On the Rouse spectrum and the determination of the
molecular weight distribution from rheological data
Wolfgang Thimm, Christian Friedrich,a) and Michael Marth
Freiburger Materialforschungszentrum, Stefan-Meier-Strasse 21,
D-79104 Freiburg im Breisgau, Germany
Josef Honerkamp
Freiburger Materialforschungszentrum, Stefan-Meier-Strasse 21,
D-79104 Freiburg im Breisgau, Germany and Universität Freiburg,
Fakultät für Physik, Hermann-Herder-Strasse 3, D-79104 Freiburg im Breisgau,
Germany
(Received 25 October 1999; final revision received 13 December 1999)
Synopsis
Maier et al. 关Maier, D., et al., J. Rheol. 42, 1153–1173 共1998兲兴 examined the reconstruction of
binary molecular weight distributions from rheological data for a series of polystyrene mixtures
using a generalized mixing rule recently introduced 关Anderssen, R. S. and D. W. Mead, J.
Non-Newtonian Fluid Mech. 76, 299–306 共1998兲兴. They found an unexpected high value for
␤ ( ␤ ⫽ 3.84⫾0.1), the mixing parameter, which is 1 for the reptation and 2 for the double reptation
or entanglement models. This result can be understood when the relaxation time spectrum is
decomposed into a Rouse and an entanglement part and only the latter is used for determination of
the molecular weight distribution. Applying a procedure which separates the Rouse processes from
the spectrum determined from measured dynamic shear moduli, ␤ values are found which are in
accordance with the double reptation theory and which give very good agreement between
molecular weight distributions determined by size-exclusion chromatography and by rheological
data evaluation. © 2000 The Society of Rheology. 关S0148-6055共00兲01202-5兴
I. INTRODUCTION
Recently, Thimm et al. 共1999a兲 derived a relation 关Eq. 共1兲兴 which relates the relaxation time spectrum h( ␶ ) with the molecular weight distribution 共MWD兲 w(m). This
relation is based on a generalized mixing rule presented and analyzed by Anderssen and
Mead 共1998兲. The derived relation, which can be used to calculate the MWD from the
estimated relaxation time spectrum, is
w共m兲 ⫽
1 ␣(1/␤ )
␤ 共 G 0 兲 1/␤
N
h̃ 共 m 兲
冉冕
⬁ h̃ 共 m ⬘ 兲
m
m⬘
dm ⬘
冊
(1/␤ ⫺1)
,
共1兲
and the inverse relation is given by
a兲
Author to whom all correspondence should be addressed; electronic mail: [email protected]
© 2000 by The Society of Rheology, Inc.
J. Rheol. 44共2兲, March/April 2000
0148-6055/2000/44共2兲/429/10/$20.00
429
THIMM ET AL.
430
FIG. 1. Error bars: Estimated spectrum of a 80 wt % 共PS 177 kg/mol兲 to a 20 wt % 共PS 60 kg/mol兲 mixture of
polystyrene on a linear scale (G 0N ⫽ 2⫻105 Pa兲.
h̃共m兲
G0N
⫽
␤
␣
冋冕
w共m兲
⬁
w共 m⬘兲
m
m⬘
dm ⬘
册
␤ ⫺1
.
共2兲
In these equations ␤ is the generalized mixing parameter, G 0N is the plateau shear modulus and ␣ is the scaling exponent in
␶ ⫽ km␣,
共3兲
with the constants k and ␣ ⬇ 3.4, which can be determined experimentally. We denote
h̃(m) ⬅ h 关 ␶ (m) 兴 . m is a dimensionless molecular weight (m ⫽ M /M 0 ), where M 0 is
the monomer molecular weight and M is the molecular weight of the polymer. These
relations are valid in a molecular weight range of m e ⬍ m ⬍ ⬁, where m e is the entanglement molecular weight. Using the data of the relaxation time spectrum 共see Fig. 1兲
estimated from dynamic shear moduli of a binary polystyrene mixture together with Eq.
共1兲 the molecular weight distribution is easily accessible, if the correct value of ␤ is
known. ␤ is 1 for the reptation 关Doi and Edwards 共1986兲兴 and 2 for double reptation
model 共or entanglement concept兲 ( ␤ ⫽ 2).
A detailed analysis of three series of different polymers revealed that the generalized
mixing parameter, ␤ , is not 2 which is favored by most theorists 关Tsenoglou 共1987兲,
1991兲; Des Cloizeaux 共1988兲兴, but lies in the range between 3 and 4 ( ␤ ⫽ 3⫺4)
关Thimm et al. 共1999b兲兴. This unexpected result, which was determined from the best
ROUSE SPECTRUM AND DETERMINATION OF MWD
431
FIG. 2. Relaxation time spectrum consisting of Rouse 共dashed line兲 and entanglement spectra 共dotted line兲
showing a typical spectrum 共a sum of the Rouse and entanglement spectra兲 plotted with double-logarithmic
axes.
agreement between the molecular weight distribution measured by size-exclusion chromatography SEC 共experiments兲 and the molecular weight distribution derived from rheological data 共model兲, was also found by Maier et al. 共1998兲 who used the same mixing
rule but a different analysis scheme to estimate the MWD. A short comparison between
the results obtained with both schemes is given in the Appendix.
In this article we give an explanation for the observed discrepancy between the theoretical predictions and the experimentally observed ␤ values. This deviation is a consequence of the improper treatment of the relaxation time spectrum’s short time tail. Therefore, in Sec. II we summarize the description of the relaxation time spectrum by the
Rouse and the entanglement models. We introduce a separation ansatz for the relaxation
time spectrum which enables us to determine that part of the spectrum which is relevant
for MWD determination 共Sec. III兲. In Sec. IV we reanalyze the experimental data again
and determine ␤ and the MWD. The conclusions are given in Sec. V.
II. ROUSE AND ENTANGLEMENT SPECTRUM
The relaxation time spectrum h( ␶ ) that is obtained by measurements will be considered as a sum of two contributions 共see Fig. 2兲:
h共␶兲 ⫽ hent共 ␶ 兲 ⫹h Rouse共 ␶ 兲 ,
共4兲
which are introduced here in Sec. II.
The spectrum h Rouse( ␶ ) that results from the Rouse model has been derived as 关Rouse
共1953兲兴
hRouse共 ␶ 兲 ⫽ ␬␶ ⫺1/2,
共5兲
for ␶ ⬍ ␶ R , where ␬ is a constant. Discussions of the Rouse model can be found, e.g., in
the books of de Gennes 共1979兲 and of Doi and Edwards 共1986兲.
The mixing rule introduced by Anderssen and Mead 共1998兲 is based just on entanglement processes. Consequently, in the relations on which Eqs. 共1兲 and 共2兲 are based, Rouse
processes are neglected. Therefore, the relaxation time spectrum, which should be used to
432
THIMM ET AL.
FIG. 3. 共a兲 Spectrum calculated from the SEC/GPC data of a 80 wt % 共PS 177 kg/mol兲 to a 20 wt % 共PS 60
kg/mol兲 mixture of polystyrene, by the analytical relation, Eq. 共2兲, with ␤ ⫽ 3.84. 共b兲 The same spectrum as in
共a兲 but with ␤ ⫽ 2.
determine the MWD from Eq. 共1兲 should contain just entanglement processes. We denote
in the following the relaxation time spectrum corresponding to Eqs. 共1兲 and 共2兲 h ent( ␶ ).
We want to illustrate which part the Rouse spectrum plays in the determination of
MWD. Therefore, a MWD determined from a SEC/共gel permeation chromatography兲
GPC examination is used to calculate the relaxation time spectrum h ent( ␶ ) according to
Eq. 共2兲. We use the MWD of a binary mixture of polystyrene 共PS兲 关Maier et al. 共1998兲兴,
shown in Fig. 5. The results obtained with ␤ ⫽ 3.84 and 2 are shown in Fig. 3. At lower
molecular weights, where the values of the MWD w(m) are zero, the calculated relaxation time spectrum h ent( ␶ ) also tends to zero 关 w(m) ⫽ 0 → h ent( ␶ ) ⫽ 0 兴 . This is
contrary to the spectrum determined from the experiment 共Fig. 1兲, because in the experimentally determined spectrum Rouse processes are present.
ROUSE SPECTRUM AND DETERMINATION OF MWD
433
It is clear from Eqs. 共2兲 and 共4兲 that if one wants to determine the MWD from
rheological data an estimate of the entanglement spectrum is necessary. In Sec. III how
such an estimate can be obtained is discussed.
III. ESTIMATION OF THE ENTANGLEMENT SPECTRUM
To construct an estimator ĥ ent( ␶ ) for h ent( ␶ ) the following procedure can be used 共the
circumflex denotes estimated quantities兲: the spectrum ĥ( ␶ ) is determined from the measured G ⬘ ( ␻ ), G ⬙ ( ␻ ) 关see Honerkamp and Weese 共1993兲兴. From this spectrum the Rouse
spectrum h Rouse( ␶ ),
ĥent共 ␶ 兲 ⫽ ĥ 共 ␶ 兲 ⫺h Rouse共 ␶ 兲 ,
共6兲
is subtracted. In the region below the entanglement relaxation time, ␶ e , the influence of
the entanglement spectrum can be considered to be negligible 共Fig. 3兲. Therefore, the
spectrum in this region can be identified with the Rouse spectrum. The constant ␬ in the
estimated Rouse spectrum, Eq. 共5兲, was adjusted from the estimated spectrum ĥ( ␶ ) in this
region.
IV. ESTIMATIONS OF PARAMETER ␤ AND MWD
By evaluating rheological data, we now examine whether the separation ansatz introduced in Sec. III really improves determination of the MWD. As the first step the parameter ␤ is determined using only the estimated entanglement spectrum ĥ ent( ␶ ). In the
second step the MWD is calculated by inserting the ␤ determined and the entanglement
spectrum ĥ ent( ␶ ) into Eq. 共1兲.
To determine ␤ the procedure introduced by Maier et al. 共1998兲 is used. In this
method the quantity ␹ , which is a measure of the accuracy of agreement between the
MWD obtained with a SEC/GPC examination and the MWD obtained from rheological
data evaluation, dependent on ␤ , is systematically calculated. Single exponential relaxation behavior is assumed. For details of this we refer the reader to their work 关Maier
et al. 共1998兲兴. ␤ is determined from the minimum in ␹ . We checked by Monte Carlo
simulation of the error in the measured data that the error of the minimum’s position is
negligible compared to the accuracy of ␤ given in this article. The results for the determination of ␤ are shown in Fig. 4. We find that for a polymethylmethacrylate 共PMMA兲
blend 共PMMA with M w ⫽ 45 and 155 kg/mol 关Fuchs et al. 共1996兲兴兲 a value of ␤
⫽ 1.6 and for a PS blend 共PS with M w ⫽ 60 000 and 177 000 g/mol 关Maier et al.
共1998兲兴兲 a value of ␤ ⫽ 2.2 describes the data best.
These values are significantly lower than the values found by Maier et al. 共1998兲 and
by Thimm et al. 共1999b兲 and are in the range of values predicted by the double reptation
or entanglement models. This observation can be explained intuitively: without subtraction of the Rouse spectrum, the short time tail of the estimated spectrum is larger than the
entanglement spectrum. Consequently, the MWD at higher molecular weights needs to be
weighted more. An analysis of Eq. 共2兲 reveals that such behavior can be obtained by
using larger values of ␤ . The value of the integral in Eq. 共2兲 changes with m from 1 to 0
and ␤ in the exponent influences the shape of the MWD.
To determine w(m) relation 共1兲 is used. The MWD for the PS mixture discussed
above determined with the value of ␤ ⫽ 2.2 is shown in Fig. 5共a兲.
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THIMM ET AL.
FIG. 4. Estimation of ␤ with the corrected entanglement spectrum for PMMA and PS gives values of ␤
⫽ 1.6 and 2.2.
For comparison, the MWD obtained with the estimated spectrum ĥ( ␶ ) and ␤
⫽ 3.84 is shown in Fig. 5共b兲. We find that the position of the peak connected to the
higher molecular weight component in particular agrees better with ␤ ⫽ 2.2 than with
the ␤ ⫽ 3.84 determination.
V. CONCLUSION
The generalized mixing rule introduced by Anderssen and Mead 共1998兲 is based on
entanglement processes. Consequently, just these processes should be used in determination of the MWD from rheological data.
We have included the effect, which is related to Rouse processes, in the evaluation of
rheological data. The estimated relaxation time spectrum is separated in an entanglement
and a Rouse part. This separation considerably improves determination of the molecular
weight distribution from rheological data.
A determination of the generalized mixing parameter ␤ , including this effect, gives
values, which are commensurate to the predictions from the established theories of
double reptation and entanglement models ( ␤ ⫽ 2). To finally conclude the material
dependence on ␤ , a larger number of materials has to be investigated.
We conclude that a highly accurate estimation of the MWD of binary linear polymer
mixtures evaluating rheological data is possible within the framework of the present
theories.
ACKNOWLEDGMENTS
The authors thank D. Maier and T. Roths for stimulating discussions and K. Fuchs and
A. Eckstein for accurately measuring the rheological data, on which this evaluation is
based. One of the authors 共W.T.兲 was supported by the Graduiertenkolleg für Strukturbildung in Makromolekularen Systemen 共Deutsche Forschungsgemeinschaft兲.
ROUSE SPECTRUM AND DETERMINATION OF MWD
435
FIG. 5. 共a兲 Estimated entanglement spectrum of a 80 wt % 共PS 177 kg/mol兲 to a 20 wt % 共PS 60 kg/mol兲
mixture of polystyrene used to determine the MWD with the generalized mixing parameter ␤ ⫽ 2.2 共error
bars兲. Solid curve: SEC/GPC determination. 共b兲 The same polymer mixture as in 共a兲 examined with ␤
⫽ 3.84 and the spectrum estimated from the dynamic moduli G ⬘ ( ␻ ),G ⬙ ( ␻ ) 共error bars兲.
APPENDIX
The analytical relation between the molecular weight distribution and the relaxation
time spectrum 关Thimm et al. 共1999a兲兴 can be used to calculate the molecular weight
distribution w(m) from rheological data. Maier et al. 共1998兲 have used a different
method to determine the MWD. They inverted the mixing rule,
G共t兲
G0N
⫽
冉冕
⬁
F共t,m兲1/␤
me
w共 m 兲
m
dm
冊
␤
,
共A1兲
436
THIMM ET AL.
FIG. 6. MWD of the binary mixture with 20% of PS 60 and 80% of PS 177 obtained by the SEC measurement
共solid line兲 and the regularization procedure 共error bars兲. The Des Cloizeaux kernel and ␤ ⫽ 2 have been used.
The evaluation of the Rouse modes has been incorporated.
in which G(t) is the relaxation shear modulus, G 0N the plateau modulus, m e the entanglement molecular weight and F(t,m) an integral kernel, which describes the relaxation
behavior of the polymer fraction with molecular weight m. Here, we concentrate on the
case in which the generalized mixing parameter has a value ␤ ⫽ 2. Maier et al. 共1998兲
examined five different integral kernels to determine the MWD: single exponential kernel, Tuminello kernel, Doi kernel, Baumgaertel, Schausberger, and Winter 共BSW兲 kernel
and Des Cloizeaux kernel 关see Maier et al. 共1998兲; Baumgaertel et al. 共1990兲; Des
Cloizeaux 共1990兲兴.
It was discussed by Thimm et al. 共1999b兲 that the results obtained with their analytical
relation and inversion of the mixing rule using the single exponential kernel give approximately the same results. Using the BSW and Des Cloizeaux kernel Maier et al. 共1998兲
obtained only poor results for determination of the MWD 关see Fig. 10共c兲 in Maier et al.
共1998, p. 1171兲兴. At this point the question arises as to whether this effect can be
attributed to the kernel’s properties or to the correction scheme. We suggest that these
poor results can only in small part be explained by their use of a spectrum that still
contains Rouse modes.
One may use the relaxation time spectrum ĥ ent( ␶ ), from which the Rouse modes have
been subtracted to calculate the relaxation shear modulus G(t). Then the Des Cloizeaux
or BSW kernel was inserted into the mixing rule and Eq. 共A1兲 was inverted to obtain the
MWD. The results are given in Figs. 6 and 7. Obviously, the third peak at the lower
molecular weight found by Maier et al. 共1998兲 corresponds to the Rouse spectrum, which
had not been subtracted, and is removed when the MWD with the corrected spectrum is
calculated. The weight of the former third peak is compensated for 共the MWD is a
normalized quantity兲 by enhanced sharpness of the remaining two peaks. Moreover, the
results calculated using h̃ ent( ␶ ) and the BSW kernel better represent the peak positions
than does the Des Cloizeaux kernel. But the locations of maxima are still poorer than
these, which are obtained using the analytical relation.
ROUSE SPECTRUM AND DETERMINATION OF MWD
437
FIG. 7. MWD of the binary mixture with 20% of PS 60 and 80% of PS 177 obtained by the SEC measurement
共solid line兲 and the regularization procedure 共error bars兲. The BSW kernel and ␤ ⫽ 2 have been used. The
evaluation of the Rouse modes has been incorporated.
The reason for this behavior can be explained by differences in the kernels. We regard
the three kernels: single exponential kernel F SE , Des Cloizeaux kernel F DC and BSW
kernel F BSW as
FSE共 t,m 兲 ⫽ e ⫺t/ ␶ (m) ,
FBSW共 t,m 兲 ⫽ a
冕
1
0
冉
duu a⫺1 exp ⫺
t
␶共m兲u
冊
共A2兲
⬇ e⫺t/x␶(m),
共A3兲
where a is fixed experimentally and x is found numerically in the range between 0.5
and 1,
FDC共 t,m 兲 ⫽
冋
8
⬁
兺
1
␲ 2 n ⑀ odd n 2
再 冋
2
exp ⫺n
t
␶共m兲
⫹
m
g
12.5m e
⬇ e ⫺2t/ ␶ (m) ,
冉
tm
12.5m e ␶ 共 m 兲
冊 册冎 册
2
共A4兲
with the function
g共x兲 ⫽
⬁
1⫺exp共⫺l2x兲
l⫽0
l2
兺
.
An extensive illustration of the different relaxation behaviors described by these kernels
corresponding to monodisperse fractions of polystyrene can be found in Fig. 8 of Maier
et al. 共1998兲.
The relation ␶ ⫽ km ␣ determines the position of the peaks in the molecular weight
distribution. Since the results obtained with the single exponential kernel agree with those
found with the analytical relation 关Thimm et al. 共1999b兲兴, we denote this prefactor k
⫽ k SE . With regard to kernels discussed above, we find that their effective prefactors
(k BSW ,k DC), which can be related to the BSW and Des Cloizeaux kernels, differ:
438
THIMM ET AL.
k BSW ⬇ xk SE and k DC ⬇ 0.5k SE . Consequently, the peaks in the MWD determined
with the Des Cloizeaux kernel are at positions which are about a factor of 2 too high.
It is the effective change in the prefactors which determines how suitable the kernels
for determination of the MWD using the method of Maier et al. 共1998兲 are.
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