Skip to main content

Sequential Machine Learning Applications of Particle Packing with Large Size Variations


We present work on the application of sequential supervised machine learning for a reduced-dimension, ballistic deposition, Monte Carlo particle packing. Calculations are carried out for a combination of three distinguishable hard spheres representing different materials. Each set of spheres has a distribution of particle sizes in order to mimic realistic milling conditions of raw ingredients. Since infinite combinations of particle size, distribution, fraction, and density exist, we employ machine learning to aid in the design optimization of new high packing density mixtures. Previously calculated binary packs of particle radius ratios of 80:1 were analyzed, but this work highlights results of ternary packs with radius ratios greater than 300:1. We demonstrate a sequential learning approach where iterative experiments are performed based on minimizing the uncertainty in the target regime of high packing density. New candidate mixtures are identified via classification rather than regression which provides superior ability to extrapolate into high packing density mixtures.

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


  1. 1.

    Elsner P, Eyerer P, Hubner C, Geibler E (1999) The importance of micromechanical phenomena in energetic materials. Propellants Explos Pyrotech 24(1):119–125

    Google Scholar 

  2. 2.

    Jackson TL, Hegab A, Kochevets S, Buckmaster J (2001) Random packs and their use in modeling heterogeneous solid propellant combustion. J Propuls Powder 17(4):883–891

    Article  Google Scholar 

  3. 3.

    Hatch RL, Lee Davis I (2006) Mechanical properties for an arbitrary arrangement of rigid spherical particles embedded in an elastic matrix. Defense Technical Information Center, p 69

  4. 4.

    Coelho D, Thovert JF, Adler PM (1997) Geometrical and transport properties of random packings of spheres and aspherical particles. Phys Rev E 55(2):1959

    CAS  Article  Google Scholar 

  5. 5.

    Kim K, Munakata T (2003) Glass transition of hard sphere systems: molecular dynamics and density functional theory. Phys Rev E 68(2):021502

    Article  Google Scholar 

  6. 6.

    Donev A, Cisse I, Sachs D, Variano EA, Stillinger FH, Connelly R, Torquato S, Chaikin PM (2004) Improving the density of jammed disordered packings using ellipsoids. Science 303(5660):990–993

    CAS  Article  Google Scholar 

  7. 7.

    Ohern CS, Silbert LE, Liu AJ, Nagel SR (2003) Jamming at zero temperature and zero applied stress: the epitome of disorder. Phys Rev E 68(1):011306

    Article  Google Scholar 

  8. 8.

    Matheson AJ (1974) Computation of a random packing of hard spheres. J Phys C Solid State Phys 7(15):2569

    Article  Google Scholar 

  9. 9.

    Visscher WM, Bolsterli M (1972) Random packing of equal and unequal spheres in two and three dimensions. Nature 239(5374):504–507

    Article  Google Scholar 

  10. 10.

    Jodrey WS, Tory EM (1985) Computer simulation of close random packing of equal spheres. Phys Rev A 32(4):2347

    CAS  Article  Google Scholar 

  11. 11.

    Tobochnik J, Chapin PM (1988) Monte Carlo simulation of hard spheres near random closest packing using spherical boundary conditions. J Chem Phys 88(9):5824–5830

    CAS  Article  Google Scholar 

  12. 12.

    He D, Ekere NN, Cai L (1999) Computer simulation of random packing of unequal particles. Phys Rev E 60(6):7098

    CAS  Article  Google Scholar 

  13. 13.

    Oern CS, Langer SA, Liu AJ, Nagel SR (2002) Random packings of frictionless particles. Phys Rev Lett 88(7):075507

    Article  Google Scholar 

  14. 14.

    Lubachevsky BD, Stillinger FH (1990) Geometric properties of random disk packings. J Stat Phys 60(5–6):561–583

    Article  Google Scholar 

  15. 15.

    Lubachevsky BD, Stillinger FH, Pinson E (1991) Disks vs. spheres: contrasting properties of random packings. J Stat Phys 64(3–4):501–524

    Article  Google Scholar 

  16. 16.

    Webb MD, Lee Davis I (2006) Random particle packing with large particle size variations using reduced-dimension algorithms. Powder Technol 167(1):10–19

    CAS  Article  Google Scholar 

  17. 17.

    McGeary RK (1961) Mechanical packing of spherical particles. J Am Ceram Soc 44(10):513–522

    CAS  Article  Google Scholar 

  18. 18.

    Hugill HR, Westman AER (1930) The packing of particles. J Am Ceram Soc 13(10):767–779

    Article  Google Scholar 

  19. 19.

    Walton SF, White HE (1937) Particle packing and particle shape. J Am Ceram Soc 20:155–166

    Article  Google Scholar 

  20. 20.

    Furnas CC (1931) Mathematical relations for beds of broken solids of maximum density. Ind Eng Chem 23(9):1052–1058

    CAS  Article  Google Scholar 

  21. 21.

    Scott GD (1960) Packing of spheres. Nature 188:908

    Article  Google Scholar 

  22. 22.

    Mason J, Bernal JD (1960) Co-ordination of randomly packed spheres. Nature 188:190

    Google Scholar 

  23. 23.

    Berryman JG (1983) Random close packing of hard spheres and disks. Phys Rev 27(2):1053–1061

    CAS  Article  Google Scholar 

  24. 24.

    Tory EM, Jodery WS (1991) Computer simulation of close random packing of equal spheres. Phys Rev 32(4):2347–2351

    Google Scholar 

  25. 25.

    Lee DI (1970) Packing spheres and its effect on the viscosity of suspensions. Paint Technol 42:579

    CAS  Google Scholar 

  26. 26.

    van Lingen JNJ, Ingenhut BLJ, ten Cate AT, Maalderink HH, Straathof MH, van Driel CA (2019) Development of propellant compositions for vat photopolymerization additive manufacturing. Propellants Explos Pyrotech 44:1–18

    Article  Google Scholar 

  27. 27.

    Parry M, Rizvi Z, Couper S, Lin F, Miyagi L, Spark TD, Brgoch J, Tehrani AM, Oliynyk AO (2018) Machine learning directed search for ultraincompressible, superhard materials. J Am Chem Soc 140(31):9844–9853

    Article  Google Scholar 

  28. 28.

    Vazquez A, Kauwe SK, Graser J, Sparks TD (2018) Machine learning prediction of heat capacity for solid inorganics. Integr Mater Manuf Innov 7:43–651

    Article  Google Scholar 

  29. 29.

    Belviso F et al (2019) Viewpoint: atomic-scale design protocols toward energy, electronic, catalysis, and sensing applications. Inorg Chem 58:14939–14980

    CAS  Article  Google Scholar 

  30. 30.

    Kauwe SK, Welker T, Sparks TD (2020) Extracting knowledge from DFT: experimental band gap predictions through ensemble learning. Integr Mater Manuf Innov 9(3):213–220

    Article  Google Scholar 

  31. 31.

    Murdock RJ, Kauwe SK, Yang AY-T, Sparks TD (2020) Is domain knowledge necessary for machine learning materials properties? Integr Mater Manuf Innov 9:221–227

    Article  Google Scholar 

  32. 32.

    Gaultois MW, Oliynyk AO, Mar A, Sparks TD, Mulholland GJ, Meredig B (2016) Perspective: web-based machine learning models for real-time screening of thermoelectric materials properties. Appl Mater 4(5):053213

    Article  Google Scholar 

  33. 33.

    Sparks TD, Gaultois MW, Oliynyk A, Brgoch J, Meredig B (2016) Data mining our way to the next generation of thermoelectrics. Scr Mater 111:10–15

    CAS  Article  Google Scholar 

  34. 34.

    Al AO, Oliynyk EA (2016) High-throughput machine-learning-driven synthesis of full-Heusler compounds. Chem Mater 20:7324–7331

    Google Scholar 

  35. 35.

    Graser J, Kauwe SK, Sparks TD (2018) Machine learning and energy minimization approaches for crystal structure predictions: a review and new horizons. Chem Mater 30(11):3601–3612

    CAS  Article  Google Scholar 

  36. 36.

    Hall J (2021) Particle packing machine learning code.

  37. 37.

    Kim SB, Park CH (2015) Sequential random k-nearest neighbor feature selection for high-dimensional data. Expert Syst Appl 42:2336–2342

    Article  Google Scholar 

  38. 38.

    Edali M, Yucel G (2019) Exploring the behavior space of agent-based simulation models using random forest metamodels and sequential sampling. Simul Model Pract Theory 92:62–81

    Article  Google Scholar 

  39. 39.

    Kauwe SK, Graser J, Murdock R, Sparks TD (2020) Can machine learning find extraordinary materials? Comput Mater Sci 174:109498

    Article  Google Scholar 

Download references


The authors gratefully acknowledge support from the NSF CAREER Award DMR 1651668.

Author information



Corresponding author

Correspondence to Jason R. Hall.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hall, J.R., Kauwe, S.K. & Sparks, T.D. Sequential Machine Learning Applications of Particle Packing with Large Size Variations. Integr Mater Manuf Innov (2021).

Download citation


  • Supervised learning
  • Particle packing
  • Ballistic deposition
  • Packing fraction