Part III: Polymer Characterization
- Chapter 6: Characterization of Molecular Weight
- Chapter 7: Polymer Solubility and Solution
- Chapter 8: Phase Transition in Polymer
Chapter 6: Characterization of Molecular Weight
• Average molecular weight
– Mn : number-average molecular weight
– Mw : weight- average molecular weight
– xn: no. avg. degree of polymerization
– xw: wt. avg. degree of polymerization
– Mo: Mw of monomer (or repeating unit)
– PI, MWD: polydispersity index = Mw / MN
Mw, Mn calculations
Mn = first moment =  C(M)M dM
 C(M) dM
Mw = 2nd moment =  C(M)M2 dM
 C(M)M dM
Definition of Mw, Mn
In integral form
Mn
c( M ) MdM

= First Moment =
 c(M )dM
c( M ) M dM

= Second Moment =
 c(M )dM
2
Mw
In discrete summation form
ni = mole fraction =
Ni
 Ni
wi = weight fraction =
nni i 11 
wwi i 11

Ni M i

N
M
N
M
W

i
i
i
M n   ni M i 

Ni
Mi ( i i ) i i i
Mn   niMi   ( i )  N i
  Ni
i  Ni
 Ni
2
N
M
 i i
2
M
N
i
i i
2
2 ni Mi i 
M

w
M

w
i
i
M w   w iMi   niMi 
 NiMi


 Ni
 Ni
Wi
i
 Ni
Wi
Wi
=
ni M i
 ni M i
Ex1. Measurements on two monodisperse fractions of a
linear polymer, A and B, yield molecular weights of 100 000
and 400 000, respectively. Mixture 1 is prepared from one
part by weight of A and two parts by weight of B. Mixture
2 contains two parts by weight of A and one of B.
Determine the weight- and number-average molecular
weights of mixtures 1 and 2
Solution. For mixture 1
1
NA 
 110 5
100000
2
NB 
 0.5 10 5
400000
NiMi 1105 105  0.5 105 4 105

Mn 

 2.0 105
1105  0.5 105
 Ni
Wi
1
2
5
Mw   ( )Mi  110  4 105  3 105
W
3
3

  

For mixture 2

 


  



5
5
5
5
2

10
10

0
.
25

10
4

10
2
5

1
.
33

10
NA 
 2 10 5 Mn 
2 10 5  0.25 105
100000
1
2
1
5
5
5
NB 
 0.25 10 5
M
w

1

10

4

10

2

10
400000
3
3

 

Ex2. Two polydisperse samples are mixed in equal weights.
Sample A has M n = 100 000 and Mw = 200 000.
Sample B has Mn = 200 000 and Mw = 400 000.
What are Mn and Mw of the mixture ?
Solution. First, let’s derive general expressions for
calculating the averages of mixtures:
W
Mn 
N
 Wi

 Ni
i
i
Where the subscript i refers to various polydisperse
components of the mixture.Now, for a given
component,
Wi
Ni 
Mni
Wi
Mn(mixture) 
 Wi / Mni 
i
i


  wx M x 

wx M x
i  i

i
Mw 

W
Wi


 i
  wx M x 
 x
i
Mwi 
Wi
 M W   Wi 
Mw(mixture) 
 
Mwi
W
  Wi 


wi
i
i
i
i
i
i
Where ( Wi / Wi ) is the weight fraction of component i in
the mixture. In this case,
Let WA =1 g and WB = 1 g. Then
WA  WB
11
Mn 

 133000
5
5
WA / M nA   WB / M nB  1/ 10  1/ 2 10

 

 WA 
 WB 
M W A  
M W B
Mw  
 W A  WB 
 WA  WB 
1
1
5
  2  10   4  10 5  300000
2
2
Note that even though the polydispersity index of each
component of the mixture is 2.0, the PI of the mixture is
greater, 2.25.
Determination of average molecular weight
• 2 catagories
(a) Absolute methods:
-Measured quantities are theoretically related to MW
Ex. Endgroup analysis (Mn)
Colligative property measurement (Mn)
Light scattering (Mw)
Ultracentrifuge (Mw)
(b) Relative methods:
-Measured quantities are related to MW
-but need calibration with one of the absolute methods
Ex. Solution viscosity (Mv)
Size-Exclusion Chromatography (MWD)
(a) Absolute methods:
-Measured quantities are theoretically related
to MW
A1. Endgroup analysis (Mn)
A2. Colligative property measurement (Mn)
A3. Light scattering (Mw)
A.4 Ultracentrifuge (Mw)
(b) Relative methods:
-Measured quantities are related to MW
-but need calibration with one of the absolute
methods
Ex.1 Solution viscosity (Mv)
Ex.2 Size-Exclusion Chromatography (MWD)
Solution viscosity (Mv)
Vis=a+bt
t = travel time
a,b = constants
Solution viscosity
 =  (S , T, polymer conc., no. of entanglements, M )
- measure using Ostwald type
Viscometer
Ublelohde type
Definition:
=
s
solution viscosity
= solvent viscosity
Specific viscosity SP
SP =  - S
S
r = relative viscosity
=  - 1 = r – 1
S
Reduced viscosity (normalized for conc.)
red = SP = (/S) – 1
C
C
get rid of entanglement effect by
reducing viscosity to zero conc.
Intrinsic viscosity
show effect of
Single polymer
coil to viscosity
[] =
(/S) – 1
c0
C
lim
ขึ ้นกับ coil dimension
= lim red
c0
[] 


MW of polymer in soln
polymer – solvent system
temp.
Get quantitative MW
fix solvent, temp.
Huggin’s equation
for r < 2 or (solution
< 2solvent)
sp
red =
= [] + k′[]2c
(Huggin’s equation)
c
where k′ is ~ 0.4 (for a variety of polymer – solvent system)
Advantage if
[] is known  can obtain relationship of red
Equivalent form of Huggin’s equation
inh = ln r = [] + k” []2c
c
where
inh = inherent viscosity
k” = k’ – 0.5
and conc.
Vis
1
2
conc.
0.1
0.5
[]
Ref: S.L. Rosen,JohnWiley & Sons 1993
(alternative definition of intrinsic viscosity)
[] =
lim   lim  ln( / s ) 
c0
inh
c0 

C


Relationship of [] vs. M
[for monodisperse sample of a certain MW]
เรี ยกว่า Mark-Houwink-Sakurada (MHS) relation
[]x = K(Mx)a
(0.5<a<1)
K, a  Look up inpolymer handbook at a specific temp.
1/ a
a

1/ a 
M
W
[

]
 x
 
x
Mv   


K
W
 


1/ a
a


M
n
M
 x x x 


n
M

x x


โดย 0.5 < a < 1, Mn<< Mv < Mw
1/ a
  n M (1 a ) 
x x


n
M


x x 

[]x = K(Mx)a
Ref: S.L. Rosen,JohnWiley & Sons 1993
Ex. Mv (viscosity average molecular weight)
• Example 1: PMMA, calculate Mv for mixture 1 and 2 in acetone at 30
oC
and compare with Mn and Mw (From experiment: a = 0.72)
Mixture 1:
1/ a




1 / 0.72
0
.
72
0
.
72
  wx  a 
2
1

M v   
  1x10 5
 4x10 5
 288,000
M x 

3
3

  W 

compare to : M n  200,000
M w  300,000
Mixture 2:
1/ a




1 / 0.72
0
.
72
0
.
72
  wx  a 
1
2

M v   
  1x10 5
 4x10 5
 187,000
M x 

3
3

  W 

compare to : M n  133,000
M w  200,000
Ex1. Measurements on two monodisperse fractions of a
linear polymer, A and B, yield molecular weights of 100 000
and 400 000, respectively. Mixture 1 is prepared from one
part by weight of A and two parts by weight of B. Mixture
2 contains two parts by weight of A and one of B.
•Example 1: PMMA, calculate Mv for mixture 1 and 2 in
acetone at 30 oC and compare with Mn and Mw (From
experiment: a = 0.72)
Solution viscosity terminology
Ref: S.L. Rosen,JohnWiley & Sons 1993
Last but Not Least!
Size-Exclusion Chromatography (MWD)
(or Gel Permeation Chomatography (GPC))
- หา Molecular weight + MWD รวดเร็ว
Porous particle
(gel)
“gel” – a cross linked polymer that is swollen by solvent
Unimodal = 1 peak
Bimodal =2 peak
“column”
large molecules come out first
big molecule
smallest come out last
small molecule
large molecules
come out first
small molecules
come out last
(go through interstices of the substrate pores)
Most common detector : differential refractometer
(measure refractive index difference)
Ref: S.L. Rosen,JohnWiley & Sons 1993
Ref: S.L. Rosen,JohnWiley & Sons 1993