Rheologica Acta

, Volume 47, Issue 7, pp 689–700

Determining polymer molecular weight distributions from rheological properties using the dual-constraint model

Authors

    • Department of Chemical EngineeringThammasat University
  • Ronald G. Larson
    • Department of Chemical EngineeringThe University of Michigan
  • Anuvat Sirivat
    • The Petroleum and Petrochemical CollegeChulalongkorn University
Original Contribution

DOI: 10.1007/s00397-008-0264-5

Cite this article as:
Pattamaprom, C., Larson, R.G. & Sirivat, A. Rheol Acta (2008) 47: 689. doi:10.1007/s00397-008-0264-5

Abstract

Although multiple models now exist for predicting the linear viscoelasticity of a polydisperse linear polymer from its molecular weight distribution (MWD) and for inverting this process by predicting the MWD from the linear rheology, such inverse predictions do not yet exist for long-chain branched polymers. Here, we develop and test a method of inverting the dual-constraint model (Pattamaprom et al., Rheol Acta 39:517–531, 2000; Pattamaprom and Larson, Macromolecules 34:5229–5237, 2001), a model that is able to predict the linear rheology of polydisperse linear and star-branched polymers. As a first step, we apply this method only to polydisperse linear polymers, by comparing the inverse predictions of the dual-constraint model to experimental GPC traces. We show that these predictions are usually at least as good, or better than, the inverse predictions obtained from the Doi–Edwards double-reptation model (Tsenoglou, ACS Polym Prepr 28:185–186, 1987; des Cloizeaux, J Europhys Lett 5:437–442, 1988; Mead, J Rheol 38:1797–1827, 1994), which we take as a “benchmark”—an acceptable invertible model for polydisperse linear polymers. By changing the predefined type of molecular weight distribution from log normal, which has two fitting parameters, to GEX, which has three parameters, the predictions of the dual-constraint model are slightly improved. These results suggest that models that are complex enough to predict branched polymer rheology can be inverted, at least for linear polymers, to obtain molecular weight distribution. Further work will be required to invert such models to allow prediction of the molecular weight distribution of branched polymers.

Keywords

Molecular weight distributionRheologyDual-constraint modelDouble-reptation model

Introduction

Polymer molecular weights (MW) and molecular weight distributions (MWD) are typically measured by gel permeation chromatography (GPC), which has long been considered the most convenient and reliable technique for these measurements. Nevertheless, GPC is sometimes expensive, or cannot be used, especially for polymers that dissolve only in very toxic solvents, or require high temperature columns. Moreover, high-molecular-weight tails, which contribute significantly to the extensional properties of polymers, are unfortunately not always detected by standard GPC columns.

During the last two decades, interest has developed in retrieving the molecular weight distributions of linear polymers from their linear viscoelasticity, by “inverting” the predictions of linear viscoelastic models (Tuminello 1986; Tsenoglou 1987; des Cloiseoux 1988; Mead 1994; Wasserman 1995; Carrot and Guillet 1997; Anderssen et al. 1997; Maier et al. 1998; Thimm et al. 2000; Léonardi et al. 2002; van Ruymbeke et al. 2002b; Cocchini and Nobile 2003). Interest in these “inverse” methods has arisen as a result of the recent development of methods for quantitative predictions of the linear viscoelastic response of polymer melts from their molecular weight distributions (Carrot et al. 1996; Pattamaprom et al. 2000; Pattamaprom and Larson 2001; Léonardi et al. 2000; van Ruybeke et al. 2002a; Likhtman and McLeish 2002). The differences among these models are mostly due to differences in the method of incorporating contour-length fluctuation and constraint release effects into the Doi–Edwards reptation model.

More recently, models have been developed for predicting the linear viscoselasticity of long-chain-branched polymers from their molecular weight distribution (Pattamaprom et al. 2000; Park et al. 2005; Das et al. 2006) raising the tantalizing possibility that in the future, the molecular weight distribution and possibly information about long-chain branching might be inferred from the linear rheology. One of these models, the “dual-constraint model” of Pattamaprom et al., accounts for polydisperse linear polymers or star-branched polymers and is simple enough to be readily inverted to predict molecular weight distribution.

The dual-constraint model includes both the direct effect of relaxation of each “test” chain in the sample and the indirect “constraint release” effect on that chain caused by the relaxation of neighboring “matrix” chains. In this model, the relaxation of the test chain occurs via reptation and contour-length fluctuation, while constraint release is introduced through double-reptation and tube dilation. The upper limit of the combined constraint release rate from reptation and tube dilation is controlled by Rouse-like motions of the tube containing the test chain, i.e., the so-called “constraint–release Rouse” processes (Viovy et al. 1991; Milner et al. 1998). The dual-constraint model has been shown to predict the linear viscoelastic response of various entangled linear polymers from their molecular weight distribution either quantitatively or at least semiquantitatively.

In this paper, we assess the prospects for inverting this dual-constraint model to obtain the MWD of linear polymers from their linear viscoelastic rheological properties. We develop an optimization scheme to determine the MW and MWD of several linear polymers, using GPC measurement of the MWD to evaluate the success of the inversion. Linear viscoelastic data from polystyrenes and polybutadienes are used to evaluate the model due to the availability of accurate GPC results for these polymers and because these polymers are believed to have no branch points. As a benchmark, the predictive capability of the dual-constraint inverse model is also compared with that of the simpler double-reptation model with the conventional Doi–Edwards kernel function. While this conventional model is not always the most accurate model for linear polymers, it is used in commercial software for inverting rheology to obtain molecular weight distribution, and is accurate enough to serve as a “minimally acceptable” model for extracting molecular weight distribution from rheology for linear polymers. Our strategy, then, is to develop a method of inverting the dual-constraint model and to test the accuracy of this model by applying it to polydisperse linear polymers, comparing its predictions to those obtained from the basic double-reptation model with Doi–Edwards kernel. If the dual-constraint model is at least as accurate as the double-reptation model with Doi–Edwards kernel for linear polymers, then we can regard it as “minimally acceptable,” and thus worth endeavoring to invert for branched polymers also. Here, however, we will only develop an inversion method for linear polymers.

Theoretical models

The dual-constraint model accounts for reptation, contour-length fluctuation, and constraint release by reptation and contour-length fluctuation of surrounding chains. This constraint release is described using the concepts of “double diffusion” and “dynamic tube dilation,” where double diffusion (sometimes also called “thermal constraint release”) includes constraint release by both reptation and contour-length fluctuation of the matrix chains. The dual-constraint model improves on the Doi–Edwards double-reptation model (Tsenoglou 1987; des Cloizeaux 1988; Mead 1994) in that the latter ignores dynamic tube dilation and accounts for contour-length fluctuations only indirectly by using an empirical power-law dependence of relaxation time on molecular weight. More recent literature (Leonardi et al. 2000; van Ruymbeke et al. 2002a; des Cloizeaux 1990), has improved upon the original Doi–Edwards double-reptation model by using a more accurate version of contour-length fluctuation involving an integral formulation with exponential time dependence. The mechanisms included in those models and in our model are quite similar, except that our model integrates an expression for relaxation mechanisms valid for both linear and star-branched polymers in a unified differential form (with the reptation shut off for star-branched polymers). Another difference is that the constraint release mechanisms included in our model include double-reptation, tube dilation, and constraint-release Rouse processes, which are important in mixtures of long chains with very short chains. In the dual-constraint model, the dilution effect is accounted for by first computing the relaxation of a chain in a fixed matrix. Then, the time-dependent tube survival probability computed from this “fixed-matrix” relaxation function is used to compute a time-dependent tube diameter. The tube diameter is then used in a second corrected calculation of the tube survival probability, ϕ(t), which thereby includes the dilution effect. Thus, the tube dilution effect is included in a two-step process, which is different from the dynamic dilution method of Ball and McLeish (1989). However, this two-step process yields very similar predictions for star polymers as are obtained from the Ball-McLeish method, and does not result in an infinite tube diameter at the terminal time, as the Ball-McLeish (or Milner and McLeish 1997) formulas do. Also, our method is easy to apply to both star and linear polymers. Thus, in our method, even a monodisperse linear polymer has a small degree of dynamic dilution, and for polydisperse linear polymers, this leads to better predictions of the dependence of the zero shear viscosity on Mw than is obtained from the simple double-reptation model, which has no dynamic dilution (Pattamaprom and Larson 2001). In the “double-reptation” method for accounting for constraint release, the relaxation function ϕ(t) is squared, where one factor of ϕ(t) accounts for relaxation of the chain, while the other factor accounts for constraint release. To include “double-reptation” constraint release in the dual-constraint model, the factor ϕ(t) accounting for constraint release is first corrected to prevent relaxation faster than allowed by constraint release Rouse motion, which gives the matrix relaxation function ϕ′(t). The modulus is then proportional to ϕ(t)ϕ′(t). The details of our method are explained elsewhere (Pattamaprom et al. 2000; Pattamaprom and Larson 2001).

Recently, the definitions of entanglement spacing and time constants were clarified (Larson et al. 2003), and mistakes made by several researchers due to confusion in the usage of these definitions were pointed out. As our earlier publications describing the dual-constraint model also contain such errors, in this publication, we will also correct these mistakes before using the model for the inverse problem of determining the molecular weight distribution from the linear viscoelastic properties.

A brief summary of the theories, the corrections, and model parameters for the dual-constraint model and a brief overview of the double-reptation model are given below.

Dual-constraint model

The dual-constraint model was originally developed by T.J. Van Dyke, R.G. Larson, D.W. Mead and M. Doi and refined by Pattamaprom et al. (2000) and Pattamaprom and Larson (2001). This model provides good predictions for monodisperse, bidisperse, and polydisperse linear polymers for a broad range of molecular weights. As mentioned above, the published model and its predictions used inconsistent definitions for the entanglement molecular weight and relaxation time constant. To correct this, we here base our definitions on Ferry’s definitions (F-definitions in Table 1 of Larson et al. 2003). The definitions used here and in the original dual-constraint model are compared in Table 1. Figure 1 shows that for linear polymers, the most significant effect of these corrections is to shift along the frequency axis the linear viscoelastic curves, which is equivalent to adjusting τe, the Rouse relaxation time of chain occupying a single entanglement spacing. Therefore, the predictions of this corrected version of the dual-constraint model are almost exactly the same as the original version except that one of the fitting parameters, τe, is reduced to 0.375 times the original value.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_Fig1_HTML.gif
Fig. 1

Relationship between the normalized viscosity \(\left( {\eta _{\text{r}} = \frac{{\eta _o }}{{G_N^o \tau _{\text{e}} }}} \right)\) and normalized molecular weight (Mr = Mw/Me) obtained from the prediction of the original dual-constraint model (solid line), from the corrected dual-constraint model (dashed line), and the adjustment of the prediction of the corrected dual-constraint model by multiplying the normalized viscosity by a factor of 0.375. This ability to overlay the curves implies that the predictions of the two version are nearly identical when τe,new = 0.375τe,old

Table 1

The corrected definitions of entanglement spacing and time constants vs. the original definitions for the dual-constraint model

Parameter

Original definitions (with mistakes)

Corrected definitions

Z Number of tube segments

\(Z = \frac{M}{{M_e^F }}\)

\(Z_{new} = \frac{5}{4}\frac{M}{{M_e^F }}\)

τd Reptation time

\(\tau _R = 3\left( Z \right)^3 \tau _e \)

\(\tau _{d,new} = 3\left( {Z_{new} } \right)^3 \tau _e \)

τR(s) Rouse relaxation time

\(\tau _R = \frac{1}{2}\left( Z \right)^2 \tau _e \) (Rouse stress time)

\(\tau _{R,new} = \left( {Z_{new} } \right)^2 \tau _e \) (Rouse rotational time)

τearly (s) early time fluctuation

\(\tau _{early} \left( \xi \right) = \left( {\frac{1}{2}} \right)\frac{{225}}{{256}}\pi ^3 Z^4 \tau _e \xi ^4 \) (using Rouse stress time)

\(\tau _{early} \left( \xi \right) = \left( 1 \right)\frac{9}{{16}}\pi ^3 Z_{new}^4 \tau _e \xi ^4 \) (apply Rouse rotational time)

U(si) activation energy for chain retraction (in fixed tube)

\(U\left( {\xi _i } \right)\frac{{15}}{8}Z\left( {1 - \xi _i } \right)^2 \)

\(U_{new} \left( {\xi _i } \right) = \frac{3}{2}Z_{new} \left( {1 - \xi _i } \right)^2 \)

τlate(s) late-time fluctuation

\(\tau _{late} \left( {\xi _i } \right) = \frac{{\tau _{R,i} }}{{c_i^2 }}\exp \left( {U\left( {\xi _i } \right)} \right)\)

\(\tau _{late} \left( {\xi _i } \right) = \frac{{\tau _{R,i} }}{{c_i^2 }}\exp \left( {U_{new} \left( {\xi _i } \right)} \right)\)

The parameters not listed in this table were already correct, based on the “F” definitions of Larson et al. (2003)

Note that in the original version of the dual-constraint model, we used the Rouse stress relaxation time for τR in the formulas for early time fluctuations and the late-time fluctuations. In the corrected version, we correct this by using the Rouse rotational time, which is double the Rouse stress time, for τR.

The dual-constraint model combines two forms of constraint release. The first is “double diffusion” which augments “double reptation” by inclusion of primitive path fluctuations as a mechanism of constraint release. The second form of constraint release is “dynamic dilution,” which was introduced by Ball and McLeish (1989) to describe the time-dependent loosening of the effective entanglement network, or widening of the tube. The upper limit of the combined constraint release rate from both mechanisms is controlled by Rouse-like motions of the tube containing the test chain; these motions are called “constraint-release Rouse” processes (Viovy et al. 1991; Milner et al. 1998)

The diffusion equation for the tube survival probability pi of chain type i including constraint release is rewritten as
$$\frac{{\partial p_i \left( {s_i ,t} \right)}}{{\partial t}} = \frac{{D_i }}{{L_i^2 }}\frac{{\partial ^2 p_i \left( {s_i ,t} \right)}}{{\partial s_i^2 }} - \frac{{p_i \left( {s_i ,t} \right)}}{{\tau _{\xi ,i} \left( {s_i } \right)}}$$
(1)
where pi(si,t) is the survival probability of a tube segment occupied by a chain of type i as a function of time t and contour distance si, where si ranges from 0 at the center of a linear polymer chain to one half at the end, or from 0 at the branch point to 1 at the end of a star branch. Note that the variable ξi in Table 1 is given by ξi = 1 − si for a star arm and is ξi = 1 − 2si for a linear molecule. Di, the curvilinear diffusion coefficient, is given by \(\frac{{L_i^2 }}{{\tau _{d,i} \pi ^2 }}\), where Li is the average contour length of the tube and τd,i is the reptation time constant (Doi and Edwards 1986). \(\tau _{\xi ,i} \) is the time constant for contour-length fluctuations in the presence of constraint release consisting of the early time (τearly) and late-time (τlate) fluctuations. \(\tau _{\xi ,i} \) is obtained from τearly and τlate using a crossover formula reported in our earlier work. Note that in Pattamaprom et al. (2000), due to a typological error, a square root is missing from the second line of the crossover formula in Eq. 4, where \(\tau _{_{\xi ,i} }^ * \) should equal \(\sqrt {\tau _{{\text{early,}}i} \cdot \tau _{{\text{late}},i}^ * } \). τearly and τlate are related to the relaxation time of one entanglement segment, τe, by the equations described in Table 1.
The overall survival probability ϕi(t) of a tube occupied by chain i can then be calculated from pi(si,t) by
$$\phi _i \left( t \right) = \int\limits_0^1 {p_i \left( {s_i ,t} \right)ds_i } $$
(2)
and the average survival probability ϕ(t) of all tubes is
$$\phi \left( t \right) = \Sigma w_i \phi _i \left( t \right)$$
(3)
where wi is the weight fraction of chains of type i.
The relaxation modulus G(t) from the contributions of reptation, contour-length fluctuations, and constraint release can be obtained from:
$$G{\left( t \right)} = G^{0}_{N} {\kern 1pt} \phi {\left( t \right)}\phi \prime {\left( t \right)}$$
(4)
where ϕ′(t) is the average survival probability modified by accounting for the constraint-release Rouse mechanism (Viovy et al. 1991). A full description of ϕi(t) and a complete explanation of the model, can be found in Pattamaprom et al. (2000) and Pattamaprom and Larson (2001). The model parameters (Table 2) are the plateau modulus \(G_N^0 \) and the Rouse relaxation time of a tube segment τe, which are both independent of the polymer molecular weight.
Table 2

Model parameters of the dual-constraint and the double-reptation models

Polymer

T (°C)

Mea

\(G_N^0 \)a (Pa)

Dual-Constraint Model

Double-reptation model (λ = KdMa; 5 Doi–Edwards terms)

τe (s)

a

Kd(s)

Polystyrene (PS)

15

16,625

2 × 105

1.875 × 10−2b

3.4

1.3 × 10−17b

0

17

1.5 × 10−3

9 × 10–19

0

Poly-butadiene (PBD)

25

1,929

1.15 × 106

3.5 × 10–7

3.4

4.5 × 10–19

28

2.8 × 10−7c

3.6 × 10−19c

aData obtained by shifting from 170 °C using the experimental shift factor aT = 14.41

bFrom Fetters et al. (1994), where \(M_e = {{\rho RT} \mathord{\left/ {\vphantom {{\rho RT} {G_N^o }}} \right. \kern-\nulldelimiterspace} {G_N^o }}\).

cData obtained from WLF shifting from 25 °C

The Doi–Edwards double-reptation model (commercial software)

In the double-reptation model as formulated by Mead, the model parameters are the plateau modulus \(G_N^0 \), the exponent a, and the empirical parameter K, where the relaxation modulus is given simply by (Mead 1994):
$$G\left( t \right) = G_N^0 \left[ {\sum\limits_i {w_i \left( t \right)p_i^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( t \right)} } \right]^{^2 } $$
(5)
Here, wi is the weight fraction of chains of type i and pi(t) is the tube survival probability of chain i at time t, which for the form of double reptation considered here, is calculated from the original theory of Doi and Edwards (1978, 1979) by
$$p_i \left( t \right) = \frac{8}{{\pi ^2 }}\sum\limits_{k = {\text{odd}}}^\infty {\frac{1}{{k^2 }}\exp \left[ {{{ - k^2 t} \mathord{\left/ {\vphantom {{ - k^2 t} {\tau _{{\text{rep}},i} }}} \right. \kern-\nulldelimiterspace} {\tau _{{\text{rep}},i} }}} \right]} $$
(6)
Since k includes odd numbers only, pi(t) can be closely approximated using a small number of summation terms. In this way, the calculation time can be dramatically shortened. In this paper, we use the first five terms because calculation using a higher number of terms does not change the prediction significantly. In the molecular weight determination using TA Instruments software, this number of summation terms is known as the number of Doi–Edwards terms. τrep,i, the longest relaxation time of chain i in a fixed tube, is obtained from the following empirical dependence on molecular weight:
$$\tau _{rep,i} = KM_{w,i}^a $$
(7)
where “a” is the power-law exponent, which is usually set to 3.4, and K is an empirical parameter. In this paper, experimental data for PS1_bm and PBD1 (which are polystyrene and polybutadiene samples described below) are used to obtain the value of K for polystyrene at 170 °C and polybutadiene at 28 °C, respectively.
For either model, the stress–relaxation modulus G(t) is combined with the contribution from high-frequency Rouse processes giving
$$G_{total} \left( t \right) = G\left( t \right) + \sum\limits_i {w_i G_{R,i} \left( t \right)} $$
(8)
where the high-frequency Rouse process GR,i(t) is given by
$$G_{R,i} \left( t \right) = \frac{1}{{N_{en,i} }}\left[ {\frac{1}{3}G_N^0 \sum\limits_{p = 1}^{N_{en,i} } {\exp \left( {\frac{{ - p^2 t}}{{2\tau _{HF,i} }}} \right) + } G_N^0 \sum\limits_{p = N_{en,i} + 1}^{N_i } {\exp \left( {\frac{{ - p^2 t}}{{\tau _{HF,i} }}} \right)} } \right]$$
(9)
and τHF,i is the Rouse stress time, which is one half the magnitude of the Rouse rotational time (τR,i). The time constant used for the first contribution on the right side of Eq. 9 is chosen to be the Rouse rotational time, following the theoretical derivation of the longitudinal mode of stress relaxation proposed by Likhtman and McLeish (2002) while continuing to use the front factor of 1/3 as in Milner and McLeish (1997). Note that, this value is twice as large as that used in Pattamaprom et al. (2000) and Pattamaprom and Larson (2001). We apologize for the missing front factor 1/Nen,i in the equation for high-frequency Rouse contribution to the modulus in Eq. 18 of Pattamaprom et al. 2000.
The total stress–relaxation modulus Gtotal(t) from either the dual-constraint or double-reptation model can then be converted to the storage modulus (G′) and the loss modulus (G″) using
$$G^\prime \left( \omega \right) = \omega \int\limits_0^\infty {G_{total} \left( t \right)\sin \left( {\omega t} \right)} \operatorname{d} t,\quad {\text{and}}\quad G^{\prime \prime } \left( \omega \right) = \omega \int\limits_0^\infty {G_{total} \left( t \right)\cos \left( {\omega t} \right)} \operatorname{d} t$$
Although the dependences of zero-shear viscosity ηo on molecular weight are similar for the double-reptation and the dual-constraint models, Fig. 2 shows that the variations of ηo with Mw/Mn for the two models are different. Therefore, the accuracy of the model predictions depends not only on the average molecular weight of the test samples but also on the molecular weight distribution. For unimodal polystyrenes and polybutadienes, Mw/Mn generally varies only within the range 1.0–2.6. Better test samples might be polymers with bimodal and multimodal molecular weight distributions, which are typically broader overall than are unimodal distributions for these polymers. Here, we compare predictions to experimental data for both unimodal and bimodal polymers, and even for a trimodal polymer.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_Fig2_HTML.gif
Fig. 2

The predictions of ηo versus Mw/Mn for polystyrene with Mw = 500 K at 170 °C using the dual-constraint model and the double-reptation model using the first five summation terms. The molecular-weight distribution is taken to be log-normal. The parameters used in both models were obtained by fitting the model predictions with a single polystyrene sample (PS1_bm – see Table 3) with Mw = 320 K and Mw/Mn = 1.2

Types of molecular weight distribution

To prevent the inverse problem from being ill-posed, we specify the functional form of the molecular weight distribution a priori. The types of distribution used here are the log-Gaussian, the generalized exponential (GEX), and the bi-log-Gaussian distributions, where the first two are unimodal distributions, and the last one is a bimodal distribution.

Unimodal distribution

For unimodal polydisperse polymers, we specify the distribution to be either a log-Gaussian or a generalized exponential (GEX) distribution, depending on the goodness of the fit. We find that the results obtained from GEX distribution are slightly improved relative to the log-Gaussian distribution. To shorten the calculation time, first, the optimization is carried out using a log-Gaussian distribution, which involves only two parameters. If the fitting using this distribution is not adequate (meaning that the overall discrepancy χ described later in Eq. 13 is still higher than 0.005), the GEX distribution, which involves three parameters, is applied to fine-tune the goodness of fit. Nevertheless, for the unimodal samples tested in this paper, the log-Gaussian distribution already yields a sufficiently low value of χ.

Log-Gaussian distribution

The log-Gaussian distribution generally fits experimental distributions of polymers produced by addition polymerization with heterogeneous catalysts such as those used to produce HDPE and polypropylene (Dealy and Wissbrun 1990). The log-Gaussian or log-normal distribution is
$$P\left( {\ln \;M} \right)\; = \;\frac{1}{{\sqrt {2\pi \sigma } }}\exp \left[ { - \frac{{\ln ^2 \left( {{M \mathord{\left/ {\vphantom {M m}} \right. \kern-\nulldelimiterspace} m}_{_0 } } \right)}}{{2\sigma ^2 }}} \right]$$
(10)
where the weight-average molecular weight MW is related to m0 by \(M_{\text{w}} = m_0 \exp \left( {{{\beta ^2 } \mathord{\left/ {\vphantom {{\beta ^2 } 4}} \right. \kern-\nulldelimiterspace} 4}} \right)\) and Mw/Mn is related to the SD σ of the distribution by \({{M_{\text{w}} } \mathord{\left/ {\vphantom {{M_{\text{w}} } {M_n }}} \right. \kern-\nulldelimiterspace} {M_n }} = \exp \left( {\sigma ^2 } \right)\). There are only two parameters needed for this distribution, which are either m0 and σ, or, equivalently, Mw and Mw/Mn. The distribution is Gaussian on a logarithmic scale with no skewness.

GEX distribution

The generalized exponential function, a.k.a. GEX distribution (Gloor 1978, 1983), has been used to incorporate skewness into the distribution, and it has been found to work well by several groups (Nobile et al. 1996; Léonardi et al. 2002; van Ruymbeke et al. 2002b). The equation for this distribution is
$$w\left( M \right) = \frac{b}{{M_o \Gamma \left( {\frac{{a + 1}}{b}} \right)}}\left[ {\frac{M}{{M_o }}} \right]^a \exp \left( { - \left[ {\frac{M}{{M_o }}} \right]^b } \right)$$
where Γ(x) is the gamma function. The corresponding Mw and Mn are
$$M_n = M_o \frac{{\Gamma \left( {\frac{{a + 1}}{b}} \right)}}{{\Gamma \left( {\frac{a}{b}} \right)}},\quad {\text{and}}\quad M_w = M_o \frac{{\Gamma \left( {\frac{{a + 2}}{b}} \right)}}{{\Gamma \left( {\frac{{a + 1}}{b}} \right)}},$$
and the polydispersity index is \(\frac{{M_w }}{{M_n }} = \frac{{\Gamma \left( {\frac{{a + 2}}{b}} \right)\Gamma \left( {\frac{a}{b}} \right)}}{{\left[ {\Gamma \left( {\frac{{a + 1}}{b}} \right)} \right]^2 }}\).

The parameters for this distribution are Mo, a, and b. The skewness and the broadness of the distribution can be reduced by increasing the values of a and b, respectively. When a is large, the distribution has no skewness and reduces to a log-normal distribution. When a and b are both large, Mo, the characteristic molecular weight, equals the weight average molecular weight, and the distribution approaches monodispersity. The distribution reduces to “the most probable distribution” when a = b = 1.

Multimodal distribution

For a multimodal distribution, the distribution is assumed to be a linear combination of log-normal distributions, as described below.
$$P\left( {\ln \;M} \right) = \sum\limits_{k = 1}^{nk} {w_k P_k } \left( {\ln M} \right)$$
(11)
Here, nk is the number of modes or peaks in the distribution and wk is the weight fraction of the mode k. P(ln M) can be obtained from Eq. 10. At present, our program incorporates only up to two modes (nk = 2) for a bi-log-Gaussian distribution. For this distribution, P(ln M) is
$$P\left( {\ln {\text{ }}M} \right){\text{ }} = {\text{ }}w_1 P_1 \left( {\ln {\text{ }}M} \right){\text{ }} + {\text{ }}w_2 P_2 \left( {\ln {\text{ }}M} \right)$$
(12)
where \(w_1 + w_2 = 1\). This distribution has five parameters, Mw1, Mw1/Mn1, Mw2, Mw2/Mn2, and w2.

Experimental data

We have collected experimental GPC traces and G′ and G″ curves for monodisperse, unimodal, and bimodal polybutadienes and polystyrenes from different laboratories and from the literature as summarized in Table 3. The polydispersity indices Mw/Mn from GPC traces for these polymers are shown in the column “GPC” of Tables 4, 5, and 6 for unimodal and bimodal polystyrenes and bimodal polybutadienes, respectively. These experimental data include some of the polystyrene benchmark standards generated by BASF, which are PS1_bm, PS2_bm, and PS3_bm. These standards were kindly produced by BASF under the leadership of Dr. Martin Laun as model samples for testing predictions of linear viscoelastic models proposed by various research groups.
Table 3

Sources of experimental data used in this paper

Polymer

Source

Sample name

Type of distribution

Rheological test temp. (°C)

Polystyrene (PS)

Wasserman and Graessley 1992

PS_M1

Polydisperse PS (mixture of several monodisperse PS)

150

PS_M2

van Ruymbeke et al. 2002a

PS1_ev

Monodisperse

170

PS9_ev

Bimodal (mixture of two unimodal polydisperse PS)

PS12_ev

BASF Laboratory

PS1_bm

Unimodal polydisperse

170

PS2_bm

Unimodal polydisperse

PS330

Unimodal polydisperse

PS3_bm

Trimodal polydisperse

Poly-butadiene (PBD)

Baumgaertel et al. 1992

PBD1

Monodisperse

28

Struglinski et al. 1985

PBD0.1

Bidisperse (mixture of two monodisperse PBD)

25

PBD0.3

PBD0.5

PBD0.7

The data consist of the linear viscoelastic response and the molecular weight distribution measured experimentally.

Table 4

Monodisperse and unimodal polydisperse polystyrenes: molecular weight distributions obtained from optimization based on the dual-constraint model and on the double-reptation model compared to the benchmark GPC data

Name

Polymer

N

GPC (experimental data)

Double-reptation (Log Normal)

Dual-constraint (Log Normal)

Dual-constraint (GEX)

MW (×103)

MWD

MW (×103)

MWD

MW (×103)

MWD

MW (×103)

a

b

PS_M1

PS (150 °C) (Wasserman and Graessley 1992)

25

356.7

2.3

310

2.3

347.5

1.8

337

2.28

0.38

PS_M2

25

398.5

2.6

345.6

2.7

392.5

2.2

378.2

4.32

0.2

PS1_ev

PS(170 °C) (van Ruymbeke et al. 2002a)

20

355.5

1.01

370

1.1

345

1.0

347.8

6.38

12.48

PS1_bm (used for parameter fitting)

PS (170 °C) (BASF Lab.)

20

320

1.2

320

1.2

317

1.2

315

4.1

1.52

PS2_bm

27

280

2.2

241.8

2.0

271

1.9

268

2.51

0.37

PS_330

95

324

2.85

297

1.98

320

1.8

322

3.53

0.34

N is the number of experimental data points.

Table 5

Discrete bidisperse and trimodal polystyrenes: molecular weight distributions obtained from optimization based on the dual-constraint model and on the double-reptation model compared to the benchmark GPC data

Name

Polymer

N

GPC (experimental)

Double-reptation results

Dual-constraint results

MW (×103)

MWD

xl

MW (×103)

MWD

xl

MW (×103)

MWD

xl

PS9_ev

PS(170 °C) (van Ruymbeke et al. 2002a)

17

191

1.02

 

220.8

1.3

 

190.1

1.01

 

887

1.09

0.5

1,033

1.12

0.32

916.5

1.16

0.5

PS12_ev

22

191.3

1.02

 

250.2

1.42

 

185.2

1.03

 

676

1.04

0.35

841.8

1.02

0.17

695

1.00

0.36

PS3_bm (trimodal)

PS (170 °C) (BASF Lab.)

23

288

2.1

184

1.45

 

168

1.15

 

1,310

1.20

0.15

1,212

1.23

0.22

Table 6

Bidisperse polybutadienes: molecular weight distributions obtained from optimization based on the dual-constraint model and based on the double-reptation model compared to the experimental data

Name

Polymer

N

GPC (experimental data)

Double-reptation results

Dual-constraint results

MW (×103)

MWD

xl

MW (×103)

MWD

xl

MW (×103)

MWD

xl

PBD0.1

PBD (25 °C) (Struglinski et al. 1985)

12

39

1.01

 

38.2

1.21

 

38

1.0

 

181

1.01

0.1

172.2

1.02

0.08

179.3

1.0

0.13

PBD0.3

12

39

1.01

 

39.9

1.28

 

38.2

1.08

 

181

1.01

0.3

182.1

1.02

0.22

169

1.05

0.33

PBD0.5

15

39

1.01

 

41.6

1.5

 

37

1.15

 

181

1.01

0.5

188.6

1.07

0.39

172

1.07

0.52

PBD0.7

15

39

1.01

 

43

1.43

 

69.8

1.94

 

181

1.01

0.7

191.7

1.05

0.6

172.9

1.01

0.64

Optimization scheme for inverting the dual-constraint model

We obtain MW and MWD of polymers from their rheological properties by matching the storage moduli (G′) and loss moduli (G″) of the linear viscoelastic oscillatory curves with those from the dual-constraint predictions. The values of MW and MWD used for the calculated G′ and G″ curves were systematically adjusted until the predicted curves superimposed on the experimental data. The adjustment was done using an optimization scheme which begins by specifying an initial guessed value of MW. The initial guess of MW was acquired from the correlation between the zero-shear viscosity η0 and the MW of monodisperse polymers predicted by the dual-constraint model (Fig. 1).

The optimization is then continued to obtain the minimum value of the overall discrepancy χ, which is measured by:
$$\chi = \quad \frac{1}{N}\left[ {\sum\limits_{i = 1}^N {\left| {\log \left( {\frac{{G_{\exp }^\prime \left( {\omega _i } \right)}}{{G_{cal}^\prime \left( {\omega _i } \right)}}} \right)} \right|} + \sum\limits_{i = 1}^N {\left| {\log \left( {\frac{{G_{\exp }^{\prime \prime } \left( {\omega _i } \right)}}{{G_{cal}^{\prime \prime } \left( {\omega _i } \right)}}} \right)} \right|} \,} \right]$$
(13)

Here, N is the total number of experimental data points. The values of N for each case are reported along with the experimental MW and MWD values in Tables 4, 5, and 6. The optimal parameters for each distribution type that minimize χ were determined by the Nelder-Mead simplex method (Nelder and Mead 1965), which is also available as a subroutine named amoeba in the book Numerical Recipes in Fortran 77 (Press et al. 1992). The optimization is terminated when the value of χ changes by less than 1% from the previous step, and the value of χ is less than 0.005. In our calculations, the number of iterations ranges from 70 to 250 for unimodal distributions and around 300–400 iterations for bimodal distributions.

Optimization results and discussion

Here, we compare the optimized MW and MWD values calculated from the dual-constraint model to those obtained from the commercial software based on the double-reptation model using the experimental GPC results as the benchmark. The comparisons are made for a number of monodisperse, bidisperse, and polydisperse polymers. The model parameters for both the double-reptation model and the dual-constraint model for polystyrenes and polybutadienes are summarized in Table 2. The time constants τe and K for polystyrenes were obtained by fitting the model predictions to experimental G′ and G″ data for a single polystyrene sample at 170 °C. In our case, the experimental data for PS1_bm was used. For polybutadienes, the time constants were obtained in the same way using the sample PBD1, a polybutadiene sample at 28 °C. The parameters at other temperatures were obtained using the experimental WLF shift factor. The results for unimodal and bimodal polymers are given in separate sections below.

Unimodal molecular weight distributions

Examples of the molecular weight distributions obtained from the predictions of the inverse dual-constraint scheme are compared in Figures 3, 4, and 5 with those of the inverse double-reptation scheme along with the GPC traces as benchmarks. The predefined molecular weight distributions for the predictions in those figures are log-normal. As can be seen, the inverse dual-constraint model provides slightly better predictions of the molecular weight distributions for all polystyrenes and bidisperse polybutadienes. We found that the inverse double-reptation model tends to overpredict the broadness of the distributions for narrow molecular weight distributions. This may be because the dual-constraint model can better describe the variation of linear viscoelastic moduli with molecular weight distribution of linear polymers.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_Fig3_HTML.gif
Fig. 3

a The best-fit linear viscoelastic oscillatory curves from the dual-constraint model (https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_IEqa_HTML.gif) and the double-reptation model (https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_IEqb_HTML.gif) compared to the experimental data (filled circles, filled triangles) for polystyrene sample PS1_bm. b The computed molecular weight distributions from the dual-constraint inversion (https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_IEqc_HTML.gif) and from double-reptation inversion (https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_IEqd_HTML.gif) corresponding to the best-fit curves in (a) compared to the experimental GPC traces of the same sample (https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_IEqe_HTML.gif). The predefined distribution is log-Gaussian. The experimental data were obtained from the BASF laboratory

https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_Fig4_HTML.gif
Fig. 4

a, b The same as in Fig. 3, except for polystyrene sample PS2_bm. The predefined distribution is log-Gaussian. The experimental data were obtained from the BASF laboratory

https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_Fig5_HTML.gif
Fig. 5

a, b The same as in Fig. 3, except for polystyrene sample PS_M1. The predefined distribution is log-Gaussian. The experimental data were obtained from Wasserman and Graessey (1992), who used a mixture of multiple discrete nearly monodisperse components to mimic a continuous molecular weight distribution. x(i) is the weight fraction of each discrete component

By switching the predefined type of molecular weight distribution from log normal, which has two fitting parameters, to GEX, which has three parameters, a better fit is obtained for the inverse dual-constraint to the experimental G′ and G″ data (as shown by the lower χ values). The results show that the predictions of the molecular weight distributions from the GEX distribution are also slightly improved further from those given by the log normal distribution, as shown in Fig. 6. The comparison of the predictions using the dual-constraint model and the double-reptation model are summarized in Table 4 for various unimodal polystyrenes.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_Fig6_HTML.gif
Fig. 6

Comparison of the dual-constraint predictions using log normal distribution with those using GEX distribution for samples a PS1_ev, b PS2_bm, c PS_M1

Bimodal and trimodal molecular weight distributions

For polymers that show upturns in the G′ curve or multiple humps in the G″ curves at frequencies lower than the Rouse frequency, the bi-log-Gaussian distribution defined in Eq. 12 is applied in the inverse calculations. Figure 7 and Table 5 shows the calculated results compared to the distributions obtained experimentally for polystyrenes at 170 °C. The experimental data were obtained from van Ruymbeke et al. (2002a) and from the BASF laboratory. As can be seen, the predictions of the inverse dual-constraint model improve upon those of the double-reptation model. Note, however, that for the trimodal distribution in Fig. 8, the inverse dual-constraint scheme with a predefined bi-log-Gaussian distribution can predict the high and low molecular weight peaks of the distribution well but fails to capture the peak or shoulder in the middle. Instead, the program compensates for the absence of a middle peak by broadening the distributions of the other peaks. The scheme could presumably be further improved by using a tri-log-Gaussian distribution instead of bi-log-Gaussian distribution. Nevertheless, very little improvement can be obtained by using a tri-modal distribution, as the optimized prediction of G′ and G″ obtained from a bi-log-Gaussian MWD distribution already fits well the experimental G′ and G″ curves (with a χ value lower than 0.001). Thus, the rheological method is not sensitive enough to detect the small middle peak in molecular weight, as the extra “humps” at intermediate frequency on the G′ and G″ curves produced by this small peak are hardly visible.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_Fig7_HTML.gif
Fig. 7

a, b The same as in Fig. 3, except for sample PS9_ev. The predefined distribution is bi-log-Gaussian. The experimental data were obtained from van Ruymbeke et al. (2002a)

https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_Fig8_HTML.gif
Fig. 8

a, b The same as in Fig. 3, but for sample PS3_bm. The pre-defined distribution is bi-log-Gaussian. The experimental data were obtained from the BASF laboratory

Other examples are bidisperse polybutadienes (PBD) taken from Struglinski et al. (1985). The bidisperse mixtures were produced by mixing two nearly monodisperse PBDs [M1 = 39,000 (Mw/Mn = 1.01) and M2 = 181,000 (Mw/Mn = 1.01)] at different ratios, where the fraction of the high molecular weight sample (xl) varies from 0.1 to 0.7. The rheological properties were measured at 25 °C. The comparisons of the predicted molecular weight distributions are shown in Fig. 9 and summarized in Table 6 for some mixtures using the model parameters in Table 2. From Table 6, it is obvious that the predictions of the inverted dual-constraint model, especially the broadness of the distribution, are closer to the actual distribution than are those of the double-reptation model, while the maxima in Mw are comparable. Nevertheless, at high xl (xl = 0.7), the double-reptation model captures the short-chain contribution better than does the dual-constraint model.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-008-0264-5/MediaObjects/397_2008_264_Fig9_HTML.gif
Fig. 9

a, b The same as in Fig. 3, except for bidisperse polybutadiene (39k/181k) at 25 °C (Struglinski et al. 1985), where the volume fraction of the high molecular weight component (xl) is 0.5 (sample PBD0.5). The predefined distribution is bi-log-Gaussian

Conclusions and perspective

We have tested the performance of the dual-constraint inversion of linear viscoelastic data for obtaining the molecular weight distributions of polybutadienes and polystyrenes by comparing it with the double-reptation inversion with the Doi–Edwards kernel. Using GPC data as the benchmark, we found that the dual-constraint inversion scheme provides predictions of MW and MWD that are generally closer to the experimental distributions than are those of the double-reptation inversion scheme for monodisperse, unimodal, and bimodal polydisperse polymers. The double-reptation model with Doi–Edwards kernel gives slightly better inversions only for monodisperse polybutadienes at high molecular weight and for bidisperse polybutadiene with high fraction of high-molecular-weight component. The calculation time on a PC of the dual-constraint inversion scheme ranges from 5 min to 2 h depending on the distribution type, which is considerably longer than that of the double-reptation inverse scheme, for which the calculation time is only about 3–5 min. In addition, the double-reptation model prediction can be improved fairly easily by using a kernel function that yields better predictions for monodisperse polymers than does the Doi–Edwards kernel, which predicts too narrow a relaxation spectrum. However, the dual-constraint model seems to be the simplest model that can predict the rheology of both linear and star-branched polymers. We have shown here, that when applied to linear polymers, the dual-constraint model gives predictions of MWD that are, overall, at least as good as those of the simplest acceptable model for linear polymers, namely, the double-reptation model with Doi–Edwards kernel. Thus, our work indicates that the dual-constraint model, or some model like it, might be used in the future to extract polydispersity from the rheology of either linear or star polymers.

Acknowledgement

The authors are grateful for the financial support from Thailand research fund grant number TRG4580064. Evelyn van Ruymbeke and BASF research laboratory, especially Dr. Martin Laun, are deeply appreciated for their kind consideration in sharing experimental data. R.G. Larson was supported by the NSF, grant numbers DMR-0096688 and DMR-0604965. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

Copyright information

© Springer-Verlag 2008