Chapter 2. Molecular Weight and Polymer Solutions
2.1 Number average and weight average molecular weight
2.2 Polymer solutions
2.3 Measurement of number average molecular weight
2.4 Measurement of weight average molecular weight
2.5 Viscometry
2.6 Molecular weight distribution
POLYMER CHEMISTRY
2.1 Number Average and Weight Average Molecular Weight
A. The molecular weight of polymers
a. Some natural polymer (monodisperse) :
All polymer molecules have same molecular weights.
b. Synthetic polymers (polydisperse) :
The molecular weights of polymers are distributed
c. Mechanical properties are influenced by molecular weight
much lower molecular weight ; poor mechanical property
much higher molecular weight ; too tough to process
optimum molecular weight ; 105 -106 for vinyl polymer
15,000 - 20,000 for polar functional group containing polymer (polyamide)
POLYMER CHEMISTRY
Acclaim 與 Arcol PPG 差異
B. Determination of molecular weight
a. Absolute method :
mass spectrometry
colligative property
end group analysis
light scattering
ultracentrifugation.
b. Relative method : solution viscosity
c. Fractionation method : GPC
POLYMER CHEMISTRY
C. Definition of average molecular weight
a. number average molecular weight ( Mn )
 Ni M i
Mn= N
i
(colligative property and end group analysis)
b. weight average molecular weight ( Mw)
WiMi
Mw= W
i
(light scattering)
POLYMER CHEMISTRY
C. Definition of average molecular weight
c. z average molecular weight ( MZ )
M Z=
NiMi3
NiMi2
(ultracentrifugation)
d. general equation of average molecular weight :
M=
( a=0 , Mn
NiMia+1
NiMia
a=1 , Mw
a=2 , Mz
)
e. Mz > Mw > Mn
POLYMER CHEMISTRY
D. Polydispersity index : width of distribution
polydispersity index (PI) = Mw / Mn ≥ 1
POLYMER CHEMISTRY
E. Example of molecular weight calculation
a. 9 moles, molecular weight (Mw) = 30,000
5 moles, molecular weight ( Mw) = 50,000
Mn=
(9 mol x 30,000 g/mol) + (5 mol x 50,000 g/mol)
= 37,000 g/mol
9 mol + 5 mol
9 mol(30,000 g/mol)2 + 5 mol(50,000 g/mol)2
Mw =
9 mol(30,000 g/mol) + 5 mol(50,000 g/mol)
= 40,000 g/mol
POLYMER CHEMISTRY
E. Example of molecular weight calculation
b. 9 grams, molecular weight ( Mw ) = 30,000
5 grams, molecular weight ( Mw ) = 50,000
Mn =
Mw =
9g+5g
(9 g/30,000 g/mol) + (5 g/50,000 g/mol)
(9 g/30,000 g/mol) + (5 g/50,000 g/mol)
9g+5g
= 35,000 g/mol
= 37,000 g/mol
POLYMER CHEMISTRY
2.2 Polymer Solutions
A. Process of polymer dissolution : two step
first step : the solvent diffuses into polymer masses to make
a swollen polymer gel
second step : swollen polymer gel breaks up to solution
POLYMER CHEMISTRY
2.2 Polymer Solutions
B. Thermodynamics of solubility :
Gibb's free energy relationship
G =H - TS
ΔG < 0 : spontaneously dissolve
T and ΔS are always positive for dissolving process.
Conditions to be negative ΔG,
ΔH must be negative or smaller than TΔS.
POLYMER CHEMISTRY
C. Solubility parameter : δ
Hmix=Vmix
E1
[( V ) -(E ) ]  
1/2
2
1/2
V2
1
2
1 2
ψ1, ψ2 = volume fraction
ΔE1/V1, ΔE2/V2 = cohesive energy densities
δ1, δ2 = solubility parameter
δ1, δ2 =
(
E
)
1/2
V
Hmix= Vmix(δ1 – δ2)212
E = Hvap- RT
δ1 =
(
H vap - RT
V
)
1/2
if δ1= δ2, then Hmix= 0
POLYMER CHEMISTRY
D. Small's and Hoy's G parameter
a. Small(designated G derived from Heat of vaporization, Table 2.1)
dG
δ = MM
( d : density , M : molecular weight of unit )
ex) polystyrene 1.05(133+28+735)
δ=
= 9.0
104
b. Hoy(designated G based on vapor pressure measurement, Table 2.1)
dG
δ = MM
ex) polystyrene :
δ=
1.05[131.5+85.99+6(117.1)]
104
= 9.3
POLYMER CHEMISTRY
E. Hydrodynamic volume of polymer molecules in solution.
to be depended on followings
a. polymer-polymer interaction
b. solvent-solvent interaction
c. polymer-solvent interaction
d. polymer structure ( branched or not )
e. brownian motion
r = end-to-end distance
s = radius of gyration
Figure 2.1 Coil molecular shape
r 2 = ro 2  2
s2 = so 2  2
2)1/2
(r
=
(ro2)1/2
The greater the value of α, the ‘better’
the solvent
α = 1, 'ideal' statistical coil.
2.2 Polymer Solutions
F. theta(θ) temperature and theta(θ) solvent
The lowest temperature at which α=1 : theta(θ) temperature blink
The solvent satisfied this condition : theta(θ) solvent point
G. Flory-Fox equation :
The relationship among hydrodynamic volumes,
intrinsic viscosity and molecular weight
[η] =
(r2)3/2
M
[η] : intrinsic viscosity
M : average molecular weight
ψ : Flory constant (3×1024/mol)
r : end-to-end distance
POLYMER CHEMISTRY
2.2 Polymer Solutions
H. Mark-Howink-Sakurada equation
: The relationship between intrinsic viscosity and molecular weight
[η] =
KMa
[η] : intrinsic viscosity
K , a : constant for specific polymer and solvent
M : average molecular weight
I. Important properties of polymer solution : solution viscosity
a. paint spraying and brushing
b. fiber spinning
POLYMER CHEMISTRY
2.3 Measurement of Number Average Molecular Weight
2.3.1 End-group Analysis
A. Molecular weight limitation up to 50,000
B. End-group must have detectable species
a. vinyl polymer : -CH=CH2
b. ester polymer : -COOH, -OH
c. amide and urethane polymer : -NH2, -NCO
d. radioactive isotopes or UV, IR, NMR detectable functional group
POLYMER CHEMISTRY
2.3 Measurement of Number Average Molecular Weight
C.
Mn =
2 x 1000 x sample wt
meq COOH + meq OH
D. Requirement for end group analysis
1. The method cannot be applied to branched polymers.
2. In a linear polymer there are twice as many end of the chain
and groups as polymer molecules.
3. If having different end group, the number of detected end group
is average molecular weight.
4. End group analysis could be applied for
polymerization mechanism identified
E. High solution viscosity and low solubility : Mn = 5,000 ～ 10,000
POLYMER CHEMISTRY
FIGURE 2.2 Schematic representation of a membrane osmometer.
2.3.2 Membrane Osmometry
A. According to van't Hoff equation
(

c
)C=0 =
RT
+ A2C
Mn
limitation of : 50,000～2,000,000
The major error arises from low-molecular-weight species diffusing
through the membrane.
FIGURE 2.3 Automatic membrane osmometer [Courtesy of Wescan Instruments, Inc.]
FIGURE 2.4. Plot of reduced osmotic pressure (/c) versus concentration (c).
/c
RT
Mn
Slope = A2
C
POLYMER CHEMISTRY
2.3.3 Cryoscopy and Ebulliometry
A. Freezing-point depression (Cryoscopy)
 Tf
RT2
+ A2C
(
)C=0 =
Hf Mn
C
Tf : freezing-point depression,
C : the concentration in grams per cubic centimeter
R : gas constant
T : freezing point
Hf: the latent heats of fusion
A2 : second virial coefficient
POLYMER CHEMISTRY
2.3.3 Cryoscopy and Ebulliometry
B. Boiling-point elevation (Ebulliometry)
Tb
RT2
+ A2C
(
)C=0 =
C
HvMn
Tb : boiling point elevation
H v : the latent heats of vaporization
We use thermistor to major temperature. (1×10-4℃)
limitation of Mn : below 20,000
POLYMER CHEMISTRY
2.3.4 Vapor Pressure Osmometry
The measuring vapor pressure difference of solvent and solution drops.
T =
(
RT2
100
)m
λ : the heat of vaporization per gram of solvent
m : molality
limitation of Mn : below 25,000
Calibration curve is needed to obtain molecular weight of polymer sample
Standard material : Benzil
POLYMER CHEMISTRY
2.3.5 Mass spectrometry
A. Conventional mass spectrometer for low molecular-weight compound
energy of electron beam : 8 -13 electron volts (eV)
POLYMER CHEMISTRY
B. Modified mass spectrometer for synthetic polymer
a. matrix-assisted laser desorption ionization mass spectrometry
(MALDI-MS)
b. matrix-assisted laser desorption ionization time-of-flight
(MALDI-TOF)
c. soft ionization
sampling : polymers are imbedded by UV laser absorbable organic
compound containing Na and K.
d. are calculated by using mass spectra.
e. The price of this mass is much more than conventional mass.
f. Up to = 400,000 for monodisperse polymers.
POLYMER CHEMISTRY
FIGURE 2.5. MALDI mass spectrum of low-molecular-weight poly(methyl methacrylate).
POLYMER CHEMISTRY
2.3.6 Refractive Index Measurement
A. The linear relationship between refractive index and 1/Mn .
B. The measurement of solution refractive index by refractometer.
C. This method is for low molecular weight polymers.
D. The advantage of the method is simplicity.
POLYMER CHEMISTRY
2.4 Measurement of Weight Average Molecular Weight
2.4.1 Light Scattering
A. The intensity of scattered light or turbidity(τ) is depend on following factors
a. size
b. concentration
c. polarizability
d. refractive index
e. angle
f. solvent and solute interaction
g. wavelength of the incident light
POLYMER CHEMISTRY
g. wavelength of the incident light
 = HcMW
32
H= 3
Hc =

No2(dn/dc)2
4No
1
+ 2A2C
MP()
C : concentration
no: refractive index of the solvent
λ : wavelength of the incident light
No : Avogadro's number
dn/dc : specific refractive increment
P() : function of the angle,θ
A2 : second virial coefficient
Zimm plot (after Bruno Zimm) : double extrapolation of concentration
and angle to zero (Fig 2.6)
POLYMER CHEMISTRY
FIGURE 2.6. Zimm plot of light-scattering data.
Hc

1
Mw
C=0
Experimental
Extrapolated
sin2/2 + kc
FUNCTIONAL
POLYMERS LAB
POLYMER CHEMISTRY
2.4.1 Light Scattering
B. Light source
High pressure mercury lamp and laser light.
C. Limitation of molecular weight( ) : 104～107
FIGURE 2.7.
Schematic of a laser
light-scattering photometer.
FUNCTIONAL
POLYMERS LAB
POLYMER CHEMISTRY
2.4.2 Ultracentrifugation
A. This technique is used
a. for protein rather than synthetic polymers.
b. for determination of Mz
B. Principles : under the centrifugal field, size of molecules are
distributed perpendicularly axis of rotation.
Distribution process is called sedimentation.
FUNCTIONAL
POLYMERS LAB
POLYMER CHEMISTRY
2.5 Viscometry
A. IUPAC suggested the terminology of solution viscosities as following.
Relative viscosity :

t
rel =  =
 : solution viscosity
to
o
o: solvent viscosity
t : flow time of solution
t o: flow time of solvent
Specific viscosity :
 - o
o
sp =
sp
Reduced viscosity :
rel =
Inherent viscosity :
inh =
Intrinsic viscosity :
c
[] = (
c
t - to
to
=
=
= rel - 1
rel - 1
c
In rel
c
sp
) =(η )C = 0
c c=o inh
POLYMER CHEMISTRY
FIGURE 2.8. Capillary viscometers : (A) Ubbelohde, and (B) Cannon-Fenske.
FUNCTIONAL
POLYMERS LAB
POLYMER CHEMISTRY
B. Mark-Houwink-Sakurada equation
[η] = KMa
log[η] = logK + alogMv
(K, a : viscosity-Molecular weight constant, table2.3)
Mw > Mv > Mn
Mv
is closer to Mw
than Mn
POLYMER CHEMISTRY
TABLE 2.3. Representative Viscosity-Molecular Weight Constantsa
Polymer
Solvent
Polystyrene
(atactic)c
Polyethylene
(low pressure)
Poly(vinyl chloride)
Polybutadiene
98% cis-1,4, 2% 1,2
97% trans-1,4, 3% 1,2
Polyacrylonitrile
Poly(methyl methacrylate-costyrene)
30-70 mol%
71-29 mol%
Poly(ethylene terephthalate)
Nylon 66
Molecular Weight
Range  10-4
8-42e
4-137e
3-61f
3-100e
Kb 103
Cyclohexane
Cyclihexane
Benzene
Decalin
Temperature,
oC
35 d
50
25
135
Benzyl alcohol
Cyclohexanone
155.4d
20
4-35e
7-13f
156
13.7
0.50
1.0
Toluene
Toluene
DMFg
DMF
30
30
25
25
5-50f
5-16f
5-27e
3-100f
30.5
29.4
16.6
39.2
0.725
0.753
0.81
0.75
1-Chlorobutane
1-Chlorobutane
M-Cresol
M-Cresol
30
30
25
25
5-55e
4.18-81e
0.04-1.2f
1.4-5f
80
26.9
9.52
67.7
17.6
24.9
0.77
240
ab
0.50
0.599
0.74
0.67
0.67
0.63
0.95
0.61
aValue
taken from Ref. 4e.
bSee text for explanation of these constants.
cAtactic defined in Chapter 3.
d temperature.
eWeight average.
fNumber average.
gN,N-dimethylformamide.
POLYMER CHEMISTRY
2.6 Molecular Weight Distribution
2.6.1 Gel Permeation Chromatography (GPC)
A. GPC or SEC (size exclusion chromatography)
a. GPC method is modified column chromatography.
b. Packing material: Poly(styrene-co-divinylbezene),
glass or silica bead swollen and porous surface.
c. Detector : RI, UV, IR detector, light scattering detector
d. Pumping and fraction collector system for elution.
e. By using standard (monodisperse polystyrene), we can obtain Mn , Mw .
POLYMER CHEMISTRY
FIGURE 2.9. Schematic representation of a gel permeation chromatograph.
POLYMER CHEMISTRY
FIGURE 2.10. Typical gel permeation chromatogram. Dotted lines represent volume “counts.”
Detector
response
Baseline
Elution volume (Vr) (counts)
POLYMER CHEMISTRY
FIGURE 2.11. Universal calibration for gel permeation chromatography. THF, tetrahydrofuran.
Log([η]M)


109



107



108
Polystyrene (linear)
Polystyrene (comb)  
 Polystyrene (star)
 Heterograft copolyner 
 Poly (methyl methacrylate) 

Poly (vinyl chloride)
Styrene-methyl methacrylate graft copolymer
Poly (phenyl siloxane) (ladder)
Polybutadiene

106
105
18
20
22
24
26
Elution volume ()5 ml counts, THF solvent)
28
30
POLYMER CHEMISTRY
FIGURE 2.12. Typical semilogarithmic calibration plot of molecular weight versus retention volume.
Molecular weight (M)
106
105
104
103
Retention volume (Vr) (counts)
POLYMER CHEMISTRY
B. Universal calibration method
[η]1M1 = [η]2M2
to be combined Mark-Houwink-Sakurada
equation
logM2 = (
1
1 + a2
K1
)log( K ) + ( 11 ++ aa1
2
2
)logM1
POLYMER CHEMISTRY
2.6.2 Fractional Solution
Soxhlet-type extraction by using mixed solvent.
Reverse GPC : from low molecular weight fraction
to high molecular weight fraction
Inert beads are coated by polymer sample.
POLYMER CHEMISTRY
2.6.3 Fractional Precipitation
Dilute polymer solution is precipitated by variable non-solvent mixture.
Precipitate is decanted or filtered
Reverse fractional solution : from high molecular weight fraction to
low molecular fraction
POLYMER CHEMISTRY
2.6.4. Thin-layer Chromatography (TLC)
Alumina- or silica gel coated plate.
Low cost and simplicity.
Preliminary screening of polymer samples or
monitoring polymerization processes.
POLYMER CHEMISTRY