Part I. POLYMER STRUCTURE
AND PROPERTIES
Chapter 2. Molecular Weight & Polymer Solutions.
2.1. Number Ave. Molecular weight
2.2. Polymer Solutions
2.3. Measurement of Number Ave. Molecular Weight
2.4. Measurement of Weight Ave. Molecular Weight
2.5. Viscometry
2.6. Molecular Weight Distribution
Chapter 2. MW & Polymer solutions.
2.1 Number Ave. & wt Ave. Molecular weight
MW
Physical properties
ex) Higher MW polymer : Tougher
????
Too high MW
“What do we mean by high molecular weight ?”
Low MW
:
High MW
Factors ?
Where the boundary ?
CH2 CH2
*
ex) Polyethylene vs Polyamide
ex) Low MW polymer
Initial processing
H
H
ex) Vinyl polymer
MW
105
~
106
N
(CH2)5 N
H
*
O
C
(CH2)4 C
OH
n
O
O
Polyamide w/ polar group
MW 15000 ~ 20000
H
N
H
(CH2)5 C
OH
What we have calculated is
something called a number average,
which is defined mathematically
below. If you’re not used to dealing
with summations, this looks
Horrible.
To give you a feel for how it works,
and also introduce a different
average- the weight average
let’s consider a ridiculous example.
Molecular weight
Number of end group
Free volume
Viscoelastic property
etc
(Example)
“Number average molecular weight”
(Example)
“Number average molecular weight”
“Weight average molecular weight”
Now, let’s say we had a sample with 5 (moles
of) chains of “length” (degree of
polymerization or DP) 100, 5 (moles of) chains
of length 150 and 5 (moles of ) chains of 200.
Nx is simply the number ( of moles) of chains
of the “x-species”. We have three species in
our sample;
Chains of DP 100, chains of DP 150 and chains
of DP 200, whose weights M, are therefore
10,000, 15,000, and 20,000, respectively.
Before proceeding, see if you can substitute
correctly into the equation opposite.
What about the weight average molecular weight ?
Now note that the total weight of species x
present is just the molecular weight of each
chain of type x multiplied by the number of
chains of this type (e.g. 5 chains, each of weight
10,000 means that Wx is 50,000):
WE can now substitute in the equation
at the top to obtain a different form of
the equation for weight average.
“Molecular Weight Distribution”
Number and weight average molecular weight
[Text P37]
9 moles having 30,000 molecular weight
and 5 moles having 50,000 molecular weight
Substituting grams for moles:
In each instance, we see that Mw is greater than Mn.
n
Please determine Mn and Mw now !!
Mn
Mw
PDI
4 moles having 1 molecular weight
and 2 moles having 5 molecular weight
2.33
3.86
1.65
4 moles having 4 molecular weight
and 2 moles having 5 molecular weight
4.33
4.38
1.01
What difference can we see after comparison ???
We have seen that ave. molecular weight is not unique. It turnes out that
there are more than two ways to define an average.
Look at the definitions of number and weight average again.
You can see that we can go from number to weight average by
multiplying each of the terms inside the summations by Mx. Higher order
averages can be constructed in the same way; e.g. the z-average
The ratios of these averages can be related to the moments of the
distribution and tell us about its breadth and “skewedness”.
Effect of MW on the Physical properties.
Mechanical
Strength
C
B
A:
*
H2
C
H2
C *
B:
*
H2
C
H2
C *
1000
2000
(Mn)A= 28000 (g/mole)
(Mn)A= 56000 (g/mole)
MW
A
Useful MW Optimum MW
20000-40000
(Min)
MW 5000-10000
Ni
A:
*
H2
C
H2
C *
N: Number of repeating unit
1000
Distribution ?
I (chain length)
** MW Distribution
ⓐ
Tensile strength
Modulus
Thermal expansion coefficient
Free Volume
Refractive index
Dielectric constant
Etc.
wt%
ⓑ
DP
What properties ?
(a) ,
(b)
* Molecular Weight ?
wt%
A
ⓐ ⓑ
wt%
DP
B
ⓐ
ⓑ
DP
Molecular Weight Distribution (MWD)
(a)
Wx
Nn
(b)
Mn Mw
Mw Mz
Mx
Mx
Mn
Polydisperse Polymer : Practical Polymer
Mn
Mw
Mz
Mn
Mw
Mz
Monodisperse polymer ?
Polydispersity Index (PDI) =
Mw
Mn
2.2 Polymer Solution
Dissolution
① Solvent molecule diffuse through the Polymer matrix
Swollen gel
② Gel breaks up and the molecules are dispersed into a true solution
Slow Process
∝ T
Network polymer ?
Choice of Solvent
Polymer Handbook
Thermodynamic Principles
Semiempirical relationship
- When a polymer dissolves spontaneously,
∆G is Negative !!
The ∆S invariably has a (+) value arising from increased conformational
mobility of the polymer chains
∴ ∆H
determine
Sign of ∆G
Heat of mixing ∆Hmax for binary system
Concentration & Energy Parameters
Cohesive Energy
Vm: total volume of the mixture
V1, V2: molar volume (molecular weight/density)
F1, F2 : volume fraction
DE1, DE2: Energy of vaporization
If (∆E/V)1/2 is replaced by the symbol d, the equation is
written more simply
Solubility Parameter
For dissolution (negative ∆G),
∆Hmix must be small
∴ (d1- d2)2 must be small!
d1 ≈ d2 similar solubility parameters
<Solvent 1, Polymer 2>
Cohesive E
: Energy needed to remove a molecule from its nearest neighbors.
≈ heat of vaporization per Volume for volatile Compound
For solvent, d1
From the latent heat of vaporization (∆Hvap )
D
vap
Since Polymers have negligible vapor pressure !!
d2
“Group
“Group molar attraction constants”
Contribution”
expect many physical properties !!
“Group Contribution”
G values are additive for a given structure
Ex) PS
*
*
n
r=d=1.05
Unit mass = 104
i) Small’s
ii) Hoy’s
?? Trouble Molecular Interaction ?
Strong Dipolar interactions (e.g. Hydrogen Bonding)
Polymer-Solvent System
How the polymer molecules behave in that solvent ?
Resultant size
Hydrodynamic volume
in solution
Depend on ① interaction btw solvent & polymer molecule
② Chain branching
③ Conformational effects
(arising from the polarity & steric bulk
of the substituent)
④ restricted rotation
Resonance
ex) Type of Polyamide
O
C
H
N
O
H
C
N
Mean Square Average Distance for Linear Polymer
Mean Square Average Radius of Gyration about COG
for a branched Polymer
Average shape of the Coiled
molecule “Spherical”
“The greater the affinity of the
solvent for polymer,
the larger will be the Sphere”
“Hydrodynamic
Volume”
Solvent
Interaction
3/22/2005
r 2 = r 02 a 2
s 2= s 02 a 2
Expansion factor
*Unperturbed dimension : No solvent Effect
r0, s0
Combination of free rotation and intramolecular
steric and polar interaction
*Expansion factor
For a linear polymer
a
a
a=1
interaction btw Solvent and Polymer
r2= 6 s2
a >1 in a good solvent !
better solvent
“ Ideal” statistical Coil
Solubility ∝ f(temp) in a given solvent
a ∝ f(temp)
“No solvent interaction”
For a given polymer in a given solvent
Lowest T (a=1)
Theta (θ) temperature
“Theta State of Polymer” in solution.
From the stand point of MW determinations,
significance of the Parameters
Dilute Solution Viscosity
“Flory-Fox Equation”
Where [ŋ] is the Intrinsic Viscosity
M is average MW.
Φ proportionality constant ~ 3X1024 mol-1
Substituting r02 a2 for r2
Since r0 & M are Constant
set k= F(ro2 M-1)3/2
at θ Temp, a=1
a ∝ f (polymer, solvent, T)
“Mark-Houwink-Sakurada” eqn.
In q solvent
“Mark-Houwink-Sakurada” equation
2.3. Measurement of Molecular Weight
No relation with the kind of molecule!!
2.3. Measurement of Number Average Molecular Weight
2.3.1. End-group analysis
: No average MW of any Linear Polymer having End-groups.
very low concentration.
Limit of MW ~ 50,000
“If high MW or low MW polymer ???”
1) Titration
2) Elemental Analysis
3) Measurement of activity of a Radioactive
-tagged end group
4) UV spectroscopic determination of an end group
w/ a characterizable chromophore
Points
1. Not valid for branched polymer unless the number of branches is
known.
2. In a linear polymer, twice as many end groups as polymer molecules.
3. If the polymer contains different groups at each end of the chain and
only one characterizable
End group is being measured, the number of this type is equal to the
number of polymer molecules.
4. Measurement of molecular weight by end-group analysis is only
meaningful when the mechanisms of initiation and termination are well
understood.
Typical Example
H3COOC
COOCH3 +
O
O
*
O
HOCH2CH2OH
+ 2CH3OH
O
n
*
to determine No average MW of the linear Polyester
Titrate the carboxyl and hydroxyl end groups by standard
methods.
*Carboxyl : Titration
*Hydroxyl : Titration
A weighed sample of
polymer is dissolved in an
appropriate solvent
(acetone)
A sample is acetylated w/
excess acetic anhydride
Titrated w/ standard base
to a phenolphthalein end
point.
Liberated acetic acid
<together w/ carboxyl end
groups> is similarly titrated
“Two steps”
From the two titrations
# of milliequivalents of carboxyl and hydroxyl
2
in the numerator
: two end groups counted per molecule
• Shortly : Difficulty !!
5000 ~10000 “Valid”
OSMOSIS
2.3.2. Membrane Osmometry
: Number ave. MW most useful!!
Definition : Osmotic Pressure
The pressure at equilibrium
(no further passage of the solvent)
Dynamic Equilibrium Method
Apply a counter pressure to the measuring tube connected
to the solution compartment
Prevents flow of Solvent and maintains equal liquid levels in the
two measuring tubes.
Static Equilibrium Method : Long period of time
for Equilibrium
∴Dynamic method
Encompass horizontal membrane
Separating solution and solvent Cells
:Measures osmotic pressure directly via a strain guage transducer
attached to a flexible diaphragm in the Solvent Cell
Semipermeable Membranes
Cell acetate, Cell nitrate, rubber, PVA
Osmotic Pressure is related to MW by the van’t Hoff equation
P : osmotic pressure. P = pgΔh
R : 0.082 L atm mol-1K-1
T ; Kelvin.
C: Conc in g/L
ρ : solvent density in g/cm3
g : acceleration due to gravity 9.81 m/s2
Δh: the difference in heights of solvent and solution in
an
A2 : 2nd virial coefficient (measure of the interaction btw
solvent & polymer)
Intrinsic parameter
Osmotic pressure
: allowing the system to reach equilibrium and measuring
the hydrostatic head that develops.
WAIT and WAIT for equl. !!!
Thin is referred to as the
“Static Equilibrium Method”
P/C : dyne L g-1 cm-1, J kg-1
A plot of Reduced
osmotic pressure (π/c) vs
concentration is linear
with the intercept equal to
RT/Mn and slope equal to
A2
What does A2 mean ?
A2 : Measure of Solvent-Polymer interaction
∴ Slope = 0 at
General
Mn ~ 50000 ~ 2x106
Tθ Theta Condition!
Versatile Method
Preferred
!! Error Source : Low-molecular-wt species diffuse through
the Membrane!
B
A
2.3.3. Cryoscopy & Ebulliometry
Cryoscopy : freezing-point depression
Ebulliometry : boiling point elevation
C : cont of cm3
T : freezing of boiling T of the solvent (K)
R : Gas constant
r: solvent density
∆Hv ∆Hf : latent heats of
fusion & vaporization per
gram of solvent
A2 : 2nd virial coeff.
Mn : No Ave. MW
!! Error source : sensitivity of the Methods of measuring
Mn < 20000
Preferred (not useful)
∆Hv ∆Hf
2.4. Measurement of weight Average MW
2.4.1. Light scattering
: useful, popular method (~osmometry)
Scattering for a pure liquid
finite nonhomogenieties in the distribution of molecules within
adjacent area that give rise to differences in density
+ solvent molecular scattering
The amplitude of intensity of the scattered light
Concentration, size, polarizability
• Refractive index depends on conc. and amplitude of vibration
Turbidity
n0 : refractive index of the solvent
l : wavelength of the incident light
N0 : Avogadro #
dn
: specific refractive increment
dc
n vs c
slope !!
General : constant (polymer, solvent, Temp)
Who is this man ?
As molecular size approaches in magnitude the l of light
Corrections must be made for interference
btw scattered light coming from different
parts of the molecules.
James Clerk Maxwell !!!
To determine molecular wt
P(θ) : function of the angle θ
A2 : 2nd-virial coeff.
Depends on the shape of the molecule in solution
Experimental data
: extrapolation to both c=0, θ=0
where p(θ) is equal to 1
Intercept corresponds to
1
Mw
Error source : Dust!
Availability : Mw 10,000 ~ 10,000,000
2.4.2. Ultracentrifugation
: Intricate and expensive instrument
: Not widely used
: applicable to natural polymer
: useful to determine Mz of synthetic P
Stimulus
Strong centrifugal field
Distribute them according to size
perpendicularly to the axis of rotation
“Sedimentation”
2.5. Viscometry
Dilute solution viscosity :
simplest popular
Not absolute method
Stand polymer solution : calibration
• Conc. : 0.5g/100ml
Folw time : sec
Temp : 30.0 ± 0.01 oc
•Viscosity Expression
Rel. viscosity (ηrel)
Specific viscosity (ηsp) :
fractional increase in η
Name by IUPAC
Dimensionless !
•Intrinsic viscosity : eliminate conc. effect
• Inherent
viscosity
: approximate indication of MW
C : g/100ml
g/cm3
Reduced, inherent, intrinsic η
dL/g
cm3/g (less)
•Intrinsic viscosity
Mark-Houwink-Sakurada eqn.
• For most common polymers
“a” varies 0.5~0.8 (random coil polymer in θ solvent)
For more rodlike extended-chain polymers,
where hydrodynamic vol is relatively large.
A
•K
1.0
10-3 ~ 0.5
“solvent”
“temperature”
• Error source : chain branching too broad MWD
solvation of polymer molecules alternating or block
2.6. MW Distribution
2.6.1. Gel permeation chromatography (GPC)
SEC : size exclusion chromatography
highly porous materials
Separates the polymer molecules
according to size,
“Molecular sieving”
Principle :
i) small molecules : diffuse into the pore
Slowly travel
ii) higher MW fractions are thus eluted first
• Detection of polymer fraction
i)
ii)
Refractive index
UV or IR detector
• Typical GPC
Detector response vs vol. of dilute polymer solvent
(retention vol.) elution vol.
• Major problem w/ calibrating a GPC column
: Few standard samples (narrow MWD)
*
Ex)
*
n
MW 600~2.5 million
Universal calibration method
[η]•M independent of polymer type
“Universal calibration parameter”
Plot of log [η]•M vs elution vol in THF
Single curve (~ linear)
• log([η]•M ) : constant for all polymers
for a given column, Temp,
elution vol
If assuming reference polymer(PS) is P1
sample polymer P2
From Mark-Houwink-Sakurada
To determine the M2 at a given
retention t,
column must be calibrated w/
standard PS fraction (same solvent,
same temp)
Semilogarithmic calibration plot : linear
Over a broad range of MW
• K, a
polymer handbook
M1 from calibration of col.
K, a
M2 can be readily calculated !
• Problem ?
2.6.2. Fractional Solution
- extract a polymer in a soxhlet apparatus
- Solvent (+Nonsolvent)
low
Dissolution of high MW polymer