Determining polymer molecular weight distributions from rheological properties using the dual-constraint model
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 distribution Rheology Dual-constraint model Double-reptation modelIntroduction
Polymer molecular weights (M_{W}) 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 M_{w} 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 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 | \(Z = \frac{M}{{M_e^F }}\) | \(Z_{new} = \frac{5}{4}\frac{M}{{M_e^F }}\) |
τ_{d} Reptation time | \(\tau _R = 3\left( Z \right)^3 \tau _e \) | \(\tau _{d,new} = 3\left( {Z_{new} } \right)^3 \tau _e \) |
τ_{R}(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) |
τ_{early} (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(s_{i}) activation energy for chain retraction (in fixed tube) | \(U\left( {\xi _i } \right)\frac{{15}}{8}Z\left( {1 - \xi _i } \right)^2 \) | \(U_{new} \left( {\xi _i } \right) = \frac{3}{2}Z_{new} \left( {1 - \xi _i } \right)^2 \) |
τ_{late}(s) late-time fluctuation | \(\tau _{late} \left( {\xi _i } \right) = \frac{{\tau _{R,i} }}{{c_i^2 }}\exp \left( {U\left( {\xi _i } \right)} \right)\) | \(\tau _{late} \left( {\xi _i } \right) = \frac{{\tau _{R,i} }}{{c_i^2 }}\exp \left( {U_{new} \left( {\xi _i } \right)} \right)\) |
Note that in the original version of the dual-constraint model, we used the Rouse stress relaxation time for τ_{R} 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 τ_{R}.
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)
Model parameters of the dual-constraint and the double-reptation models
Polymer | T (°C) | M_{e}^{a} | \(G_N^0 \)^{a} (Pa) | Dual-Constraint Model | Double-reptation model (λ = K_{d}M^{a}; 5 Doi–Edwards terms) | |
---|---|---|---|---|---|---|
τ_{e} (s) | a | K_{d}(s) | ||||
Polystyrene (PS) | 15 | 16,625 | 2 × 10^{5} | 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 × 10^{6} | 3.5 × 10^{–7} | 3.4 | 4.5 × 10^{–19} |
28 | 2.8 × 10^{−7c} | 3.6 × 10^{−19c} |
The Doi–Edwards double-reptation model (commercial software)
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
GEX distribution
The parameters for this distribution are M_{o}, 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, M_{o}, 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
Experimental data
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 |
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 (×10^{3}) | MWD | MW (×10^{3}) | MWD | MW (×10^{3}) | MWD | MW (×10^{3}) | 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 |
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 (×10^{3}) | MWD | x_{l} | MW (×10^{3}) | MWD | x_{l} | MW (×10^{3}) | MWD | x_{l} | |||
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 |
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 (×10^{3}) | MWD | x_{l} | MW (×10^{3}) | MWD | x_{l} | MW (×10^{3}) | MWD | x_{l} | |||
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 M_{W} 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 M_{W} 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 M_{W}. The initial guess of M_{W} was acquired from the correlation between the zero-shear viscosity η_{0} and the M_{W} of monodisperse polymers predicted by the dual-constraint model (Fig. 1).
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 M_{W} 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 τ_{e} 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
Bimodal and trimodal molecular weight distributions
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 M_{W} 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.
Notes
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).
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