Chapter 2. Molecular Weight and Polymer Solutions

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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
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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)
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• 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

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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
?Wi
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c. z average
molecular weight ( MZ )                      
   MZ
(ultracentrifugation) d.
general equation of average molecular weight
M                
               ( a0 , Mn           
a1 , Mw            a2 , Mz         )  
e. Mz gt Mw gt Mn   
C. Definition of average molecular weight
?NiMi3
?NiMia1
?NiMia
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D. Polydispersity index   width of distribution
 
polydispersity index (PI) Mw / Mn 1
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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)
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E. Example of molecular weight calculation
 
b. 9 grams, molecular
weight ( Mw ) 30,000    
5 grams, molecular weight ( Mw )
50,000
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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
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2.2 Polymer Solutions
B. Thermodynamics of solubility
Gibb's free energy
relationship                          


  ?G ??H - T?S   
?G lt 0
spontaneously dissolve    T
and ?S are always positive for dissolving
process.    Conditions to be
negative ?G,    ?H must be
negative or smaller than T?S.
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C.  Solubility parameter d
                 ?HmixVmix(
)1/2-( ) 1/22?1?2    

  ?1, ?2 volume fraction   
?E1/V1, ?E2/V2 cohesive energy densities
   d1, d2 solubility
parameter                                  
       d1, d2   
(                  )1/2
?Hmix Vmix(d1
d2)2?1?2 ?E
?Hvap- RT d1 (
)1/2                    
 if d1 d2, then Hmix 0
      
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D. Small's and Hoy's
G parameter   a.
Small(designated G derived from Heat of
vaporization, Table 2.1)
d                      
        ( d density , M
  molecular weight of unit )  
ex) polystyrene          
d
9.0
b. Hoy(designated G
based on vapor pressure measurement, Table 2.1)
                            
d
ex) polystyrene           

d
d?G
M
M
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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
The greater the value of a, the better the
solvent  a 1, 'ideal' statistical coil.
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2.2 Polymer Solutions
F. theta(?) temperature and
theta(?) solvent        The
lowest temperature at which a1 theta(?)
temperature  blink The
solvent satisfied this condition theta(?)
solvent  point
G. Flory-Fox equation
The relationship among hydrodynamic
volumes,     intrinsic
viscosity and molecular weight        

?
intrinsic viscosity                     
M average
molecular weight                      
? Flory
constant (31024/mol)                      
r
end-to-end distance
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2.2 Polymer Solutions
H. Mark-Howink-Sakurada equation
The relationship between
intrinsic viscosity and molecular weight
                                
?
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
? KMa
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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 -CHCH2    b. ester polymer
-COOH, -OH    c. amide and urethane polymer
-NH2, -NCO    d. radioactive isotopes or UV, IR,
NMR detectable functional group
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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
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FIGURE 2.2 Schematic representation of a membrane
osmometer.
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2.3.2 Membrane Osmometry   A. According to
van't Hoff equation                       
limitation of 50,0002,000,000     The
major error arises from low-molecular-weight
species diffusing     through the membrane.
FIGURE 2.3 Automatic membrane osmometer Courtesy
of Wescan Instruments, Inc.
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FIGURE 2.4. Plot of reduced osmotic pressure
(?/c) versus concentration (c).
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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
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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. (110-4?)     limitation
of Mn below 20,000
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2.3.4 Vapor Pressure Osmometry  
The measuring vapor pressure difference of
solvent and solution  drops.          

 ? 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
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2.3.5 Mass spectrometry   A.
Conventional mass spectrometer for low
molecular-weight compound   energy of electron
beam 8 -13 electron volts (eV)
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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.
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FIGURE 2.5. MALDI mass spectrum of
low-molecular-weight poly(methyl methacrylate).
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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.
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 2.4 Measurement of Weight Average Molecular
Weight 2.4.1 Light Scattering   A. The
intensity of scattered light or turbidity(t) 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
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g. wavelength of the incident light
   C   concentration                
          no refractive index of the solvent   
     ? wavelength of the incident light    
No  Avogadro's number     dn/dc  specific
refractive increment      P(?)  function of the
angle,?     A2 second virial coefficient
     Zimm plot (after Bruno Zimm) double
extrapolation of concentration 
and angle
to zero (Fig 2.6)
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FIGURE 2.6. Zimm plot of light-scattering data.
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2.4.1 Light Scattering
B. Light source    High pressure mercury lamp
and laser light. C. Limitation of molecular
weight( ) 104107
FIGURE 2.7. Schematic of a laser
light-scattering photometer.
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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.
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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  
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FIGURE 2.8. Capillary viscometers (A)
Ubbelohde, and (B) Cannon-Fenske.
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B. Mark-Houwink-Sakurada
equation                     


? KMa log?
logK alogMv
(K, a viscosity-Molecular
weight constant, table2.3)                 
Mv        is
closer to Mw       than Mn       
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TABLE 2.3. Representative Viscosity-Molecular
Weight Constantsa
Solvent Cyclohexane Cyclihexane Benzene Decalin
Benzyl alcohol Cyclohexanone Toluene Toluene DMFg
DMF 1-Chlorobutane 1-Chlorobutane M-Cresol M-Cr
esol
Temperature, oC 35 d 50 25 135 155.4d 20 30 30
25 25 30 30 25 25
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 5-5
5e 4.18-81e 0.04-1.2f 1.4-5f
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
ab 0.50 0.599 0.74 0.67 0.50 1.0 0.725 0.753 0.
81 0.75 0.67 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.

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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 .
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FIGURE 2.9. Schematic representation of a gel
permeation chromatograph.
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FIGURE 2.10. Typical gel permeation chromatogram.
Dotted lines represent volume counts.
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FIGURE 2.11. Universal calibration for gel
permeation chromatography. THF, tetrahydrofuran.
Log(?M)

?
?
?
?
?
?
?
?
?
?
Polystyrene (linear) Polystyrene
(comb) Polystyrene (star) Heterograft
copolyner Poly (methyl methacrylate) Poly (vinyl
chloride) Styrene-methyl methacrylate graft
copolymer Poly (phenyl siloxane)
(ladder) Polybutadiene
?
?
?
?
?
?
?
?
?
?
?
18 20 22 24
26 28 30
Elution volume ()5 ml counts, THF solvent)
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FIGURE 2.12. Typical semilogarithmic calibration
plot of molecular weight versus retention volume.
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B. Universal calibration method        
 to be combined Mark-Houwink-Sakurada
equation
?1M1 ?2M2
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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.
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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
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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.