Chapter 8. Measurement of molecular weight and size

Chapter 8. Measurement of molecular weight and size

8.1. End-group analysis

8.2. Colligative property measurement 8.3. Osmometry

8.4. Gel-permeation chromatography 8.5. Ultracentrifugation

8.6. Light-scattering methods

8.7. Solution viscosity and molecular size

Molecular weights of polymers can be determined by chemical or

physical methods of functional-group analysis. They are measurement of the colligative properties, light scattering, or ultracentrifugation;

or by measurement of dilute-solution viscosity.

Molecular weights can be calculated without reference to callibration by 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 to infinite dilution, and operate in a Θ solvent in which have ideal-solution behaviour.

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

The sum of all molecular species of the number of moles N_i of each species:

∑

^∞

=1 i

N

i

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

∑

∑

^∞

=

∞

=

=

=

1 i

i i 1

i

i

N M

w w

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

∑

∑

∑

^∞

∞

=

∞

=

=

^{i 1}

i i n

N M N N

M w

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 becomes too small to measured with precision (>25,000).

Condensation polymers

End-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 an alcoholic or phenolic solvent.

• Amino groups in PA are titrated with acid

• Hydroxyl groups is reacted with a titratable reagent

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 polymers

End-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.

The equations for this method

n v

2 b

0

c M

1 ρ∆H

RT c

lim

∆T ⁼

→ _{f n}

2 f

0

c M

1 ρ∆H

RT c

lim

∆T ⁼

→

0 n

c M

RT c

lim

π ⁼

→

where: ∆T_b, ∆T_f, and π are the boiling-point elevation, freezing-point depression, and osmotic pressure, respectively,

ρ is the density of the solvent,

∆H_v and ∆H_f are the enthalpies of vaporization and fusion, respectively, of the solvent per gram,

c is the solute concentration (gr/cm³),

M_n is the number-average molecular weight.

8.3. Osmometry

A basic equation for a solvent: ∆µ₁ = RT ln a₁ = −πV₁ 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

c 1 2 χ

1 V

1 Mυ M

RT c

π ₂

1 3 12

1 2

where: υ is the specific volume of the polymer, and

χ₁₂ is the interaction energy per solvent molecules / kT

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:

 



 

 + + +

= A c A c ....

M RTc 1

π

_{2 3} ²

n

where A₂ is second virial coeffiient,

A₃ is the third virial coefficient,



 



 −

= ₁₂

1 2

2 χ

2 1 V A υ







= 

1 3

3 V

υ 3 A 1

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

The experimental procedures to determine osmotic pressure are relatively simple but very time consuming.

A pure solvent and a dilute solution of the polymer in the same solvent are placed on

opposite sides of a semipermeable membrane (cellulose or a cellulose derivative).

The differences chemical potentials between solvent and the polymer solution causes solvent

to 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.

Vapour-pressure osmometry

When a polymer is added to a solvent, the vapour pressure of the solvent will be lowered due to the decrease in solvent activity. The relation of

the difference in vapour pressure between solvent and solution,

∆p and the number-average molecular weight of the polymer, M_n is:

n o 1 o 1 0

c M

V p c

lim

^_^ ∆p^_^{ = −}

→

where: ∆p = p₁ – p₁^o,

p₁^o and V₁^o are the vapour pressure and molar volume of the pure solvent, respectively,

p₁ is vapour pressure of the solution.

n VPO 0

c M

K c

∆R  =



 





→

Condensation of solvent vapour onto the solution results the temperature of the solution thermistor increases until the vapour pressure of the solution equals to that of the solvent.

K_VPO id the calibration constant obtained by measuring R for a low mol-weight standard whose mol weight is precisely known.

8.4. Gel-permeation chromatography (GPC)

One of the most widely used methods for routine determination of molecular weight and molecular-weight distribution is GPC.

This method based on the principle of size-exclusion chromatography to separate samples of polydisperse polymers into fractions of

narrower molecular-weight distribution.

The equipment:

Several small-diameter columns (L = 30 – 50 cm) are packed with small highly porous beads (∅ = 10 – 10⁷ Â). Pure pre-filtered solvent is

continuously pumped through the columns at a constant flow rate (1 – 2 mL/min). Then, a small amount (1 – 5 mL) of a dilute polymer solution 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.

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

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

flow-rate vs elution volume (V_r)

For a given polymer, solvent, temperature, pumping rate, and column packing size, V_r is related to molecular weight.

In calculation of molecular-weight averages, the peak height ~ W_i. A proper calibration curve should be measured to relate V_i to W_i, direct calculation of all molecular weights (M_n, M_w, M_z) are possible.

commercially available software

8.5. Ultracentrifugation method

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

An ultracentrifuge consists of an Al rotor (ø ∼1-2 inch) that is rotated at high speed in an evacuated chamber. The solution being

centrifuged 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 based on measurements of refractive index or absorption.

The solvents must have difference both density and refractive index from the polymer the density differences allow the

sedimentation and the refractive index differences allow the measurement.

In the sedimentation equilibrium experiment, the ultracentrifuge is operated at a low speed of rotation for times up 1 or 2 weeks under constant conditions. A thermodynamic equilibrium is reached in which the polymer is distributed in the cell according to its molecular weight and molecular-weight distribution.

The force on a particle:

ρ)m υ

r(1 ω

f =

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.

For an ideal solution in the equilibrium condition:

( ) (

1²

)

2 2 2

1 2

w

1 υ ρ ω r r

c ln c

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 temperature near Θ and the slope depends on the second virial coefficient.

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

Centrifuge

8.7. Light-scattering methods

Light-scattering techniques are important in polymer research but it is not routinely used for molecular-weight determination because of the difficulty and expense of sample preparation and the specialized

facilities required.

Light-scattering method dilute polymer solution, Small-angle neutron scattering solid samples.

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

( ) ^{M P} ( ) ^{θ 2A c ...}

1 θ

R K

2 w

c

= + +

where:

2 4

2 0 2

dc dn N λ

n

K 2π 



 



= 

( ) ( )

V I

r θ θ i

R

2

= (Rayleigh ratio)

K is a function of the refractive index, n_o, 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,

N_A 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 the incident beam.

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

( ) [ ^e ( ^{1 υ} ) ]

υ θ 2

P =

₂ ^−υ

− −

^{where: }



 



 



 



= 

2 sin θ λ s

16 πn

υ ^{2 2}

2

and,

P(θ) is the particle scattering function which incorporates the effect of chain size and conformation on the angular dependence of scattered light intensity,

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

For linear-chain polymers:

6 s r

2

2

=

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

P(θ) and R(θ) are important to determine the M_w.

In practice, 2 approaches can be used to determine P(θ).

a. Dissymmetry method

This method requires the measurement of the scattered intensity at 3 angles, typically 45°, 90°, 135°and at several different dilute polymer concentrations.

( ( )

^o

)

o

135 i

45 z = i

where z is normally concentration dependent,

z (I = 0°) is determined by plotting (z-1)^-1 vs c graph.

b. Zimm method

This method requires Zimm plot. This method can determine the chain conformation better compared the dissymmetry method.

However, Zimm plot requires measurements at more angles than the dissymmetry method.

( ) ^{2 2A c}

sin θ λ s

πn 3

1 16 M

1 θ

R K

2 2

2 2

w

c

+

 





 



 



 



 



 

 + 

=

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

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:

( ) ^{M 2A c}

1 R θ

K

2 w

c

= +

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

Other scattering methods:

• Dynamic light scattering,

• Neutron scattering,

• Light scattering from very large particles.

8.8. Solution viscosity and molecular size

A method that is widely use for routine molecular-weight determination is 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:

[ ] ^{η = K M}

^a_v The Mark-Houwink-Sakurada equation

∑

∑

=

=

+

= _N

1 i

i i N

1 i

a 1 i i v

M N

M N

and M See Table 3.6, Fried

for K values

where M_v is the viscosity-average molecular-weight,

K and a are empirical Mark-Houwink constants that are specific for a given polymer, solvent, and temperature,

a is thermodynamic constant

(0.5 a Θ solvent; and 1.0 a therm-good solvent, M_v = M_w.

The value of M_v normally lies between the values of M_n and M_w obtained by osmometry and light-scattering methods respectively.

Intrinsic viscosity is indicated by Huggins equation:

[ ] ^{η k} [ ] ^{η c}

c

η

2

H

i

= +

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

s s

i

η

η η η −

=

where η_i is the relative viscosity-increment,

η and η_s are viscosities of the dilute polymer-solution and pure solvent, respectively.

It is defined that η_red ≡ η_i/c and called as the reduced viscosity.

The η_i can be obtained from the intercept of linear graph of η_i/c vs c.

In practice, reduced-viscosity is obtained at different concentrations not by direct measurement of solution and solvent viscosities but

by measurement of the time required for a dilute solution (t) and pure solvent (ts) to fall from one fiducial mark to another in small glass

capillary.

There are 2 types of capillary viscometers: the Ostwald-Fenske and Ubbelohde.

s s

i

t

t η t −

The previous equation becomes:

=

Another important equation: 2

[ ]

^2/3

Θ η

r M 



 



= 

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

Θ is the Flory constant (= 2.1 X 10²¹ dL/g cm³).