4. DETERMINATION OF MOLECULAR WEIGHT OF
4.1Molecular weight of polymers can be determined with the following techniques.
i)End group analyses
vi)High Pressure Liquid Chromatography(HPLC)[Gel permeation Chromatography or Size exclusion chromatography].
4.2Viscosity 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
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
(lnrel )/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 polymer- solvent system, k’ is not sensitive to molecular weight.
All three equation yield linear plots with the intercept equal to  at c= 0
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 is given by the KUHN-MARK-HOUWING-SAKURADA EQUATION :
 = K
Extensive tables of constants K and a are available in the literature.
The VISCOSITY AVERAGE MOLECULAR WEIGHT is defined as
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 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 lies between the number average () and weight average molecular weight ()
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
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 :
- The narrow molecular weight distribution standards method.
- The universal calibration method.
- 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)
 is the limiting viscosity number,
Mvis 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.
Universal calibration curve of a polymer is given below.
Retention Volume Vr
4.6CALCULATION 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 and from ;
where NiMi = Wi
and the polydispersity index from /
HI = /
Following table can be use for the determination of related parameters in the equations.
Table : Calculation of molecular weight of averages
Retention Recorded Molecular weight Number of molecules Value of
Volume VRi division Mi from calibration of the ith kindNiMi
(ml or counts) Hi = NiMi curveNi = Hi/Mi
Molecular weightRecorded Recorded divisionSum up to given Mi
Mi From calibration Division Normalized(Wi%)
curve Hi =NiMi Wi(%)
M1 N1M1W1 = W1
M2 N2M2W2 = W2
Mn MnNn W2 = W1+ W2+ W3..
Wi(%) = 100 Wi(%) =100