Chapter 3
Despair Inc.
POLYMERIZATION KINETICS REVIEW
Characterization of polymers: Length and time
scales are important
[Reproduced from G. M. Kavanagh and S. B. Ross-Murphy, “Rheological characterisation of polymer
gels”, Prog. Polym. Sci. 23, 533 (1998).]
Polymer solubility
• In solvent, other polymer(s) or plasticizer
• Non-Newtonian properties: Viscosity, rheology
• Need to understand the configuration,
conformation, and dynamics of macromolecules
– Statistical mechanics
• Series of mathematical models
• Tied to experiment (viscosity, light scattering)
Thermodynamics of Mixing: (Free energy)
DG = DH -TDS
For solution DG < 0
-TDS
Entropy > 0, so –TDS < 0
DH can be negative or positive
Polymers are
more sensitive to
DH
Spinoidal decomposition of a solution into two phases
• Quenching (rapid cooling) of solution
• Polymer rich & polymer poor phases
Careful removal of polymer poor phase “microporous”
Foams.
Any solution process is governed by the free energy relationship
DG = DH－TDS
DG＜0  polymer dissolves spontaneously
DS＞0  arising from increased conformational mobility of
the polymer chain
For a binary system,
D mix
DE1
DE 2
 Vmix [(
) (
) ] 1 2
V1
V2
1
2
DHmix: heat of mixing
M .W .
(

)
V1,V2: molar volumes
d
1, 2: volume fractions
DE1,DE2 : energies of vaporization
DE1/V1, DE2/V2: cohesive energy densities
1
2 2
Enthalpy of Mixing
 DHmix can be a positive or negative quantity
– If A-A and B-B interactions are stronger than A-B interactions, then
DHmix > 0 (unmixed state is lower in energy)
– If A-B interactions are stronger than pure component interactions,
then DHmix < 0 (solution state is lower in energy)
•
An ideal solution is defined as one in which the interactions between all
components are equivalent. As a result,

DHmix = HAB - (wAHA + wBHB) = 0
for
an ideal mixture
•
In general, most polymer-solvent interactions produce DHmix > 0, the
exceptional cases being those in which significant hydrogen bonding
between components is possible.
– Predicting solubility in polymer systems often amounts to
considering the magnitude of DHmix > 0.
– If the enthalpy of mixing is greater than TDSmix, then we know that
the lower Gibbs energy condition is the unmixed state.
DE1 2
1  (
)
V1
1
DE 2 2
2  (
)
V2
1
1, 2: solubility parameters
DHmix = Vmix(1－2)212
∴ (1－2)  0; DHmix  small
if (1－2) = 0  DHmix ＝ 0
then, DG = －TDS  DG ＜ 0
Under this condition, solubility is governed solely by entropy effects
i.e., For an ideal solvent  s  p
Cohesive energy density (DE/V; 2) 
● energy needed to remove a molecule from its nearest
neighbors.
● analogous to heat of vaporization per volume for a volatile
compound.
For 1 (solvent) 
calculated directly from the latent heat of vaporization (DHvap)
DH vap  RT 2
DE 2
1  ( )  (
)
V
V
1
1
For 2 (polymer) 
estimated from group molar attraction constants G
(due to their negligible vapor pressure)
G
● Consider intermolecular forces between molecular groups
● Derived from low M.W. compounds
● by Small  heat of vaporization
● by Hoy  vapor pressure measurements
Table 2.1. Representative group molar attraction constantsa
G[(cal/cm3)1/2mol-1]
Group
Small
Hoy
214
147.3
CH2
133
131.5
CH
28
85.99
-93
32.03
126.5
CH3
C
CH2
CH
190
19
C6H5 (phenyl)
735
CH
CO2
117.1
(aromatic)
C O (ketone)
(ester)
84.51
275
262.7
310
326.6
a
Values taken from Refs. 5 and 6.
Which set one uses is normally determined by the method used for determining 1
for the solvent.
 
G d  G

V
M
(V: molar volume, d: density, M: MW of repeat unit)
CH 2 CH
Small’s system  
C6H5
Hoy’s system
 
1.05(133  28  735)
 9.0
104
1.05[131.5  85.99  6(117.1)]
 9.3
104
For taking into account the strong intermolecular dipolar forces,
solubility parameters  d (dispersion forces)
p (dipole-dipole attraction)
H (hydrogen bonding)
Hydrodynamic volume  polymer size in solution
●Interaction between solvent and polymer molecules
●Chain branching
●Conformational effects (arising from the polarity and steric
bulkiness of the substituent groups)
Restricted rotation caused by resonance
O
C NH
O
C NH
DHmix and the Solubility Parameter
• The most popular predictor of polymer solubility is the solubility
parameter, i. Originally developed to guide solvent selection in
the paint and coatings industry, it is widely used in spite of its
limitations.
• For regular solutions in which intermolecular attractions are
minimal, DHmix can be estimated through:
DH1,2  DU1,2  1 2 (1   2 )2
cal / cm3
• where DU1,2 = internal energy change of mixing per unit volume,
•
i = volume fraction of component i in the proposed
mixture,
•
i = solubility parameter of component i: (cal/cm3)1/2
• Note that this formula always predicts DHmix > 0, which holds only
for regular solutions.
Determining the Solubility Parameter
The conditions of greatest polymer solubility exist when the
solubility parameters of polymer and solvent match.
If the polymer is crosslinked, it cannot dissolve but
only swell as solvent penetrates the material.
The solubility parameter
of a polymer is therefore
determined by exposing
it to different solvents,
and observing the  at
which swelling is
maximized.
Solubility (Hildebrant) Parameters of
Select Materials
Material
Acetone
Benzene
Tetrahydrofuran
Carbon tetrachloride
n-Decane
Dibutyl amine
Mineral spirits
Methanol
Toluene
Water
Xylene
3 1/2
 (cal/cm )
9.9
9.2
9.5
8.6
6.6
8.1
6.9
14.5
8.9
23.4
8.8
Material
Poly(butadiene)
Poly(ethylene)
Poly(methylmethacrylate)
Poly(tetrafluoroethylene)
Poly(isobutylene)
Poly(styrene)
Cellulose triacetate
Nylon 6,6
Poly(vinyl chloride)
Poly(acrylonitrile)
If D < 1, polymer will dissolve
3 1/2
 (cal/cm )
8.4
7.9
9.45
6.2
7.85
9.10
13.6
13.6
10.5
12.4
Solubility Parameters of Select Materials
Partial Miscibility of Polymers in Solvents
•Idealized representation of three generalized possibilities for the dependence of the
Gibbs free energy of mixing, DGm, of a binary mixture on composition (volume
fraction of polymer, 2) at constant P and T.
•I. Total immiscibility;
•II. Partial miscibility;
•III.
Total miscibility.
•Curve II represents the intermediate case of partial miscibility whereby the mixture
will separate into two phases whose compositions () are marked by the volumefraction coordinates, 2A and 2B, corresponding to points of common tangent to the
free-energy curve.
Factors Influencing Polymer-Solvent
Miscibility
Based on the Flory-Huggins treatment of polymer solubility, we can
explain the influence of the following variables on miscibility:
1. Temperature: The sign of DGmix is determined by the FloryHuggins interaction parameter, c.
As temperature rises, c decreases thereby improving solubility.
Upper solution critical temperature (UCST) behaviour is
explained by Flory-Huggins theory, but LCST is not.
2. Molecular Weight:
Increasing molecular weight reduces the
configurational entropy of mixing, thereby
reducing solubility.
3. Crystallinity: A semi-crystalline polymer has a more positive
DHmix = HAB - xAHA - xBHB due to the heat of
fusion that is lost upon mixing.
Theta Temperature
Partial Miscibility of Polymers in Solvents
• Phase diagrams for the
polystyrene-acetone system
showing both UCSTs and
LCSTs.
• Molecular weights of the
polystyrene fractions are
indicated.
Polymer-Solvent Miscibility
Phase diagrams for four samples of
polystyrene mixed with cyclohexane
plotted against the volume fraction of
polystyrene. The molecular weight of
each fraction is given.
The dashed lines show the
predictions of the Flory-Huggins
theory for two of the fractions.
2
Practical Use of Polymer TDs
Fractionation
1 phase clear solution
•
•
•
•
•
•
Consider solution in poor solvent
of two polymers, p1 and p2.
Flory-Huggins tells us that if p2
has higher molecular weight it
should precipitate more readily
than p1
add non-solvent until solution
becomes turbid
heat, cool slowly and separate
precipitate
finite drop in temperature always
renders finite range of molecular
weight insoluble
some p2 will also remain soluble!
1
1

 2c
N 1  
T
2 phase
cloudy p1
p2
2
volume fraction polymer
Fractionation of Polymers using Precipitation
Industrial Relevance of Polymer Solubility
Polymer
Diblock
copolymers
Solvent
Motor Oil
Effect
Colloidal suspensions
dissolve at high T,
raising viscosity
Application
Multiviscosity motor
oil (10W40)
Poly(ethylene
oxide)
Water
Reduces turbulent flow
Heat exchange
systems, lowers
pumping costs
Polyurethanes,
cellulose esters
Esters, alcohols, Solvent vehicle
various
evaporates, leaving film
for glues
Varnishes, shellac
and adhesives
Poly(vinyl chloride)
Dibutyl
phthalate
Plasticizes polymer
Lower polymer Tg,
making ТvinylУ
Polystyrene
Poly(phenylene
oxide)
Mutual solution,
toughens polystyrene
Impact resistant
objects, appliances
Polystyrene
Triglyceride oils
Phase separates upon
oil polymerization
Oil-based paints,
tough, hard coatings
Polymer Solubility
• When two hydrocarbons such as dodecane and 2,4,6,8,10pentamethyldodecane are combined, we (not surprisingly)
generate a homogeneous solution:
• It is therefore interesting that polymeric analogues of these
compounds, poly(ethylene) and poly(propylene) do not mix, but
when combined produce a dispersion of one material in the
other.
n
n
Polymer solutions: Definitions
• Configuration: arrangement of atoms through bonds
– Inter-change requires bond breaking and making (>200
kJ/mol)
– Different compositions of polymers
– Different isomers or stereochemistries
• Conformation: within the constraints of a configuration, the
possible arrangement(s) of atoms in space
– Interchange requires bond rotations, but no bonds are
broken or made (can have non-bonding , ie. hydrogen
bonding)
What are the dimensions of a
polymer?
• In a solution, in the solid state, in its melt, in
vacuum?
Polyethylene coil
Fully stretched Polyethylene
Factors Affecting Macromolecule
Dimensions
Polymer Conformations
Flexible coil
Rigid rod
And everything in between
How to describe conformations
with models: The freely jointed
chain
• Simplest measure of a chain is the length along the backbone
– For n monomers each of length l, the contour length is nl
l
1
2
3
. ..
n
First, Most Primative Model: Freely Jointed Chain
Any bond
angles &
orientations
are possible
A useful measure of the size of
macromolecules: end-to-end distance r
•
For an isolated polymer in a solvent the end-toend distance will change continuously due to
molecular motion
– But many conformation give rise to the same
value of r, and some values of r are more
likely than others e.g.,
• Only one conformation with r = nl - a fully
extended chain
• Many conformation have r = 0, (cyclic
polymers)
– Define the root mean square end-to-end
distance
– Permits statistical treatments
r2
1
2
But we still don’t
know the shape;
need something
more
Free rotation model: Infinite number of conformations
Random Walk Model for linear polymers
Example of a “low resolution”
polymer coil RW. Contour length =
lct, unit vectors to describe the
length = e(l), a = vectors that
connect the junction points on the
chain, r = end-to-end distance of the
chain
A freely jointed chain in 2D
from a random walk of 50
steps.
# of steps based on
number atoms in chain
Complications: excluded
volume & steric limitations
What does
this all
mean??
An ideal polymer chain with 106 repeat units (not
unusual), each unit about 6Å will have:
• a rms end-to-end distance R of 600 nm
• a contour length of 600 μm
A better measure of Polymer dimensions:
Radius of Gyration of a Polymer Coil
The radius of gyration Rg is defined as the RMS
distance of the collection of atoms from their
common centre of gravity.
R
For a solid sphere of radius R;
2
Rg 
R  0.632R
5
For a polymer coil with rms end-to-end distance R ;
Rg
1 2
Rg  R
6
1/ 2
a 1/ 2
 N
6
Can be related directly to statistical distributions from polymer characterizations
Restrict the bond angles to what we see in the
polymer: Valence Angle Model
Valence angle model
•
Simplest modification to the freely jointed chain model
–
–
–
•
Introduce bond angle restrictions
Allow free rotation about bonds
Neglecting steric effects (for now)
If all bond angles are equal to q,
 r 2  fa  nl 2
(1 - cosq )
(1  cosq )
indicates that the result is for the valence angle model
•
E.g. for polyethylene q = 109.5° and cos q ~ -1/3, hence,
 r 2  fa  2nl 2
Finite Number of Conformations due to torsional &
steric interactions: Restricted Rotation Angle
The energy barrier between gauche
and trans is about 2.5 kJ/mol
RT~8.31*300 J/mol~2.5 kJ/mol
Equivalent Freely Jointed Chain Model
C∞
C∞ is a function of the stiffness of the chain. Higher is stiffer
Steric parameter and the characteristic ratio
• In general
 r 2 0  s 2 nl 2
(1 - cosq )
(1  cosq )
– where s is the steric parameter, which is usually determined
for each polymer experimentally
– A measure of the stiffness of a chain is given by the
characteristic ratio
C   r 2 0 / nl 2
– C typically ranges from 5 - 12
Polymer Conformations
Flexible coil
<r2>1/2 ~ N1/2
Rigid rod
<r2>1/2 ~ N
The shape of the polymers can therefore only be usefully
described statistically.
Calculate size of macromolecules!
Excluded volume
• Freely jointed chain, valence angle and rotational isomeric
states models all ignore
– long range intramolecular interactions (e.g. ionic polymers)
– polymer-solvent interactions
• Such interactions will affect
– Define
where
r2
ar
 a r2 r 2
r2
0
is the expansion parameter
Space filling?
The random walk and the steric limitations
makes the polymer coils in a polymer melt
or in a polymer glass “expanded”.
However, the overlap between molecules
ensure space filling
Expansion Factors: Measure of solvent -Polymer Interactions
2
r : mean-square average distance between chain ends
for a linear polymer
s : square average radius of gyration about the center of
gravity for a branched polymer
2
Better solvent  stronger interaction between solvent and
polymer  larger hydrodynamic volume
2
r  ro a
2
2
2
s  so a 2
2
r0, s0: unperturbed dimension (i.e. the size of the
macromolecule exclusive of solvent effects)
a : an expansion factor
2
2
r  6s (for a linear polymer)
a
2
1
2
2
1
2
(r )
(ro )
greater a  better solvent
a = 1  ideal statistical coil
Solubility vary with temperature in a given solvent
 a is temperature dependent
Theta (θ) temperature (Flory temperature) 
● For a given polymer in a given solvent, the lowest
temperature at which a = 1
● q state / q solvent
Polymer in a q state
• having a minimal solvation effect
• on the brink of becoming insoluble
• further diminution of solvation effect  polymer precipitation
The expansion parameter ar Solvent
Effects on the Macromolecules
•
ar depends on balance between i) polymer-solvent and ii)
polymer-polymer interactions
– If polymer-polymer are more favourable than polymer-solvent
• ar < 1
• Chains contract
• Solvent is poor
– If polymer-polymer are less favourable than polymer-solvent
• ar > 1
• Chains expand
• Solvent is good
– If these interactions are equivalent, we have theta condition
• ar = 1
• Same as in amorphous melt
The theta temperature
•
For most polymer solutions ar depends on temperature, and increases
with increasing temperature
•
At temperatures above some theta temperature, the solvent is good,
whereas below the solvent is poor, i.e.,
T>q
ar > 1
T=q
ar = 1
T<q
ar < 1
Often polymers will precipitate out of solution,
rather than contracting
What determines whether or not a polymer is soluble?
Solubility of Polymers
Encyclopedia of Polymer Science, Vol 15, pg 401 says it best...
A polymer is often soluble in a low molecular weight liquid if:
•the two components are similar chemically or are so constituted that
specific attractive interactions such as hydrogen bonding take place
between them;
•the molecular weight of the polymer is low;
•the bulk polymer is not crystalline;
•the temperature is elevated (except in systems with LCST).
The method of solubility parameters can be useful for identifying potential
solvents for a polymer.
Some polymers that are not soluble in pure liquids can be dissolved in a multicomponent solvent mixture.
Binary polymer-polymer mixtures are usually immiscible except when they
possess a complementary dissimilarity that leads to negative heats of mixing.
Another way of looking at Solubility:
Thermodynamics of Mixing
mA moles
material A
mB moles
material B
DGmix < 0
A-B solution
+
DGmix > 0
DGmix (Joules/gram) is defined by:
DGmix = DHmix -T DSmix
immiscible blend
where DHmix = HAB - (xAHA + xBHB)
DSmix = SAB - (xASA + xBSB)
and xA, xB are the mole fractions of each material.
Polymer-solvent: volume fraction of polymers
Thermodynamics of Mixing: Small
Molecules
Ethanol(1) / n-heptane(2) at 50ºC
Ethanol(1) / chloroform(2) at 50ºC
Ethanol(1) / water(2) at 50ºC
Solvent-Solvent Solutions vs Polymer-Solvent Solutions
Solvents can easily replace one another.
Polymers are thousands of solvent sized monomers
connected together. Monomers can not be placed
randomly!
• Entropy of mixing is generally positive
• Enthalpy is also often positive: smaller the better
Ideal = 0
Flory-Huggins Theory
•
•
Flory-Huggins theory, originally derived for small molecule systems, was
expanded to model polymer systems by assuming the polymer consisted of
a series of connected segments, each of which occupied one lattice site.
Assuming segments are randomly distributed and that all lattice sites are
occupied, the free energy of mixing per mole of lattice sites is
DGmix=RT[(A/NA)lnA+(B/NB)lnB+cFHAB]
 i is the volume fraction of polymer i
 Ni is the number of segments in polymer i
 cHF is the Flory-Huggins interaction parameter
(3)
Gibbs Energy of Mixing: Flory-Huggins Theory
Combining expressions for the enthalpy and entropy of mixing generates
the free energy of mixing:
DG mix  DH mix  T DSmix
 RT c2 n1 x1  RT (n1 ln 1  n2 ln 2 )
 RT (n1 ln 1  n2 ln 2  c2 n1 x1 )
The two contributions to the Gibbs energy are configurational
entropy as well as an interaction entropy and enthalpy
(characterized by c ; “chi” pronounced “Kigh”).
Note that for complete miscibility over all concentrations, c for the
solute-solvent pair at the T of interest must be less than 0.5.
• If c > 0.5, then DGmix > 0 and phase separation occurs
• If c < 0.5, then DGmix < 0 over the whole composition range.
• The temperature at which c = 0.5 is the theta temperature.
For a mixture of polymer and solvent or two polymers
Mixing is always entropically good.
Gibbs Energy of Mixing: Flory-Huggins Theory
Combining expressions for the enthalpy and entropy of mixing generates
the free energy of mixing:
DG mix  DH mix  T DSmix
 RT c2 n1 x1  RT (n1 ln 1  n2 ln 2 )
 RT (n1 ln 1  n2 ln 2  c2 n1 x1 )
The two contributions to the Gibbs energy are configurational
entropy as well as an interaction entropy and enthalpy
(characterized by c ; “chi” pronounced “Kigh”).
Note that for complete miscibility over all concentrations, c for the
solute-solvent pair at the T of interest must be less than 0.5.
• If c > 0.5, then DGmix > 0 and phase separation occurs
• If c < 0.5, then DGmix < 0 over the whole composition range.
• The temperature at which c = 0.5 is the theta temperature.
Case 2: Exothermic mixing (negative H)
c < 0.5
(2EAB < EAA + EBB)
No Phase separation: Completely Miscible
Case 3: Endothermic mixing (positive H)
c > 0.5
(2EAB > EAA + EBB)
Partial Miscibility: two phases of mixed compositions
Start with homogeneous mixture
composition c at free energy of F.
Splits into two phases.
Blends are not easy to make:
• Blends are not easy to discover-often
copolymers are needed to make blend with
another homopolymer
• Phase separation gets out of hand in
immiscible systems without compatibilizers.
• Some blends must be made with solvent.
Polymer Mixture
Immiscible
Some miscible blends
Polymer blend
poly(ethylene terephthalate) with poly(butylene terephthalate)
Some miscible blends
PMMA & poly(vinylidene fluoride)
Some miscible blends
Mini Cooper / Cooper S, radiator grille
& PETE
with
Immiscible Blends
Immiscible Blends
1:2 (PTV:PCBM)
1:3 (PTV:PCBM)
1:4 (PTV:PCBM)
1:6 (PTV:PCBM)
1:10 (PTV:PCBM)
C12H25
S
n
1. Linear polymer: copolymers
Morphology of diblock Copolymers:
Random:
AABABBAAABABABB
Alternating: ABABABABABABABA
Block:
AAAAAAAABBBBBBB
PMMA-PS-PB ternary blend
Dilute Solution Viscosity
•
The “strength” of a solvent for a given polymer not only effects
solubility, but the conformation of chains in solution.
– A polymer dissolved in a “poor” solvent tends to aggregate while a
“good” solvent interacts with the polymer chain to create an
expanded conformation.
– Increasing temperature has a similar effect to solvent strength.
•
•
•
The viscosity of a polymer
solution is therefore dependent
on solvent strength.
•

•

•
•
Consider Einstein’s equation:
hhs(1+2.5)
where h is the viscosity
hs is the solvent viscosity
and  is the volume fraction of
dispersed spheres.
Molecular Weight and Polymer Solutions
Number Average and Weight Average Molecular Weight
Mn number average molecular weight
Mw weight average molecular weight
Determine M. W. of small molecules
Mass spectrometry
Cryoscopy (freezing-point depression)
Ebulliometry (boiling-point elevation)
Titration
Determine M. W. of polymers
Osmometry  Mn
Light scattering  Mw
Ultracentrifugation  Mw
End-group analysis  Mn
Measurement of Number Average Molecular Weight
End-group analysis
Titration
polyester (-COOH, -OH), polyamide (-C(O)NH-), polyurethanes(
isocyanate), epoxy polymer (epoxide), acetyl-terminated
polyamide (acetyl)
Elemental analysis
Radioactive labeling
UV / NMR / (IR)
Upper limit M. W.  50,000 (due to the low concentraction of
end groups)
preferred range: 5,000-10,000
Not applicable to branched polymers
Analysis is meaningful only when the mechanisms of initiation
and termination are well understood
Membrane Osmometry
● based on colligative properties
● most useful method for Mn
(range: 50,000 to 2,000,000)
● major error source 
arising from the loss of low
MW species
● obtained Mn values are
generally higher than other
colligative measurement
method
Static equilibrium method 
measure the hydrostatic head (Dh)
after equilibrium
Dynamic equilibrium method 
measure the counter pressure
needed to maintain equal liquid levels
h
Osmotic pressure () 
related to M n by the van’t Hoff equation
extrapolated to zero concentration (C; g/L) to obtain the intercept

RT
PV  nRT
( ) C 0 
 A2 C
W
C
Mn
V  t RT
Mn

V
RT

Wt M n

1 RT

C Mn

C

RT
Mn
RT
C
Mn
RT

C  A2C 2  A3C 3  ...
Mn

A2 = v2V-1(0.5 – c)

RT
( ) C 0 
 A2C
C
Mn
Cryoscopy and Ebulliometry
Thermodynamic relationships
 DT f

 C

RT 2


 A2C
C  0 DH f M n
RT 2
 DTb 

 A2 C


C

 C 0 DH v M n
for freezing-point depression
for boiling-point elevation
T:
DHf :
DHv:
A2 :
:
freezing point or boiling point (solvent)
latent heat of fusion (per gram)
latent heat of vaporization (per gram)
second virial coefficient
solvent density
• Limited by the sensitivity of measuring DTf, DTb
• As MW  higher; DTf, DTb  smaller
• Upper limit － Mn = 40,000
• Preferred － Mn ＜ 20,000
Vapor Pressure Osmometry
● For Mn ＜ 25,000
● No membrane needed
• Thermodynamic principle similar to membrane osmometry
Measurement method
→ Add a drop of solvent and solution,
on a pair of matched themistors,
in an insulated chamber saturated with solvent vapor
then, → Condensation heats the solution thermistor
(until the vapor pressure of the solution equal to that of the
pure solvent)
Temperature change is measured (by resistance change of
the thermistor) and related to solution molality
 RT 2 
m
DT  
 100 
 : heat of vaporization per gram (solvent)
m : molarity
Wt / M n
n
m=
=
g g
Mass Spectrometry
MALDI-MS (MALDI-TOF)
MALDI-MS (matrix-assisted laser desorption ionization mass
spectrometry)
(TOF － time-of-flight)
● Imbeds polymer in a matrix of low MW organic compound
● Irradiates the matrix with UV laser
● Matrix transfer the absorbed energy to polymer and vaporize the
polymer
● Integrated peak areas 
the number of ions
● Mn, Mw can be calculated
Soft ionization method
 field desorption (FD-MS)
 laser desorption (LD-MS)
 electrospray ionization (ESI-MS)
Refractive Index Measurement
Most suitable for low MW polymers
n
1
Mn
Measurement of Weight Average Molecular Weight (Mw)
Light Scattering
• Most widely used method for measuring absolute Light
loses energy by absorption, conversion to heat, and scattering
• Light scattering is caused by inhomogeneous distribution of
molecules  due to the irregular change in density and
refractive index (because of the fluctuation in composition)
• Scattering intensity depends on concentration, size and
polarizability of the molecules
• Refractive index also depends on concentration and
amplitude of vibration
LS adds optical effects  Size
q = 0 in phase
Is maximum
2
q > 0 out of phase,
Is goes down
Is  1
q R
3
2
g
I
 e-l
I0
Scattering causes turbidity ()
  Kc M w = Rq
I0: intensity of incident light
I : intensity of scattered light
l : length of sample solution
 : turbidity
ciMi
  i  KciMi  Kc
c
W i /V M i
ciMi
 Kc
 Kc
ci
W i /V 
W i M i
 Kc
 Kc M w
W i
32 3 n 0 2 (dn /dc) 2
K
3
4 N 0
n0
: solvent refractive index
: wavelength of light
N0: Avogadro’s number
n: refractive index of solution

dn/dc (specific refractive increment)
• is a constant for a given polymer,

solvent, and temperature
• measured from the slope of n vs. c
Debye equation
Kc
1

 2A2c
Rq
Mw
(A2: the second virial coefficient)

As molecular size approaches the wavelength of the light
• Interference between scattered light coming from different
parts occurs
• Correction is needed
Turbidity ( or DRq) associated with large particles
• measured at different concentrations (c) and angles (q)
 Kc

1

 2A2c 

 DRq M w P(q )

P(q): Particle scattering factor
P(q) is a function of q; depending on the shape of molecule in solution
For a monodisperse of randomly coiling polymers
P(q )  (
2
v
)[
e
 (1  v)]
2
v
v  (16 )(
2
s
2
2
q
) sin ( )
2
s
2
s 2: mean-square radius of gyration of
the molecue
s: the wavelength in solution
s 

n
Zimm Plot
 Kc 
Plot   vs.
 Rq 
[sin2(q/2)+kc]
k is an arbitrary constant
Extrapolated to both zero concentration and zero angle 

intercept =
1
Mw
 Kc 
 
 Rq 

(P(q) = 1 at c = 0 and q = 0)
Light scattering photometer
• Suitable range of Mw: 10,000－10,000,000
• Dust-free solution is required
• Light source
Old: high-pressure Hg lamps plus filter
New: laser
60
0,64
50
Rg = 0,0074.(Mw)
Linear PS
Rg (nm)
40
poly(macromonomer)
Linear PS
30
20
0,54
Rg=0,0126.(Mw)
10
2
R = 0,99
0
0
200000
400000
600000
Mw ddl (g/mol)
800000
1000000
1200000
Ultracentrifugation
• Most expensive
• Extensively used with proteins
• Determining M z
• Sedimentation rate is proportional to molecular mass
• Distributed according to size along the perpendicular
direction
• Concentration gradients within the polymer solution are
observed by refractive index measurements and
interferometry
Viscometry
• Simplest and most widely used
• Not an absolute number
• Can be calibrated with the absolute MW (by light scattering)
of fractionated polymer samples
• Typical conditions: 0.5 g/100 mL; 30.0±0.01℃
Viscometers:
(a) Ubbelohde
(b) Cannon-Fenske
Ubbelohde type
• more convenient
• not necessary to have exact
volumes of solution
• additional solvent can be added
(for dilution)
• dust particles can greatly alter the
flow time (via the capillary tube)
Viscosity
h
t


h0 t0
h rel
Relative viscosity (viscosity ratio), hrel
● ratio of solution viscosity to solvent viscosity
● viscosity units (poises) or flow times cancel out (→ dimensionless)
Specific viscosity (hsp)
h  h0 t  t 0

 h rel  1
● fractional increase in viscosity h sp  h
t0
0
● dimensionless
Inherent viscosity (hinh; dL/g)
● determined only by a single solution
at a specific concentration
● an approximate indication of MW
hinh
ln h rel

C
 h sp 
Intrinsic viscosity ([h]; dL/g)

[h ]  
 hinh C 0
● most useful
 C  C 0
● eliminating the concentration effect
● by dividing hsp with C and then extrapolating to zero C
a
Table 2.2. Dilute Solution Viscosity Designations
Common Name
Relative viscosity
Specific viscosity
Reduced viscosity
Inherent viscosity
Intrinsic viscosity
a
IUPAC Name
Viscosity ratio
－
Viscosity number
h rel 
h sp 
Limiting viscosity
number
h0
t0

h0
t  t0
t0
h sp

C
 hrel  1
hrel  1
C
hinh 
 h sp
[h ]  
 C

t

h  h0
hred 
Logarithmic viscosity
number
h
ln hrel
C


 (hinh )C 0
 C 0
Concentrations (most commonly expressed in grams per 100 mL of solvent) of about 0.5 g/dL
M v viscosity average molecular weight
1 a

[h ]  K M v (Mark-Houwink-Sakurada) M    N i M i
v
 N M
i
i
log[ h ]  log K  a log M

a
Mw (or log M n )  measure a and K
• in general, M n < M < M
(closer to M w )
v
w
• plot log [h] vs log
• better results are obtained if the MW of fractionated
samples are used (here, M n  M  M w )
v
● typically, a  0.5－0.8, K  10-3－0.5
● for randomly coiled polymer in a q solvent
 N M 
a = 0.5; M v   i

N
M

i
i 
1.5
i
1
0.5
 N M 1.5 2
 i i 
 N i M i 
• for a rodlike extended chain
polymer  a = 1.0;
< Mw
 N i M i 2 
  Mw
M v  


N
M
i
i 





1
a
In polymers a values have solvent dependence
[η] = KMa
Linear polymers: a = 0.5-1.0
Rigid rod: a = 1.0-2.0
Branched polymers: a = 0.21-0.28
Rigid spherical particles: a = 0
Complication in applying Mark-Houwink-Sakurada relationship
• chain branching
• broad MWD sample
• solvation of polymer
• backbone sequences (alternating or block)
• chain entanglement (when MW is extremely large)
Viscosity measurements based on mechanical shearing
• for concentrated polymer solutions or undiluted polymer system
• more applicable to the flow properties of polymers
Dynamic Viscosity
Informally: resistance to flow
Formally: Ratio of Shearing
stress (F/A) to the velocity
gradient (dvx/dz)
Force: Pa = N m-2 = kg m s-2
Force/Area = Pa m-1 = kg m-1 s-2
Viscosity = Pa s = kg m s-1
Other units (Poise = dyne s cm-2)
1 Pa s = 10 Poise
1 centipoise = 1 millipascal second (mPa s)
EXAMPLES
• Water 1 mPa s
• Honey 10,000 mPa s
• Peanut Butter 200,000 mPa s
Kinematic viscosity is the
dynamic viscosity divided by the
density (typical units cm2/s,
Stokes, St).
Flow Regimes of Typical Polymer Processing
 s  Sedimentation
Extrusion
Chewing
10 3
Pipe flow
Injection molding
Spraying
Rolling
High-speed coating
10
10 1
Lubrication
10 3
105
103
101
101
103
105
 (s-1 )
10 7
Typical viscosity curve of a polyproprene (PP),
melt flow rate (230 C/2.16 Kg) of 8 g/10 min at 230 C
with indication of the shear rate regions of different
conversion techniques.
[Reproduced from M. Gahleitner, “Melt rheology
of polyolefins”, Prog. Polym. Sci. 26, 895 (2001).]