calculation of molecular weight distribution using
TRANSCRIPT
© 2021 The Korean Society of Rheology and Springer 65
Korea-Australia Rheology Journal, 33(1), 65-78 (February 2021)DOI: 10.1007/s13367-021-0006-0
www.springer.com/13367
pISSN 1226-119X eISSN 2093-7660
Calculation of molecular weight distribution using extended Cole-Cole
model and quadratic mixing rule
Junghaeng Lee1, Sangmo Kim
2 and Kwang Soo Cho
1,*1Department of Polymer Science and Engineering, Kyungpook National University, Daegu, Korea
2S-OIL, Seoul, Korea
(Received April 24, 2020; final revision received December 8, 2020; accepted December 30, 2020)
We suggest a numerical method to calculate molecular weight distribution from linear viscoelastic data. Thecalculation method consists of three components: (1) a viscoelastic model of a monodisperse polymer asa function of molecular weight; (2) the mixing rule connecting viscoelastic data of monodisperse andpolydisperse polymers through molecular weight distribution; (3) an algorithm which calculates the molec-ular weight distribution from the chosen mixing rule. Since we cannot measure the relaxation modulus ofall monodisperse samples, we need an accurate monodisperse model for any molecular weight. It is knownthat a dynamic test is more reliable than a relaxation test, while the mixing rule needs relaxation modulus.Hence, we should have a smart numerical method that can convert dynamic data to relaxation modulus withthe minimum conversion error. If we use the numerical method, then we have to generate numerical datafrom the model. Then it takes quite a long time. On the other hand, if we have a monodisperse model withthe analytical relaxation spectrum, then calculation time can be reduced dramatically. Since the conversionfrom relaxation modulus to dynamic modulus suffers from smaller errors than the reverse conversionbecause of ill-posedness of the interconversion, the analytical conversion can be implemented more quicklyat an acceptable level of errors. This paper proposes a new method satisfying the requirements.
Keywords: molecular weight distribution, extended Cole-Cole model, quadratic mixing rule, fixed point
iteration, continuous relaxation time distribution
1. Introduction
There is a report Park et al. (2015) that the rheological
method for calculating molecular weight distribution
(MWD) is more reliable than gel permeation chromatog-
raphy (GPC) in some polymer systems such as ultrahigh
molecular weight polyethylene. It does not mean that the
rheological method is always better than GPC. The rhe-
ological method is complimentary to GPC in some cases.
Since we think that most rheological methods have merits
and demerits, it is a meaningful attempt to improve the
rheological method.
For this reason, many rheological methods for calculat-
ing MWD have been studied (Anderssen and Mead, 1998;
Carrot and Guillet, 1997; Des Cloizeaux, 1988; Friedrich
et al., 2009; Guzman et al., 2005; Lang 2017; Léonardi et
al., 2002; Maier et al., 1998; McGrory and Tuminello,
1990; Mead, 1994; Nobile and Cocchini, 2008; Park et al.,
2015; Pattamaprom et al., 2008; Thimm et al., 1999;
2000; Tsenoglou, 1987; Tuminello, 1986; Van et al., 2002;
Wasserman, 1995; Wu, 1985). In order to calculate a MWD,
we need an equation connecting the linear viscoelastic
functions of the polydisperse polymer and its ingredient
polymers with narrow MWD. This equation is called a
mixing rule, and the generalized mixing rule is proposed
by Anderssen and Mead (1998):
, (1)
where represents the relaxation modulus of the
monodisperse polymer of molecular weight M, is the
weight fraction of the monodisperse chains, b is a mixing
rule parameter, MC is the critical molecular weight,
is the maximum molecular weight used to calculate the
MWD. Since the double reptation theory predicts ,
Eq. (1) is a generalization.
Maier et al. (1998) determined from the binary
mixture of nearly monodisperse polystyrenes (PS’s) with
different molecular weights. The determination of the
optimum depends mainly on the choice of kernel func-
tion:
, (2)
where is the plateau modulus. In the determination of
, they did not use . They determined by
using regression fitting to for chosen values
of . The validity of the model for monodisperse polymers
is tested by the experimental data of nearly mono-
disperse polymer samples with polydisperse index (PDI)
≤1.1. Hence, their regression might deteriorate the results.
Use of regression for the determination of
m max
1
( ) ( , ) forN
i i CG t G t M M M M
m( , )G t M
maxM
2
3.84
o
( , )( , ) m i
i
N
G t MF t M
G
o
NG
( , )F t M
F1/
M t 1
o[ ( ) ]N
G t G
F1/
M t *Corresponding author; E-mail: [email protected]
Junghaeng Lee, Sangmo Kim and Kwang Soo Cho
66 Korea-Australia Rheology J., 33(1), 2021
seems to originate from excessive confidence in GPC
MWDs. If a perfect sample of monodisperse polymer is
injected into GPC, no Dirac delta function is obtained
from any GPC. Since GPC MWDs contain an inevitable
error, it is meaningless to use it to estimate the viscoelastic
properties of monodisperse samples, which are necessary
for the calculation of MWD via the mixing rule. However,
the GPC MWD of polydisperse samples, of course, must
be necessary for checking the validity of the calculation.
Bae and Cho (2015) used the relaxation modulus calcu-
lated from the dynamic data of nearly monodisperse sam-
ples rather than and showed that agrees
well with experimental data of binary and quartic blends
of nearly monodisperse PS’s. The support for the validity
of can also be found in Ruymbeke et al. (2002) and
Pattamaprom et al. (2008). Hence, we will use .
In the choice of the model for the monodisperse poly-
mer, it is of importance whether the model can fit the
experimental data of nearly monodisperse samples with
various molecular weights. Many researchers do not seem
to care about the importance of the monodisperse model
because they used unrealistic models such as the single
Maxwell model (Lang, 2017; Tsenoglou, 1987; Tuminello,
1986; Maier et al., 1998; Mead, 1994). The calculation of
the MWD from rheological data needs a monodisperse
model as accurate as possible. The parameter estimation is
usually made by the dynamic data, while the mixing rule
needs the relaxation modulus for MWD calculation. It is
advantageous to use a monodisperse model which has an
analytic relation between relaxation and dynamic moduli
because such a model improves the calculation time and
prevents errors that may occur during the interconversion
between the two types of modulus. Assume that an accu-
rate monodisperse model is available in the form of relax-
ation modulus and we have to determine the model
parameters from dynamic modulus data. Then the process
of the determination of the model parameters involves
additional errors due to the conversion. If an accurate
model is given in the form of dynamic moduli (or the
Laplace transform of creep compliance), then we can
determine the model parameters with quite small errors.
Note that the conversion from dynamic moduli to the
relaxation modulus suffers from smaller errors than the
reverse conversion (Cho et al., 2017). Furthermore, if the
monodisperse model allows analytical conversion, we can
get a more accurate relaxation modulus of monodisperse
polymers.
Hence, we focus on the development of such a mono-
disperse model first. Although Pattamaprom et al. (2008)
showed that molecular models could accurately describe
the experimental data of nearly monodisperse samples,
such models suffer from complicated conversion between
relaxation and dynamic moduli. Hence, we will modify
the model suggested by Marin and Graessley (1977) which
provides analytical conversion.
The discrete version of the mixing rule is a linear equa-
tion of . Although the regression for Eq. (1) looks
simple, the solution must satisfy both of the following
constraints:
. (3)
Furthermore, it is well known that the calculation of is
a kind of ill-posed problem (Maier et al., 1998; Mead,
1994; Park et al., 2015; Thimm et al., 1999). In order to
calculate the MWD, we will modify the fixed-point iter-
ation method (FPI) which was developed for the contin-
uous relaxation time spectrum by Cho and Park (2013).
Although there are many algorithms to calculate relax-
ation time spectrum (Bae and Cho, 2015; Honerkamp and
Weese, 1989; Honerkamp and Weese, 1993; Stadler and
Bailly, 2009) and there are even programs available for
free (Takeh and Shanbhag, 2013; Shanbhag, 2019), the
fixed-point iteration is a simple iterative method and always
gives non-negative values of the spectrum if and only if
the initial spectrum is non-negative. Furthermore, it has
the ability of regularization. Since solving the mixing rule
for MWD is similar to the calculation of the relaxation
spectrum, we adopted FPI which avoids the occurrence of
negative values of . Nonlinear regularization prevents
the negative , too. However, we guess that FPI is much
simpler and faster than nonlinear regularization.
Hence, this paper consists of three parts: the validity of
the monodisperse model, the MWD calculation algorithm
based on the fixed-point iteration and the results and dis-
cussion.
2. Linear Viscoelastic Model for Monodisperse Poly-mers
2.1. Extended Cole-Cole modelMarin and Graessley (1997) suggested a phenomeno-
logical model motivated from the Cole-Cole model which
can be expressed by the Laplace transform of creep com-
pliance as follows
. (4)
Since the compliance of the glassy state Jg is negligibly
small (~109 Pa1), we will omit it. The zero-shear viscos-
ity, and the two retardation times 1 and 2 must
depend on molecular weight strongly compared with the
compliances J1 and J2. It is a reasonable assumption that
the two exponents 1 and 2 are nearly independent of
molecular weight (Marin and Graessley, 1977). The omis-
sion of Jg in Eq. (4) implies that the model seems to con-
sist of 7 parameters. However, as for monodisperse
polymers, the zero-shear viscosity and the steady-state
F1/
M t 2
2
2
0
1
1 and 0
N
1 2
1 2
o 1 2
1( )
1 ( ) 1 ( )g
J JsJ s J
s s s
�
o
o
Calculation of molecular weight distribution using extended Cole-Cole model and quadratic mixing rule
Korea-Australia Rheology J., 33(1), 2021 67
compliance satisfy
,
(5)
where and are fitting parameter dependent on the
kind of polymer, MC is the first critical molecular weight,
MC is the second critical molecular weight and .
Many experimental results show that (Rubinstein
and Colby, 2003), and there have been theories to explain
this behavior using molecular models (Doi and Edwards,
1986; Watanabe, 1999). It is known that the first critical
molecular weight is approximately two times of the entan-
glement molecular weight (Me), while the second critical
molecular weight is about six times as high as (Cho
et al., 2004).
Equation (5) is a phenomenological formula for the
experimental behavior such that zero-shear viscosity is
proportional to for and to M for
and the steady-state compliance is proportional to for
and converge to a specific value for
(Rubinstein and Colby, 2003).
We will introduce the molecular weight dependence to
and in Eq. (4) by use of . We consider
is nearly independent of molecular weight because it rep-
resents the relaxation process for the short length scale.
Finally, we model the molecular weight dependencies of
the parameters of Eq. (4) as follows
(6)
Equations (4) and (6) indicate that the model has nine
material parameters such as , , , , , , r
and . The empirical parameter r is chosen because
is not valid for some polymers and the mea-
surement of is usually less precise than that of . The
determination of the parameters suffers from the wide
range of the values of them as well as the dynamic moduli
and frequencies vary in logarithmic scale. Hence, conven-
tional methods for nonlinear regression based on the gra-
dient method are not effective. For this problem, we
adopted the Monte Carlo method (Kim et al., 2018) which
is similar to simulated annealing (Aarts and Korst, 1989)
without annealing process. The exponent is introduced
to check the validity of the regression method since it is
well known that .
Since available experimental data is often given by
dynamic moduli instead of , we need
, (7)
where i is the imaginary unit , and
are storage and loss compliances, and and
are storage and loss moduli, respectively. Application of
Eq. (7) to Eq. (4) yields
. (8)
Dynamic modulus is obtained from Eq. (8) as follows
.
(9)
2.2. Determination of parameter of extended Cole-
Cole modelWe used the data of Schausberger et al. (1985) and Jeon
(2010). The reference temperature is 180°C. The samples
are all PS with narrow MWD (PDI < 1.1), as listed in
Table 1.
The dynamic moduli of the PS’s in Table 1 are shown in
Figs. 1 and 2 and the regression results are listed in Table
2.
As expected, is obtained. The entanglement
molecular weight was determined as 12,300 g/mol
which is little bit lower than but close to the values in the
literature (16,625 g/mol (Pattamaprom, 2008); 17,000 g/
mol (Rubinstein and Colby, 2003); 14,900 g/mol (Vega et
o
eJ
1
e o
o o e( ) 1 , ( )
1
C
C C C
M M M MM J M J
M M M M
e
o J
3.4
3.4
eM
o
3.4M
CM M
CM M
M
CM M
CM M
1J
2J
o
e( )J M
2
1
e
o o
1 2
e
e e
( ) 1 ,
( ) for 1, 2,1
( ) ,
2 , .
C C
C
k k
C
C C
M MM
M M
M MJ M J k
M M
MM
M
M M M rM
1
2
2
eM
e
o
1J
2J
e6
CM M
o
eJ o
3.4
( )J s�
i (i ) ( ) i ( )
1( ) i ( )
( ) i ( )
J J J
G GJ J
�
2i 1 ( )J ( )J
( )G ( )G
2
21
2
21o
1 ( ) cos2( )
1 2( ) cos ( )2
( ) sin1 2( )
1 2( ) cos ( )2
k
k k
k
k k
k
k
k
kk
k k
k
k
k
kk
k k
J J
J J
2 2 2 2
( ) ( )( ) , ( )
( ) ( ) ( ) ( )
J JG G
J J J J
3.46
eM
Table 1. Specification of monodisperse polystyrene samples.
Sample Mw [kg/mol] PDI
PS 1 34 1.05
PS 2 65 1.06
PS 3 125 1.06
PS 4 292 1.07
PS 5 757 1.07
PS 6 2540 1.05
mPS 111k 111 1.07
mPS 79K 79 1.04
Junghaeng Lee, Sangmo Kim and Kwang Soo Cho
68 Korea-Australia Rheology J., 33(1), 2021
al., 2004)). As for the steady-state compliance, literature
values (Montfort et al., 1986) are about 105 Pa1 which
has the same order of in Table 2. According to Fuchs
et al. (1996), it is known that . Then we can
generate dynamic moduli of monodisperse PS for all
molecular weights we need using these parameters obtained
above. It should be remarked again that our phenomeno-
logical model is easier than molecular models in the con-
version between relaxation and dynamic moduli, as shown
in the Appendix.
The terminal constants such as and can be
obtained from experimental data through the following
. (10)
On the other hand, the model implies that .
We compared the terminal constants from experimental
data and the model prediction, Eq. (6) in Fig. 3. The zero-
shear viscosity agrees with the model entirely while the
steady-state compliance shows considerable deviation. It
is interesting that irrespective of the deviation in the
steady-state compliance, the fit of dynamic moduli is suc-
1J
o o
e N2.4J G
o
o
eJ
o
o e 2 20 0
o
( ) ( )lim , lim
G GJ
o
e 1 2J J J
Fig. 1. (Color online) Comparison of regression results and
experimental data of storage modulus of monodisperse polysty-
rene measured by Schausberger et al. (1985) and Jeon (2010).
Symbol and solid lines are experimental and regression data,
respectively.
Fig. 2. (Color online) Comparison of regression results and
experimental data of loss modulus of monodisperse polystyrene
measured by Schausberger et al. (1985) and Jeon (2010). (Sym-
bol and solid lines are experimental and regression data, respec-
tively.)
Table 2. Monte Carlo regression results (Reference temperature
180°C).
k = 1 k = 2
[Pa1] 2.18 × 105
5.10 × 106
0.36 0.59
[s] 2.44 × 10-4
3.46
[Pa·s] 33.90
r 3.52
Me [kg/mol] 12.30
kJ
k
2
e
o
Fig. 3. (Color online) Comparison of the zero-shear viscosity [a] and the steady-state compliance [b] calculated from the extended Cole-
Cole model and the experimental data of Figs. 1 and 2.
Calculation of molecular weight distribution using extended Cole-Cole model and quadratic mixing rule
Korea-Australia Rheology J., 33(1), 2021 69
cessful. The overestimation in Fig. 3b would be explained
by the sensitivity of the steady-state compliance on MWD
(Doi and Edwards, 1986; Kwon et al., 2019). In Ref.
(Kwon et al., 2019), our group considered the effect of the
PDI on and using the log-normal distribution
MWD. Figures 20 and 21 of Ref. (Kwon et al., 2019)
show the calculation result of and using a quadratic
mixing rule for the log-normal distribution MWD. In
those two figures, we can see that the sensitivity of on
MWD is more significant than . Zero shear viscosity is
almost constant when the PDI is less than 1.2, while the
increment of is much bigger than even if the PDI
starts to deviate slightly from 1. In addition, Fig. 7.5 of
Doi and Edwards (1986) shows severe scatters of the
experimental data of (monodisperse samples). It sup-
ports numerical simulation. That is, we can see that is
more affected by PDI than unless PDI is precisely 1.
Also, the extended Cole-Cole model describes well the
linear viscoelasticity of monodisperse PS (Figs. 1 and 2).
One may also guess that the overestimation indicates the
problem of the model prediction. To check the model pre-
diction, we impose a constraint of .
The value is chosen from the experimental
data of Fig. 3b. This constraint makes Fig. 3b better and
Fig. 3a maintains its accuracy while the fitting quality of
Figs. 1 and 2 slightly decreases. Since the accuracy of Fig.
3b is not more important for the calculation of MWD than
the accuracy of Figs. 1 and 2, we will not use the con-
straint.
The parameter is much smaller than the result
of Cho et al. (2004) . This discrepancy seems to con-
tribute to the inaccuracy in Fig. 3b. However, the inaccu-
racy of Fig. 3b does not deteriorate the fitting quality of
Figs. 1 and 2.
As similar to the results of Marin and Graessley (1977),
the exponents 1 and 1 have the values of 0.36 and 0.59,
respectively. These parameters control the shapes of the
relaxation time spectrum as well as the retardation time
spectrum.
We need to consider the entire molecular weight range
beyond Mr. Mr is the transition molecular weight at which
the exponent of changes from 3.4 to 3.0. According to
Vega et al. (2004), Mr evaluated from the polydisperse
polymer is about 400-500 kg/mol which is about 15 times
of Mc. However, they estimated Mr from monodisperse
samples and they could not find Mr up to M = 2500 kg/
mol. This result corresponds with our result in Fig. 3a.
Then we do not have to consider scaling relation beyond
Mr because, in this paper, the effect of molecular weight
beyond 2500 kg/mol is less than 1%. It will be shown in
Table 3.
o o
eJ
o
o
eJ
o
eJ
o
o
eJ
o
o
eJ
o
eJ
o
5 1
1 21.8 10 PaJ J
5 11.8 10 Pa
3.52r
6r
o
Table 3. Composition of polydisperse polystyrene samples.
Mw [kg/mol]M1 Weight
Fraction
M2 Weight
Fraction
2.89 0.001 0.001
5.57 0.002 0.002
9.10 0.004 0.004
19.60 0.008 0.008
37.90 0.03 0.03
96.40 0.15 0.148
190 0.26 0.257
355 0.358 0.353
706 0.14 0.139
1090 0.039 0.038
2890 0.008 0.01
3840 0 0.007
4480 0 0.003
total 1 1
Fig. 4. (Color online) Dynamic moduli of M1 and M2 mixtures (symbols). The lines are those calculated from the calculated MWD
and the monodisperse model.
Junghaeng Lee, Sangmo Kim and Kwang Soo Cho
70 Korea-Australia Rheology J., 33(1), 2021
3. Algorithm for Solving MWD from the MixingRule
3.1. Experimental dataWasserman and Graessley (1992) made two mixtures of
monodisperse PS’s with the compositions listed in Table
3. We digitized the dynamic moduli of M1 and M2 which
are shown in Fig. 4. To convert the dynamic moduli to the
relaxation modulus, we adopted the fixed-point iteration
of Cho and Park (2013). From the spectra of Fig. 5, we
calculated the relaxation moduli of M1 and M2, as shown
in Fig. 6 (Ferry, 1980). The vertical lines in Fig. 5 are the
Davies-Anderssen lines (Davies and Anderssen, 1997)
which indicate the effective range of relaxation time for a
given range of frequency of the experimental data.
By carefully looking at Fig. 6 one recognizes that the
relaxation modulus from calculated MWD shows better fit
for than for where can be interpreted as
the relaxation time at which the movement of the polymer
chain first feels the entanglement effect. It is because the
quadratic mixing rule is valued for the regime of fully
developed entanglement. Although the distinction about
is very small in Fig. 6, the application of of the
whole time range of Fig. 6 gives MWD mostly different
from the MWD calculated from with . It will
be shown in Fig. 11.
To obtain from the monodisperse model, we
used the Fuoss-Kirkwood relation (Fuoss and Kirkwood,
1941):
. (11)
et
et
e
e ( )G t
( )G te
t
( , )G t M
1 2 2
Im (i ) Re (i )H G G
Fig. 5. (Color online) Relaxation time spectrum of polydisperse
PS sample: [a] M1 sample and [b] M2 sample. Dashed line indi-
cates Davies-Anderssen line (Davis and Anderssen, 1997).
Fig. 6. (Color online) Relaxation moduli of M1 and M2 calcu-
lated from the spectra of Fig. 5 (symbols) and those calculated
from the calculated MWD and the monodisperse model (lines).
The vertical line represents the entanglement relaxation time
. We calculate MWD using data from the time domain
above . The agreement between symbols and lines reveals
that our algorithm is acceptable.
150
e
℃
150
e
℃
Calculation of molecular weight distribution using extended Cole-Cole model and quadratic mixing rule
Korea-Australia Rheology J., 33(1), 2021 71
Long manipulation of equations in the Appendix, we
finally have
. (12)
Then Eq. (12) gives the relaxation modulus of the ingre-
dient PS. The validity of Eq. (12) is checked, as shown in
Fig. 7. We calculated the relaxation time spectra of the
monodisperse PS’s in Figs. 1 and 2 in two ways, such as
fixed-point iteration and the use of Eq. (12). Analytically
calculated spectra agree with the numerically calculated
ones very well. Now we can calculate the relaxation mod-
uli of the monodisperse PS’s in Table 3. The relaxation
moduli are stored in the computer program to be used in
the calculation of the MWD of M1 and M2 mixtures.
Note that the parameters obtained from the data of Figs.
1 and 2 are those for 180°C while the dynamic modulus
data of Wasserman and Graessley (1992) are determined
at the reference temperature of 150°C. Since the shift fac-
tor is nearly independent of molecular weight, we
obtained the Williams-Landel-Ferry (WLF) parameters
from Schausberger et al. (1986) and Jeon (2010). The value
of the shift factor from 180°C to 150°C is .
Because we have to remove time region below the
Rouse regime (Thimm et al., 2000), it needs to calculate
which is the entanglement relaxation time at 150℃.
We can derive using the result of Schausberger et al.
(1985) and the shift factor obtained above. We determine
by the use
(13)
Its value is about 103 s. This value is shown as a dotted
line in Fig. 6. Then we use the relaxation modulus of M1
and M2 data for times longer than 103 s.
3.2. Modified FPI for MWD calculationAs mentioned in the Introduction, we will calculate
. Here, we used the Greek alphabet in order to dis-
tinguish the index for time . The Greek index denotes
the molecular weight. Since the discrete mixing rule of
Eq. (1) is a linear equation, the sum of the square error to
be minimized is
, (14)
where is the number of times and N is the number of
molecular weights. The normal equation of Eq. (14) is
given by
, (15)
where
.(16)
Directly solving Eq. (15) might result in negative weight
fractions for some molecular weights because of the errors
imbedded in and . Hence, we consider
the following iteration equation:
. (17)
This is the fixed-point iteration (Cho and Park, 2013).
Note that n indicates the iteration number. If the initial
MWD then all is not negative. To meet the
constraints of Eq. (3), we did rescaling such that
(18)
for each iteration step.
Since the molar mass of the monomer of PS is about
104 g/mol, as for the interval of molecular weight of
, N is approximately 5 million.
The use of such large N is not practical. For the first iter-
ations, we choose small and the molecular weights
, (19)
where and . The ini-
tial MWD of the first iteration is taken as the uniform dis-
tribution such that
. (20)
Although various initial conditions could be chosen, we
adopt the simplest one: uniform distribution. If the initial
MWD is closer to the real one, then the iteration would be
Re
2 2
Re Im Re
2( , )
( ) 4
VH M
U V V
10 Tlog 1.65a
150
e
℃
150
e
℃
150
e
℃
3.4
o
e o e
e
MJ
M
it
2
1 1
11 ( , )
( )
TN N
m i
i i
G t MG t
TN
1
N
S g
1 1
( , ) ( , ) ( , ),
( ) ( )
T TN N
m i m im i
i ii i
G t M G t M G t MS g
G t G t
( , )m i
G t M
( )i
G t
( 1) ( )
( )
1
log logn n
N
n
g
S
(0)0
( )n
1
n
n
N
n
3 610 g/mol 5 10 g/molM
10N
max min
min
log( )log log ( 1), 1,2, ,
1
M MM M N
N
min1000 g/molM
6
max5 10 g/molM
(0) 1, 1,2, ,N
N
Fig. 7. (Color online) Relaxation time spectra obtained from the
extended Cole-Cole model and FPI method.
Junghaeng Lee, Sangmo Kim and Kwang Soo Cho
72 Korea-Australia Rheology J., 33(1), 2021
finished earlier. As iteration goes, the MWD evolves to
lower the sum of the square error . If approaches to a
certain value, we stop the first iteration and generate a
new initial MWD for the second iteration process, which
has the doubled partition of molecular weight ( ),
by the interpolation of the last MWD of the first iteration
process. We repeat this procedure until decreases suffi-
ciently, as shown in Fig. 8.
We tested the number of molecular weights at the first
iterations and observed as a function of iteration number
, as shown in Fig. 9. Before , the sum of
square error reached its saturate values. Of course, the
higher N, the lower the saturated sum of square error.
We set at the first iterations. At every 200 iter-
ations, we doubled the number of molecular weights, as
shown in Fig. 10. As mentioned before, the initial MWD
of the first iterations is the uniform one . When
the first doubling of N ( ), is calculated
from the interpolation explained in Fig. 8. At ,
we used and the initial MWD is the inter-
polation of . As for the first three doublings, we can
find a dramatic reduction of while after the fourth dou-
bling, continuous reduction of is observed.
Since FPI is an iteration algorithm, it needs a stopping
criterion. As shown in Fig. 9, we can stop the iteration
when a further increase of iteration does not give any sig-
nificant reduction of the sum of square error . One may
select the stopping criterion as . As
shown in Fig. 10, decreases and decreases as both N and
increases. We stopped the partition of molecular
2N N
itrN
itr40N
10N
(0) 110
itr200N
(200)
itr400N
40N
(400)
(399)
itr 1 itr
1N N
itrN
Fig. 8. (Color online) The modified FPI method is conducted by [1]. The initial MWD is taken by 1/N. [2] MWD after the Nitr times
iteration. [3] Addition of new components to the midpoints of neighbor molecular weight components. Doubling of the number of
molecular weights. (The interpolated MWD will be used as the initial MWD for the next iterations) [4] MWD calculated from the next
iterations (Nitr times again). This procedure is repeated until the sum of square error is lower than a certain value.
Fig. 9. (Color online) The results of the first iteration. As the
number of molecular weights increases, the sum of square error
at a high iteration number decreases. The saturated values of
were achieved at Nitr < 40.
Calculation of molecular weight distribution using extended Cole-Cole model and quadratic mixing rule
Korea-Australia Rheology J., 33(1), 2021 73
weight at 1280 since converged sufficiently. Although
higher N would give a smaller value for , we get to know
that the change in the shape of MWD becomes negligible.
Hence the users of this algorithm have to decide to stop
the iteration and the partitioning by observing the change
of the shape of MWD.
4. Results and Discussion
Figure 11 shows the final MWD’s for M1 and M2 mix-
tures, and . The symbols are the
weight fraction listed in Table 3 and the solid lines are the
MWD calculated from with and the dotted
lines are the MWD calculated from over the whole
range of time in Fig. 6. As the dotted lines show, imagi-
nary peaks appear at for both M1 and M2.
On the other hand, with gives acceptable
results. This reveals that the quadratic mixing rule is valid
for a fully entangled regime. It can also be said that our
strategy, the stepwise doubling of molecular weight points,
is quite effective, although the raw data were obtained
from the digitization of the graphs in Wasserman and
Graessley (1992).
Without a careful look, the two viscoelastic data of M1
and M2 would look similar, although M2 shows longer
relaxation at low-frequency range. Such longer relaxation
of M2 is originated from the small portion of higher
molecular weight, as shown in Table 3. Hence, we need to
magnify the high-molecular weight regime of Fig. 11.
Figure 12 is the magnification of Fig. 11 for high molec-
ular weights. Our algorithm shows considerable deviation
from the real MWD (symbols). The calculated MWD for
M2 shows higher values than that of M1. This qualitative
tendency agrees with the real MWD.
It is noteworthy that the mean relaxation time of mon-
odisperse polymers is longer than , if their molec-
ular weight exceeds 2100 kg/mol. However, the time
range of for both M1 and M2 is .
Thus, the discrepancy of Fig. 12 is natural because the vis-
coelastic data for is no longer enough to pre-
dict the MWD in the region of .
Theoretically, the molecular weight of a single polymer
chain must be the integer multiple of the molecular weight
of the monomer (say Mo). Hence, an extremely accurate
device for MWD, if existing, gave a train of peaks located
at the integer multiples of the molecular weight of the
monomer. However, GPC gives a continuous and smooth
MWD. If rheological data were free from any error and
the MWD calculation were perfect, then the calculated
itr1600N 1280N
( )G te
t
( )G t
2 kg/molM
( )G te
t
43 10 s
( )G t4 4
2 10 s 3 10 st
43 10 st
2100 kg/molM
Fig. 10. (Color online) The effect of partitioning. At every 200
iterations, the number of molecular weight N is doubled. Dou-
bling of N reduces dramatically until the number of the dou-
bling is three. However, such dramatic reduction becomes
continuous if the number of doubling exceeds four.
Fig. 11. (Color online) The final MWD’s with N = 1280. The
symbols are the original MWD of each polydisperse sample
listed in Table 3 and the solid lines are the MWD calculated from
G(t) with and the dotted lines are the MWD calculated
from G(t) over the whole range of time in Fig. 6.e
t
Junghaeng Lee, Sangmo Kim and Kwang Soo Cho
74 Korea-Australia Rheology J., 33(1), 2021
MWD with the extreme partition such that
must be a train of peaks. Cho and Park (2013) showed that
the application of FPI to a single Maxwell model gives the
relaxation spectrum which becomes the Dirac delta func-
tion as the iteration increases. Hence, one may expect that
the application of FPI to the MWD would give a train of
peaks if the partition of molecular weight is the extreme
one. However, it cannot be implemented because of the
memory problem in our computer. Furthermore, the effect
of the errors in rheological data prevents the prediction of
the train of peaks.
As for a binary mixture of nearly monodisperse samples,
we can take a narrower molecular weight range than M1
and M2 data. Hence, the partition of molecular weight can
be closer to the extreme one. Then we can confirm
whether the calculated MWD looks like a train of peaks.
We test our algorithm for simulated data and two exper-
imental data (Maier et al., 1998 and Jeon, 2010). Figure
13 shows the calculation results of the MWD. As expected,
we have two sharp peaks similar to the delta function. The
1 oM M M
Fig. 12. (Color online) The magnification of Fig. 11 for high
molecular weights. The symbols are the original MWD of each
polydisperse sample and the blue-dotted line is M1 and the red-
solid line is M2.
Fig. 13. (Color online) Calculation results of MWD of binary mixtures: [a] Simulated MWD with 20% of PS (60 kg/mol) and 80%
of PS (177 kg/mol), [b] Experimental data of Maier et al. (1998), and [c] Experimental data of Jeon (2010). Symbols are original MWD
and solid lines are the calculation results of MWD.
Fig. 14. (Color online) Relaxation moduli corresponding to Fig 13. Symbols are conversion results from dynamic moduli and solid lines
are the calculation results from obtained MWD.
Calculation of molecular weight distribution using extended Cole-Cole model and quadratic mixing rule
Korea-Australia Rheology J., 33(1), 2021 75
calculated MWD for the simulated data agrees with the
original MWD (discrete one), while the calculated MWD
for the two experimental data does not agree with the
positions and heights of the peaks of the original MWD.
The discrepancy for the experimental data must originate
from the errors in the raw data. On the other hand, the
relaxation moduli calculated from the calculated MWD
agrees well with the relaxation moduli calculated from the
measured data of the dynamic modulus (Fig. 14). This
shows that a conversion from relaxation modulus to
MWD is an ill-posed problem as strong as that in the con-
version from dynamic modulus to relaxation spectrum.
Note that Maier et al. (1998) used the regularization
method (LRG) which gives a MWD similar to that of
GPC because of the features of LRG. Although the multi-
mode Maxwell model gives a viscoelastic spectrum like a
train of peaks, LRG gives overlapped broadened peaks.
On the other hand, the application of FPI to the spectrum
gives a train of sharp peaks for the multi-mode Maxwell
model. The width of peaks of the FPI, of course, depends
on the errors in the raw data. Since we believe that FPI is
more precise than LRG in the calculation of the spectrum,
we expect the application of FPI to be superior to the
LRG.
We generate 40 copies of Fig. 6a (the relaxation mod-
ulus of M1 with ) by adding 3% statistical errors in
order to show how stable our algorithm is. The errors are
the random variables generated from the normal distribu-
tion with the mean of zero and the standard deviation of
unity. We calculated the error bars of MWD as the stan-
dard deviation of the MWD’s calculated from the 40 cop-
ies of the relaxation modulus. As shown in Fig. 15, the
error bar is narrower than the results of a regularization
method (see Fig. 2 of Thimm et al. (1999)). Similar results
are obtained for M2. Figure 15 is the result from
and Fig. 2 of Thimm et al. (1999) is the MWD of a binary
blend of nearly monodisperse PS. It is remarkable that the
error bars for are much broader than those for
. This result agrees with that the quadratic mixing
rule is valid for .
Our algorithm is designed for the concept that MWD of
homopolymer is basically discrete because the molecular
weight of the polymer should always be an integer mul-
tiple of the molecular weight of the monomer except for
the terminal group. As for PS, the molecular weight of the
monomer is about 100 g/mol. Hence if we take the range
of molecular weight as then,
the partition of molecular weight is given by .
The number of molecular weight N is too high to give rise
to memory problems for a conventional personal com-
puter even if stepwise doubling (see Figs. 8 and 10) are
applied. If the quadratic mixing rule and the monodisperse
model are perfect, we expected that our algorithm would
give such discrete MWD for the extreme partition of
molecular weight. However, since the premises are not
perfect, the goal cannot be achieved. Furthermore, because
of the slight polydispersity of nearly monodisperse sam-
ples, the MWD of Table 3 is not really the discrete MWD.
Table 3 is just the mixing ratio. Hence, we need an
approximation based on continuous MWD:
et
80N
CM M
CM M
CM M
31 kg/mol 5 10 kg/molM
55 10N
Fig. 15. (Color online) The effect of the errors of G(t) on cal-
culated MWD. 40 copies of the G(t) of Fig. 6[a] are generated by
the addition of 3% statistical errors. The errors are generated
from the normal distribution with the mean of zero and the stan-
dard deviation of unity. The symbols represent the MWD of M1
without error. The error bars represent the standard deviation
from the MWD calculated from the 40 copies. It is remarkable
that error bars for M > MC are narrower than those for M < MC.
Note that the quadratic mixing rule is valid for M > MC.
Fig. 16. (Color online) Comparison of GPC data and the calcu-
lated MWD for the polydisperse PS produced by Aldrich. The
line represents the GPC data and the symbols represent the cal-
culated MWD. The vertical line means
which corresponds to . Since the time interval of G(t)
is , MWD cannot be calculated accurately for
molecular weight higher than the vertical lines by a rheological
method.
1330kg molM
7000s
3 310 s 7 10 st
Junghaeng Lee, Sangmo Kim and Kwang Soo Cho
76 Korea-Australia Rheology J., 33(1), 2021
. (21)
We expect that Eq. (21) will allow us to compare our
results with GPC MWD.
We applied the algorithm to commercial PS purchased
from Aldrich. The weight-average molecular weight
( ) of the PS is about 350 kg/mol and the number-aver-
age molecular weight ( ) is about 170 kg/mol. We mea-
sure dynamic modulus at 130, 150, 180, 200, 240 and
270°C and master curve were obtained by Time-tempera-
ture superposition (TTS). Figure 16 shows the comparison
of GPC data and the calculated MWD using Eq. (21) for
the commercial PS. The line represents the GPC data and
the symbols represent the calculated MWD. The vertical
line means and the mean relaxation time
of monodisperse PS is about 7000 s at .
Since the time interval of the relaxation modulus
is allowed by the TTS, the rheological
data is not sufficient to predict correct MWD, as shown in
Fig. 16. Although the weight-average molecular weight of
the PS is about 350 kg/mol (maker’s data), the portion of
is significantly high. Our measurement
of MWD gives the while the calculated
MWD gives . Figure 16 shows that the
time interval of relaxation modulus
gives a radical decrease of MWD at . We
could not check the ability of our method because of the
lack of experimental data.
Since we obtained MWD, we can recover relaxation and
dynamic moduli of M1 and M2 mixtures using the mon-
odisperse model, as shown in Figs. 4 and 6, respectively.
Recovered dynamic moduli were calculated by using the
relaxation time spectrum obtained from the recovered
relaxation modulus. Good agreement between symbols
and lines reveals that our algorithm is quite acceptable
even though the raw data of dynamic moduli were obtained
from the digitalization of the graphs in Wasserman and
Graessley (1992).
5. Conclusions
Most linear viscoelastic data are measured by dynamic
test while mixing rule is expressed by relaxation modulus.
The extended Cole-Cole model has a good agreement
with the dynamic modulus of monodisperse polystyrenes
for all molecular weights we have. Thus, we can deter-
mine the parameters of the model from the experimental
data of monodisperse PS and calculate relaxation modulus
for any molecular weight. Since the model has an analyt-
ical equation for the relaxation spectrum, we can make
faster calculations than the use of any numerical method
of the conversion from dynamic modulus to relaxation
modulus.
It is a difficult problem to calculate MWD from the mix-
ing rule without any assumed model for MWD because
the weight fraction in the mixing rule must satisfy the con-
straints of Eq. (3). To solve the problem, we adopted the
modification of the fixed-point iteration which was orig-
inally developed for the continuous relaxation spectrum.
To reduce the computation load, we adopted a stepwise
doubling of the number of molecular weights. The calcu-
lated MWD was compared with known data. The com-
parison reveals that our algorithm is acceptable.
As for commercial polymer samples, previous algo-
rithms (Wasserman, 1995) without any MWD model give
a calculated MWD which is qualitatively similar to the
GPC MWD. Compared with these algorithms, we have a
strong point on the monodisperse model. Hence, we expect
that our algorithm will give, at least, similar accuracy.
Although we could not compare our algorithm with the
nonlinear regularization algorithm directly, it is reported
that the accuracy of FPI for the relaxation spectrum is
equivalent to nonlinear regularization (Bae and Cho, 2015).
Hence, we expect that our algorithm works as well as the
nonlinear regularization (Honerkamp and Weese, 1993).
Acknowledgment
This work was supported by the Mid-Career Researcher
Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education, Sci-
ence and Technology (2017R1A2B1005506).
Appendix
Omitting glassy compliance ( ), the expression of
the extended Cole-Cole model can be written as
. (A.1)
Replacement of s by gives Eq. (8). In order to calcu-
late the relaxation time spectrum by use of the Fuoss-
Kirkwood relation, we need the following
. (A.2)
Then Eq. (8) yields
, (A.3)
1 1
1 1 1
d ( ) ( )( ) ; ( )
d log log ( )
n
n n
n
n n
w W M W MW M M
M M M
wM
nM
1330 kg molM
1330 kg molM
3 310 s 7 10 st
1000 kg/molM
400 kg/molw
M
230 kg/molw
M
3 310 s 7 10 st
930 kg molM
0g
J
2
1o
1( )
1 ( ) k
k
k k
JsJ s
s s
�
i
Re Im
Re Im
(i ) ( ) i ( )
(i ) ( ) i ( )
J U U
J V V
2
Re
1
2
Im
1
Re Im
2
Im
1o
1 1 cos2( )
2(1 ) 2
sin 2( )
2
( ) ( )
1 (1 )(1 cos2 )( )
2 (1 )
k k
k
k k k
k k
k
k k
k k k
k
k k k
zU J
z d
zU J
d
V U
z zV J
d z
Calculation of molecular weight distribution using extended Cole-Cole model and quadratic mixing rule
Korea-Australia Rheology J., 33(1), 2021 77
where
. (A.4)
Substituting of Eq. (A.2) with (A.3) into Eq. (9) and using
Eq. (11) yields the spectrum of the extended Cole-Cole
model:
. (A.5)
Then relaxation modulus can be calculated by Eq. (A.5)
as follows
. (A.6)
References
Aarts, E. and J. Korst, 1989, Simulated Annealing and Boltzmann
Machines: A Stochastic Approach to Combinatorial Optimiza-
tion and Neural Computing, Wiley.
Anderssen, R.S. and D.W. Mead, 1998, Theoretical derivation of
molecular weight scaling for rheological parameters, J. Non-
Newton. Fluid Mech. 76, 299-306.
Bae, J.E., and K.S. Cho, 2015, Logarithmic method for contin-
uous relaxation spectrum and comparison with previous meth-
ods, J. Rheol. 59, 1081-1112.
Carrot, C. and J. Guillet, 1997, From dynamic moduli to molec-
ular weight distribution: A study of various polydisperse linear
polymers, J. Rheol. 41, 1203-1220.
Cho, K.S., K.H. Ahn, S.J. Lee, 2004, Simple method for deter-
mining the critical molecular weight from the loss modulus, J.
Polym. Sci.: Polym. Phys. 42, 2724-2729.
Cho, K.S., and G.W. Park, 2013, Fixed-point iteration for relax-
ation spectrum from dynamic mechanical data, J. Rheol. 57,
647-678.
Cho, K.S., M.K. Kwon, J. Lee, and S. Kim, 2017, Mathematical
analysis on linear viscoelastic identification, Korea-Aust.
Rheol. J. 29, 249-268.
Davies, A.R. and R.S. Anderssen, 1997, Sampling localization in
determining the relaxation spectrum, J. Non-Newton. Fluid
Mech. 73, 163-179.
Des Cloizeaux, J., 1988, Double reptation vs simple reptation in
polymer melts, Europhys. Lett. 5, 437-442.
Doi, M. and S.F. Edwards, 1986, The Theory of Polymer Dynam-
ics, Oxford University Press, New York.
Ferry, J.D., 1980, Viscoelastic properties of polymers, John Wiley
& Sons
Friedrich, C., R.J. Loy, and R.S. Anderssen, 2009, Relaxation
time spectrum molecular weight distribution relationships,
Rheol. Acta 48, 151-162.
Fuchs, K., Chr. Friedrich, and J. Weese, 1996, Viscoelastic prop-
erties of narrow-distribution poly(methyl methacrylates), Mac-
romolecules 29, 5893-5901.
Fuoss R.M., J.G. Kirkwood, 1941, Electrical properties of solids,
J. Am. Chem. Soc. 63, 385-394.
Guzman, J.D., J.D. Schieber, and R. Pollard, 2005, A regular-
ization-free method for the calculation of molecular weight dis-
tributions from dynamic moduli data, Rheol. Acta 44, 342-351.
Honerkamp, J., and J. Weese, 1989, Determination of the relax-
ation spectrum by a regularization method, Macromolecules
22, 4372-4377.
Honerkamp, J., and J. Weese, 1993, A nonlinear regularization
method for the calculation of relaxation spectra, Rheol. Acta
32, 65-73.
Jeon, H.C., 2010, Effect of polydispersity on the viscoelasticity
of model homo-polystyrene melts, MS Thesis, Kyungpook
National University, Korea.
Kim, S, J. Lee, S. Kim, and K.S. Cho, 2018, Application of
Monte Carlo method to nonlinear regression of theological
data, Korea-Aust. Rheol. J. 30, 21-28.
Kwon, M.K., J. Lee, K.S. Cho, S.J. Lee, H.C. Kim, S.W. Jeong,
and S.G. Lee, 2019, Scaling analysis on the linear viscoleas-
ticity of Cellulose 1-ethyl-3-methyl Imidazolium Acetate solu-
tions, Korea-Aust. Rheol. J. 31, 123-139.
Lang, C., 2017, A Laplace transform method for molecular mass
distribution calculation from rheometric data, J. Rheol. 61,
947-954.
Léonardi, F., A. Allal, and G. Marin, 2002, Molecular weight dis-
tribution from viscoelastic data: The importance of tube
renewal and Rouse modes, J. Rheol. 46, 209-224.
Maier, D., A. Eckstein, Cr. Friedrich, and J. Honerkamp, 1998,
Evaluation of models combining rheological data with the
molecular weight distribution, J. Rheol. 42, 1153-1173.
Marin, G., and W.W. Graessley, 1977, Viscoelastic properties of
high molecular weight polymers in the molten state, Rheol.
Acta 16, 527-533.
McGrory, W.J. and W.H. Tuminello, 1990, Determining the
molecular weight distribution from the stress relaxation prop-
erties of a melt, J. Rheol. 34, 867-890.
Mead, D.W., 1994, Determination of molecular weight distribu-
tion of linear flexible polymers from linear viscoelastic mate-
rial functions, J. Rheol. 38, 1797-1827.
Montfort, J.P., G. Marin, and Ph. Monge, 1986, Molecular weight
distribution dependence of the viscoelastic properties of linear
polymers: the coupling of reptation and tube-renewal effects,
Macromolecules 19, 1979-1988.
Nobile, M.R. and F. Cocchini, 2008, A generalized relation
between mwd and relaxation time spectrum, Rheol. Acta 47,
509-519.
Park, J.W., J. Yoon, J. Cha, and H.S. Lee, 2015, Determination of
molecular weight distribution and composition dependence of
monomeric friction factors from the stress relaxation of ultra-
high molecular weight polyethylene gels, J. Rheol. 59, 1173-
1189.
Pattamaprom, C., R.G. Larson, and A. Sirivat, 2008, Determining
polymer molecular weight distributions from rheological prop-
erties using the dual-constraint model, Rheol. Acta 47, 689-
700.
Rubinstein, M. and R.H. Colby, 2003, Polymer Physics, Oxford
University Press, New York.
21 2 cos2
( ) ,2
k
k k k k
k
k k k
d z z
z
Re
2 2
Re Im Re
1 2 ( )
( ) ( ) 2 ( )
VH
U V V
1
Re
2 21 1 1
Re Im Re
2 ( )exp( )( ) d log
( ) ( ) 2 ( )
V tG t
U V V
Junghaeng Lee, Sangmo Kim and Kwang Soo Cho
78 Korea-Australia Rheology J., 33(1), 2021
Schausberger, A., G. Schindlauer, and H. Janeschitz-Kriegl, 1985,
Linear elastico-viscous properties of molten standard polysty-
rene, Rheol. Acta 24, 220-227.
Shanbhag, S., 2019, pyReSpect: A computer program to extract
discrete and continuous spectra from stress relaxation experi-
ments, Macromol. Theor. Simul. 3, 1900005.
Stadler, F.J., and C. Bailly, 2009, A new method for the calcu-
lation of continuous relaxation spectra from dynamic mechan-
ical data, Rheol. Acta 48, 33-49.
Takeh, A., and S. Shanbhag, 2013, A computer program to
extract the continuous and discrete relaxation spectra from
dynamic viscoelastic measurements, Appl. Rheol. 23, 1-10.
Thimm, W., C. Friedrich, M. Marth, and J. Honerkamp, 1999, An
analytical relation between relaxation time spectrum and
molecular weight distribution, J. Rheol. 43, 1663-1672.
Thimm, W., C. Friedrich, M. Marth, and J. Honerkamp, 2000, On
the Rouse spectrum and the determination of the molecular
weight distribution from rheological data, J. Rheol. 44, 429-
438.
Tsenoglou, C., 1987, Viscoelasticity of Binary Homopolymer
Blends, ACS Polym. Prepr. 28, 185-186.
Tuminello, W.H., 1986, Molecular weight and molecular weight
distribution from dynamic measurements of polymer melts,
Polymer Engineering and Science 26, 1339-1347.
Van Ruymbeke, E., R. Keunings, and C. Bailly, 2002, Determi-
nation of the molecular weight distribution of entangled linear
polymers from linear viscoelasticity data, J. Non-Newtonian
Fluid Mech. 105, 153-175.
Vega, J.F., S. Rastogi, G. W.M. Peters, and H.E.H. Meijer, 2004,
Rheology and reptation of linear polymers. Ultrahigh molec-
ular weight chain dynamics in the melt, J. Rheol. 48, 663-678.
Wasserman, S.H. and W.W. Graessley, 1992, Effects of polydis-
persity on linear viscoelasticity in entangled polymer melts, J.
Rheol. 36, 543-572.
Wasserman, S.H., 1995, Calculating the molecular weight distri-
bution from linear viscoelastic response of polymer melts, J.
Rheol. 39, 601-625.
Watanabe, H., 1999, Viscoelasticity and dynamics of entangled
polymers, Prog. Polym. Sci. 24, 1253-1403.
Wu, S., 1985, Polymer molecular-weight distribution from dynamic
melts viscosity, Polym. Eng. and Sci. 25, 122-128.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.