Measurement: Molecular Weight and Polymer Solutions
2.3 Measurement of Number Average Molecular Weight
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
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
Determine M. W. of polymers
c. amide and urethane polymer : -NH2, -NCO
Osmometry ⇒ Mn
Light scattering ⇒ Mw
Ultracentrifugation ⇒ Mw
End-group analysis ⇒ Mn
d. radioactive isotopes or UV, IR, NMR detectable functional group
2.3 Measurement of Number Average Molecular Weight
C.
Mn =
2 x 1000 x sample wt
End-group analysis
meq COOH + meq OH
Titration
polyester (-COOH, -OH), polyamide (-C(O)NH-), polyurethanes
(isocyanate), epoxy polymer (epoxide), acetyl-terminated
polyamide (acetyl)
Elemental analysis
Radioactive labeling
UV / NMR / (IR)
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
Membrane Osmometry
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
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
2.3.2 Membrane Osmometry
● based
Static equilibrium method ⎯
measure the hydrostatic head (∆h)
after equilibrium
Dynamic equilibrium method ⎯
measure the counter pressure
needed to maintain equal liquid levels
Summary of Measurement of Number Average
Molecular Weight
A. According to van't Hoff equation
h
(
π
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.]
Semipermeable membrane
FIGURE 2.4. Plot of reduced osmotic pressure (π/c) versus concentration (c).
Osmotic pressure (π) ⎯
related to M n by the van’t Hoff equation
extrapolated to zero concentration (C; g/L) to obtain the intercept
π
RT
( ) C =0 =
+ A2 C
C
Mn
π/c
PV = nRT
πV =
RT
Mn
π
V
RT
=
Wt M n
π
1 RT
=
C Mn
π
C
Slope = A2
Wt
RT
Mn
=
RT
Mn
RT
C
Mn
RT
C + A2C 2 + A3C 3 + ...
π=
Mn
π=
C
A2 = v2V-1(0.5 – χ)
2.3.3 Cryoscopy and Ebulliometry
π
RT
( ) C =0 =
+ A2C
C
Mn
2.3.3 Cryoscopy and Ebulliometry
A. Freezing-point depression (Cryoscopy)
B. Boiling-point elevation (Ebulliometry)
∆ Tf
RT2
+ A2C
(
)C=0 =
C
ρ∆Hf Mn
∆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
(
∆Tb
C
)C=0 =
RT2
ρ∆HvMn
+ A2C
∆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
Vapor Pressure Osmometry
2.3.4 Vapor Pressure Osmometry
The measuring vapor pressure difference of solvent and solution drops.
∆T =
2
RT
( λ100
)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
● 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
∆T = ⎜⎜
⎝ λ100 ⎠
λ : heat of vaporization per gram (solvent)
m : molarity
m=
W / Mn
n
= t
1000g 1000g
2.3.5 Mass spectrometry
A. Conventional mass spectrometer for low molecular-weight compound
energy of electron beam : 8 -13 electron volts (eV)
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
Soft ionization method
z field desorption (FD-MS)
z laser desorption (LD-MS)
z electrospray ionization (ESI-MS)
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.
FIGURE 2.5. MALDI mass spectrum of low-molecular-weight poly(methyl methacrylate).
●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 th
e
polymer
●Integrated peak areas →
the number of ions
● Mn, Mw can be calculated
2.4 Measurement of Weight Average Molecular Weight
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.
g. wavelength of the incident light
τ = HcMW
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
32π
H= 3
Hc =
τ
No2(dn/dc)2
λ4No
1
+ 2A2C
MP(θ)
C : concentration
no: refractive index of the solvent
λ : wavelength of the incident light
No : Avogadro's number
dn/dc : specific refractive increment
P(θ) : 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)
FIGURE 2.6. Zimm plot of light-scattering data.
2.4.1 Light Scattering
B. Light source
High pressure mercury lamp and laser light.
C. Limitation of molecular weight( ) : 104 107
Hc
τ
FIGURE 2.7.
Schematic of a laser
light-scattering photometer.
C=0
Experimental
Extrapolated
1
Mw
sin2θ/2 + kc
2.4.2 Ultracentrifugation
A. This technique is used
a. for protein rather than synthetic polymers.
b. for determination of Mz
2.5 Viscometry
A. IUPAC suggested the terminology of solution viscosities as following.
Relative viscosity :
η
t
ηrel = η =
η : solution viscosity
to
o
ηo: solvent viscosity
t : flow time of solution
t o: flow time of solvent
Specific viscosity :
η - ηo
ηo
ηsp =
B. Principles : under the centrifugal field, size of molecules are
distributed perpendicularly axis of rotation.
Reduced viscosity :
ηsp
ηrel =
c
Distribution process is called sedimentation.
Inherent viscosity :
c
FIGURE 2.8. Capillary viscometers : (A) Ubbelohde, and (B) Cannon-Fenske.
=
= ηrel - 1
ηrel - 1
c
In ηrel
c
ηinh =
ηsp
[η] = (
Intrinsic viscosity :
t - to
to
=
c
)c=o=(ηinh)C = 0
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
TABLE 2.3. Representative Viscosity-Molecular Weight Constantsa
Polymer
Solvent
Polystyrene
(atactic)c
Cyclohexane
Cyclihexane
Benzene
Decalin
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
Temperature, Molecular Weight
oC
Range × 10-4
8-42e
35 d
50
4-137e
3-61f
25
3-100e
135
Benzyl alcohol
Cyclohexanone
155.4d
4-35e
20
7-13f
Toluene
Toluene
DMFg
DMF
30
30
25
25
5-50f
5-16f
5-27e
3-100f
1-Chlorobutane
1-Chlorobutane
M-Cresol
M-Cresol
30
30
25
25
Kb× 103
80
26.9
9.52
67.7
156
13.7
0.50
1.0
30.5
29.4
16.6
39.2
0.725
0.753
0.81
0.75
17.6
24.9
0.77
240
5-55e
4.18-81e
0.04-1.2f
1.4-5f
2.6 Molecular Weight Distribution
ab
0.50
0.599
0.74
0.67
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.
0.67
0.63
0.95
0.61
c. Detector : RI, UV, IR detector, light scattering detector
d. Pumping and fraction collector system for elution.
aValue
taken from Ref. 4e.
text for explanation of these constants.
cAtactic defined in Chapter 3.
dθ temperature.
eWeight average.
fNumber average.
gN,N-dimethylformamide.
bSee
e. By using standard (monodisperse polystyrene), we can obtain Mn , Mw .
FIGURE 2.9. Schematic representation of a gel permeation chromatograph.
FIGURE 2.10. Typical gel permeation chromatogram. Dotted lines represent volume “counts.”
Detector
response
Baseline
Elution volume (Vr) (counts)
FIGURE 2.11. Universal calibration for gel permeation chromatography. THF, tetrahydrofuran.
FIGURE 2.12. Typical semilogarithmic calibration plot of molecular weight versus retention volume.
Log([η]M)
106
Molecular weight (M)
×∆+
××
∆
108
∆
∆
∆
109
+
Polystyrene (linear)
Polystyrene (comb) × ∆
+ Polystyrene (star)
∆ Heterograft copolyner +
× Poly (methyl methacrylate)
∆
106
∆
107
∆+
×
Poly (vinyl chloride)
Styrene-methyl methacrylate graft copolymer
Poly (phenyl siloxane) (ladder)
Polybutadiene
104
103
105
18
20
22
24
26
Elution volume ()5 ml counts, THF solvent)
105
28
30
Retention volume (Vr) (counts)
B. Universal calibration method
2.6.2 Fractional Solution
[η] 1M1 = [η] 2M2
to be combined Mark-Houwink-Sakurada
equation
Soxhlet-type extraction by using mixed solvent.
Reverse GPC : from low molecular weight fraction
to high molecular weight fraction
logM2 = (
1
1 + a2
)log( KK1 ) + (
2
1 + a1
1 + a2
)logM1
2.6.3 Fractional Precipitation
Inert beads are coated by polymer sample.
2.6.4. Thin-layer Chromatography (TLC)
Dilute polymer solution is precipitated by variable non-solvent mixture.
Alumina- or silica gel coated plate.
Precipitate is decanted or filtered
Low cost and simplicity.
Reverse fractional solution : from high molecular weight fraction to
low molecular fraction
Preliminary screening of polymer samples or
monitoring polymerization processes.