# Scaling laws as a tool of materials informatics

- 148 Downloads
- 1 Citations

## 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.

## Keywords

Dimensional Analysis Problem Parameter Numerical Constant Dimensionless Group Partial Similarity## Preview

Unable to display preview. Download preview PDF.

## References

- 1.W.H. Hunt, “Materials Informatics: Growing from the Bio World,”
*JOM*, 58(7) (2006), p. 88.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar - 5.D. Cebon and M.F. Ashby, “Engineering Materials Informatics,”
*MRS Bulletin*, 31(12) (2006), pp. 1004–1012.Google Scholar - 6.C.B. Geller et al., “A Computational Search for Ductilizing Additives to Mo,”
*Scripta Materialia*, 52(3) (2005), pp. 205–210.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.Google Scholar - 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.CrossRefGoogle Scholar - 24.K.M.K. Yip, “Model Simplification by Asymptotic Order of Magnitude Reasoning,”
*Artificial Intelligence*, 80(2) (1996), pp. 309–348.CrossRefGoogle 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.Google Scholar - 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar - 37.B.C. Kenney, “On the Validity of Empirical Power Laws,”
*Stochastic Hydrology and Hydraulics*, 7 (1993), pp. 179–194.CrossRefGoogle 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].Google Scholar - 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.CrossRefGoogle 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.Google Scholar - 45.T. Nakamura et al., “Universal Scaling Law in Human Behavioral Organization,”
*Phys. Rev. Lett.*, 99 (2007), p. 138103.CrossRefGoogle Scholar - 46.D. Brockmann, L. Hufnagel, and T. Geisel, “The Scaling Laws of Human Travel,”
*Nature*, 439 (2006), pp. 462–465.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar - 50.A.-L. Barabasi and R. Albert, “Emergence of Scaling in Random Networks,”
*Science*, 286(5439) (15 October 1999), pp. 509–512.CrossRefGoogle 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.CrossRefGoogle Scholar - 54.Z. Xu and R. Gencay, “Scaling, Self-Similarity and Multifractality in Fx Markets,”
*Physica A*, 323 (2003), pp. 578–590.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar - 59.J. Huang et al., “Capillary Wrinkling of Floating Thin Polymer Films,”
*Science*, 317 (2007), p. 650.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar - 63.M.M. Kokar, “A Procedure of Identification of Laws in Empirical Sciences,”
*Systems Science*, 7(1) (1981).Google Scholar