Skip to main content
Log in

Scaling laws as a tool of materials informatics

  • Overview
  • Materials Informatics 2008
  • Published:
JOM Aims and scope Submit manuscript

Abstract

This paper discusses the utility of scaling laws to materials informatics and presents the algorithm Scaling LAW (SLAW), useful to obtain scaling laws from statistical data. These laws can be used to extrapolate known materials property data to untested materials by using other more readily available information. This technique is independent of a characteristic length or time scale, so it is useful for a broad diversity of problems. In some cases, SLAW can reproduce the mathematical expression that would have been obtained through an analytical treatment of the problem. This technique was originally designed for mining statistical data in materials processing and materials behavior at a system level, and it shows promise for the study of the relationship between structure and properties in materials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. W.H. Hunt, “Materials Informatics: Growing from the Bio World,” JOM, 58(7) (2006), p. 88.

    Article  Google Scholar 

  2. K.F. Ferris, L.M. Peurrung, and J. Marder, “Materials Informatics: Fast Track to New Materials,” Advanced Materials & Processes, 165(1) (2007), pp. 50–51.

    Google Scholar 

  3. Z.K. Liu, L.Q. Chen, and K. Rajan, “Linking Length Scales Via Materials Informatics,” JOM, 58(11) (2006), pp. 42–50.

    Article  CAS  Google Scholar 

  4. P.F. Mendez and F. Ordøõez, “Scaling Laws from Statistical Data and Dimensional Analysis,” Journal of Applied Mechanics, 72(5) (2005), pp. 648–657.

    Article  Google Scholar 

  5. D. Cebon and M.F. Ashby, “Engineering Materials Informatics,” MRS Bulletin, 31(12) (2006), pp. 1004–1012.

    CAS  Google Scholar 

  6. C.B. Geller et al., “A Computational Search for Ductilizing Additives to Mo,” Scripta Materialia, 52(3) (2005), pp. 205–210.

    Article  CAS  Google Scholar 

  7. J.B. Fourier, Théorie Analytique De La Chaleur (Paris: Firmin Didot, 1822).

    Google Scholar 

  8. E. Buckingham, “On Physically Similar Systems; Illustrations of the Use of Dimensional Equations,” Physics Review, 4(4) (1914), pp. 345–376.

    Article  Google Scholar 

  9. Y. Le Page, “Data Mining in and around Crystal Structure Databases,” MRS Bulletin, 31 (2006), pp. 991–994.

    Google Scholar 

  10. C.C. Fischer et al., “Predicting Crystal Structure by Merging Data Mining with Quantum Mechanics,” Nature Materials, 5(8) (2006), pp. 641–646.

    Article  CAS  Google Scholar 

  11. P.W. Bridgman, Dimensional Analysis, first edition (New Haven, CT: Yale University Press, 1922), p. 113.

    Google Scholar 

  12. A.E. Ruark, “Inspectional Analysis: A Method Which Supplements Dimensional Analysis,” Journal of the Mitchell Society, 51 (1935), pp. 127–133.

    Google Scholar 

  13. C.J. Geankoplis, Transport Processes and Separation Process Principles: (Includes Unit Operations), 4th edition (Upper Saddle River, NJ: Prentice Hall Professional Technical Reference, 2003).

    Google Scholar 

  14. B.R. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, first edition (New York: John Wiley & Sons, 1960).

    Google Scholar 

  15. A. Bejan, Convection Heat Transfer, 3rd edition (Hoboken, NJ: Wiley, 2004).

    Google Scholar 

  16. J. Szekely and N.J. Themelis, “Chapter 16: Similarity Criteria and Dimensional Analysis,” Rate Phenomena in Process Metallurgy (New York: John Wiley & Sons, 1971), pp. 557–597.

    Google Scholar 

  17. M.M. Denn, Process Fluid Mechanics, first edition, Prentice-Hall International Series in the Physical and Chemical Engineering Sciences, ed. N.R. Amundson (Englewood Cliffs, NJ: Prentice-Hall, 1980).

    Google Scholar 

  18. W.M. Deen, Analysis of Transport Phenomena (New York: Oxford University Press, 1998).

    Google Scholar 

  19. S.J. Kline, Similitude and Approximation Theory (New York: Springer-Verlag, 1986).

    Google Scholar 

  20. J.A. Dantzig and C.L. Tucker, Modeling in Materials Processing (Cambridge, U.K.: Cambridge University Press, 2001).

    Google Scholar 

  21. P.J. Sides, “Scaling of Differential Equations: Analysis of the Fourth Kind,” Chemical Engineering Education (Summer 2002), pp. 232–235.

  22. M.M. Chen, “Scales, Similitude, and Asymptotic Considerations in Convective Heat Transfer,” Annual Review of Heat Transfer, ed. C.L. Tien (New York: Hemisphere Pub. Corp., 1990), pp. 233–291.

    Google Scholar 

  23. G. Astarita, “Dimensional Analysis, Scaling, and Orders of Magnitude,” Chemical Engineering Science, 52(24) (1997), pp. 4681–4698.

    Article  CAS  Google Scholar 

  24. K.M.K. Yip, “Model Simplification by Asymptotic Order of Magnitude Reasoning,” Artificial Intelligence, 80(2) (1996), pp. 309–348.

    Article  Google Scholar 

  25. W.B. Krantz, Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation (Hoboken, NJ: John Wiley & Sons, 2007).

    Google Scholar 

  26. P.F. Mendez, “Advanced Scaling Techniques for the Modeling of Materials Processing,” Sohn International Symposium-Advanced Processing of Metals and Materials: Volume 7: Industrial Practice, ed. F. Kongoli and R.G. Reddy (Warrendale, PA: TMS, 2006), pp. 393–404.

    Google Scholar 

  27. G. Bradshaw, P. Langley, and H.A. Simon, “Bacon 4: The Discovery of Intrinsic Properties,” Third Nat. Conf. of the Canadian Society for Computational Studies of Intelligence (Toronto, Ont., Canada: CSCSI, 1980).

    Google Scholar 

  28. T. Washio and H. Motoda, “Extension of Dimensional Analysis for Scale-Types and Its Application to Discovery of Admissible Models of Complex Processes,” 2nd Int. Workshop on Similarity Method (1999), pp. 129–147.

  29. M.M. Kokar, “Determining Arguments of Invariant Functional Descriptions,” Machine Learning, 1(4) (December 1986), pp. 403–422.

    Google Scholar 

  30. T. Washio, M. Motoda, and Y. Niwa, “Enhancing the Plausibility of Law Equation Discovery,” Proc. 17th International Conference on Machine Learning (San Francisco, CA: Morgan Kaufmann Publishers Inc., 2000), pp. 1127–1134.

    Google Scholar 

  31. C.C. Li and Y.C. Lee, “A Statistical Procedure for Model-Building in Dimensional Analysis,” International Journal of Heat and Mass Transfer, 33(7) (1990), pp. 1566–1567.

    Article  CAS  Google Scholar 

  32. V.G. Dovi et al., “Improving the Statistical Accuracy of Dimensional Analysis Correlations for Precise Coefficient Estimation and Optimal-Design of Experiments,” International Communications in Heat and Mass Transfer, 18(4) (1991), pp. 581–590.

    Article  Google Scholar 

  33. G.A. Vignaux, “Dimensional Analysis in Operations-Research,” New Zealand Operational Research, 14(1) (1986), pp. 81–92.

    Google Scholar 

  34. G.A. Vignaux and J.L. Scott, “Simplifying Regression Models Using Dimensional Analysis,” Australian & New Zealand Journal of Statistics, 41(1) (1999), pp. 31–41.

    Article  Google Scholar 

  35. G.A. Vignaux, “Some Examples of Dimensional Analysis in Operations Research and Statistics” (Presentation at the 4th International Workshop on Similarity Methods, Stuttgart, Germany: University of Stuttgart, 2001).

    Google Scholar 

  36. B.B. Hicks, “Some Limitations of Dimensional Analysis and Power Laws,” Boundary-Layer Meteorology, 14 (1978), pp. 567–569.

    Article  Google Scholar 

  37. B.C. Kenney, “On the Validity of Empirical Power Laws,” Stochastic Hydrology and Hydraulics, 7 (1993), pp. 179–194.

    Article  Google Scholar 

  38. G.I. Barenblatt, Cambridge Texts in Applied Mathematics: Scaling, Self-Similarity, and Intermediate Asymptotics, 1st edition, (New York: Cambridge University Press, 1996).

    Google Scholar 

  39. G.I. Barenblatt, Cambridge Texts in Applied Mathematics: Scaling (Cambridge, U.K.: Cambridge University Press, 2003).

    Google Scholar 

  40. M. Taylor et al., “100 Years of Dimensional Analysis: New Steps toward Empirical Law Deduction,” Submitted to New Journal of Physics (IOP) (2007) arXiv:0709.3584v3 [physics.class-ph].

  41. W.R. Stahl, “Dimensional Analysis in Mathematical Biology I. General Discussion,” Bulletin of Mathematical Biology (Springer), 23(4) (1961), pp. 355–376.

    Google Scholar 

  42. W.R. Stahl, “Dimensional Analysis in Mathematical Biology II,” Bulletin of Mathematical Biology (Springer), 24(1) (1962), pp. 81–108.

    Google Scholar 

  43. V.B. Kokshenev, “Observation of Mammalian Similarity through Allometric Scaling Laws,” Physica a-Statistical Mechanics and Its Applications, 322(1–4) (2003), pp. 491–505.

    Article  Google Scholar 

  44. R.K. Azad et al., “Segmentation of Genomic DNA through Entropic Divergence: Power Laws and Scaling,” Physical Review E, 65(5) (2002), art. no.-051909.

  45. T. Nakamura et al., “Universal Scaling Law in Human Behavioral Organization,” Phys. Rev. Lett., 99 (2007), p. 138103.

    Article  Google Scholar 

  46. D. Brockmann, L. Hufnagel, and T. Geisel, “The Scaling Laws of Human Travel,” Nature, 439 (2006), pp. 462–465.

    Article  CAS  Google Scholar 

  47. T. Faug et al., “Varying Dam Height to Shorten the Run-out of Dense Avalanche Flows: Developing a Scaling Law from Laboratory Experiments,” Surveys in Geophysics, 24 (2003), pp. 555–568.

    Article  Google Scholar 

  48. V.M. Arunachalam and D.B. Muggeridge, “Ice Pressures on Vertical and Sloping Structures through Dimensional Analysis and Similarity Theory,” Cold Regions Science and Technology, 21(3) (2003), pp. 231–245.

    Article  Google Scholar 

  49. K.R. Housen, R.M. Schmidt, and K.A. Holsapple, “Crater Ejecta Scaling Laws-Fundamental Forms Based on Dimensional Analysis,” Journal of Geophysical Research, 88(B3) (1983), pp. 2485–2499.

    Article  Google Scholar 

  50. A.-L. Barabasi and R. Albert, “Emergence of Scaling in Random Networks,” Science, 286(5439) (15 October 1999), pp. 509–512.

    Article  Google Scholar 

  51. J.M. Carlson and J. Doyle, “Power Laws, Highly Optimized Tolerance and Generalized Source Coding,” Physical Review Letters, 84(24) (2000), pp. 56–59.

    Google Scholar 

  52. F.J. Jong and W. Quade, “Dimensional Analysis for Economists,” Contributions to Economic Analysis (Amsterdam: North Holland Pub. Co., 1967), p. 223.

    Google Scholar 

  53. J. Chave and S. Levin, “Scale and Scaling in Ecological and Economic Systems,” Environmental and Resource Economics, 26 (2003), pp. 527–557.

    Article  Google Scholar 

  54. Z. Xu and R. Gencay, “Scaling, Self-Similarity and Multifractality in Fx Markets,” Physica A, 323 (2003), pp. 578–590.

    Article  Google Scholar 

  55. S. Newcomb, “Note on the Frequency of Use of the Different Digits in Natural Numbers,” American Journal of Mathematics, 4 (1881), pp. 39–40.

    Article  Google Scholar 

  56. F. Benford, “The Law of Anomalous Numbers,” Proceedings of the American Philosophical Society, 78(4) (1938), pp. 551–572.

    Google Scholar 

  57. Scaling Laws. SLAW Homepage, http://illposed. usc.edu/~pat/SLAW.

  58. J.-W. Park, P.F. Mendez, and T.W. Eagar, “Strain Energy Distribution in Ceramic to Metal Joints,” Acta Materialia, 50 (2002), pp. 883–899.

    Article  CAS  Google Scholar 

  59. J. Huang et al., “Capillary Wrinkling of Floating Thin Polymer Films,” Science, 317 (2007), p. 650.

    Article  CAS  Google Scholar 

  60. P. Mazzatorta et al., “The Importance of Scaling in Data Mining for Toxicity Prediction,” Journal of Chemical Information and Computer Sciences, 42(5) (2002), pp. 1250–1255.

    Article  CAS  Google Scholar 

  61. Y. Li, “Predicting Materials Properties and Behavior Using Classification and Regression Trees,” Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing, 433(1–2) (2006), pp. 261–268.

    Google Scholar 

  62. K. Rajan, “Materials Informatics,” Materials Today, 8(10) (2005), pp. 38–45.

    Article  CAS  Google Scholar 

  63. M.M. Kokar, “A Procedure of Identification of Laws in Empirical Sciences,” Systems Science, 7(1) (1981).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Patricio F. Mendez.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mendez, P.F., Furrer, R., Ford, R. et al. Scaling laws as a tool of materials informatics. JOM 60, 60–66 (2008). https://doi.org/10.1007/s11837-008-0036-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11837-008-0036-9

Keywords

Navigation