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MOLECULAR WEIGHTS AND THEIR CHARACTERIZATION
I. Molecular weight dispersion
A. Most bulk polymer samples have a variety of different sized molecules.
B. Ambiguity must be resolved to create a measure to compare samples.
II. Number-average molecular weight – Mn
W  Ni Mi
Mn 

A. Definition
N
 Ni
1. Mi – molecular weight for a specific total length of polymer molecules.
2. Ni – number of molecules with a specific molecular weight.
B. Measured with colligative properties, osmosis
III. Weight-average molecular weight – Mw
Mi Wi  Ni Mi2

Mw 

A. Definition
W
 Ni Mi
B. Measured with light scattering
IV. Z-average molecular weight – Mz
Ni M3i

Mz 
A. Definition
 Ni Mi2
B. Measured with ultracentrifugation
V. Viscosity-average molecular weight – Mv
A. Definition
Mv 
1 a
N M
N M
1 a
i
i
i
i
1. The value of “a” comes from Mark-Houwink equation:   KMav
a. Theoretically, a = 0.5 for random coil in  solvent.
b. a = 2.0 for a rigid rod.
2. “a” is an experimentally determined parameter
3. Therefore, Mv is not definition from first principles
B. Measured with viscometry
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VI. Polydispersity
A. The different molecular weights can be ordered according to their values.
Mn < M v < Mw < Mz
B. Often polydispersity is measured with the polydispersity index.
M
PDI  w
Mn
C. PDI can very from 1 to approximately 20.
D. Consequences of polydispersity
1. Increased solubility
2. Decreased strength
3. Degradation of polymer first occurs in low M molecules
Stolen from: http://openlearn.open.ac.uk/mod/resource/view.php?id=196637
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VII. Molecular Weight Separations
A. Fractionation
1. The solubility of polymer is inversely proportional to its molecular
weight.
2. Adding a nonsolvent to a polymer solution cause higher molecular weight
molecules to precipitate.
3. Relatively monodisperse fractions can be collected by carefully adding
increasing amounts of nonsolvent.
B. Gel Permeation Chromatography (GPC)
1. Also known as Size-Exclusion Chromatography (SEC)
2. Crosslinked polystyrene beads (Sephadex) have pore sizes that can be
controlled through processing conditions.
3. High pressure pumps force eluent through column.
4. Smaller molecules become entangled in pores and proceed slowly through
the column. Larger molecules move quicker through the column.
5. Fractionation is possible with a series of columns with Sephadex beads of
differing porosity.
VIII. Molecular Weight Characterizations
A. Osmometry
1. van’t Hoff equation
cRT

 Bc2 

Mn
 RT

 Bc 
c Mn
a. plot of  vs. c yields RT/Mn as the intercept
- find number average molecular weight at infinite dilution (ideal
solution)
cRT
 RT
 Bc 2 
 lim 
c

0
Mn
c Mn
- ideal solution also exists at theta temperature where B(T)= 0
- at θ-temperature, configuration of coil at random-walk.

b. c is concentration is g/L
2. very accurate and sensitive technique
3.
classical methods are slow (several weeks); however modern methods are
relatively quick (15 minutes)
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B. End group analysis
1. Use titrimetry to measure number of end groups
m
Mn 
n
2. End groups than can be titrated include carboxylate, hydroxyl, amino,
ethylene oxide, etc…
C. Ebulliometry, Cryometry, Vapor pressure lowering
1. Boiling point elevation, freezing point depression, vapor pressure lowering
are not very concentration-sensitive colligative properties
2. Rarely used in molecular weight determination.
D. Refractometry
1. Refractive index varies with concentration
2. Convenient for quick measurements
3. Need monodisperse standards to create calibration curve.
E. Light Scattering
1. Measure intensity of scattered light from a polymer solution over very
large angles.
2. Debye and Zimm formulated a theory of light scattering in polymer
solutions that relates to the osmotic pressure for polymer molecules
assumed to be spheres.
Hc
1   

 
R    RT  c 
3. The optical constant, H, is related to refractive index, n0, the refractive
index gradient with concentration, n c , and the wavelength of the
incident radiation, λ. NA – Avogadro’s number, 1 = 3.1416…
22 n 2  n 
H  1 40  
N A   c 
2
- the refractive index gradient depends on the polymer/solvent pair
- quality of the molecular weight determination depends on the quality of
refractive index gradient
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4. The Debye/Zimm theory becomes more sophisticated when the light
scattering intensities depend on the shape as well as the size of the
polymer molecule.

cRT
 Bc 2 
Mn

 RT

 2Bc 
c M n
Hc
1

 2Bc 
R    R  solvent  M w P   
- R(solvent) is a correction factor for scattering of the solvent molecules
5. Rayleigh ratio, R(), is the fraction of light scattered at angle  per unit
length.
i   r 2
R   
(for dilute gases)
I0 1  cos  
i() – intensity of scattered light per unit volume
r – distance away from scattering center
6. When the scattering is limited to small angle scattering the Rayleigh ratio
reduces to
3
R   
161
7.  is the turbidity from Lambert’s law (Beer’s law with turbidity)
I
 e l
I0
 - turbidity, l – length of cell
8. P() is called the scattering form-factor and is related to the radius of
gyration of the polymer particles.
P   


2  2 2
 R 2 K2
R K  1 e g 
4  g

R K 
4
g
K
41

sin

2
9. For small angles, P(), expands as a series to
P    1 
R g2 K 2
3
 1  R g2
1612

sin 2
2
3
2
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10. The basic light scattering equation for polymer solutions has two limits.
Hc
1

 2Bc 
R    R  solvent  M w P   
a. The zero concentration limit
1
 Hc 
lim 


c 0 R   

 Mw
 1  41 2 2 2 
1  
 R g sin 
2
 3   



i. for a specific angle, measure Hc/R for several concentrations of
polymer.
ii. use the Hc/R vs. concentration data to find the intercept (c = 0)
iii. make several other plots for different angles and find intercepts.
iv. plot the intercepts as a Hc/R vs.  plot to find weight-average
molecular weight and radius of gyration.
b. The zero scattering angle limit
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 Hc 
lim 

 2Bc 

0 R   
M
w


i. for a specific concentration, measure Hc/R for several small
angles of scattering of polymer.
ii. use the Hc/R vs. angle data to find the intercept ( = 0)
iii. make several other plots for different concentrations and find
intercepts.
iv. plot the intercepts as a Hc/R vs. c plot to find weight-average
molecular weight and radius of gyration.
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 = 45
 = 60
 = 75
 = 90
 = 105
 = 120
 = 135
 = 150
Kc/R() 106
 = 0
11. Zimm Plot
a. The two separate extrapolation processes above (c = 0,  = 0) can be
simplified into a single double-extrapolation plot called a Zimm plot.
c = 0.0006 g/cm3
c = 0.0005 g/cm3
c = 0.0004 g/cm3
c = 0.0003 g/cm3
c = 0.0002 g/cm3
c = 0.0001 g/cm3
c = 0 g/cm3
100c + sin2(/2)
b. The plot is made by plotting Hc/R vs. the sum of concentration and
angle, (sin2(/2)). An appropriate scale factor for the concentration
must be used to ensure that all data plotted can be seen. (Otherwise
data points may overlap. Thus the need for an “appropriate” scale
factor.)
c. With the Zimm plot, the intercept yields the number-average molecular
weight, whereas, two slopes can be found from the double extrapolation
to yield the radius of gyration and the second virial coefficient.
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F. Dynamic Light Scattering
1. Scattering off of polymer particles causes an interference pattern.
a. Scattering function
G    1  e
2Dq 2

.
D – diffusion constant, q – wavevector of incident light
2. Diffusion of particles causes interference pattern to change.
a. Stokes-Einstein relationship
D
kT
(assuming hard sphere model)
6R
b. Delay time
Dq 2   1
3. Molecular weight is related to the size of the polymer particle, R.
4. Technique can be used to explain morphological changes of polymer
particles.
G. Viscometry
1. Definitions
a. Relative viscosity, r

.
0
0 – viscosity of pure solvent.
r 
b. Specific viscosity, sp
sp 
  0
.
0
c. Reduced viscosity, red
red 
sp
c
.
d. Inherent viscosity, inh

inh  ln  r
 c
e.

.

Intrinsic viscosity, []
 sp 
.
c


  lim

c 0
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2. Viscosity of dilute polymer solutions with constant concentration increase
with increasing MW.
  K  M w 
a
– Mark – Houwink equation
a. Because K constants can’t be determined theoretically, viscometry is
useful for relative measurements or measurements against standards.
i. Often MW standards (measured with other methods such as light
scattering) are used to create calibration curves.
ii. Viscometry yields very quick determinations, thus the bother of
creating standard solutions may have a high benefit.
b. The shape of the molecule significantly impacts the viscosity of the
polymer solution by affecting the exponent in the Mark – Houwink
equation.
i. a = 0.5
ii. a = 0
iii. a = 2
random coil
hard spheres
rigid rods
3. Volume of molecule is considered in the Flory equation.
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  
  r02  2 
M
r0 – end-to-end distance in  solvent
3
 r02  2 – hydrodynamic volume
 – Flory expansion factor
 – empirical fitting parameter
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H. Ultracentrifugation
1. Sedimentation
a) Forces
i) Gravitational
Fg  mg
m – mass of polymer particle
ii) Buoyancy
Fb  mv2g
v 2 – specific volume of polymer particle
 - density of solvent
ii) Friction force
Ff  fu
u – velocity of particle
b) At terminal velocity of particle, sum of the forces equals zero.
Fg  Fb  Ff  mg  mv2g  fu  0
u
m 1  v2  g
f
i) Define sedimentation constant.
s
1  v2  u ter
f
g
ii) Unit of the sedimentation constant is the Svedberg.
iii) s has dimensions of time. 1 S = 10-13 s
2. Centrifugation
a) Method I (with sedimentation and diffusion coefficients)
i.) Centrifugation is similar to sedimentation replacing gravitational
acceleration with centrifugal acceleration. g  2 x
 – angular velocity
x – distance from rotation axis
1  v2  
u ter
f
2 x
m 1  v2  2 x
u
f
s
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ii) The Stokes-Einstein equations, which examine viscosity
theoretically from a microscopic perspective yield an expression
for the frictional force on a smooth sphere in terms of the thermal
energy and the diffusion coefficient.
Ff 
kTu
kT
 fu  f 
D
D
iii.) Substitute for f.
m 1  v2  2 x
f
u
f
u
 m
 2 
s
f
1  v2  x 1  v2
kTs
RTs
m
 Mz 
D 1  v2 
D 1  v2 
iv.) Spinning causes a relatively distinct boundary to occur between
the high concentration portion and a low concentration portion.
v.) This boundary moves toward the cell bottom over time.
dx
xb
t
 x 
u ter
dx
2
dt
s 2  2
  s  dt  
 2st  ln  b 
x 
x x
x
0
x b ,i
 b,i 
vi.) Centrifugation is similar to sedimentation replacing gravitational
acceleration with centrifugal acceleration. g  2 x
vii.) Thus a plot of ln(xb) versus time yields the sedimentation constant.
1 d ln x b
s 2

dt
viii.) From the sedimentation constant and a known diffusion constant,
the z-average molecular weight can be determined.
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b) Method II (with sedimentation coefficient only)
i) Determination of the diffusion coefficient can be difficult; thus a
method without making a diffusion constant determination can be
preferred.
ii.) The flux of polymer particles away from the rotation axis due to
centrifugal forces is in quasi-equilibrium with the flux of particles
moving toward the rotation axis due to diffusion. (Quasiequilibrium implies that the boundary is changing very slowly in
comparison of the time scale of the fluxes.)
 diff   cent.
cent.  uc
dc
 diff  D
dx
dc
 D
 uc 
dx
u dc 1

D dx c
iii.) Substitute for the diffusion coefficient and terminal velocity into
the equation from the balance of forces above.
u
m 1  v2  2 x
f
m
 m
f
u
kT
u
 2 
 2
1  v2  x 1  v2  D  x
kT
u
kT
dc 1
 

2
2
1  v2  x D 1  v2  x dx c
Mz 
RT
dc 1

2
1  v2  x dx c
2
dc M 1  v2  

x dx
c
RT
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iv.) Integration of the equation yields a usable expression.
2
2
2
2
 c 2  M z 1  v2    x 2  x1 
dc M 1  v2  

x
dx

ln

 
c

RT
2RT
 c1 
v.) Clever substitution leads to a linear plot where the molecular
weight is found by measuring how the boundary changes with
concentration with a plot of ln c vs. x2.

c 
dc
  d ln c  ln c2  ln c1  ln  2 
c
 c1 
x 22  x12 

1
2
 x dx  2  d  x   2
2
d ln c M z 1  v2  

RT
d  x2 
 Mz 
d ln c
RT
2
2
d  x  1  v2  
I. MALDI-MS (Matrix – assisted laser deposition/ionization mass spectroscopy)
1. Polymer and “matrix substance” are dissolved in solvent.
2. Solution is deposited on probe and dried.
3. Matrix molecules absorb UV light.
a. Vibrational relaxation causes polymer to desorb from surface.
b. Desorbed polymer molecules are carried into mass spectrometer.