Mi=i * M0, where M0 is the molecular weight of monomer.
Wi: the total weight of a polymer chain corresponding to a degree of
polymerization i;
and
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Molecular Weight Distribution
• Closely related is the Number average degree of polymerization
M0 is the molecular weight of monomer.
Number average MW is directly measured by techniques like:
GPC, vapor pressure osmometry, viscosity, and end-group NMR
Some of these techniques will be discussed in the section after exam 1.
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Molecular Weight Distribution
• Weight average molecular weight:
Mw
Ni: the number of molecules of length (degree of polymerization) i.
Mi: the molecular weight of a polymer chain corresponding to a degree
of polymerization i;
and
, where M0 is the molecular weight of monomer.
Wi: the total weight of a polymer chain corresponding to a degree of
polymerization i;
and
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Molecular Weight Distribution
• Closely related is the Weight average degree of polymerization
M0 is the molecular weight of monomer.
Weight average MW is directly measured by techniques like:
light scattering, small angle neutron scattering, x-ray scattering,
sedimentation velocity
Some of these techniques will be discussed in the section after exam 1.
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Molecular Weight Distribution
• Viscosity average molecular weight: M v
æ
1+a ö
N
M
i
i
çå
÷
i=1
Mv = ç N
÷
ç åN M ÷
i
i ÷
çè
ø
i=1
N
1/a
a is a viscosity parameter
between 0.5 and 1
• z-average MW
Mv
N
Mz =
3
N
M
å i i
Mz
i=1
N
2
N
M
å i i
i=1
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Measure Viscosity of Polymers
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Exercise-1
• Calculate the following based on the molecular weight distribution
shown below:
• The number average molecular weight
• The weight average molecular weight
120
i
Ni Mi
7 20 749
8 70 856
9 110 963
10 90 1070
11 80 1177
100
80
60
40
20
0
7
8
9
10
11
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Polydispersity Index (PDI)
• PDI serves as a measure of the breadth of the distribution
Perfectly monodisperse
PDI = 1.0
Living polymerization
1.0 < PDI < 1.1
Monodisperse (general term)
1.0 < PDI < 1.1
Free radical polymerization
1.5 < PDI < 2.0
Polydisperse
Condensation polymerization
PDI ~ 2
PDI > 1.1
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Exercise-2
• Calculate the following based on the molecular weight distribution
shown below:
• PDI
Mz
W, g
i
Mi
Wi
1
12,500
0.6
2
17,500
1.9
3
22,500
2.7
4
4.1
W5
5
27,500
5
32,500
5.3
W4
6
37,500
6.9
7
42,500
7.8
8
47,500
9.6
10
W8
W7
W6
W3
W2
W1
M1 M2 M3 M4
10,000
M5
30,000
M6
M7
M8
50,000
Molecular weight, g/mole
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Solution
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Exercise-3
•
•
•
The single-parameter Flory distribution is given as
W(X)=X(lnp)2pX
where X represents the degree of polymerization and p represents the
fractional monomer conversion in a step-growth polymerization. Using this
equation, obtain expressions for the number-average and weight-average
degrees of polymerization.
•
•
HINT: The geometric series is given as.
a + ar + ar2 + ar3 + ... = a/(1-r)
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Solution
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Exercise-4
•
Given the following chain lengths for poly(ethylene terephthalate), determine the
number average molecular weight, weight average molecular weight, and
polydispersity index. The molecular weight of the PET repeat unit is 192 g/mol.
Fraction
N
n
1
1
12
2
1
25
3
1
175
4
1
186
5
1
192
6
1
194
7
1
199
8
1
200
9
1
202
10
1
212
Sum
10
Molecular
weight
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NM
NM2
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Molecular Weight Dependent Properties
1. Glass Transition Temperature
2. Surface Tension
G
  ( )T , P
A
Reversible work required to create a
unit surface area at constant T, P and
composition, n. A is the surface area.
   
Glass Transition Temperature
(Tg): Long- range, main-chain
cooperative motions.
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k
M
2/3
n
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Molecular Weight Dependent Properties
3. Viscosity
 ~ Mw
~ Mw
(M w  M C )
3.4
(M w  M C )
Entanglement,
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 ~M
3.4
w
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Molecular Weight Dependent Properties
3. Viscosity
• Mark-Houwink equation
æ N
ö
1+a
ç å N i Mi ÷
÷
M v = ç i=1N
ç
÷
N
M
çè å i i ÷ø
i=1
1/a
æ N a
ö
ç å Mi N i Mi ÷
÷
= ç i=1N
ç
÷
N
M
çè å i i ÷ø
i=1
1/a
æ N a ö
ç å M i wi ÷
÷
= ç i=1
ç W ÷
çè
÷ø
1/a
æ éh ù ö
= ç ë û÷
çè K ÷ø
1/a
N
éëh ùû =
å éëh ùû w
i
i=1
N
åw
i=1
i
wi
é
ù
= å ëh û
iW
N
i=1
i
wi = total weight of the mer
W = total weight of sample
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Solution viscosity
Quantity
Units
Common Name
𝝶
cP
Solution viscosity
𝝶s
cP
Solvent viscosity
𝝶r=𝝶/𝝶s
--
Relative viscosity
𝝶sp=(𝝶-𝝶s)/𝝶s=𝝶r-1
--
Specific viscosity
𝝶red=𝝶sp/c=𝝶r-1/c
dL/g
Reduced viscosity
𝝶inh=ln(𝝶r)/c
dL/g
Inherent viscosity
dL/g
Intrinsic viscosity
[𝝶]=lim 𝝶𝑟𝑒𝑑 = lim 𝝶𝑖𝑛ℎ
𝑐→0
𝑐→0
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Exercise-4
•
•
Example 5.4. The following data were obtained for a sample of PMMA, in
acetone at 30ºC:
𝝶r
c (g/100 mL)
1.170
0.275
1.215
0.344
1.629
0.896
1.892
1.199
For PMMA in acetone at 30ºC, 𝜂 = 5.83 × 10−5 𝑀𝑣
and 𝑀𝑣 for the sample and K
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Determine 𝜂
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Solution
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Summary
N
åN M
i
N
åW
i
i
• Molecular weight
M n = i=1N
= N i=1
• Number average
Ni
(Wi / M i )
å
å
• Weight average
i=1
i=1
• PDI
• Molecular weight dependent properties
N
N
2
• Glass transition temperature
N
M
å i i åWi M i
i=1
• Surface tension
M w = i=1
=
N
N
• Viscosity
NM
W
å
i=1
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i
i
å
i=1
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i