Chemical Engineering 160/260Polymer Science and Engineering

Lecture 5 - Indirect Measures ofLecture 5 - Indirect Measures ofMolecular Weight: Intrinsic ViscosityMolecular Weight: Intrinsic Viscosityand Gel Permeation Chromatographyand Gel Permeation Chromatography

January 26, 2001January 26, 2001

End-to-end Distance of a Random Coil

rIn general, we will express the end-to-end distance in termsof the root-mean-square statistical average.

⟨ ⟩r2 1 2/

Freely Jointed Model Polymer Chain:Mean-squared End-to-end Distance

r rr li

i

n

==∑

1

r r r l l l l li j

i ji

ii j

i j n

2 2

0

2= • = • = + •∑ ∑ ∑< < ≤

r r r r r r

,

⟨ ⟩ = ⟨ ⟩ + ⟨ • ⟩∑ ∑

< < ≤

r l l lii

ii j n

j2 2

0

2r r

⟨ ⟩ = + ⟨ • ⟩ =

<∑r nl l l nlii j

j2 2 22

r r

rr

Average over allconfigurations.

There is no bondcorrelation.

Vector sum

Outline

!! DefinitionsDefinitions

!! Equivalent sphere and Einstein relationshipEquivalent sphere and Einstein relationship

!! Effect of concentrationEffect of concentration

!! Mark-Mark-HouwinkHouwink-Sakurada equation-Sakurada equation

!! Gel permeation chromatographyGel permeation chromatography

!! Molecular weight summaryMolecular weight summary

Definitions

Relative viscosity

Specific viscosity

Reduced viscosity

Inherent viscosity

Intrinsic viscosity

ηηηrel

o o

solution

solvent

t solution

t solvent= =

( )( )

( )( )

ηη η

ηηsp

o

orel

solution solvent

solvent=

−= −

( ) ( )( )

1

ηη

inhrel

c=

ln

ηη

redsp

c=

ηηsp

cc

≡ [ ]

=0

Equivalent Sphere

Einstein Relationship

η η ωφ= +[ ]o 1

For a suspension containing volume fraction φ of suspended material with shape factor ω, Einstein found that

where η is the viscosity of the suspension and ηo is theviscosity of the suspending fluid. For a sphere orrandom coil, ω = 2.5.

If the polymer is treated as an equivalent hydrodynamicsphere with volume Ve and radius Re we obtain

ηspp

e

n

VV=

2 5. V Re e=

43

3πn

V

cN

Mp A

v

=

ηηsp

c

A e

vc

N V

M

≡ [ ] =

=0

2 5.

Excluded Volume

That portion of the solution volume that is inaccessible topolymer chain segments due to prior occupancy by otherchain segments is known as the excluded volume.

As a consequence of the volume exclusion, the overallspatial dimensions of a real polymer chain must increaserelative to those predicted by the simple chain models.

One can compensate for chain expansion due to excludedvolume through placing the polymer in a poor solventsuch that interactions between polymer segments andsolvent molecules are thermodynamically unfavorable.Such a solvent, at a given temperature, is a theta solvent.

Excluded Volume of a Flexible Chain

⟨ ⟩ = ⟨ ⟩r r o2 2 2α

α = expansion factor

For large α, α ∝ M0 10.

Effect of solvent on chain dimensions: Good solvent - chain expansion Poor solvent - chain contraction Theta solvent - chain contraction exactly compensates for excluded volume effect

⟨ ⟩ = ⟨ ⟩

=

r r o2 2

1αA2 = 0 for theta solvent

Expansion Factor

V

M

R

M

R

MMe

v

e

v

e

vv= =

43

43

3 2 3 2

1 2π π/

/

~ constant

Define an expansion factor for a coil in a good solventcompared to a theta solvent.

R Re eo= α

ηπ

α[ ] =

2 5

43

2 3 2

1 2 3./

/NR

MMA

eo

vv

αηη

3 = [ ][ ]Θ

Interrelationships Among Parameters

For dilute solutions,

ln ln( )η η η

ηrel sp sp

sp= + ≅ − +12

2

L

lnηηrel

cc

= [ ]=0

ηηsp

cc

≡ [ ]

=0

Compare this to

Effect of Concentration in Dilute Solution

Huggins Equation

Kraemer Equation

For many polymers in good solventskH = 0.4 +/- 0.1 kK = 0.05 +/- 0.05

ηη ηsp

Hck c= [ ] + [ ]2

lnηη ηrel

Kck c= [ ]− [ ]2

Intrinsic Viscosity and Molecular Weight

An empirical relationship that works well for correlatingintrinsic viscosities and molecular weights of fractionatedsamples is the Mark-Houwink-Sakurada equation.

η[ ] =i iaKM

For a sample at infinite dilution,

η η η ηsp i i i H i i i i ia

ic k c c KM c( ) = [ ] + [ ] ≅ [ ] =2 2

ηsp ia

i iK M c= ∑

For the unfractionated solution,

Intrinsic Viscosity and Molecular Weight

The observed intrinsic viscosity may be obtained from

ηη

[ ] = →

=

∑lim( )c

cK M

c

csp

ia

i

i0

Sincec

c

c

ci i

ii

=∑ the weight ratio is

w

w

w

wi i

ii

=∑

η[ ] = =∑∑ ∑

K w M

wK W M

i ia

i

ii

ii iaand

Viscosity Average Molecular Weight

M W Mv i ia

i

a≡ ( )∑

1/

Define the viscosity average molecular weight by

η[ ] = KMva

The intrinsic viscosity then is related to molecular weight by

Typically, 0.5 < a < 0.8 for flexible polymers, witha = 0.5 for theta conditions and increasing with increasing solvent quality. Also, a = 1 for semicoils and a = 2 for rigid rods.

Mark-Houwink-Sakurada Parameters

Polymer Solvent Temp. Mol. Wt. 100K a

PMMA chloroform 25 C 80,000-1,400,000

0.48 0.80

methylethylketone

25 C 80,000-1,400,000

0.68 0.72

acetone 25 C 80,000-1,400,000

0.75 0.70

PS benzene 20 C 1,200-140,000

1.23 0.72

toluene 20-30 C 20,000-2,000,000

1.05 0.72

methylethylketone

20-40 C 8,000-4,000,000

3.82 0.58

Gel Permeation Chromatography

A mixture of different size solute molecules is eluted through a column of porous particles. Larger moleculesare swept through unhindered, while small moleculesare retarded in the pores.

GPC Calibration

ηηsp

c

A e

c

N V

M

≡ [ ] =

=0

2 5.

The relationship between intrinsic viscosity and hydrodynamicvolume is the basis for a “universal” calibration procedure.

ln ln . ln lim( )η[ ]( ) = ( ) + →[ ]M N c VA e2 5 0

Ve

[η]M

GPC Calibration

For equal elution volumes of two different polymers,

η η[ ] = = [ ] =+ +1 1 1 1

12 2 2 2

11 2M K M M K Ma a

ln ln lnMa

aM

a

K

K21

21

2

1

2

11

11

=++

++

η[ ] = +i i i i

aM K M i 1

Use the Mark-Houwink-Sakurada relation

Molecular Weight Summary: Averages

M W Mv i ia

i

a≡ ( )∑

1/

Mcc

M

w

M

n

ii

i

ii

i

ii

= =∑∑ ∑

1

Mc M

c

n M

n Mw

i ii

ii

i ii

i ii

= =∑∑

∑∑

2

Mn M

n Mz

i ii

i ii

=∑∑

3

2

Mn M

nn

i ii

ii

=∑∑

M M M Mz w v n> > >

Molecular Weight Summary:“Most Probable” Distribution

x

x

M

Mpw

n

w

n

= = + →1 2

x x p pp

pwx

x= −( ) =

+−

−∑ 2 1 2111

x xp ppn

x

x= −( ) =

−−∑ 1 1

11

N p pxx= −( )−1 1 W xp px

x= −( )−1 21

Molecular Weight Summary: Osmometry

lim( )cc

RT

Mn

→

=0

π

πc

RTM

A c A cn

= + + +

12 3

2 L

Under theta conditions, A2 = 0. For a particular polymer,the temperature at which this occurs depends on the solvent.

Mean-squared Radius of Gyration

sm s

m

m s

m n ns

i i

n

i

n

i

n

i

n2

2

0

0

2

0 2

011

1= =

+=

+

∑

∑

∑∑( )

A1

An

Ao

A n-1

A n-2

s1

sn

sn-2s0

sn-1Center of mass

⟨ ⟩ ≡s Rg2 1 2/

Molecular Weight Summary:Light Scattering

Hc

R MA c

w( )θ θ

= +

=0

2

12

Hc

R MR

c o wg( ) ©

sinθ

πλ

θ

= +

+

=

11

13

42

22 2 L

RI w

I Vo s

( )θ θ=2

Hn

dn

dcN

o

A

=

2 2 22

4

π

λ

Hc

R RT c T( )θ∂π∂

=

1

Molecular Weight Summary:Intrinsic Viscosity

η[ ] = KMva

ηη η

ηηsp

o

orel

solution solvent

solvent=

−= −

( ) ( )( )

1

ηηsp

cc

≡ [ ]

=0

Molecular Weight Summary:Gel Permeation Chromatography

ηηsp

c

A e

c

N V

M

≡ [ ] =

=0

2 5.