4. DETERMINATION OF MOLECULAR WEIGHT OF
POLYMERS
4.1 Molecular weight of polymers can be determined with the
following techniques.
i)
ii)
iii)
iv)
v)
vi)
End group analyses
Osmotic pressure
Colligative properties
Viscosimetry
Light scattering
High Pressure Liquid Chromatography(HPLC)[Gel
permeation Chromatography or Size exclusion
chromatography].
4.2 Viscosity Measurements for Dilute Polymers
In dilute solutions we find the application of the following
viscosity - related quantities by viscosimetric studies.
The main type capillary viscometer are given below.
(i)
The VISCOSITY RATIO( The relative viscosity) rel
which is given by the ratio of the outflow time for the
solution (t) to the outflow time for the pure solvent (to).
rel = t/to (dimensionless)
(ii)
The SPECIFIC VISCOSITY (sp), which is the relative
increment in viscosity of the solution over the viscosity of
the solvent,
sp = ( - o)/ o = rel - 1(dimensionless)
(iii)
The VISCOSITY NUMBER (The REDUCED SPECIFIC
VISCOSITY (red) is the specific viscosity taken per unit
concentration (c)
red = sp /c = (rel – 1)/c
(in deciliter per gram(CGS) or cubic meters per kilogram(SI)(
where c is the concentration of polymer( in gram per deciliter
(CGS) or kilogram per cubic meter (SI))
(iv)
The
LIMITING
VISCOSITY NUMBER (The
INTRINSIC VISCOSITY) [] is the viscosity
number(the reduced specific viscosity) extrapolated to
c=0



[] = lim   sp   lim   rel 
c 0  c  c 0  c 
An extrapolation to infinite dilution require measurement of the
viscosity at several concentration( at least four concentration,
e.g. 0.05, 0.1, 0.15, 0.20 g Per 100 ml). The sample concentration
should not be too large because additional effects may then arise
from intermolecular forces and entanglements between
chains(for very large molecular weight).
There are several empirical equations for calculation of the
limiting viscosity number (intrinsic viscosity).
(i)
The HUGGINS EQUATION
sp /c = [ ] +k’[ ]2c ;
(ii)
The KRAMER EQUATION
(lnrel )/c = [ ] – k’’[ ]2c ;
(iii)
The SUHULZ-BLASCHKE EQUATION
sp /c = [ ] +k’’’[ ]sp ;
where k’, k’’, and k’’’ are constants for a given polymer at a
given temperature in a given solvent. k’ and k’’ are related by
the equation.
k’ - k’’  0.5
The value of k’ is usually in the range 0.3  k’ 0.4 and
increases as solvent power decreases. For a given polymersolvent system, k’ is not sensitive to molecular weight.
All three equation yield linear plots with the intercept equal to
[ ] at c= 0
spl /c
rel /c

Concentration
Figure : Plot of sp /c or rel /c versus concentration
4.3 VISCOSITY AVERAGE MOLECULAR WEIGHT
For polydisperse linear polymers the VISCOSITY
AVERAGE MOLECULAR WEIGHT M V is given by the
KUHN-MARK-HOUWING-SAKURADA EQUATION :
[ ] = K M V
a
Extensive tables of constants K and a are available in the
literature.
a
The VISCOSITY AVERAGE MOLECULAR WEIGHT M V is
defined as
M v  [ N i M i1a /  N i M i ]1 / a
Where K and a are constants for a given polymer at a given
temperature in a given solvent.
A typical plot of log [ ] versus log M V for a given polymer in a
given solvent at a given temperature is shown in figure below. K
and a constants can be calculate from the intercept and slope of
straight line.
The viscosity average molecular weight M V lies between
the number average ( M n ) and weight average molecular weight
( Mw )
Mn  MV  M w
4.4 Gel Permeation Chromatography(GPC)(Size exclusion
Chromatography)(SEC)
GEL PERMEATION CHROMATOGRAPHY(GPC) is a
chromatographic technique which uses highly porous, non-ionic
gel beads for the separation of polydisperse polymers in
solution.
Present theories and models of GPC fractionation indicate
that the hydrodynamic volume of the molecule governs the
separation, not the molecular weight.
The general concept of the fractionation mechanism is that
the largest macromolecules of the solute cannot penetrate the
pores within the cross-linked gel beads, and thus elute first(their
retention volume is larger).
The GPC CHROMATOGRAM is presented as a plot of
detector response versus retention volume(VR) see Figure.
Instead of detector response and retention volume(VR), however,
it is usual to measure heights(Hi) above the baseline and counts
respectively.
The GPC chromatogram of a sample should be normalized
before its shape is compared with the standard chromatogram
because it is almost impossible to inject exactly the same ampunt
of sample into the chromatograph each time.
Figure: A typical GPC chromatogram
The following schedule should be followed in comparing sample
and reference standard chromatogram.
(i)
First the baseline must be constructed. Normally this is
done by joining portions of the chromatograms occurring
prior to the sample appearance and after the final peaks
have been eluted.
(ii) Each 2.5 or 5 ml count is divided into between two and five
equal parts and the chromatogram peak height (Hi) at each
division is measured and recorded.
(iii) The chromatogram height (Hi) for each equal division of
retention volume (VRi) is divided by the sum of all such
heights.
n
Hi = H i /  H i 
i 1
Hi
H1  H 2  H 3  H 4  H 5
This procedure is called NORMALIZATION OF THE GPC
CHRAMATOGRAM
THE RESULTING ELUTING VALUE REPRESENT THE
WEIGHT FRACTION OF MACROMOLECULES ELUTING
AT THE RETENTION VOLUME.
4.5 CALIBRATION METHODS
Three calibration methods are generally used :
1.
The narrow molecular weight distribution standards
method.
2.
The universal calibration method.
3.
The broad molecular weight distribution (polydisperse)
standards method.
The narrow molecular weight distribution standards method.
The main disadvantages o this method is the difficulty of
preparing standards of narrow molecular weight distribution
for most polymers only polystyrene and polyethylene is
commercially available over the whole range of molecular
weight. Reliance on a molecular weight(or size) calibration
performed with a given polymer to interpret the chromatograms
of other polymers can lead to serious errors.
A primary calibration curve of polystyrene are given below.
The universal calibration method.
The limiting viscosity number (intrinsic viscosity) [] is
related to the hydrodynamic volume of the macromolecules
through the following equation(sometimes called the EISTAEIN
VISCOSTY LAW)
[]Mv=(h)Vh
[] is the limiting viscosity number,
Mv is the viscosity average molecular weight,
(h) is a function related to the hydrodynamic behavior of
the macromolecules(constant for a particular solvent and
a given temperature and
Vh is the hydrodynamic volume of the macromolecule.
Log[]M
Universal calibration curve of a polymer is given below.
Retention Volume Vr
4.6 CALCULATION OF MOLECULAR WEIGHT
AVERAGES BY GPC
The basic data from GPC are recorded in the from of a
chromatogram see chromatogram figure. The peak height (Hi)
as a function of the retention volumes (Vr) are proportional to
some value NiMi, where Ni is the number or molecules of the ith
kind with molecular weight Mi.
Using corresponding values of Mi = f(VRi) from the calibration
curve it is easy to calculate the values Ni and NiMi and to obtain
the sums necessary to calculate M n and M w from ;
 Ni Mi
Mn 
Ni
 N i M i2
Mw 
Ni Mi
where NiMi = Wi
and the polydispersity index from M w / M n
HI = M w / M n
Following table can be use for the determination of related
parameters in the equations.
Table : Calculation of molecular weight of averages
Retention
Recorded
Volume VRi division
(ml or counts) Hi = NiMi
(mm)
Molecular weight Number of molecules Value of
Mi from calibration of the ith kind
NiMi
curve
Ni = Hi/Mi
VRi
Hi
Mi
Ni
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 Ni Mi
Molecular weight
Mi From calibration
curve
Recorded Recorded division
Division
Normalized
Hi =NiMi Wi(%)
N1M1
M2
N2M2
Mn
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 Ni
M1
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NiMi
N1 M1
W1 =
 N1 M1
N M
W2 = 2 2
 N2M2
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MnNn
 Ni Mi
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 Ni Mi
Sum up to given Mi
(Wi%)
W1
W2
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NnMn
W1+ W2+ W3..
 NnMn
Wi(%) = 100 Wi(%) =100
W2 =