Chapter 8. Measurement of molecular weight and size 8.2. …staff.ui.ac.id/system/files/users/ariadne.laksmidevi/material/fispol8.pdf · 8.7. Light-scattering methods Light-scattering

Sem 1 2006/2007 Fisika Polimer

Ariadne L. Juwono

Chapter 8. Measurement of molecularweight and size8.1. End-group analysis8.2. Colligative property measurement 8.3. Osmometry8.4. Gel-permeation chromatography 8.5. Ultracentrifugation8.6. Light-scattering methods8.7. Solution viscosity and molecular size


Ariadne L. Juwono

Molecular weights of polymers can be determined by chemical orphysical methods of functional-group analysis. They are measurementof the colligative properties, light scattering, or ultracentrifugation;or by measurement of dilute-solution viscosity.

Molecular weights can be calculated without reference to callibrationby another method.Dilute-solution viscosity is not a direct measure of molecular weight and empirically related to molecular weight for many systems.

All methods require the solubility of polymer, involve extrapolation toinfinite dilution, and operate in a Θ solvent in which have ideal-solutionbehaviour.


Ariadne L. Juwono

Typical polymers consist of mixtures of many molecular species andmolecular weight methods always provide average values.

The sum of all molecular species of the number of moles Ni of eachspecies:

∑∞

=1i

iN

The total weight of the sample, w, = the sum of the weights of each molecular species

∑∑∞

=

∞

=

==1i

ii

1i

i MNww

The number-average molecular weight, Mn molecular weight as weight of sample per mole

∑

∑

∑∞

=

∞

=∞

=

==

1i

i

1i

ii

1i

i

n

N

MN

N

wM


Ariadne L. Juwono

8.1. End-group analysis

The end-group analysis needs the information about the number of determinable groups per molecule. This method is not suitable for high molecular weight because the fraction of end groups becomestoo small to measured with precision (>25,000).

Condensation polymersEnd-group analysis in condensation polymers usually involves chemical methods of analysis for functional groups.Examples:• Carboxyl groups in PE and in polyamides are titrated with base in analcoholic or phenolic solvent.

• Amino groups in PA are titrated with acid • Hydroxyl groups is reacted with a titratable reagent


Ariadne L. Juwono

8.2. Colligative property measurement

The relations between the colligative properties and molecular weight for infinitely dilute solutions in a fact that the activity of the solute in a solution becomes equal to its mole fraction as the solute concentration becomes sufficiently small.

This method is based on •Vapour-pressure lowering,•Boiling-point elevation (ebulliometry),•Freezing-point depression (cryoscopy), and•Osmotic pressure (osmometry).

Addition polymersEnd-group analysis in addition polymers does not have general procedure because of the variety of type and origin of the end groups.Analysis may be made for initiator, elements, radioactive atoms.


Ariadne L. Juwono

The equations for this method

nv

2

b

0c M

1

ρ∆H

RT

c

∆Tlim =

→ nf

2

f

0c M

1

ρ∆H

RT

c

∆Tlim =

→

n0c M

RT

c

πlim =

→

where: ∆Tb, ∆Tf, and π are the boiling-point elevation, freezing-pointdepression, and osmotic pressure, respectively,ρ is the density of the solvent,∆Hv and ∆Hf are the enthalpies of vaporization and fusion, respectively, of the solvent per gram,c is the solute concentration (gr/cm3),Mn is the number-average molecular weight.


Ariadne L. Juwono

8.3. Osmometry

A basic equation for a solvent: 111 πValn RT∆µ −==

where: π is the osmotic pressure,∆µ1 is the chemical potential,a1 is the activity,V1 is the molar volume of the solvent.

Using Flory-Huggins equation for solvent activity, it gives:

+

+

−

+= ...c

V

Mυ

3

1cχ

2

1

V

Mυ1

M

RT

c

π 2

1

3

12

1

2

where: υ is the specific volume of the polymer, andχ12 is the interaction energy per solvent molecules / kT


Ariadne L. Juwono

The classical van’t Hoff equation for the osmotic pressure of an ideal,dilute solution (when χ12 = ½):

M

RT

c

π=

The osmotic pressure becomes:

+++= ....cAcA

M

1RTcπ 2

32

n

where A2 is second virial coeffiient,

A3 is the third virial coefficient,

−= 12

1

2

2 χ2

1

V

υA

=

1

3

3V

υ

3

1A

In a dilute solution with c < 1g/dL, cn can be neglected. A plot of π/RTcvs c a straight line with an intercept, Mn and slope A2.


Ariadne L. Juwono

The experimental procedures to determine osmotic pressure arerelatively simple but very time consuming.A pure solvent and a dilute solution of the polymer in the same solvent are placed onopposite sides of a semipermeable membrane(cellulose or a cellulose derivative).The differences chemical potentials between solvent and the polymer solution causes solventto pass through the membrane and raise the liquid head of the solution reservoir.

In the equilibrium pressure:π= ρgh

where ρ is the solvent density.


Ariadne L. Juwono

Vapour-pressure osmometry

When a polymer is added to a solvent, the vapour pressure of the solventwill be lowered due to the decrease in solvent activity. The relation ofthe difference in vapour pressure between solvent and solution,∆p and the number-average molecular weight of the polymer, Mn is:

n

o

1

o

1

0c M

Vp

c

∆plim −=

→

where: ∆p = p1 – p1o,

p1o and V1

o are the vapour pressure and molar volume of the pure solvent, respectively,p1 is vapour pressure of the solution.


Ariadne L. Juwono

n

VPO

0c M

K

c

∆R=

→

Condensation of solvent vapour onto the solution results the temperatureof the solution thermistor increases until the vapour pressure of the solutionequals to that of the solvent.

KVPO id the calibrationconstant obtained bymeasuring R for a lowmol-weight standardwhose mol weightis precisely known.


Ariadne L. Juwono

8.4. Gel-permeation chromatography (GPC)

One of the most widely used methods for routine determination ofmolecular weight and molecular-weight distribution is GPC.This method based on the principle of size-exclusion chromatographyto separate samples of polydisperse polymers into fractions ofnarrower molecular-weight distribution.

The equipment:Several small-diameter columns (L = 30 – 50 cm) are packed with smallhighly porous beads (∅ = 10 – 107 Â). Pure pre-filtered solvent iscontinuously pumped through the columns at a constant flow rate (1 – 2 mL/min). Then, a small amount (1 – 5 mL) of a dilute polymersolution is injected by syringe into the solvent stream and carried through the columns. The smallest polymer molecules are able to penetrate deeply into the bead pores but the largest may be completely excluded.


Ariadne L. Juwono


Ariadne L. Juwono

The process is repeated until all polymer molecules have been elutedout of the column in descending order of molecular weight.

The concentration of polymer molecules in each eluting fraction can bemonitored by means of a polymer-sensitive detector, such as IR or UVdevice. The detector is usually a differential refractometer (differ therefractive index between the pure solvent and polymer solution).

flow-rate vs elution volume (Vr)

For a given polymer, solvent, temperature, pumping rate, and columnpacking size, Vr is related to molecular weight.

In calculation of molecular-weight averages, the peak height ~ Wi.A proper calibration curve should be measured to relate Vi to Wi,direct calculation of all molecular weights (Mn, Mw, Mz) are possible.

commercially available software


Ariadne L. Juwono


Ariadne L. Juwono

8.5. Ultracentrifugation method

Ultracentrigugation techniques are the most intricate of the methodsfor determining the molecular-weights of high polymers. This methodis useful for biological materials, such as protein molecules.

An ultracentrifuge consists of an Al rotor (ø ∼1-2 inch) that is rotatedat high speed in an evacuated chamber. The solution beingcentrifuged is held in a small cell within the rotor near its periphery.The rotor is driven electrically or by oil or air turbine.The concentration of polymer is determined by optical methods basedon measurements of refractive index or absorption.The solvents must have difference both density and refractive indexfrom the polymer the density differences allow thesedimentation and the refractive index differences allow the measurement.


Ariadne L. Juwono

In the sedimentation equilibrium experiment, the ultracentrifuge isoperated at a low speed of rotation for times up 1 or 2 weeks underconstant conditions. A thermodynamic equilibrium is reached in whichthe polymer is distributed in the cell according to its molecular weightand molecular-weight distribution.The force on a particle:

ρ)mυr(1ωf 2 −=

where: ω is the angular velocity of rotation, r is the distance of the particle from the axis of rotation,υ is the partial specific volume of the polymer,ρ is the density of the solution,m is the mass of the particle.


Ariadne L. Juwono

For an ideal solution in the equilibrium condition:

( ) ( )2

1

2

2

2

1

2

wrrωρυ1

cc

ln 2RT

M−−

=

where: c1 and c2 are the concentrations at 2 points r1 and r2 in the cell.

For non-ideal solution, the process is held at Θ temperature.The molecular weight is a linear function of concentration at temperaturenear Θ and the slope depends on the second virial coefficient.

The disadvantage of sedimentation equilibrium experiment is taking quite long time to reach equilibrium.


Ariadne L. Juwono

Centrifuge


Ariadne L. Juwono

8.7. Light-scattering methods

Light-scattering techniques are important in polymer research but it isnot routinely used for molecular-weight determination because of thedifficulty and expense of sample preparation and the specializedfacilities required. Light-scattering method dilute polymer solution,Small-angle neutron scattering solid samples.

The basic principles of light-scattering measurements of dilute polymerbased on a fundamental relationship:

( ) ( ).....c2A

θPM

1

θR

K2

w

c ++=

where:2

4

A

2

0

2

dc

dn

λN

n2πK

= ( ) ( )

VI

rθiθR

0

2

= (Rayleigh ratio)


Ariadne L. Juwono

K is a function of the refractive index, no, of the pure solvent,n is the specific refraction index,dn/dc is the specific refractive increment of the dilute polymer solution,λ is the wave-length,NA is Avogadro’s number,I0 is the intensity of the incident light beam,i (θ) is the intensity of the scattered light measured at a distance of r

from the scattering volume, V, and at an angle θ with respect to theincident beam.

For a monodisperse system of randomly-coiling molecules in dilutesolution, a relationship of:

( ) ( )[ ]υ1eυ

2θP υ

2−−= − where:

=

2

θsins

λ

πn16υ 22

2


Ariadne L. Juwono

and,P(θ) is the particle scattering function which incorporates the effect of

chain size and conformation on the angular dependence of scatteredlight intensity,

<s2> is the mean-square radius of gyration.

For linear-chain polymers:

6

rs

2

2 =

where : r is the mean-square end-to-end distance.

P(θ) and R(θ) are important to determine the Mw.In practice, 2 approaches can be used to determine P(θ).


Ariadne L. Juwono


Ariadne L. Juwono

a. Dissymmetry method

This method requires the measurement of the scattered intensity at3 angles, typically 45°, 90°, 135°and at several different dilute polymerconcentrations.

( )( )o

o

135i

45iz =

where z is normally concentration dependent,z (I = 0°) is determined by plotting (z-1)-1 vs c graph.


Ariadne L. Juwono

b. Zimm method

This method requires Zimm plot. This method can determine the chainconformation better compared the dissymmetry method. However, Zimm plot requires measurements at more angles than thedissymmetry method.

( )c2A

2

θsin s

λ

πn

3

161

M

1

θR

K2

22

2

w

c +

+=

where 1/Mw is obtained from the intercept of the linear curve ofmeasurement on the graph of Kc/R(θ) vs <s2> sin2 (θ /2).


Ariadne L. Juwono

Low-angle Laser light-scattering (LALLS)

The high intensity of Laser sources permits scattering measurements at much smaller angles (2°– 10°), with the Laser λ = 6328 Â.This method apply the Debye equation:

( )c2A

M

1

θR

K2

w

c +=

where 1/Mw is obtained from the intercept of the linear curve of graph of Kc/R(θ) vs c.

Other scattering methods:• Dynamic light scattering,• Neutron scattering,• Light scattering from very large particles.


Ariadne L. Juwono

8.8. Solution viscosity and molecular size

A method that is widely use for routine molecular-weight determinationis based on the determination of intrinsic viscosity, η, of a polymer in solution through measurements of solution viscosity.

The fundamental relationship between η and molecular-weight is:

[ ] a

vMKη = The Mark-Houwink-Sakurada equation

∑

∑

=

=

+

=N

1i

ii

N

1i

a1

ii

v

MN

MN

MandSee Table 3.6, Friedfor K values


Ariadne L. Juwono

where Mv is the viscosity-average molecular-weight,K and a are empirical Mark-Houwink constants that are specificfor a given polymer, solvent, and temperature,a is thermodynamic constant(0.5 a Θ solvent; and 1.0 a therm-good solvent,Mv = Mw.

The value of Mv normally lies between the values of Mn and Mw obtainedby osmometry and light-scattering methods respectively.

Intrinsic viscosity is indicated by Huggins equation:

[ ] [ ] cηkηc

η 2

Hi +=

where kH is a Huggins coefficient for a specific polymer, solvent,and temperature.


Ariadne L. Juwono

s

si

η

ηηη

−=

where ηi is the relative viscosity-increment,η and ηs are viscosities of the dilute polymer-solution and puresolvent, respectively.

It is defined that ηred ≡ ηi/c and called as the reduced viscosity.The ηi can be obtained from the intercept of linear graph of ηi/c vs c.

In practice, reduced-viscosity is obtained at different concentrationsnot by direct measurement of solution and solvent viscosities butby measurement of the time required for a dilute solution (t) and puresolvent (ts) to fall from one fiducial mark to another in small glasscapillary. There are 2 types of capillary viscometers: the Ostwald-Fenske andUbbelohde.


Ariadne L. Juwono

s

si

t

ttη

−=The previous equation becomes:

Another important equation: [ ] 2/3

2

Θ

η Mr

=

where: r is the mean-square end-to-end distance,Θ is the Flory constant (= 2.1 X 1021 dL/g cm3).

Chapter 8. Measurement of molecular weight and size 8.2. …staff.ui.ac.id/system/files/users/ariadne.laksmidevi/material/fispol8.pdf · 8.7. Light-scattering methods Light-scattering

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