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

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## 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 rheology Bayesian inference Markov chain Monte Carlo Polymer blend Tube model Linear viscoelasticity Modeling Bayesian analysis Inverse problem## Notes

### 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.

### References

- Anderssen R (2005) Inverse problems: a pragmatist’s approach to the recovery of information from indirect measurements. ANZIAM J 46:C588–C622MathSciNetGoogle Scholar
- Anderssen RS, Mead DW (1998) Theoretical derivation of molecular weight scaling for rheological parameters. J Non-Newton Fluid Mech 76:299–306MATHCrossRefGoogle Scholar
- Ball RC, Mcleish TCB (1989) Dynamic dilution and the viscosity of star polymer melts. Macromolecules 22:1911–1913CrossRefADSGoogle Scholar
- Carrot C, Guillet J (1997) From dynamic moduli to molecular weight distribution: a study of various polydisperse linear polymers. J Rheol 41(5):1203–1220. doi:10.1122/1.550815 CrossRefADSGoogle Scholar
- Costeux S, Wood-Adams P, Beigzadeh D (2002) Molecular structure of metallocene-catalyzed polyethylene: rheologically relevant representation of branching architecture in single catalyst and blended systems. Macromolecules 35:2514–2528CrossRefADSGoogle Scholar
- Das C, Inkson NJ, Read DJ, Kelmanson MA, McLeish TCB (2006) Computational linear rheology of general branch-on-branch polymers. J Rheol 50(2):207–234CrossRefADSGoogle Scholar
- Dealy JM, Larson RG (2006) Molecular structure and rheology of molten polymers, 1st edn. Hanser, MunichGoogle Scholar
- Doerpinghaus PJ, Baird DG (2003) Separating the effects of sparse long-chain branching on rheology from those due to molecular weight in polyethylenes. J Rheol 47(3):717–736. doi:10.1122/1.1567751 CrossRefADSGoogle Scholar
- Doi M, Edwards SF (1986) The theory of polymer dynamics. Clarendon, OxfordGoogle Scholar
- Frenkel D, Smit B (2002) Understanding molecular simulation: from algorithms to applications, 2nd edn. Academic, OrlandoGoogle Scholar
- de Gennes PG (1975) Reptation of stars. J Phys (Paris) 36:1199–1203Google Scholar
- Graham RS, Likhtman AE, McLeish TCB, Milner ST (2003) Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release. J Rheol 47(5):1171–1200CrossRefADSGoogle Scholar
- Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4):711–732. doi:10.1093/biomet/82.4.711 MATHCrossRefMathSciNetGoogle Scholar
- Green PJ, Mira A (2001) Delayed rejection in reversible jump metropolis-hastings. Biometrika 88(4):1035–1053. http://www.jstor.org/stable/2673700 MATHMathSciNetGoogle Scholar
- Janzen J, Colby RH (1999) Diagnosing long-chain branching in polyethylenes. J Mol Struct 486:569–584CrossRefADSGoogle Scholar
- Larson RG (2001) Combinatorial rheology of branched polymer melts. Macromolecules 34:4556–4571CrossRefADSGoogle Scholar
- Larson RG, Zhou Q, Shanbhag S, Park SJ (2007) Advances in modeling of polymer melt rheology. AIChE J. 53(3):542–548CrossRefGoogle Scholar
- Lee JH, Fetters LJ, Archer LA, Halasa AF (2005) Tube dynamics in binary polymer blends. Macromolecules 38(9):3917–3932. doi:10.1021/ma040080h CrossRefADSGoogle Scholar
- Léonardi F, Allal A, Marin G (1998) Determination of the molecular weight distribution of linear polymers by inversion of a blending law on complex viscosities. Rheol Acta 37(3):199–213. doi:10.1007/s003970050108 CrossRefGoogle Scholar
- Likhtman AE, McLeish TCB (2002) Quantitative theory for linear dynamics of linear entangled polymers. Macromolecules 35:6332–6343CrossRefADSGoogle Scholar
- Lohse DJ, Milner ST, Fetters LJ, Xenidou M, Hadjichristidis N, Mendelson RA, Garcia-Franco CA, Lyon MK (2002) Well-defined, model long chain branched polyethylene. 2. Melt rheological behavior. Macromolecules 35(8):3066–3075. doi:10.1021/ma0117559 CrossRefADSGoogle Scholar
- MacKay C (1991) Bayesian methods for adaptive modelling. Ph.D. thesis, California Institute of TechnologyGoogle Scholar
- Maier D, Eckstein A, Friedrich C, Honerkamp J (1998) Evaluation of models combining rheological data with the molecular weight distribution. J Rheol 42(5):1153–1173. doi:10.1122/1.550952 CrossRefADSGoogle Scholar
- McLeish TCB, Allgaier J, Bick DK, Bishko G, Biswas P, Blackwell R, Blottiere B, Clarke N, Gibbs B, Groves DJ, Hakiki A, Heenan RK, Johnson JM, Kant R, Read DJ, Young RN (1999) Dynamics of entangled H-polymers: theory, rheology, and neutron-scattering. Macromolecules 32:6734–6758CrossRefADSGoogle Scholar
- Mead DW (1994) Determination of molecular-weight distributions of linear flexible polymers from linear viscoelastic material functions. J Rheol 38:1797–1827CrossRefADSGoogle Scholar
- Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092. doi:10.1063/1.1699114 CrossRefADSGoogle Scholar
- Milner ST, McLeish TCB (1997) Parameter-free theory for stress relaxation in star polymer melts. Macromolecules 30:2159–2166CrossRefADSGoogle Scholar
- Milner ST, McLeish TCB (1998) Arm-length dependence of stress relaxation in star polymer melts. Macromolecules 31:7479–7482CrossRefADSGoogle Scholar
- Mira A (2001) On Metropolis-Hastings algorithms with delayed rejection. Metron 59:3–4MathSciNetGoogle Scholar
- Murray I, Ghahramani I (2005) A note on the evidence and Bayesian Occam’s razor. Gatsby Unit Tech Rep, pp 1–4Google Scholar
- Nobile MR, Cocchini F (2001) Evaluation of molecular weight distribution from dynamic moduli. Rheol Acta 40(2):111–119. doi:10.1007/s003970000141 CrossRefGoogle Scholar
- Park SJ, Shanbhag S, Larson RG (2005) A hierarchical algorithm for predicting the linear viscoelastic properties of polymer melts with long-chain branching. Rheol Acta 44:318–330CrossRefGoogle Scholar
- Robertson CG, García-Franco CA, Srinivas S (2004) Extent of branching from linear viscoelasticity of long-chain-branched polymers. J Polym Sci B Polym Phys 42(9):1671–1684. doi:10.1002/polb.20038 CrossRefADSGoogle Scholar
- van Ruymbeke E, Keunings R, Bailly C (2002) Determination of the molecular weight distribution of entangled linear polymers from linear viscoelasticity data. J Non-Newton Fluid Mech 105(2–3):153–175. doi:10.1016/S0377-0257(02)00080-0 MATHCrossRefGoogle Scholar
- van Ruymbeke E, Keunings R, Bailly C (2005) Prediction of linear viscoelastic properties for polydisperse mixtures of entangled star and linear polymers: modified tube-based model and comparison with experimental results. J Non-Newton Fluid Mech 128(1):7–22CrossRefGoogle Scholar
- van Ruymbeke E, Stéphenne V, Daoust D, Godard P, Keunings R, Bailly C (2005) A sensitive method to detect very low levels of long chain branching from the molar mass distribution and linear viscoelastic response. J Rheol 49(6):1503–1520. doi:10.1122/1.2048743 CrossRefGoogle Scholar
- Shanbhag S, Park SJ, Zhou Q, Larson RG (2007) Implications of microscopic simulations of polymer melts for mean-field tube theories. Mol Phys 105(2):249–260CrossRefADSGoogle Scholar
- Shroff RN, Mavridis H (1999) Long-chain-branching index for essentially linear polyethylenes. Macromolecules 32(25):8454–8464. doi:10.1021/ma9909354 CrossRefADSGoogle Scholar
- Stark H, Woods J (1986) Probability, random processes, and estimation theory for engineers. Prentice-Hall, Upper Saddle RiverGoogle Scholar
- Swendsen RH, Wang JS (1986) Replica Monte Carlo simulation of spin-glasses. Phys Rev Lett 57(21):2607–2609. doi:10.1103/PhysRevLett.57.2607 CrossRefPubMedMathSciNetADSGoogle Scholar
- Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics, PhiladelphiaMATHGoogle Scholar
- Thimm W, Friedrich C, Marth M, Honerkamp J (2000) On the rouse spectrum and the determination of the molecular weight distribution from rheological data. J Rheol 44(2):429–438. doi:10.1122/1.551094 CrossRefADSGoogle Scholar
- Tuminello WH (1986) Molecular-weight and molecular-weight distribution from dynamic measurements of polymer melts. Polym Eng Sci 26:1339–1347CrossRefGoogle Scholar
- Vega J, Aguilar M, Peon J, Pastor D, Martinez-Salazar J (2002) Effect of long chain branching on linear-viscoelasticmelt 766 properties of polyolefins. e-Polymers 046:1–35Google Scholar
- Wasserman S (1995) Calculating the molecular weight distribution from linear viscoelastic response of polymer melts. J Rheol 39(3):601–625CrossRefADSGoogle Scholar
- Wolpert RL, Ickstadt K (2004) Reflecting uncertainty in inverse problems: a Bayesian solution using Lévy processes. Inverse Probl. 20(6):1759–1771. http://stacks.iop.org/0266-5611/20/1759 MATHCrossRefMathSciNetADSGoogle Scholar
- Wood-Adams PM, Dealy JM (2000) Using rheological data to determine the branching level in metallocene polyethylenes. Macromolecules 33(20):7481–7488. doi:10.1021/ma991534r CrossRefADSGoogle Scholar
- Wood-Adams PM, Dealy JM, deGroot AW, Redwine OD (2000) Effect of molecular structure on the linear viscoelastic behavior of polyethylene. Macromolecules 33:7489–7499CrossRefADSGoogle Scholar
- Wu S (1985) Polymer molecular-weight distribution from dynamic melt viscoelasticity. Polym Eng Sci 25(2):122–128CrossRefGoogle Scholar