Chemical Engineering 160/260Polymer Science and Engineering

Lecture 4 - Direct Measures ofLecture 4 - Direct Measures ofMolecular Weight: Molecular Weight: OsmometryOsmometry and and

Light ScatteringLight Scattering

January 24, 2001January 24, 2001

Outline

!! Chemical potential of dilute solutionsChemical potential of dilute solutions

!! Raoult’s Raoult’s LawLaw

!! Osmotic pressureOsmotic pressure

!! Van’t Hoff Van’t Hoff EquationEquation

!! Virial Virial expansionexpansion

!! Light scatteringLight scattering

Experimental Approaches

!! DirectDirect measures of molecular weight may bemeasures of molecular weight may beobtained from obtained from osmometryosmometry, light scattering, and, light scattering, andultracentrifugationultracentrifugation..

!! IndirectIndirect measures of molecular weight, such as measures of molecular weight, such asviscometry viscometry and gel permeation chromatography,and gel permeation chromatography,yield relative estimates that must be calibrated.yield relative estimates that must be calibrated.

Chemical Potential of Dilute Solutions

The chemical potential of a solvent in a solution is

µ µs so s

soRT

P

P= + ln

= chemical potential for pure solvent at T

= vapor pressure above the solution at T

= vapor pressure above the pure solvent at T

µ µs so

sRT a= + ln

µso

sa

P

P

s

so

= activity of solvent

If the solvent vapor obeys the ideal gas law, we have

P Ps so

s so

<

<µ µ

Raoult’s Law

In general, the activity is related to vapor pressure by

aP

Pss

so=

If the solution is sufficiently dilute, Raoult’s Lawwill be obeyed

a xP

Ps ss

so= =

= mole fraction of solvent in solutionxs

µ µs so

sRT x= + ln

Osmotic Pressure

Semipermeable membrane

Po + π

Solution Pure solvent

Po

The osmotic pressure π is the additional pressure thatmust be imposed to keep solvent and solution sectionsat the same level. This static method requires a long time to reach equilibrium.

Semipermeable Membrane Construction

!! Membranes for different solventsMembranes for different solvents

"" Organic - cellulose, gel cellophaneOrganic - cellulose, gel cellophane

"" Aqueous - cellulose acetate, nitrocelluloseAqueous - cellulose acetate, nitrocellulose

"" Corrosive - glassCorrosive - glass

!! Membrane porosityMembrane porosity

"" Must consider pore size and its distributionMust consider pore size and its distribution

!! Membrane conditioningMembrane conditioning

"" Membranes shipped in Membranes shipped in isopropanolisopropanol/water/water

"" Programmed sequence of solvent mixturesProgrammed sequence of solvent mixtures

"" Dried-out membranes should be discardedDried-out membranes should be discarded

Derivation of Van’t Hoff EquationAt constant temperature, the chemical potential dependsupon both pressure and composition.

µs pf P x= ( , ) x xp s= −1

The total derivative of the chemical potential is then

dP

dPx

dxss

T x

s

p T P

p

p

µ∂µ∂

∂µ∂

=

+

, ,

∂µ∂

∂µ∂

s

T x

s

p T P

pPdP

xdx

p

= −

, ,

If no solvent flow occurs, dµs = 0

Derivation of Van’t Hoff Equation

The chemical potential is defined by

µ∂∂s

s T P x

G

Np

=

, ,

= moles of solvent= moles of polymer= Gibbs free energy

N

N

G

s

p

Partial differentiation with respect to P yields

∂µ∂

∂∂ ∂

s

T x sP

G

P Np

=

,

2

Derivation of Van’t Hoff EquationThe volume of the system is given by

∂∂G

PV

T x p

=

,

Partial differentiation with respect to Ns yields

∂∂ ∂

∂∂

2G

N P

V

NV

s s T x P

s

p

=

=

, ,

= partial molarvolume of solvent

Vs

Since the order of differentiation is immaterial, we have

∂µ∂

s

T xsP

Vp

=

,

Derivation of Van’t Hoff Equation

Recall the chemical potential

µ µs so

sRT x= + ln

x xs p= −1dx

dxs

p

= −1

∂µ∂

s

s T P sx

RT

x

=

,

Note that

Partial differentiation with respect to xp yields

and

∂µ∂

∂µ∂

s

p T P

s

s T P

s

p px x

dx

dx

RT

x

=

= − −

, ,1

Derivation of Van’t Hoff Equation

∂µ∂

∂µ∂

s

T x

s

p T P

pPdP

xdx

p

= −

, ,

Vs −−RT

xp1

Recall that for conditions of no solvent flow, dµ = 0 and

Substitution leads to

V dP RTdx

xs

P

Pp

p

x

o

o p+

∫ ∫=−

π

10

Derivation of Van’t Hoff EquationIf the partial molar volume is independent of pressure, we have

π = − −( )RT

Vx

spln 1

For the very dilute solutions that obey Raoult’s Law,

ln 1

2 3

2 3

−( ) = − − − − ≅ −x xx x

xp pp p

pL

For the case where Ns >> Np,

xN

N N

N

Npp

s p

p

s

=+

≅ N V V V solutions s s= ≅ ( )

Derivation of Van’t Hoff Equation

Substitution yields π = RTN

V solutionp

( )

Convert to concentration units of mass/volumeand take the limit as concentration goes to zero

limπc

RT

Mn

= Mn

= number averagemolecular weight

At last, we have the Van’t Hoff Equation. Note thatit is applicable only at infinite dilution.

Compare this expression to the ideal gas law.

Virial Expression for Osmotic PressureIn order to account for concentration effects in polymersolutions, a virial expression is often used.

πc

RTM

A c A cn

= + + +

12 3

2 L

A2 = second virial coefficientA3 = third virial coefficient

Alternative expressions:

π π

π

c cc c

c

RT

MBc Cc

c

n

=

+ + +[ ]

=

+ +

=02 3

2

2

1 Γ Γ L

B RTART

Mn

= =

2 2Γ

The forms are equivalent if:

Effect of Solvent Quality onOsmotic Pressure

πc

c

RT

Mn→

solvent quality

Θ

A2 > 0 for a good solventA2 = 0 for a theta solventA2 < 0 for a poor solvent

Light Scattering

We will only summarize the results -- for background,see, e.g., Allcock and Lampe, 2nd ed., pp 348-363.

Io

Iθ

wVsVs =scattering volume

Light scattering arises from fluctuations in refractiveindex, which can be related to the osmotic pressure.

Rayleigh’s Ratio, R(θ)

Hc

R RT c T( )θ∂π∂

=

1

Hn

dn

dcN

o

A

=

2 2 22

4

π

λ

RI w

I Vo s

( )θ θ=2

w = scattering distanceVs = scattering volume

λ = incident wavelengthno = refractive index at λNA = Avogadro’s number

Relationship to Weight Average MolecularWeight and Second Virial Coefficient

Hc

R R solvent M PA

w( ) ( ) ( )θ θ−= +

12 2

P(θ) is the single chain form factor, which accountsfor the finite size of the macromolecule relative tothe wavelength of incident light.

Single Chain Form Factor for a Random Coil

For particles larger than λ/20, P(θ) is not unity.For small angles (in the Guinier region) and arandom coil structure,

PR k

R k R kg

g g( ) exp( )θ = − − −[ ]{ }214 4

2 2 2 2

Rg is the radius of gyration, which is a measure of thethree-dimensional structure of the random coil.

k =

42

πλ

θsin This is also referred to as q.

Zimm Plot

Extrapolations to zero angle and zero concentration allow for determination of Mw, A2, and Rg.

Hc

R MA c

w( )θ θ

= +

=0

2

12

Eliminate shape effects:

Hc

R MR

c o wg( ) ©

sinθ

πλ

θ

= +

+

=

11

13

42

22 2 L λ

λ©= o

on

Eliminate intermolecular interactions:

Zimm Plot

c = 0

θ = 0

c

θ

1002

2c +

sinθ

316

Hc

Rπ θ( )