Skip to main content

A hybrid lattice Boltzmann-molecular dynamics-immersed boundary method model for the simulation of composite foams

Abstract

Small fillers (e.g., carbon fibers) are commonly added to polymer foams to create composite foams that can improve foam properties such as thermal and electrical conductivity. Understanding the motion and orientation of fillers during the foaming process is crucial because these can affect the properties of composite foams significantly. In this work, a hybrid lattice Boltzmann method-molecular dynamics-immersed boundary method model is presented for simulating the foaming process of polymer composites. The LBM model resolves the foaming process, and the MD model accounts for filler dynamics. These two solvers are coupled by a direct forcing IBM. This solver can simulate composite foaming processes involving many bubbles and filler particles, including rigid and deformable 3D particles, and rigid, deformable, and fragile fibers. The solver relaxes most simplifying assumptions of earlier polymer composite models, allowing for a better understanding of filler motion and interaction with growing bubbles.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Notes

  1. Freely available at https://github.com/mehdiataei/mesh2lammps.

References

  1. Adloo A, Sadeghi M, Masoomi M, Pazhooh HN (2016) High performance polymeric bipolar plate based on polypropylene/graphite/graphene/nano-carbon black composites for PEM fuel cells. Renew Energy 99:867–874

    Article  Google Scholar 

  2. Advani SG, Tucker CL III (1987) The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol 31(8):751–784

    Article  Google Scholar 

  3. Ameli A, Jung P, Park CB (2013) Electrical properties and electromagnetic interference shielding effectiveness of polypropylene/carbon fiber composite foams. Carbon 60:379–391

    Article  Google Scholar 

  4. Ameli A, Jung P, Park CB (2013) Through-plane electrical conductivity of injection-molded polypropylene/carbon-fiber composite foams. Compos Sci Technol 76:37–44

    Article  Google Scholar 

  5. Ameli A, Wang S, Kazemi Y, Park CB, Pötschke P (2015) A facile method to increase the charge storage capability of polymer nanocomposites. Nano Energy 15:54–65

    Article  Google Scholar 

  6. Arefmanesh A, Advani SG, Michaelides EE (1990) A numerical study of bubble growth during low pressure structural foam molding process. Polym Eng Sci 30(20):1330–1337

    Article  Google Scholar 

  7. Ataei M, Shaayegan V, Costa F, Han S, Park CB, Bussmann M (2021) LBfoam: an open-source software package for the simulation of foaming using the Lattice Boltzmann Method. Comput Phys Commun 259:107698

    MathSciNet  Article  Google Scholar 

  8. Ataei M, Shaayegan V, Wang C, Costa F, Han S, Park CB, Bussmann M (2019) Numerical analysis of the effect of the local variation of viscosity on bubble growth and deformation in polymer foaming. J Rheol 63(6):895–903

    Article  Google Scholar 

  9. Bay RS, Tucker CL III (1992) Fiber orientation in simple injection moldings. Part I: theory and numerical methods. Polym Compos 13(4):317–331

    Article  Google Scholar 

  10. Bay RS, Tucker CL III (1992) Fiber orientation in simple injection moldings. Part II: experimental results. Polym Compos 13(4):332–341

    Article  Google Scholar 

  11. Bryning M, Islam MF, Kikkawa JM, Yodh A (2005) Very low conductivity threshold in bulk isotropic single-walled carbon nanotube-epoxy composites. Adv Mater 17:1186–1191

    Article  Google Scholar 

  12. Chang E, Ameli A, Alian A, Mark LH, Yu K, Wang S, Park CB (2020) Percolation mechanism and effective conductivity of mechanically deformed 3-dimensional composite networks: Computational modeling and experimental verification. Compos Part B Eng 207:108552

    Article  Google Scholar 

  13. Dang Z, Zheng M, Zha J (2016) 1D/2D carbon nanomaterial-polymer dielectric composites with high permittivity for power energy storage applications. Small 12(13):1688–701

    Article  Google Scholar 

  14. Dao M, Li J, Suresh S (2006) Molecularly based analysis of deformation of spectrin network and human erythrocyte. Mater Sci Eng C 26(8):1232–1244

    Article  Google Scholar 

  15. Favaloro AJ, Tseng HC, Pipes RB (2018) A new anisotropic viscous constitutive model for composites molding simulation. Compos Part A Appl Sci Manuf 115:112–122

    Article  Google Scholar 

  16. Fedosov DA, Caswell B, Karniadakis GE (2010) A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys J 98(10):2215–2225

    Article  Google Scholar 

  17. Fedosov DA, Caswell B, Karniadakis GE (2010) Systematic coarse-graining of spectrin-level red blood cell models. Comput Methods Appl Mech Eng 199(29–32):1937–1948

    MathSciNet  MATH  Article  Google Scholar 

  18. Feng ZG, Michaelides EE (2002) Hydrodynamic force on spheres in cylindrical and prismatic enclosures. Int J Multiphase Flow 28(3):479–496

    MATH  Article  Google Scholar 

  19. Feng ZG, Michaelides EE (2002) Interparticle forces and lift on a particle attached to a solid boundary in suspension flow. Phys Fluids 14(1):49–60

    MATH  Article  Google Scholar 

  20. Feng ZG, Michaelides EE (2004) The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems. J Comput Phys 195(2):602–628

    MATH  Article  Google Scholar 

  21. Folgar F, Tucker CL III (1984) Orientation behavior of fibers in concentrated suspensions. J Reinf Plast Compos 3(2):98–119

    Article  Google Scholar 

  22. Gawale A, Kulkarni A, Pratley M (2017) Multiscale modeling approach for short fiber reinforced plastic couplings. SAE Int J Mater Manuf 10(1):78–82

    Article  Google Scholar 

  23. Genheden S, Essex JW (2015) A simple and transferable all-atom/coarse-grained hybrid model to study membrane processes. J Chem Theory Comput 11(10):4749–4759

    Article  Google Scholar 

  24. Goldberg N, Ospald F, Schneider M (2017) A fiber orientation-adapted integration scheme for computing the hyperelastic tucker average for short fiber reinforced composites. Comput Mech 60(4):595–611

    MathSciNet  MATH  Article  Google Scholar 

  25. Guo Z, Shi B, Wang N (2000) Lattice BGK model for incompressible Navier–Stokes equation. J Comput Phys 165(1):288–306

    MathSciNet  MATH  Article  Google Scholar 

  26. Haghgoo M, Ansari R, Hassanzadeh-Aghdam M, Nankali M (2019) Analytical formulation for electrical conductivity and percolation threshold of epoxy multiscale nanocomposites reinforced with chopped carbon fibers and wavy carbon nanotubes considering tunneling resistivity. Compos Part A Appl Sci Manuf 126:105616

    Article  Google Scholar 

  27. Hasheminejad SM, Sanaei R (2007) Effects of fiber ellipticity and orientation on dynamic stress concentrations in porous fiber-reinforced composites. Comput Mech 40(6):1015–1036

    MATH  Article  Google Scholar 

  28. He X, Luo LS (1997) Lattice Boltzmann model for the incompressible Navier–Stokes equation. J Stat Phys 88(3):927–944

    MathSciNet  MATH  Article  Google Scholar 

  29. Inamuro T (2012) Lattice Boltzmann methods for moving boundary flows. Fluid Dyn Res 44(2):024001

    MathSciNet  MATH  Article  Google Scholar 

  30. Isaincu A, Dan M, Ungureanu V, Marşavina L (2021) Numerical investigation on the influence of fiber orientation mapping procedure to the mechanical response of short-fiber reinforced composites using Moldflow, Digimat and Ansys software. In: Materials today: proceedings

  31. Iwan A, Malinowski M, Paściak G (2015) Polymer fuel cell components modified by graphene: electrodes, electrolytes and bipolar plates. Renew Sustain Energy Rev 49:954–967

    Article  Google Scholar 

  32. Jack DA, Schache B, Smith DE (2010) Neural network-based closure for modeling short-fiber suspensions. Polym Compos 31(7):1125–1141

    Google Scholar 

  33. Jeffery GB (1922) The motion of ellipsoidal particles immersed in a viscous fluid. Proc R Soc Lond Ser A Contain Pap Math Phys Character 102(715):161–179

    MATH  Google Scholar 

  34. Jewett AI, Stelter D, Lambert J, Saladi SM, Roscioni OM, Ricci M, Autin L, Maritan M, Bashusqeh SM, Keyes T et al (2021) Moltemplate: a tool for coarse-grained modeling of complex biological matter and soft condensed matter physics. J Mol Biol 433:166841

    Article  Google Scholar 

  35. Jia L, Yan D, Yang Y, Zhou D, Cui C, Bianco E, Lou J, Vajtai R, Li B, Ajayan P, Li Z (2017) High strain tolerant EMI shielding using carbon nanotube network stabilized rubber composite. Adv Mater Technol 2:1700078

    Article  Google Scholar 

  36. Kloss C, Goniva C, Hager A, Amberger S, Pirker S (2012) Models, algorithms and validation for opensource DEM and CFD-DEM. Prog Comput Fluid Dyn Int J 12(2–3):140–152

    MathSciNet  Article  Google Scholar 

  37. Köbler J, Schneider M, Ospald F, Andrä H, Müller R (2018) Fiber orientation interpolation for the multiscale analysis of short fiber reinforced composite parts. Comput Mech 61(6):729–750

    MathSciNet  MATH  Article  Google Scholar 

  38. Lyu J, Zhao X, Hou X, Zhang Y, Li T, Yan Y (2017) Electromagnetic interference shielding based on a high strength polyaniline-aramid nanocomposite. Compos Sci Technol 149:159–165

    Article  Google Scholar 

  39. Mao W, Alexeev A (2014) Motion of spheroid particles in shear flow with inertia. J Fluid Mech 749:145

    Article  Google Scholar 

  40. Martin JJ, Riederer MS, Krebs MD, Erb RM (2015) Understanding and overcoming shear alignment of fibers during extrusion. Soft Matter 11(2):400–405

    Article  Google Scholar 

  41. Montgomery-Smith S, Jack D, Smith DE (2011) The fast exact closure for Jeffery’s equation with diffusion. J Non-Newton Fluid Mech 166(7–8):343–353

  42. Okamoto M, Nam PH, Maiti P, Kotaka T, Nakayama T, Takada M, Ohshima M, Usuki A, Hasegawa N, Okamoto H (2001) Biaxial flow-induced alignment of silicate layers in polypropylene/clay nanocomposite foam. Nano Lett 1(9):503–505

    Article  Google Scholar 

  43. Oliveira M, Sayeg IJ, Ett G, Antunes R (2014) Corrosion behavior of polyphenylene sulfide-carbon black-graphite composites for bipolar plates of polymer electrolyte membrane fuel cells. Int J Hydrogen Energy 39:16405–16418

    Article  Google Scholar 

  44. Pang H, Xu L, Yan D, Li Z (2014) Conductive polymer composites with segregated structures. Prog Polym Sci 39:1908–1933

    Article  Google Scholar 

  45. Pang H, Xu L, Yan DX, Li ZM (2014) Conductive polymer composites with segregated structures. Prog Polym Sci 39(11):1908–1933

    Article  Google Scholar 

  46. Payandehpeyman J, Mazaheri M, Khamehchi M (2020) Prediction of electrical conductivity of polymer-graphene nanocomposites by developing an analytical model considering interphase, tunneling and geometry effects. Compos Commun 21:100364

    Article  Google Scholar 

  47. Peskin CS (1977) Numerical analysis of blood flow in the heart. J Comput Phys 25(3):220–252

    MathSciNet  MATH  Article  Google Scholar 

  48. Pivkin IV, Karniadakis GE (2008) Accurate coarse-grained modeling of red blood cells. Phys Rev Lett 101(11):118105

    Article  Google Scholar 

  49. Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117(1):1–19

    MATH  Article  Google Scholar 

  50. Radzuan NAM, Sulong AB, Sahari J (2017) A review of electrical conductivity models for conductive polymer composite. Int J Hydrogen Energy 42:9262–9273

    Article  Google Scholar 

  51. Rahman WAWA, Sin LT, Rahmat AR (2008) Injection moulding simulation analysis of natural fiber composite window frame. J Mater Process Technol 197(1–3):22–30

    Article  Google Scholar 

  52. Reasor DA Jr, Clausen JR, Aidun CK (2012) Coupling the lattice-Boltzmann and spectrin-link methods for the direct numerical simulation of cellular blood flow. Int J Numer Methods Fluids 68(6):767–781

    MathSciNet  MATH  Article  Google Scholar 

  53. Ren F, Shi Y, Ren P, Si X, Wang H (2017) Cyanate ester resin filled with graphene nanosheets and NiFe2O4 reduced graphene oxide nanohybrids for efficient electromagnetic interference shielding. Nano 12:1750066

    Article  Google Scholar 

  54. Rizvi R, Naguib H (2013) Porosity and composition dependence on electrical and piezoresistive properties of thermoplastic polyurethane nanocomposites. J Mater Res 28(17):2415

    Article  Google Scholar 

  55. Sandler JKW, Kirk JE, Kinloch I, Shaffer M, Windle A (2003) Ultra-low electrical percolation threshold in carbon-nanotube-epoxy composites. Polymer 44:5893–5899

    Article  Google Scholar 

  56. Sasayama T, Inagaki M (2017) Simplified bead-chain model for direct fiber simulation in viscous flow. J Non-Newton Fluid Mech 250:52–58

    MathSciNet  Article  Google Scholar 

  57. Schwarz S, Kempe T, Fröhlich J (2015) A temporal discretization scheme to compute the motion of light particles in viscous flows by an immersed boundary method. J Comput Phys 281:591–613

    MathSciNet  MATH  Article  Google Scholar 

  58. Seil P, Pirker S (2016) LBDEMcoupling: open-source power for fluid-particle systems. In: International conference on discrete element methods, pp 679–686. Springer

  59. Semlali Aouragh Hassani FZ, Ouarhim W, Zari N, Bensalah MO, Rodrigue D, Bouhfid R, Qaiss Aek (2019) Injection molding of short coir fiber polypropylene biocomposites: prediction of the mold filling phase. Polym Compos 40(10):4042–4055

    Article  Google Scholar 

  60. Shaayegan V, Ameli A, Wang S, Park CB (2016) Experimental observation and modeling of fiber rotation and translation during foam injection molding of polymer composites. Compos Part A Appl Sci Manuf 88:67–74

    Article  Google Scholar 

  61. Shokri P, Bhatnagar N (2007) Effect of packing pressure on fiber orientation in injection molding of fiber-reinforced thermoplastics. Polym Compos 28(2):214–223

    Article  Google Scholar 

  62. Smid M (2003) Computing intersections in a set of line segments: the Bentley–Ottmann algorithm

  63. Tan J, Sinno TR, Diamond SL (2018) A parallel fluid-solid coupling model using LAMMPS and Palabos based on the immersed boundary method. J Comput Sci 25:89–100

    Article  Google Scholar 

  64. Tang H, Wang P, Zheng P, Liu X (2016) Core-shell structured BaTiO\(_3\) polymer hybrid nanofiller for poly(arylene ether nitrile) nanocomposites with enhanced dielectric properties and high thermal stability. Compos Sci Technol 123:134–142

    Article  Google Scholar 

  65. Tang ZH, Li YQ, Huang P, Fu YQ, Hu N, Fu SY (2021) A new analytical model for predicting the electrical conductivity of carbon nanotube nanocomposites. Compos Commun 23:100577

    Article  Google Scholar 

  66. Tran MP, Detrembleur C, Alexandre M, Jerome C, Thomassin JM (2013) The influence of foam morphology of multi-walled carbon nanotubes/poly (methyl methacrylate) nanocomposites on electrical conductivity. Polymer 54(13):3261–3270

    Article  Google Scholar 

  67. Tschisgale S, Kempe T, Fröhlich J (2018) A general implicit direct forcing immersed boundary method for rigid particles. Comput Fluids 170:285–298

    MathSciNet  MATH  Article  Google Scholar 

  68. Van Der Spoel D, Lindahl E, Hess B, Groenhof G, Mark AE, Berendsen HJ (2005) GROMACS: fast, flexible, and free. J Comput Chem 26(16):1701–1718

    Article  Google Scholar 

  69. Wang C, Shaayegan V, Ataei M, Costa F, Han S, Bussmann M, Park CB (2019) Accurate theoretical modeling of cell growth by comparing with visualized data in high-pressure foam injection molding. Eur Polym J 119:189–199

    Article  Google Scholar 

  70. Wang, J, Jin X (2010) Comparison of recent fiber orientation models in Autodesk Moldflow insight simulations with measured fiber orientation data. In: Polymer processing society 26th annual meeting. Citeseer

  71. Wang J, O’Gara JF, Tucker CL III (2008) An objective model for slow orientation kinetics in concentrated fiber suspensions: theory and rheological evidence. J Rheol 52(5):1179–1200

  72. Wang S, Ameli A, Shaayegan V, Kazemi Y, Huang Y, Naguib H, Park CB (2018) Modelling of rod-like fillers’ rotation and translation near two growing cells in conductive polymer composite foam processing. Polymers 10(3):261. https://www.mdpi.com/2073-4360/10/3/261

  73. Wang S, Huang Y, Chang E, Zhao C, Ameli A, Naguib HE, Park CB (2021) Evaluation and modeling of electrical conductivity in conductive polymer nanocomposite foams with multiwalled carbon nanotube networks. Chem Eng J 411:128382

    Article  Google Scholar 

  74. Wang S, Huang Y, Zhao C, Chang E, Ameli A, Naguib H, Park CB (2020) Theoretical modeling and experimental verification of percolation threshold with MWCNTs’ rotation and translation around a growing bubble in conductive polymer composite foams. Compos Sci Technol 199:108345

  75. Yamamoto S, Matsuoka T (1993) A method for dynamic simulation of rigid and flexible fibers in a flow field. J Chem Phys 98(1):644–650

    Article  Google Scholar 

  76. Yamamoto S, Matsuoka T (1995) Dynamic simulation of fiber suspensions in shear flow. J Chem Phys 102(5):2254-2260

    Article  Google Scholar 

  77. Yarin A, Gottlieb O, Roisman I (1997) Chaotic rotation of triaxial ellipsoids in simple shear flow. J Fluid Mech 340:83–100

    MathSciNet  MATH  Article  Google Scholar 

  78. Yu A, Pak AJ, He P, Monje-Galvan V, Casalino L, Gaieb Z, Dommer AC, Amaro RE, Voth GA (2021) A multiscale coarse-grained model of the SARS-CoV-2 virion. Biophys J 120(6):1097–1104

    Article  Google Scholar 

  79. Zare Y, Rhee KY (2017) A simple methodology to predict the tunneling conductivity of polymer/CNT nanocomposites by the roles of tunneling distance, interphase and CNT waviness. RSC Adv 7(55):34912–34921

    Article  Google Scholar 

  80. Zhao B, Zhao C, Li R, Hamidinejad SM, Park CB (2017) Flexible, ultrathin, and high-efficiency electromagnetic shielding properties of Poly(Vinylidene Fluoride)/Carbon composite films. ACS Appl Mater Interfaces 9(24):20873–20884

    Article  Google Scholar 

  81. Zhao C, Mark LH, Kim S, Chang E, Park CB, Lee PC (2021) Recent progress in micro-/nano-fibrillar reinforced polymeric composite foams. Polym Eng Sci 61(4):926–941. https://doi.org/10.1002/pen.25643

Download references

Acknowledgements

We thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and Autodesk Inc. for their financial support. Computations were performed on the Niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Markus Bussmann.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Conversion between dimensionless physical and lattice parameters

Appendix A: Conversion between dimensionless physical and lattice parameters

Dimensionless numbers are indicated with “*” superscript. Given a length scale \(\delta x\), a timescale \(\delta t\), a mass scale \(\delta m\), and a substance scale \(\delta n\), the following dimensionless numbers can be defined to convert lattice parameters to physical parameters, and vice versa.

$$\begin{aligned} \rho ^*= & {} \rho \frac{\delta x^3}{\delta m}\\ \nu ^*= & {} \rho \frac{\delta t}{\delta x^2}\\ q^*= & {} \rho \frac{\delta x^3}{\delta n}\\ D^*= & {} \rho \frac{\delta t}{\delta x^2}\\ \mathbf {g}^*= & {} \mathbf {g} \frac{\delta t^2}{\delta x}\\ L_f^*= & {} L_f \frac{1}{\delta x} \\ k_s^*= & {} k_s \frac{\delta t^2}{\delta m} \\ k_b^*= & {} k_b \frac{\delta t^2}{\delta m \delta x^2} \\ k_v^*= & {} k_b \frac{\delta t^2}{\delta m \delta x^2} \\ k_d^*= & {} k_d \frac{\delta t^2}{\delta m} \\ k_a^*= & {} k_a \frac{\delta t^2}{\delta m} \end{aligned}$$

where the dimensionless lattice parameters \(\tau ^*\) and \(\tau ^*_g\) are:

$$\begin{aligned} \tau ^*= & {} \frac{6\nu ^* + 1}{2} \\ \tau _g^*= & {} \frac{6D^* + 1}{2} \end{aligned}$$

Note that because of the relationship between pressure, density, and the speed of sound (\(p = \rho c_s^2\)), for a given pressure and density \(\delta t\) and \(\delta x\) cannot be changed independently. Refer to “Koerner, et al. Springer Science & Business Media, 2008” for more details.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ataei, M., Pirmorad, E., Costa, F. et al. A hybrid lattice Boltzmann-molecular dynamics-immersed boundary method model for the simulation of composite foams. Comput Mech 69, 1177–1190 (2022). https://doi.org/10.1007/s00466-021-02136-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-021-02136-9

Keywords

  • Lattice Boltzmann method
  • Fiber reinforced
  • Polymer foam composites
  • Molecular dynamics
  • Numerical methods