Direct Measurements Of Average Molecular Weights z M n and M w can be measured directly without knowing the full MWD; not M z z Primary versus Secondary methods Membrane Intrinsic Osmometry viscosity ( N M ) GPC / SEC Light scattering ( W M ) Ultracentrifugation ( Z M ) for biological polymers
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Direct Measurements Of Average Molecular Weights
Mn and Mw can be measured directly without knowing the full MWD; not Mz
• Bottom line for M analysis c = mass concentration
M = molar mass
So Rq can be equated with M
OK For gas phase scattering
After rearrangement of
terms for
Applies in situation with no interference (external or internal).
Molecular Weight Determination
Light Scattering
– Only true for relatively small molecules
– Applies in situation with no__internal interference____________
• polymers having a polymer coil size < /20 (~25 nm for = 500 nm).
• Low angle light scattering
• At bigger angles if the polymer coil size is > /20, then one must deal with internal interferences and non-ideal solutions
cAMR
Kc22
1
q
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For polymers “Zimm equation” in solution
Example
• The excess Rayleigh ratio Rq of cellulose acetate in dioxane was measured as a function of concentration by Low Angle Light Scattering measurements. Data are given in the Table. If the RI of dioxane is 1.4199, refractive index increment for CA in dioxane is 6.297 x 10-2 cm3/g and the wavelength of light was 6328 A, calculate the MW and second virial coefficient
C x 103 (g cm-3) Rq x 105(cm-1)
0.5034 0.239
1.0068 0.440
1.5102 0.606
2.0136 0.790
2.517 0.902
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cAMR
Kc22
1
q
Molecular Weight Determination
Light Scattering
57
Light scattering - Internal Interferences:
For higher angles there is a big difference between the two path lengths destructive interferences
q
Molecular Weight Determination
Light Scattering
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q
q
Large angles: destructive interferences Small angles: less affected by destructive interferences
cAPMR
Kc
w
22)(
1
qq
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Molecular Weight Determination Light Scattering
Light scattering - Interferences:
Small molecules
polymers Scattering Intensity
Note P(0 1
Molecular Weight Determination
Light Scattering
Variation of Pq with molecular weight and angle for PS
Disymmetry Factor for Different Types
of Polymer Molecules
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X axis
R/
Molecular Weight Determination
Light Scattering
• P(q) is the form factor which depends on the size and shape of the molecule
• The form factor of polymer coils was derived by Debye in 1947.
– It handles the intra-particle interferences needs to work with low polymer concentration (c < 10 g.L1 = 1 wt%).
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22
2
2
)2/(sin3
161
)(
1GR
P q
q
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Molecular Weight Determination Light Scattering
Light scattering – Radius of Gyration: RG
G
Mi
point Mi along the chain at ri distance from G
G: Center of mass of the polymer coil
ri
i
iG rn
R22 1
Molecular Weight Determination
Light Scattering
• The ratio Kc/Rq depends on polymer concentration (c) and the angle of observation (q).
• In practice, one prepares a set of solutions at different polymer concentrations.
• The light scattered by each polymer solution is monitored at different observation angles
.....32
1
)(
1 2
32 cAcAMPR
cK
wqq
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Light scattering – Non ideal solutions:
Zimm Plot!
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Molecular Weight Determination
Light scattering – Zimm plot:
.....32
1
)(
1 2
32 cAcAMPR
cK
wqq
sin2(q/2) + bc
qR
cK C (#6) C (#5) C (#4) C (#3) C (#2) C (#1)
q1 q2 q3 q4 q5 q6 q7
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Molecular Weight Determination
Light scattering – Zimm plot:
.....32
1
)(
1 2
32 cAcAMPR
cK
wqq
sin2(q/2) + bC
qR
cK C (#6) C (#5) C (#4) C (#3) C (#2) C (#1)
q1 q2 q3 q4 q5 q6 q7
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Molecular Weight Determination
Light scattering – Zimm plot:
.....32
1
)(
1 2
32 cAcAMPR
cK
wqq
sin2(q/2) + bc
qR
cK
q1 q2 q3 q4 q5 q6 q7
C (#6) C (#5) C (#4) C (#3) C (#2) C (#1) C 0
w
GM
RR
cK 1....)2/(sin
3
161 22
2
2
q
q
slope = 162RG2/(Mw 32
1/Mw
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Molecular Weight Determination
Light scattering – Zimm plot:
.....32
1
)(
1 2
32 cAcAMPR
cK
wqq
sin2(q/2) + bc
qR
cK
q 0, q1 q2 q3 q4 q5 q6 q7
C (#6) C (#5) C (#4) C (#3) C (#2) C (#1) C 0
Slope = 2A2
1/Mw
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Molecular Weight Determination
Light scattering – Zimm plot:
Zimm plot of poly(vinyl acetate) in butanone at 25 oC
What is Mw?
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Molecular Weight Determination
Light scattering – Numerical Application: What is Mw?
Mw1 = 0.8 106 mol.g1 Mw = 1.25 106 g.mol1
slope = 162RG2/(Mw 32
Multi Angle LS
Zimm Plot • Very useful for determining multiple pieces of
info – Mw
– Rg
– A2
• BUT. – Very laborious !!!
– Very sensitive to dust etc.
– Time consuming.
• Used for fundamental studies rather than as a quality control or routine research tool.
MV methods
Viscosity of dilute polymer solutions higher than that of pure solvent
A polymer solution has a higher viscosity than the solvent, because:
Solvent trapped in-between the coils can not attain the velocities which the liquid would have
- polymer coil has the same effect on the
viscosity of the mixture as a sphere
Viscosity increase depends on T, nature of solvent and polymer, C, and the sizes of polymer molecules
- Mv depends to some extent on the solvent
used
2
Dilute Solution Viscosity
• Viscosity is the quantity that describes a fluid's resistance to flow.
• Viscosity of polymer solutions depends on – Concentration
– Solvent
– Temperature
– Molecular weight
• Can be used to determine molecular weights
– Viscosity Average MW
2
3 3
Molecular Weight Determination
Viscometry:
capillary
V hydrodynamic volume
The concept of the equivalent hydrodynamic sphere:
Solvent molecules located inside the polymer coil move almost in
unison, like polymer beads, as though the solvent molecules were
bound to the polymer.
Viscometer
4
Molecular Weight Measurement Viscometry
• Secondary method
• A polymer solution has a higher viscosity than pure solvent.
– Solvent trapped in coils cannot attain velocities of free solvent
• A given polymer sample has only 1 or – May have more than one – Because “a” varies with solvent
• The broader the MWD the more may vary with solvent.
• What happens with branched polymers and copolymers? – Copolymer composition and microstructure has an effect
on polymer-solvent interactions – Branching gives a more compact structure for a given MW.
• For a given MW, viscosity will be lower for branched compared to linear (see later for GPC).
19
NM WM
vM
vM
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Example
• The data shown were obtained for polystyrene dissolved in cyclohexane, when viscosity measurements were made at the q temperature of 308K.
• Solvent flow time = 100 s
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c (g cm-3) t (s)
0.001 109.5
0.002 120
0.003 135
0.004 144
Determine the average MW if K = 8.6 x 10-2 Ans 1.1M
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Example
• The following data were obtained for the intrinsic viscosity of polystyrene fractions in C2H4 Cl2 at 22oC using LS as the measurement of MW. Evaluate the MHS constants.
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[] (cm3/g)
260 278 142 138 12.2 4.05
Mw X 10-4
178 157 56.2 48.0 1.55 0.308
Measuring Polymer Molecular Weight Gel Permeation Chromatography
What about distributions???
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Gel Permeation Chromatography
• Previous methods give molecular weight averages.
• Gel Permeation Chromatography (GPC).
– Gives molecular weight distributions
• Based on separation of polymer sizes by differential flow through a stationary bed of particles.