1. 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. 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 ; for vinyl polymer
    15, ,000 for polar functional group containing polymer (polyamide)
POLYMER CHEMISTRY

3. B. Determination of molecular weight
Absolute method :
mass spectrometry
colligative property
end group analysis
light scattering
ultracentrifugation.
b. Relative method : solution viscosity
c. Fractionation method : GPC
POLYMER CHEMISTRY

4. C. Definition of average molecular weight
a. number average molecular weight ( Mn )
                       Mn=
(colligative property and  end group analysis)
b. weight average molecular weight ( Mw)
                 
Mw=
(light scattering)
 i i
Ni N M
WiMi
Wi
POLYMER CHEMISTRY

5. c. z average molecular weight ( MZ )
                     
   MZ=
(ultracentrifugation)
d. general equation of average molecular weight :
M =
                         
     ( a=0 , Mn            a=1 , Mw            a=2 , Mz         )
e. Mz > Mw > Mn   
C. Definition of average molecular weight
NiMi3
NiMi2
NiMia+1
NiMia
POLYMER CHEMISTRY

6. D. Polydispersity index : width of distribution
polydispersity index (PI) = Mw / Mn ≥ 1
POLYMER CHEMISTRY

7. E. Example of molecular weight calculation
a. 9 moles, molecular weight (Mw) = 30,000
   
5 moles, molecular weight ( Mw) = 50,000
(9 mol x 30,000 g/mol) + (5 mol x 50,000 g/mol)
Mn=
= 37,000 g/mol
9 mol + 5 mol
9 mol(30,000 g/mol)2 + 5 mol(50,000 g/mol)2
Mw =
= 40,000 g/mol
9 mol(30,000 g/mol) + 5 mol(50,000 g/mol)
POLYMER CHEMISTRY

8. E. Example of molecular weight calculation
b. 9 grams, molecular weight ( Mw ) = 30,000
   
5 grams, molecular weight ( Mw ) = 50,000
= 35,000 g/mol
Mn =
9 g + 5 g
(9 g/30,000 g/mol) + (5 g/50,000 g/mol)
Mw =
= 37,000 g/mol
POLYMER CHEMISTRY

9. 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

10. 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

11. C. Solubility parameter : δ
                
Hmix=Vmix[( )1/2-( ) 1/2]212
   
  ψ1, ψ2 = volume fraction
   ΔE1/V1, ΔE2/V2 = cohesive energy densities
   δ1, δ2 = solubility parameter
                    
                    δ1, δ2   = (                  )1/2
Hmix= Vmix(δ1 – δ2)212
E = Hvap- RT
δ1 = ( )1/2
                   
 if δ1= δ2, then Hmix= 0        V1
E1 V2
E2 V E V
H vap - RT
POLYMER CHEMISTRY

12. D. Small's and Hoy's G parameter
a. Small(designated G derived from Heat of vaporization, Table 2.1)
δ =
                            
 ( d : density , M :  molecular weight of unit )
ex) polystyrene          
δ = = 9.0
b. Hoy(designated G based on vapor pressure measurement, Table 2.1)
ex) polystyrene :          
dG M M
104
1.05( )
dG M
104
1.05[ (117.1)]
= 9.3
POLYMER CHEMISTRY

13. E. Hydrodynamic volume of polymer molecules in solution.
  to be depended on followings
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 = ro22
s2= so22
 =
(r2)1/2
(ro2)1/2
The greater the value of α, the ‘better’
the solvent
 α = 1, 'ideal' statistical coil.

14. 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] : intrinsic viscosity
                     M : average molecular weight
                      ψ : Flory constant (3×1024/mol)
                      r : end-to-end distance
[η] =
(r2)3/2 M
POLYMER CHEMISTRY

15. 2.2 Polymer Solutions H. Mark-Howink-Sakurada equation
: The relationship between intrinsic viscosity and molecular weight
                                
[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
[η] = KMa
POLYMER CHEMISTRY

16. 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

17. 2.3 Measurement of Number Average Molecular Weight
2 x 1000 x sample wt
Mn = C.
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

18. FIGURE 2.2 Schematic representation of a membrane osmometer.

19. 2.3.2 Membrane Osmometry  RT ( )C=0 = + A2C c Mn
A. According to van't Hoff equation
                 
    
limitation of : 50,000～2,000,000
    The major error arises from low-molecular-weight species diffusing
    through the membrane. ( c 
)C=0 = Mn RT
+ A2C
FIGURE 2.3 Automatic membrane osmometer [Courtesy of Wescan Instruments, Inc.]

20. FIGURE 2.4. Plot of reduced osmotic pressure (/c) versus concentration (c).Mn RT C
Slope = A2
POLYMER CHEMISTRY

21. 2.3.3 Cryoscopy and Ebulliometry
A. Freezing-point depression (Cryoscopy)
      
      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 ( C
Tf
)C=0 =
Hf Mn
RT2
+ A2C
POLYMER CHEMISTRY

22. 2.3.3 Cryoscopy and Ebulliometry
  B. Boiling-point elevation (Ebulliometry)
      
     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 ( C
Tb
)C=0 =
HvMn
RT2
+ A2C
POLYMER CHEMISTRY

23. 2.3.4 Vapor Pressure Osmometry
The measuring vapor pressure difference of solvent and solution  drops.
         
 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
T = (
100
RT2 )m
POLYMER CHEMISTRY

24. 2.3.5 Mass spectrometry
A. Conventional mass spectrometer for low molecular-weight compound
  energy of electron beam : electron volts (eV)
POLYMER CHEMISTRY

25. 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

26. FIGURE 2.5. MALDI mass spectrum of low-molecular-weight poly(methyl methacrylate).
POLYMER CHEMISTRY

27. 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

28.  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

29. g. wavelength of the incident light  
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)
 = HcMW
32 3
H =
4No
No2(dn/dc)2  Hc =
MP() 1
+ 2A2C
POLYMER CHEMISTRY

30. FIGURE 2.6. Zimm plot of light-scattering data.
sin2/2 + kc  Hc Mw 1
C=0
Experimental
Extrapolated
POLYMER CHEMISTRY

31. 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.
POLYMER CHEMISTRY

32. 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.
POLYMER CHEMISTRY

33. 2.5 Viscometry
A. IUPAC suggested the terminology of solution viscosities as following.
  Relative viscosity :  
 : solution viscosity
                                        o: solvent viscosity
                                         t : flow time of solution
                                        t o: flow time of solvent   
Specific viscosity :
Reduced viscosity :
Inherent viscosity :
Intrinsic viscosity : 
rel = o  = to t
rel - 1
sp = o
 - o = to
t - to c
rel =
sp
inh =
In rel
[] = (
)c=o=(ηinh)C = 0
POLYMER CHEMISTRY

34. FIGURE 2.8. Capillary viscometers : (A) Ubbelohde, and (B) Cannon-Fenske.
POLYMER CHEMISTRY

35. B. Mark-Houwink-Sakurada equation [η] = KMa log[η] = logK + alogMv
             
      
[η] = KMa
log[η] = logK + alogMv
(K, a : viscosity-Molecular weight constant, table2.3)
           
    
Mv        is closer to Mw       than Mn       
Mw > Mv > Mn
POLYMER CHEMISTRY

36. TABLE 2.3. Representative Viscosity-Molecular Weight Constantsa
Polymer
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-co-styrene)
30-70 mol%
71-29 mol%
Poly(ethylene terephthalate)
Nylon 66
Solvent
Cyclohexane
Cyclihexane
Benzene
Decalin
Benzyl alcohol
Cyclohexanone
Toluene
DMFg
DMF
1-Chlorobutane
M-Cresol
Temperature, oC
35 d 50 25
135
155.4d 20 30
Molecular Weight
Range  10-4
8-42e
4-137e
3-61f
3-100e
4-35e
7-13f
5-50f
5-16f
5-27e
3-100f
5-55e e f
1.4-5f ab
0.50
0.599
0.74
0.67
1.0
0.725
0.753
0.81
0.75
0.63
0.95
0.61
Kb 103 80
26.9
9.52
67.7
156
13.7
30.5
29.4
16.6
39.2
17.6
24.9
0.77
240
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

37. 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

38. FIGURE 2.9. Schematic representation of a gel permeation chromatograph.
POLYMER CHEMISTRY

39. Elution volume (Vr) (counts) Baseline Detector response
FIGURE Typical gel permeation chromatogram. Dotted lines represent volume “counts.”
Elution volume (Vr) (counts)
Baseline
Detector
response
POLYMER CHEMISTRY

40. Heterograft copolyner Poly (methyl methacrylate) Poly (vinyl chloride)
FIGURE Universal calibration for gel permeation chromatography. THF, tetrahydrofuran.
Log([η]M)
109
108
107
106
105          
Polystyrene (linear)
Polystyrene (comb)
Polystyrene (star)
Heterograft copolyner
Poly (methyl methacrylate)
Poly (vinyl chloride)
Styrene-methyl methacrylate graft copolymer
Poly (phenyl siloxane) (ladder)
Polybutadiene           
POLYMER CHEMISTRY
Elution volume ()5 ml counts, THF solvent)

41. FIGURE 2.12. Typical semilogarithmic calibration plot of molecular weight versus retention volume.
Retention volume (Vr) (counts)
106
105
104
103
Molecular weight (M)
POLYMER CHEMISTRY

42. )log( ) + ( )logM1 B. Universal calibration method
       
 to be combined Mark-Houwink-Sakurada equation
[η]1M1 = [η]2M2
logM2 = (
1 + a2 1
)log( K2 K1
) + (
1 + a1
)logM1
POLYMER CHEMISTRY

43. 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

44. 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

45. 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