### Abstract

Heterogeneous materials abound in nature and man-made situations. Examples include porous media, biological materials, and composite materials. Diverse and interesting properties exhibited by these materials result from their complex microstructures, which also make it difficult to model the materials. Yeong and Torquato introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions. In this first part of a series of two papers, we collect the known necessary conditions on the standard two-point correlation function S2 (r) and formulate a conjecture. In particular, we argue that given a complete two-point correlation function space, S2 (r) of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space. We provide examples of realizable two-point correlation functions and suggest a set of analytical basis functions. We also discuss an exact mathematical formulation of the (re)construction problem and prove that S2 (r) cannot completely specify a two-phase heterogeneous material alone. Moreover, we devise an efficient and isotropy-preserving construction algorithm, namely, the lattice-point algorithm to generate realizations of materials from their two-point correlation functions based on the Yeong-Torquato technique. Subsequent analysis can be performed on the generated images to obtain desired macroscopic properties. These developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two-point correlation functions. We will mainly focus on basic principles in this paper. The algorithmic details and applications of the general scheme are given in the second part of this series of two papers.

Original language | English (US) |
---|---|

Article number | 031110 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 76 |

Issue number | 3 |

DOIs | |

State | Published - Sep 11 2007 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*76*(3), [031110]. https://doi.org/10.1103/PhysRevE.76.031110

**Modeling heterogeneous materials via two-point correlation functions : Basic principles.** / Jiao, Y.; Stillinger, F. H.; Torquato, S.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 76, no. 3, 031110. https://doi.org/10.1103/PhysRevE.76.031110

}

TY - JOUR

T1 - Modeling heterogeneous materials via two-point correlation functions

T2 - Basic principles

AU - Jiao, Y.

AU - Stillinger, F. H.

AU - Torquato, S.

PY - 2007/9/11

Y1 - 2007/9/11

N2 - Heterogeneous materials abound in nature and man-made situations. Examples include porous media, biological materials, and composite materials. Diverse and interesting properties exhibited by these materials result from their complex microstructures, which also make it difficult to model the materials. Yeong and Torquato introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions. In this first part of a series of two papers, we collect the known necessary conditions on the standard two-point correlation function S2 (r) and formulate a conjecture. In particular, we argue that given a complete two-point correlation function space, S2 (r) of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space. We provide examples of realizable two-point correlation functions and suggest a set of analytical basis functions. We also discuss an exact mathematical formulation of the (re)construction problem and prove that S2 (r) cannot completely specify a two-phase heterogeneous material alone. Moreover, we devise an efficient and isotropy-preserving construction algorithm, namely, the lattice-point algorithm to generate realizations of materials from their two-point correlation functions based on the Yeong-Torquato technique. Subsequent analysis can be performed on the generated images to obtain desired macroscopic properties. These developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two-point correlation functions. We will mainly focus on basic principles in this paper. The algorithmic details and applications of the general scheme are given in the second part of this series of two papers.

AB - Heterogeneous materials abound in nature and man-made situations. Examples include porous media, biological materials, and composite materials. Diverse and interesting properties exhibited by these materials result from their complex microstructures, which also make it difficult to model the materials. Yeong and Torquato introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions. In this first part of a series of two papers, we collect the known necessary conditions on the standard two-point correlation function S2 (r) and formulate a conjecture. In particular, we argue that given a complete two-point correlation function space, S2 (r) of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space. We provide examples of realizable two-point correlation functions and suggest a set of analytical basis functions. We also discuss an exact mathematical formulation of the (re)construction problem and prove that S2 (r) cannot completely specify a two-phase heterogeneous material alone. Moreover, we devise an efficient and isotropy-preserving construction algorithm, namely, the lattice-point algorithm to generate realizations of materials from their two-point correlation functions based on the Yeong-Torquato technique. Subsequent analysis can be performed on the generated images to obtain desired macroscopic properties. These developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two-point correlation functions. We will mainly focus on basic principles in this paper. The algorithmic details and applications of the general scheme are given in the second part of this series of two papers.

UR - http://www.scopus.com/inward/record.url?scp=34548861273&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548861273&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.76.031110

DO - 10.1103/PhysRevE.76.031110

M3 - Article

AN - SCOPUS:34548861273

VL - 76

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 3

M1 - 031110

ER -