Part III: Polymer Characterization

-

Chapter 6: Characterization of Molecular Weight - Chapter 7: Polymer Solubility and Solution - Chapter 8: Phase Transition in Polymer

Chapter 6: Characterization of Molecular Weight

• Average molecular weight – M n : number-average molecular weight – M w : weight- average molecular weight – x n : no. avg. degree of polymerization – x w : wt. avg. degree of polymerization – M o : Mw of monomer (or repeating unit) – PI, MWD: polydispersity index = M w / M N

M

w

, M

n

calculations

M n = first moment =   C(M)M dM C(M) dM M w = 2 nd moment =   C(M)M 2 dM C(M)M dM

Definition of M w , M n

In integral form

M M n w

= First Moment =  

c

= Second Moment =  (

M c

(

c



M

) (

M c

(

MdM

)

dM

)

M M

2

dM

)

dM

In discrete summation form

n i = mole fraction = 

N N i i

w i = weight fraction = 

W i W i



M

M n M

M

w

n n i i n



w

   1 1    n i M i

n



i



M



i

i

w

i

i

i

w i i

  1 1 (   i M N

i

i i n  i M  2 i ( )  i

i



M i

i M ) i  

i

N i

n i

 

M

i 

i

N 2 N i i M  M i 2 i

i



i

   i 

N



N

W i

i

N i

N i N M M i i i

2

i

 

W i i



N i

= 

n i n M i M i i

Ex1.

Measurements on two monodisperse fractions of a linear polymer, A and B, yield molecular weights of 100 000 and 400 000, respectively. Mixture 1 is prepared from one part by weight of A and two parts by weight of B. Mixture 2 contains two parts by weight of A and one of B. Determine the weight- and number-average molecular weights of mixtures 1 and 2

Solution. For mixture 1

N A

 1 100000  1  10  5

N B

M n M w    2 400000   NiMi Ni   0 .

 ( Wi W ) Mi



5   1  10 1 3 10 1  5  5  1 10

  

 10 5  5 3  0 .

5 4   10  5 0 .

5  10 5



10   5 3



4  10 5  10 5



 2 .

0  10 5

For mixture 2

N A N B

  2 100000 1  400000  2  10  5

M

0 .

25  10  5

n





2  10  5

M w

10 2   10 2 3   5 1   0 .

25  10  5



4  10 0 5 .

25  1 3  10 4   5 10 5  10 5   2



 1 .

33  10 5  10 5

Ex2.

Two

polydisperse

samples are mixed in equal weights.

Sample A has

M n

= 100 000 and

M w

Sample B has

M n

= 200 000 and

M w

= 200 000.

= 400 000. What are

M n

and

M w

of the mixture ?

Solution. First, let’s derive general expressions for calculating the averages of mixtures: 

M n



W N



i



i Wi Ni

Where the subscript i refers to various polydisperse components of the mixture.Now, for a given component,

M w



Ni M wi

  

w x M w

(

mixture

) 

Wi



i



W i M



i wi W i W i

   

i

 

i Wi Wi

 

M wi

i

/ 

Wi

in the mixture. In this case, Let W A

M n

 =1 g and W B

W A



W B



W A

/

nA W B

= 1 g. Then /

M nB



  1 / 10 5 1    1 1 / 2  10 5   133000

M w

   

W A W

1 2 

A

2  

W B

10 5  

M



W A

1 2    4  

W A



W B

10  5

W B

  

M W B

300000 Note that even though the polydispersity index of each component of the mixture is 2.0, the PI of the mixture is greater, 2.25.

Determination of average molecular weight

• 2 catagories (a) Absolute methods: -Measured quantities are theoretically related to MW Ex. Endgroup analysis (Mn) Colligative property measurement (Mn) Light scattering (Mw) Ultracentrifuge (Mw) (b) Relative methods: -Measured quantities are related to MW -but need calibration with one of the absolute methods Ex. Solution viscosity (Mv) Size-Exclusion Chromatography (MWD)

(a) Absolute methods:

-Measured quantities are theoretically related to MW A1. Endgroup analysis (Mn) A2. Colligative property measurement (Mn) A3. Light scattering (Mw) A.4 Ultracentrifuge (Mw)

(b) Relative methods:

-Measured quantities are related to MW -but need calibration with one of the absolute methods Ex.1 Solution viscosity (M v ) Ex.2 Size-Exclusion Chromatography (MWD)

Solution viscosity (M

v

)

Vis=a+bt t = travel time a,b = constants

Solution viscosity

 =  (  S , T, polymer conc., no. of entanglements, M ) - measure using Ostwald type Viscometer Ublelohde type

Definition

:  = solution viscosity  s = solvent viscosity

Specific viscosity



SP

 SP =   S  S =   S  r = relative viscosity - 1 =  r – 1

Reduced viscosity

(normalized for conc.)

 red =  SP = (  /  S ) – 1 C C get rid of entanglement effect by reducing viscosity to zero conc.

Intrinsic viscosity

show effect of [  ] = lim c  coil to viscosity = lim  red c  0 ( 0  /  S ) – 1 [  ]    M W of polymer in sol polymer – n solvent system temp.

fix solvent, temp.

ขึ้นกับ coil dimension Get quantitative MW

Huggin’s equation

for   r < 2 or (  solution red = = [ c  < 2 ] + k′[  ] 2  solvent c ) (Huggin’s equation) where k′ Advantage if is ~ 0.4

( for a variety of polymer – solvent system) [  ] is known  can obtain relationship of  red and conc.

Equivalent form of Huggin’s

equation  inh   ] + k” [  ] 2 c where c  inh = inherent viscosity k” = k’ – 0.5

Vis conc.

1 2 0.1

0.5

[  ] Ref: S.L. Rosen,JohnWiley & Sons 1993

(alternative definition of intrinsic viscosity) [  ] = lim c  0  inh  lim c  0   ln( / s C )   Relationship of [  ] vs. M [for monodisperse sample of a certain MW] เรียกว่า Mark-Houwink-Sakurada (MHS) relation [  ] x K, a = K(M x ) a  (0.5

M v  [ ]  1 / K a     M x a W x W   1 / a     M x  n a x n x M M x x   1 / a  โดย 0.5 < a < 1, M n << M v < M w  n  x n M x x ( 1  a ) M x  1 / a

[  ] x = K(M x ) a Ref: S.L. Rosen,JohnWiley & Sons 1993

Ex. M v (viscosity average molecular weight) • Example 1: PMMA, calculate M v for mixture 1 and 2 in acetone at 30 o C and compare with M n and M w (From experiment: a = 0.72) Mixture 1: M v  compare      to : w W x M n   M x a   1 / a  200 , 000  1 3 1 x 10 0 .

72  2 3  4 x 10 5  0 .

72  1 / 0 .

72 M w  300 , 000  288 , 000 Mixture 2: M v  compare      to : w W x M   M x a   1 / a n  133 , 000  2 3 1 x 10 0 .

72  1 3  4 x 10 5  0 .

72  1 / 0 .

72 M w  200 , 000  187 , 000

Ex1.

Measurements on two monodisperse fractions of a linear polymer, A and B, yield molecular weights of 100 000 and 400 000, respectively. Mixture 1 is prepared from one part by weight of A and two parts by weight of B. Mixture 2 contains two parts by weight of A and one of B. •Example 1: PMMA, calculate Mv for mixture 1 and 2 in acetone at 30 oC and compare with Mn and Mw (From experiment: a = 0.72)

Solution viscosity terminology

Ref: S.L. Rosen,JohnWiley & Sons 1993

Last but Not Least!

Size-Exclusion Chromatography (MWD)

(or Gel Permeation Chomatography (GPC)) หา Molecular weight + MWD รวดเร็ว Porous particle (gel) “gel” – a cross linked polymer that is swollen by solvent

Unimodal = 1 peak Bimodal =2 peak

“column”

big molecule large molecules come out first smallest come out last small molecule large molecules come out first small molecules come out last (go through interstices of the substrate pores) Most common detector : differential refractometer (measure refractive index difference)

Ref: S.L. Rosen,JohnWiley & Sons 1993

Ref: S.L. Rosen,JohnWiley & Sons 1993