Rheologica Acta

, Volume 49, Issue 4, pp 411–422

Analytical rheology of blends of linear and star polymers using a Bayesian formulation

Authors

    • Department of Scientific ComputingFlorida State University
Original Contribution

DOI: 10.1007/s00397-010-0443-z

Cite this article as:
Shanbhag, S. Rheol Acta (2010) 49: 411. doi:10.1007/s00397-010-0443-z

Abstract

A Bayesian data analysis technique is presented as a general tool for inverting linear viscoelastic models of branched polymers. The proposed method takes rheological data of an unknown polymer sample as input and provides a distribution of compositions and structures consistent with the rheology, as its output. It does so by converting the inverse problem of analytical rheology into a sampling problem, using the idea of Bayesian inference. A Markov chain Monte Carlo method with delayed rejection is proposed to sample the resulting posterior distribution. As an example, the method is applied to pure linear and star polymers and linear–linear, star–star, and star–linear blends. It is able to (a) discriminate between pure and blend systems, (b) accurately predict the composition of the mixtures, in the absence of degenerate solutions, and (c) describe multiple solutions, when more than one possible combination of constituents is consistent with the rheology.

Keywords

Analytical rheologyBayesian inferenceMarkov chain Monte CarloPolymer blendTube modelLinear viscoelasticityModelingBayesian analysisInverse problem

Introduction

The presence of small amounts of long-chain branching (LCB) in polymer melts often improves their processability due to superior shear thinning and strain hardening properties. Diagnosing these trace levels of LCB (~1/10,000 backbone atoms) in polymer melts is a long-standing industrial problem since standard analytical techniques such as solution-based chromatography, light scattering, nuclear magnetic resonance (NMR), etc. suffer from sensitivity and lack of resolution (Shroff and Mavridis 1999; Wood-Adams et al. 2000; Janzen and Colby 1999; Dealy and Larson 2006). Many of these analytical methods require dissolving the polymer in a solvent at ambient temperatures—which is nontrivial for several commercially important polymers.

Rheology, on the other hand, is extraordinarily sensitive to details of molecular structure such as molecular weight and branching and is relatively easy to perform experimentally, which makes it an attractive candidate as a tool for analysis (Janzen and Colby 1999; Dealy and Larson 2006; Larson 2001; Vega et al. 2002; Lohse et al. 2002; Costeux et al. 2002; Doerpinghaus and Baird 2003; Robertson et al. 2004). Indeed, this is the promise of “analytical rheology”, where the rheological response of an arbitrary polymeric sample is used to infer its composition.

Contemporary models of branched polymers in melts have become increasingly accurate (de Gennes 1975; Doi and Edwards 1986; Ball and Mcleish 1989; Milner and McLeish 1997, 1998; McLeish et al. 1999; Likhtman and McLeish 2002; Graham et al. 2003; Park et al. 2005; Lee et al. 2005; van Ruymbeke et al. 2005a; Das et al. 2006; Larson et al. 2007), especially in the linear viscoelastic regime, although several gaps in our understanding persist (Larson et al. 2007; Shanbhag et al. 2007). The sensitivity of rheology to fine details of molecular structure, combined with the predictive power of current models, implies that the errors due to imperfections in the models are often less serious than errors due to the unreliable detecting power of standard analytical techniques. The possibility of combining melt rheology with solution chromatography (Janzen and Colby 1999; Shroff and Mavridis 1999; Wood-Adams and Dealy 2000; van Ruymbeke et al. 2005b), or other parallel techniques, offers another route for enhancing the fidelity of predictions.

In this paper, we frame analytical rheology as an inverse modeling problem, which involves reversing a “forward model”, such as a contemporary linear viscoelastic model for branched polymers that uses structural information to predict the rheology. Although the effects of LCB are often more pronounced in nonlinear rheology, the corresponding mathematical and computational models are not as mature as those for the linear regime. However, the method proposed in this paper is general, so that when quantitative nonlinear models become available in the future, they can be easily incorporated into the prediction framework.

Even the modest formulation considered in this paper is nontrivial because inverse problems are often ill-posed and are beleaguered by problems of existence, uniqueness, and continuous dependence of solutions. Thus, two different polymer samples may have nearly identical rheological responses, and small perturbations in sample parameters may lead to very different responses (Tarantola 2005; Anderssen 2005). As our specific forward model, we will use the open-source branch-on-branch hierarchical model (BoB) of Das et al. (2006), which has been used to quantitatively predict the rheology of commercial-grade polyolefins, containing a mixture of molecules of widely varying topology. Like most current mean-field models, it is based on the tube theory which replaces the environment of obstacles surrounding a test chain in a melt with a hypothetical tube and uses the notion of hierarchical relaxation to incorporate the processes of reptation, contour-length fluctuations, and constraint release in a self-consistent manner (Larson 2001; Park et al. 2005; Das et al. 2006). Due to their generality, these hierarchical models lack the mathematical structure that has been exploited in the past to invert the viscoelastic spectra of mixtures of linear chains (Wu 1985; Tuminello 1986; Wasserman 1995; Mead 1994; Carrot and Guillet 1997; Maier et al. 1998; Anderssen and Mead 1998; Léonardi et al. 1998; Thimm et al. 2000; Nobile and Cocchini 2001; van Ruymbeke et al. 2002; Wolpert and Ickstadt 2004).

A common philosophy guiding the inversion of the linear viscoelastic spectrum seeks to obtain the best or optimal sample parameters, which minimize the “distance” from the experimental data, often in a least-squared sense. This automatically assumes uniqueness and does not inform us of other nearby or far-off degenerate solutions that may only be slightly off. Besides, the number of species in the unknown experimental mixture must be known or assumed a priori—since expanding the set of parameters by increasing the number of species will always improve the quality of the fit.

In this paper, a Bayesian data analysis method is suggested—to solve the inverse problem of using rheological data to infer the composition and structure. As its engine, it uses a Markov chain Monte Carlo (MCMC) algorithm, which overcomes many of the problems associated with other techniques, and provides a robust framework for the systematic introduction of additional information such as knowledge of polymerization or chromatography in a systematic manner using “priors” to narrow the set of possible solutions. The MCMC method uses importance sampling to sample the distribution of solutions, thereby characterizing all possible solutions and associating them with a probability of likelihood, instead of trying to pick “the” optimal solution. Therefore, if multiple solutions exist, as is often the case, they can be all be described. Importance sampling confines the Markov chain to important regions of the solution space and is therefore far more efficient than an exhaustive search. Three unique advantages of the method proposed are (a) its ability to handle systems with an unknown number of species (Green 1995), (b) a built-in Occam’s razor, which prefers “simple” solutions to complex ones (MacKay 1991; Murray and Ghahramani 2005), and (c) the ability to characterize multiple solutions.

Model and methods

As a first step, instead of using actual experimental data to infer molecular structure, we add noise to predictions made using the forward model and treat that as “experimental data”. This is motivated by two factors, viz., (a) this ensures existence of a solution to the inverse problem—since we know at least one solution by construction—and (b) this allows us to test how good the proposed method to invert the forward model is. This is described in detail in “Generation of experimental data” section. The Bayesian formulation is described next (“Bayesian formulation” section), in which Bayes rule is used to transform the inverse problem of estimating system parameters from the experimental data generated to a problem of sampling a distribution of states. A MCMC algorithm is then constructed to sample this distribution. Finally, the process of interpreting and summarizing the results of the simulation is described.

Generation of experimental data

We considered two monodisperse linear polymers, L1 and L2 of molecular weight 50K and 200K, respectively, and two monodisperse symmetric three-armed star polymers S1 and S2, with arm molecular weights of 25K and 75K, respectively. We constructed all possible binary blends (\(^4C_2 = 6\)) of these four polymers, which are labeled L1L2, S1S2, S1L1, S1L2, S2L1, and S2L2. For each of these blend systems, we considered 25%, 50%, and 75% mixtures of the two components, where the percentages refer to weight fractions. Thus, we considered a total of four pure samples and 18 (6 ×3) blend samples as shown in Table 1.
Table 1

Blend systems considered in this paper

Label

Species 1

Species 2

w1

w2

L1L2-25

L1

L2

0.25

0.75

L1L2-50

L1

L2

0.50

0.50

L1L2-75

L1

L2

0.75

0.25

S1S2-25

S1

S2

0.25

0.75

S1S2-50

S1

S2

0.50

0.50

S1S2-75

S1

S2

0.75

0.25

S1L1-25

S1

L1

0.25

0.75

S1L1-50

S1

L1

0.50

0.50

S1L1-75

S1

L1

0.75

0.25

S1L2-25

S1

L2

0.25

0.75

S1L2-50

S1

L2

0.50

0.50

S1L2-75

S1

L2

0.75

0.25

S2L1-25

S2

L1

0.25

0.75

S2L1-50

S2

L1

0.50

0.50

S2L1-75

S2

L1

0.75

0.25

S2L2-25

S2

L2

0.25

0.75

S2L2-50

S2

L2

0.50

0.50

S2L2-75

S2

L2

0.75

0.25

L1 and L2 are linear polymers of molecular weight 50K and 200K, respectively. S1 and S2 are symmetric three-armed star polymers with arm molecular weights of 25K and 75K, respectively. wi refers to the weight fraction of species i in the blend

For a particular sample, we obtained the linear viscoelastic data G(ω) and G(ω) with ω ∈ [10 − 3 − 108 radians/s] using the BoB program. The value of the dilution exponent was set to unity, and parameters appropriate for polyisoprene at 300 K were used (density of 0.9 g/cc, equilibration time τe=13.5 μs, entanglement molecular weight \(M_{e}=\text{4,900}\)), although these particular choices do not affect the results presented in this paper, as the same parameters were used in the forward and reverse steps. To generate experimental data, we added white noise to the G(ω) and G(ω) obtained above, by selecting a Gaussian random variable x, with zero mean and unit standard deviation (x = N(0,1)) and readjusted \(G_{\text{exp}}^{\prime}(\omega)={G^{\prime}(\omega)}(1+0.1x)\), and \(G_{\text{exp}}^{\prime\prime}(\omega)={G^{\prime\prime}(\omega)}(1+0.1x)\), at each ω for which the storage and loss moduli were computed. These experimental data \(G_{\text{exp}}^{*}(\omega)\) are denoted by d in the section below.

It should be pointed out that this method of generating experimental data is not completely rigorous, in the sense that the resulting \(G_{\text{exp}}^{*}(\omega)\) only approximately obey the Kramers–Kronig relationship. Perhaps, a more rigorous method of generating synthetic experimental data would be to add noise to G(t) before taking the Fourier transform, or, as was pointed out by a reviewer, to try and mimic the process of data acquisition in a rheometer (see Appendix).

Bayesian formulation

As our forward model\(\mathbf{m}(\overline{\theta})\), we used the branch-on-branch model (Das et al. 2006), where \(\overline{\theta}\) is shorthand for system parameters. For a single species, \(\overline{\theta} = \{M, s\}\), where M and s are molecular weight and species type (linear or star). All the individual species considered in this paper are strictly monodisperse, and hence, the nature and degree of polydispersity do not need to be specfied. For two species, \(\overline{\theta} = \{M_{1}, M_{2}, w_{1}, s_{1}, s_{2} \}\), where the subscripts identify the particular species and w refers to the weight fraction. Thus, \(\mathbf{m}(\overline{\theta})\) is the complex modulus G*(ω) predicted by the model for a particular \(\overline{\theta}\).

The prior probability \(\pi(\overline{\theta})\) summarizes our knowledge of the system parameters \(\overline{\theta}\), before we consider the data. In general, if GPC, light scattering, knowledge of the polymerization process, or other data are available, they can be used to propose an informative prior, such as \(\pi(\overline{\theta}) = \pi(M_1) \pi(M_2) \pi(w_1)\), where π denotes the histogram or probability distribution. If no such information is available, it can be used to propose a noninformative prior with \(\pi(\overline{\theta}) = 1\), as is done in throughout this paper.

The likelihood function \(\pi(\mathbf{d}|\overline{\theta})\) is a cost function that penalizes the error or “distance” \(\epsilon(\overline{\theta}) = ||\mathbf{d}-\mathbf{m}(\overline{\theta})|| \) between the experimental and predicted complex moduli for a given \(\overline{\theta}\). Using a logarithmic least-squared criterion as our choice, we define \(\pi(\mathbf{d}|\overline{\theta}) = e^{-\alpha \epsilon(\overline{\theta})}\), where
$$ \epsilon(\overline{\theta}) = \frac{1}{2} \sum \left( \log \frac{{G^{\prime}(\omega)}}{G_{\text{exp}}^{\prime} (\omega)} \right)^{2} + \frac{1}{2} \sum \left( \log \frac{{G^{\prime\prime}(\omega)}}{G_{\text{exp}}^{\prime\prime} (\omega)} \right)^{2} \label{eqn:error} $$
(1)
and the summation extends over all the discrete values of ω, at which the quantities are computed. We use logarithms in the likelihood function to prevent the high-frequency Rouse response from dominating the error. Obviously, the equation above is just one of several possible choices. For example, a cost function which preferentially penalizes deviations in the low-frequency regime could also be justified on physical grounds since properties like zero-shear viscosity are dictated by terminal rheology. The value of α controls how heavily deviations from the experimental data are penalized and is set to unity in this paper. It functions as the “inverse temperature” in Metropolis Monte Carlo (MC) (Metropolis et al. 1953) and may be exploited in the future to propose “replica-exchange” type of moves to enhance sampling (Swendsen and Wang 1986).
Using Bayes theorem (Stark and Woods 1986), the posterior probability modifies the prior probability in light of the data according to:
$$ \pi(\overline{\theta}|\mathbf{d}) = \frac{\pi(\mathbf{d}|\overline{\theta}) \pi(\overline{\theta})}{\sum_{\overline{\theta}} \pi(\mathbf{d}|\overline{\theta}) \pi(\overline{\theta})} \label{eq:bayesgen} $$
(2)
where the denominator sums over all possible values of the model parameter space \(\overline{\theta}\) and is similar to a partition function in molecular systems (Frenkel and Smit 2002). Since noninformative priors are assumed in this paper,
$$ \pi(\overline{\theta}|\mathbf{d}) \propto \pi(\mathbf{d}|\overline{\theta}) = e^{-\alpha \epsilon(\overline{\theta})}. \label{eq:bayesspec} $$
(3)

Sampling the posterior distribution

Since we employ importance sampling to explore the posterior distribution \(\pi(\overline{\theta}|\mathbf{d})\) (Eq. 3), the denominator in Eq. 2 need not be directly computed (Frenkel and Smit 2002). To generate Markov chain of samples \(\overline{\theta}_{i}\) from the target distribution \(\pi(\overline{\theta}|\mathbf{d})\), we will employ the Metropolis–Hastings–Green Monte Carlo algorithm (Green 1995). While implemented simplistically in the present study, the primary motivation for choosing this algorithm is that it allows us to sample the posterior even when the number of unknowns in \(\overline{\theta}\) is itself unknown, i.e., when we are unsure of the number of species N in the sample. In other words, it offers a framework for preserving detailed balance under dimension hopping. An inherent advantage of Bayesian inference methods is that they inherently prefer a small number of components, as models with a larger number of parameters while being flexible spread their probability mass (MacKay 1991; Murray and Ghahramani 2005). For simplicity, dimension hopping moves are not used; instead, separate Markov chains are used to sample dimensions corresponding to N = 1 (pure) and N = 2 (blends). This has three advantages: (a) it highlights the “Occam’s razor” inherent in this model, (b) it is simple to understand and implement, and (c) it is more efficient to parallelize such a construction.

Thus, in principle, the set of trial moves is the only remaining ingredient to be specified. There are two distinct sets of moves: local moves which change the M and w without changing s and species transformation moves which transform the architecture of one of the species. While we only studied systems with N ≤ 2, the generalization of these moves for greater N is trivial.

Local move

For N = 1, a local move attempts to displace the current \({\overline{\theta}} = \{ M, s \}\) to \({\overline{\theta}}^{\prime} = \{ M^{\prime}, s \}\), by taking a log-space random step in the molecular weight by setting M = βM, where β is drawn from the distribution q(β) = 1/[2βln (ρ)], for 1/ρ ≤ β ≤ ρ. This shall be accomplished by drawing a random number x from the uniform distribution x ∈ U(0,1) and setting β = ρ2x − 1. In this study, ρ = 1.2 shall be used. The condition of detailed balance requires that this move \({\overline{\theta}} \rightarrow {\overline{\theta}}^{\prime}\) be accepted with a probability given by,
$$ \textrm{acc}({\overline{\theta}} \rightarrow {\overline{\theta}}^{\prime}) = \beta e^{-\alpha[\epsilon({\overline{\theta}}^{\prime}) - \epsilon({\overline{\theta}}) ]} $$
(4)
For N = 2, a local move attempts to displace \(\overline{\theta}=\{M_{1}, M_{2}, w_{1}, s_{1}, s_{2} \}\) to \(\overline{\theta}^{\prime}=\{M_{1}^{\prime}, M_{2}^{\prime}, w_{1}^{\prime}, s_{1}, s_{2} \}\) by taking a similar log-space random step in the molecular weights by setting \(M_{i}^{\prime}\) = \(\beta_{i} M_{i}^{\prime}\), where βi ∈ q(β) = 1/[2βln (ρ)] for i = 1 and 2. For the weight fractions, we apply a random linear displacement, by setting \(w_{1}^{\prime} = w_{1} + U(-\Delta w,\Delta w)\), where U denotes a uniform distribution. In this study, Δw = 0.1 was used. The acceptance probability is given by:
$$ \textrm{acc}({\overline{\theta}} \rightarrow {\overline{\theta}}^{\prime}) = \beta_{1} \beta_{2} e^{-\alpha[\epsilon({\overline{\theta}}^{\prime}) - \epsilon({\overline{\theta}}) ]} $$
(5)

Species transformation move

This move contains two parts. In the first part, \({\overline{\theta}} \rightarrow {\overline{\theta}}_{0}\), the type (ex. linear) of a randomly selected molecule i ∈ (0,N) is transformed to another randomly determined type (ex. star). The weight fraction wi and molecular weight Mi are left unaltered. It is very likely that such a trial move will be rejected. To improve the odds of acceptance, we use a delayed rejection scheme, which starts from \({\overline{\theta}}_{0}\) and proposes a series of local moves described above, to explore the phase space around the proposed configuration (Green and Mira 2001). Thus, a trial species transformation move involves proposing \({\overline{\theta}} \rightarrow {\overline{\theta}}_{0} \rightarrow {\overline{\theta}}_{1} \rightarrow {\overline{\theta}}_{2} \rightarrow \cdots {\overline{\theta}}_{n}\), where the first move consists of a species change and subsequent “delayed rejection” moves are similar to local moves proposed above. In this study, the search for a “good” configuration was capped at n = 10. If we let \(\epsilon_{i} = \epsilon({\overline{\theta}}_{i})\) and \(\epsilon=\epsilon({\overline{\theta}})\), the algorithm for N = 1 or 2, fashioned after Mira (2001), is:

  1. 1.

    Generate a \({\overline{\theta}}_{0}\) from \({\overline{\theta}}\) by transforming the architecture of one of the species, keeping the molecular weight and weight fraction unchanged.

     
  2. 2.

    Draw a uniform random number u ~U(0,1). If \(u < e^{-\alpha (\epsilon_{0} - \epsilon)}\), accept the move and quit.

     
  3. 3.

    If the move is rejected, set \(\epsilon^{*}=\epsilon_{0}\), βR = 1, β* = 1, and i = 1.

     
  4. 4.

    Propose a local move, \({\overline{\theta}}_{i-1} \rightarrow {\overline{\theta}}_{i}\). Let β = ∏ βi and βR = ββR.

     
  5. 5.

    Set \(x_{1} = (\beta^{R}/\beta^{*}) e^{-\alpha (\epsilon_{i} - \epsilon^{*})}\) and \(x_{2} = \beta^{R} e^{-\alpha (\epsilon_{i} - \epsilon_{0})}\).

     
  6. 6.

    If x2 ≥ 1, accept the move \({\overline{\theta}} \rightarrow {\overline{\theta}}_{i}\) and quit.

     
  7. 7.

    If x1 ≥ 1, draw a uniform random number u ~U(0,1). If u < (x1x2 − x2)/(x1 − x2), accept the move and quit. If the move is rejected, set β* = βR and \(\epsilon^{*}=\epsilon_{i}\)

     
  8. 8.

    Let i = i + 1.

     
  9. 9.

    If i > n, keep the old configuration \({\overline{\theta}}\) and quit. Otherwise, go to step 4.

     

Measurement and postprocessing

For each of the pure and blend systems (4+18=22), two separate Markov chains with N = 1 and N = 2 were used to simulate a total of 1,000 Monte Carlo trial steps (MCS). This process was repeated 50 times for each of the 44 (22 systems × 2 different Ns) Markov chains, with a randomly selected initial configuration. The overall computational cost including preprocessing and postprocessing of all the systems was ten CPU days (about 5 h per Markov chain per single processor). For a given N, the initial conditions were determined randomly, with the molecular weight being chosen with uniform probability between zero and 500K. After every 100 local moves, a species transformation move with delayed rejection was attempted. The data were sampled every ten MCS.

Figure 1 depicts the error \(\epsilon({\overline{\theta}}_{i})\) during the evolution of five Markov chains corresponding to the pure linear sample L1, with N = 1, and is representative of all other samples. The burn-in period—the time it takes to forget the initial configuration and characterized by a rapid decrease in error—was quite rapid. Usually, it took less than 10% of the 1,000 MCS and was ignored in the analysis. For a given experimental system, the burn-in period was eliminated by first combining all the samples explored using the N = 1 and N = 2 Markov chains and finding the minimum error εmin. The system parameters \({\overline{\theta}}_{\rm min}\) corresponding to this error (\(\epsilon({\overline{\theta}}_{\rm min})=\epsilon_{\rm min}\)) offer the best possible match with the experimental data in the space of parameters explored. If one were interested in a best-fit solution, this would be the only \({\overline{\theta}}\) of interest. To eliminate the burn-in period, only those samples \({\overline{\theta}}\) with \(\epsilon({\overline{\theta}}) < 10 \epsilon_{min}\) were considered. Further, since we seek to characterize distributions of \({\overline{\theta}}\) that are approximately consistent with the data, we define another cutoff, εcut = εmin + 0.1 σε, where σε is the standard deviation of the error of the postburn-in samples as identified above. Thus, the samples with ε < 10 εmin constitute the universe of samples, and ε < εcut represent the fraction of these samples that are most consistent with the experimental data.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-010-0443-z/MediaObjects/397_2010_443_Fig1_HTML.gif
Fig. 1

The evolution of the error for five Markov chains corresponding to the pure linear system L1, with N = 1. The minimum error εmin = 2.82, and the cutoff error εcut = 3.21. There is a sharp drop in the error initially as the system tries to improve upon the randomly chosen initial configuration. In one of the samples, the error does not fall all the way towards the minima because the wrong architecture (star) was initially selected. However, after 300 MCS, it changes its architecture via a delayed rejection move to the correct architecture (linear)

In general, Markov chains with N = 2 will always be able to match the experimental data at least as well as N = 1 since the phase space of N = 1 is a proper subset of the phase space of N = 2. This is obvious since one can always set w1 = 1 regardless of M2 and s2 or, alternatively, set M1 = M2 and s1 = s2, regardless of w1, to effectively produce a one component system. However, the volume of the total phase space for N = 2 is much larger than that for N = 1. To discriminate between the number of components in a given experimental system, we need to quantify the fraction of phase space that the samples corresponding to an error of less than εcut occupy, for a given N.

To rephrase the idea in less abstract language, consider a completely different problem for the moment. Suppose that a certain number N > 1 of dice are rolled and the sum of the faces is calculated. Let us suppose that this sum is 7. What is the relative probability that N = 2 dice were rolled and not N = 3? Enumerating all the possibilities, it can be figured out that the probability of obtaining a sum of 7 is 6/36 with N = 2 and 15/216 with N = 3. Although there are more ways of obtaining a sum of 7 using N = 3 dice, the fraction of “phase space” that is consistent with the experimental observation of 7 is 36/216 and 15/216, for N = 2 and 3, respectively. Thus, the relative likelihood of N = 2 over N = 3 is 36/15=2.4. This illustrates the origin of why a Bayesian analysis prefers simple solutions over complex ones. It does so, despite the fact that volume of phase space consistent with the experimental data increases as the number of dice or species is increased because the volume of overall phase space increases faster.

Unfortunately, we cannot simply take the ratio of number of samples with ε < εcut and ε < 10 εmin to quantify the fraction of phase space that samples corresponding to an error of less than εcut occupy, for a particular N. Due to importance sampling, the Markov chain spends a disproportionately large time in regions of low ε. We can obtain an approximate formula for the fraction of phase space by deweighting the samples with \(\epsilon({\overline{\theta}})\), under the assumption that the phase space is reasonably explored. Thus, the standard histogram reweighting formula gives us,
$$ f(N) = \frac{\sum\limits_{\textrm{cut}} e^{\alpha \epsilon({\overline{\theta}})}}{\sum\limits_{\textrm{all}} e^{\alpha \epsilon({\overline{\theta}})}} $$
(6)
where the numerator includes all the samples \({\overline{\theta}}\) with \(\epsilon({\overline{\theta}})< \epsilon_{\rm cut}\) and the denominator includes all the samples \({\overline{\theta}}\) with \(\epsilon({\overline{\theta}})< 10 \epsilon_{\rm min}\). The ratio f(N = 1)/f(N = 2) can then be used to quantify the probability that the experimental sample contains only one species.

Results

Pure species

The predictions for the pure species are presented in Fig. 2. Histograms obtained for N = 1 species are depicted in red and for N = 2 species are depicted in black. In all the results presented in this paper, histogram reweighting has been performed. Several things stand out from the figures. First, the probability that the experimental data corresponds to a single species is larger than a blend, as evidenced by the near invisibility of the N = 2 data on this scale. These probabilities computed as f(N = 1)/[f(N = 1) + f(N = 2)] are 93%, 99%, 91%, and 100% for L1, L2, S1, and S2, respectively. Secondly, the maxima in M and w are distributed around the correct values. Although it is not explicitly shown in this figure, the correct architecture is also identified. Thirdly, although it is not particularly obvious due to the scale, the distribution of w for N = 2 is skewed toward one dominant component (w1 ≈ 1 and w2 ≈ 0, or vice-versa). This is clear if we plot merely the N = 2 data and not let the N = 1 data dwarf it.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-010-0443-z/MediaObjects/397_2010_443_Fig2_HTML.gif
Fig. 2

Pure linear and star polymers. The red bars denote the histograms obtained with N = 1, and the largely invisible black bars denote histograms with N = 2 or blends

Linear–linear and star–star blends

The results for linear–linear and star–star blends are depicted in Fig. 3. In all of these six cases, the ratio of f(N = 1)/[f(N = 1) + f(N = 2)] ≈ 0, indicating that the experimental data can only be explained by a blend system. For the linear–linear blends and S1S2-25, the MCMC algorithm correctly identifies the blend constituents. For the remaining two star–star blends, S1S2-50 and S1S2-75, the method assigns, respectively, 56% and 60% probability that the mixture is composed of two star molecules. The histograms depicted in Fig. 3b, c correspond to this fraction of the phase space. With the remaining probability, the algorithm suggests that the experimental system could be described by a star–linear blend. This is the first example of the simulation capturing degeneracy of solutions. It will be shown later (when discussing S1L1 since degeneracy issues are the most severe for this sample) that this degenerate solution also describes the experimental data quite well.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-010-0443-z/MediaObjects/397_2010_443_Fig3_HTML.gif
Fig. 3

Linear–linear and star–star blends. For the star–star blends S1S2-50 and S1S2-75, the histogram considers only star–star samples that the algorithm produces

Star–linear blends

The star–linear blends are considered next, starting with S2-L1 and S2-L2, as shown in Fig. 4. For all experimental systems considered in this figure, the algorithm correctly identifies them as star–linear blends with a probability of 93% or greater. Hence, this is a clear cut result, not plagued by any type of degeneracy.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-010-0443-z/MediaObjects/397_2010_443_Fig4_HTML.gif
Fig. 4

Star–linear blends with S2 as the star polymer

Similar results are observed for S1L2, as shown in Fig. 6a–c. The algorithm correctly identifies it as a star–linear blend with a probability of 85% or greater. The next system, S1L1, is the most challenging since the linear viscoelastic response does not vary significantly with composition as shown in Fig. 5. This results in a large number of degenerate solutions.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-010-0443-z/MediaObjects/397_2010_443_Fig5_HTML.gif
Fig. 5

The experimental linear viscoelastic data for S1L1 does not vary significantly with composition. Consequently, there are a number of solutions to the inverse problem as discussed in the text

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Fig. 6

Star–linear blends with S1 as the star polymer

For S1L1-25, the method suggests that the experimental data correspond to a star–linear blend with 45% probability and to a linear–linear blend with 55% probability. The histogram shown in Fig. 6d corresponds to a star–linear blend. The histogram for the linear–linear blend (not shown) is distributed around its minimum which occurs near \({\overline{\theta}} = \{M_{1}=50K, M_{2}=95K, w_{1}=0.85\), and w2 = 0.15 }. The histograms corresponding to S1L1-50 and S1L1-75 are even more messy, in the sense that they do not have sharp peaks, especially for w. However, in either of these cases, one can discern peaks in the molecular weight distribution near the expected values of 25K and 50K. For S1L1-50, the method suggests that the experimental data correspond to a star–linear blend with 51% probability, to a linear–linear blend with 27% probability, and to a star–star blend with 21% probability. Similarly, for S1L1-75, the method suggests that the experimental data correspond to a star–linear blend with 56% probability, to a linear–linear blend with 22% probability, and to a star–star blend with 22% probability.

Let us examine the results for S1L1-75, more closely, although much of same applies to samples that have a significant fraction of degenerate solutions, such as S1L1-25, S1L1-50, and S1S2-50 and S1S2-75. In Fig. 7, the black line is identical to the experimental data for S1L1-75 shown in Fig. 5. As mentioned above, the MCMC algorithm suggests that the experimental data could come from a linear–linear or a star–star blend with nonnegligible probabilities. Instead of looking at the distribution of molecular weights and weight fractions, let us look at the “best solutions” explored by the Markov chain for each of the three different types of binary blends. The best star–linear blend corresponded to a star of arm molecular weight 25.0K and weight fraction 0.73 and a linear of molecular weight 52.1K and weight fraction 0.27. Similarly, best linear–linear blend corresponded to molecular weights 95.7K and 50K and weight fractions 0.34 and 0.66, respectively. The best star–star blend corresponded to arm molecular weights 27.6K and 20.9K and weight fractions 0.53 and 0.47, respectively. These three “best of class” solutions are shown in Fig. 7. This figure demonstrates that this algorithm can correctly capture all the degenerate solutions since all the “best solutions” for the different architectures explain the data equally well. It is useful to point out that the probabilities associated with different blend architectures mentioned above are proportional to the volume of phase space near the corresponding best of class solution.
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Fig. 7

For S1L1-75, the MCMC algorithm suggests a star–linear blend with 56% probability, a linear–linear blend with 22% probability, and a star–star blend with 22% probability. The best solutions for these three types of blends, namely linear–linear, star–linear, and star–star (filled circles, or different colors) are shown together with the experimental data and indicate that these indeed constitute degenerate solutions

Summary and perspective

A Bayesian data analysis method which translates the inverse problem of analytical rheology into a sampling problem was described. Using Markov chain Monte Carlo method, the posterior distribution was sampled. Trial moves including local moves and species transformation moves, which employed the idea of delayed rejection to enhance acceptance probabilities, were proposed. The algorithm was applied to mixtures of linear and star polymers. Important results include the ability of the method to (a) diagnose whether a mixture contained one or two components, (b) recover the composition of the original mixture used to generate the experimental data, and (c) describe multiple solutions, when more than one possible combination of constituents is able to produce similar rheology.

This study constitutes a first attempt at developing a general practical algorithm to invert rheological models of branched polymers. The algorithm and the results presented in this manuscript address some of the key drawbacks associated with other methods. The Bayesian formulation, presented in this paper, does not depend on the particular choice of the forward model. From an engineering perspective, the forward model does not even have to be a zero-parameter model, an ideal that most hierarchical models, including BoB, seek. It can include “human intelligence” components or heuristics, such as “use a value of p2 = 1, for H-polymers, but a value of p2 = 1/40 for asymmetric stars with short arm length less than 5 entanglements long”, based on the meticulous testing that these models have been subjected to. Indeed, much like weather prediction algorithms, a composite model comprising different forward models can easily be synthesized, to emphasize the strengths of the constituent forward models and make the composite model more reliable than the most reliable of its constituents. Further, it can easily be generalized to include complementary techniques such as chromatography or polymerization, either by directly incorporating them in the likelihood function, proposing informative priors, or to discard implausible degenerate solutions.

Acknowledgements

I am grateful to Dr. Chinmay Das for making the source code of BoB freely available and for diligently responding to my queries. I am also thankful to Michael Ericson, Ron Larson, Zuowei Wang, and Evelyne van Ruymbeke for helpful discussions.

Copyright information

© Springer-Verlag 2010