Chapter 2. Molecular Weight and Polymer Solutions

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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

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

a. 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

MN

Wi

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WiMi

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   

NiMi3

NiMia+1

NiMia

NiMi2

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C. Definition of average molecular weight

  polydispersity index (PI) = Mw / Mn ≥ 1

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D. Polydispersity index :  width of distribution

E. Example of molecular weight calculation  

a. 9 moles, molecular weight (Mw) = 30,000     5 moles, molecular weight ( Mw) = 50,000

Mn=9 mol + 5 mol

(9 mol x 30,000 g/mol) + (5 mol x 50,000 g/mol)= 37,000 g/mol

Mw = 9 mol(30,000 g/mol) + 5 mol(50,000 g/mol)

9 mol(30,000 g/mol)2 + 5 mol(50,000 g/mol)2

= 40,000 g/mol

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  b. 9 grams, molecular weight ( Mw ) = 30,000     5 grams, molecular weight ( Mw ) = 50,000

E. Example of molecular weight calculation

= 35,000 g/molMn =9 g + 5 g

(9 g/30,000 g/mol) + (5 g/50,000 g/mol)

Mw =(9 g/30,000 g/mol) + (5 g/50,000 g/mol)

9 g + 5 g= 37,000 g/mol

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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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B. Thermodynamics of solubility : Gibb's free energy relationship

                         

  G =H - TS  

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.

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2.2 Polymer Solutions

C.  Solubility parameter : δ                 

Hmix=Vmix[( )1/2-( ) 1/2]212   

  ψ1, ψ2 = volume fraction

   ΔE1/V1, ΔE2/V2 = cohesive energy densities

   δ1, δ2 = solubility parameter                     

                    δ1, δ2   = (                  )1/2

Hmix= Vmix(δ1 – δ2)212

E = Hvap- RT

δ1 = ( )1/2

                   

 if δ1= δ2, then Hmix= 0       

V1

E1

V2

E2

V

H vap - RT

VE

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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)                             

δ =

ex) polystyrene :          

δ =

dG

MM

104

1.05(133+28+735)

dG

MM

1041.05[131.5+85.99+6(117.1)]

= 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

The greater the value of α, the ‘better’ the solvent  α = 1, 'ideal' statistical coil.

r 2 = ro22

s2= so22

= (r2)1/2

(ro2)1/2

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

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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

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2.2 Polymer Solutions

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

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Mn =2 x 1000 x sample wt

meq COOH + meq OHC.

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

2.3 Measurement of Number Average Molecular Weight

FIGURE 2.2 Schematic representation of a membrane osmometer.

2.3.2 Membrane Osmometry   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.]

/c

Mn

RT

C

Slope = A2

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

(C

Tf )C=0 = Hf Mn

RT2

+ A2C

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  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

(CTb )C=0 = HvMn

RT2

+ A2C

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2.3.3  Cryoscopy and Ebulliometry

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 = ( 100RT2

)m

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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).

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(τ) is depend on following factors     a. size     b. concentration     c. polarizability     d. refractive index     e. angle     f. solvent and solute interaction

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g. wavelength of the incident light

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

323H = 4No

No2(dn/dc)2

Hc =

MP()1

+ 2A2C

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FIGURE 2.6. Zimm plot of light-scattering data.

sin2/2 + kc

Hc

Mw

1

C=0

ExperimentalExtrapolated

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B. Light source    High pressure mercury lamp and laser light. C. Limitation of molecular weight( ) : 104～ 107

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2.4.1 Light Scattering

FIGURE 2.7. Schematic of a laser light-scattering photometer.

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 : 

rel = o

=

to

t

rel - 1sp = o

- o =to

t - to =

c

rel = csp = c

rel - 1

inh = cIn rel

[] = (csp )c=o=(η inh)C = 0 POLYMER CHEMISTRY

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

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B. Mark-Houwink-Sakurada equation                      [η] = KMa

log[η] = logK + alogMv

(K, a : viscosity-Molecular weight constant, table2.3)                 

Mv        is closer to Mw       than Mn       

Mw > Mv > Mn

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TABLE 2.3. Representative Viscosity-Molecular Weight Constantsa

Polymer

Polystyrene(atactic)c

Polyethylene(low pressure)Poly(vinyl chloride)

Polybutadiene98% cis-1,4, 2% 1,297% trans-1,4, 3% 1,2Polyacrylonitrile

Poly(methyl methacrylate-co-styrene)30-70 mol%71-29 mol%Poly(ethylene terephthalate)Nylon 66

Solvent

CyclohexaneCyclihexaneBenzeneDecalin

Benzyl alcoholCyclohexanone

TolueneTolueneDMFg

DMF

1-Chlorobutane1-ChlorobutaneM-CresolM-Cresol

Temperature, oC35 d

5025135

155.4d

20

30302525

30302525

Molecular WeightRange 10-4

8-42e

4-137e

3-61f

3-100e

4-35e

7-13f

5-50f

5-16f

5-27e

3-100f

5-55e

4.18-81e

0.04-1.2f

1.4-5f

Kb 103

80 26.9 9.52 67.7

156 13.7

30.5 29.4 16.6 39.2

17.6 24.9 0.77240

ab

0.500.5990.740.67

0.501.0

0.7250.7530.810.75

0.670.630.950.61

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 .

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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.”

Elution volume (Vr) (counts)

BaselineDetectorresponse

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FIGURE 2.11. Universal calibration for gel permeation chromatography. THF, tetrahydrofuran.

Log([η]M)

109

108

107

106

105

18 20 22 24 26 28 30

Polystyrene (linear)Polystyrene (comb)Polystyrene (star)Heterograft copolynerPoly (methyl methacrylate)Poly (vinyl chloride)Styrene-methyl methacrylate graft copolymerPoly (phenyl siloxane) (ladder)Polybutadiene

Elution volume ()5 ml counts, THF solvent)

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FIGURE 2.12. Typical semilogarithmic calibration plot of molecular weight versus retention volume.

Retention volume (Vr) (counts)

106

105

104

103

Mol

ecu

lar

wei

ght

(M)

B. Universal calibration method        

 to be combined Mark-Houwink-Sakurada equation

[η]1M1 = [η]2M2

logM2 = ( 1 + a2

1 )log( K2 K1 ) + (1 + a2

1 + a1 )logM1

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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.

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