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Molecular Weight Distribution (MWD) of Linear Polymers

Objective: Calculation of the Molecular Weight Distribution (MWD) from dynamic mechanical data G', G"(w).

The module is intended to estimate the Molecular Weight Distribution, w(M), of a polymer melt from dynamic mechanical data in the terminal zone. The MWD dependent kernel introduced by Timm et al. is used to correlate MWDs to the linear viscoelastic behavior. In particular the relation between the relaxation time spectrum h(l) and w is considered

 

 

where a 3.4 is the scaling exponent, and the constant k is a material parameter in the scaling relation for the relaxation time as a function of the molecular weight and .

The ill-posed problem of the inversion is solved by constraining the MWD function to a Generalized Exponential (GEX) function.

 

To begin with the MWD determination, we open the IRIS data file (w,G',G") of the linear polymer, see below:

The program asks for a modulus value (optional) and then calculated MWD. The results looks like this:

 

Calculated generalized exponential distribution

Reference: Cocchini F, Nobile MR (2003) Rheol Acta 42:232

**** Input Parameters ****

mixing rule exponent (beta) : 2.00 [-]

scaling law exponent (alfa) : 3.40 [-]

scaling law constant (k) : 8.00e-018 [s/(g/mole)^alpha]

Plateau modulus (GN0) : 2.00e+005 [Pa]

MWD distribution : GEX

Rouse start-up threshold frequency : 8.10e-001 [rad/s]

**** Output Parameters ****

Standard deviation on fit : 3.91e+000 [%]

Plateau modulus (GN0) : 2.00e+005 [Pa]

Zero-shear viscosity (eta0) : 8.40e+006 [Pa s]

Steady-state compliance (je0) : 2.50e-004 [1/Pa]

GEX a constant : 1.1210e+000 [-]

GEX b constant : 5.1180e-001 [-]

GEX M0 constant : 1.7000e+004 [g/mole]

Average Mn : 1.1182e+005 [g/mole]

Average Mw : 3.3459e+005 [g/mole]

Average Mz : 6.7015e+005 [g/mole]

Mw/Mn : 2.9921e+000 [-]

Mz/Mn : 5.9930e+000 [-]

Mz/Mw : 2.0029e+000 [-]

 

MOLECULAR WEIGHT DISTRIBUTION

 

No. M w(M)

[g/mole] [-]

1 1.000e+004 1.076e-002

2 1.259e+004 1.593e-002

3 1.585e+004 2.333e-002

4 1.995e+004 3.369e-002

5 2.512e+004 4.793e-002

6 3.162e+004 6.704e-002

7 3.981e+004 9.199e-002

8 5.012e+004 1.236e-001

9 6.310e+004 1.620e-001

10 7.943e+004 2.067e-001

11 1.000e+005 2.557e-001

12 1.259e+005 3.057e-001

13 1.585e+005 3.516e-001

14 1.995e+005 3.871e-001

15 2.512e+005 4.058e-001

16 3.162e+005 4.026e-001

17 3.981e+005 3.753e-001

18 5.012e+005 3.264e-001

19 6.310e+005 2.623e-001

20 7.943e+005 1.930e-001

21 1.000e+006 1.286e-001

22 1.259e+006 7.658e-002

23 1.585e+006 4.022e-002

24 1.995e+006 1.833e-002

25 2.512e+006 7.125e-003

26 3.162e+006 2.315e-003

27 3.981e+006 6.148e-004

28 5.012e+006 1.301e-004

29 6.310e+006 2.133e-005

30 7.943e+006 2.625e-006

31 1.000e+007 2.337e-007

 

 

 

 

GENERALIZED EXPONENTIAL DISTRIBUTION

 

References

Nobile MR, Cocchini F (2001) Evaluation of molecular weight distribution from dynamic moduli. Rheol Acta 40:111-119

Cocchini F, Nobile MR (2003) Constrained inversion of rheological data to molecular weight distribution for polymer melts. Rheol Acta 42:232-242

Thimm W, Friedrich C, Marth M (1999) An analytical relation between relaxation time spectrum and molecular weight distribution. J Rheol 43: 1663-1672

 



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