Theoretical prediction of effective properties for multiphase material systems is very important not only to analysis and optimization of material performance, but also to new material designs. This review first examines the issues, difficulties and challenges in prediction of material behaviors by summarizing and critiquing the existing major analytical approaches dealing with material property modeling. The focus then shifts to some recent advances in numerical methodology that are able to predict more accurately and efficiently the effective physical properties of multiphase materials with complex internal microstructures. A random generation-growth algorithm is highlighted for reproducing multiphase microstructures, statistically equivalent to the actual systems, based on the geometrical and morphological information obtained from measurements and experimental estimations. Then a high-efficiency lattice Boltzmann solver for the corresponding governing equations is described which, while assuring energy conservation and the appropriate continuities at numerous interfaces in a complex system, has demonstrated its numerical power in yielding accurate solutions. Various applications are provided to validate the feasibility, effectiveness and robustness of this new methodology by comparing the predictions with existing experimental data from different transport processes, accounting for the effects due to component size, material anisotropy, internal morphology and multiphase interactions. The examples given also suggest even wider potential applicability of this methodology to other problems as long as they are governed by the similar partial differential equation(s). Thus, for given system composition and structure, this numerical methodology is in essence a model built on sound physics principles with prior validity, without resorting to ad hoc empirical treatment. Therefore, it is useful for design and optimization of new materials, beyond just predicting and analyzing the existing ones.

Predictions of effective physical properties of complex multiphase materials

The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. However, the gap in the band structure of the exact multiplicative Kohn–Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS density-functional theory. Here, we give a simple proof of a theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approximate functional if the GKS potential operator is continuous and the density change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from metageneralized gradient approximations (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen molecules, solid aluminum arsenide, and solid argon provide numerical illustrations.

https://www.pnas.org/content/pnas/114/11/2801.full.pdf

Understanding band gaps of solids in generalized Kohn–Sham theory

Electromagnetic properties of small-particle composites

Effective thermal conductivity of composite spheres in a continuous medium with contact resistance

http://www.physics.fudan.edu.cn/tps/people/jphuang/Mypapers/PR-1.pdf

Enhanced nonlinear optical responses of materials: Composite effects

Piezoelectric composites consisting of spherically anisotropic piezoelectric inclusions (i.e., piezoceramic material) in an infinite nonpiezoelectric matrix under a uniform electric field are theoretically investigated. Analytical solutions for the elastic displacements and the electric potentials are derived exactly. Taking account of the coupling effects of elasticity, permittivity, and piezoelectricity, formulas are derived for the effective dielectric and piezoelectric responses in the dilute limit. A piezoelectric response mechanism is revealed, in which the effective piezoelectric response vanishes irrespective of how much spherically anisotropic piezoelectric inclusions are inside. Moreover, the effective coupled responses of the piezoelectric composites show that the effective dielectric responses decrease (increase) as the inclusion elastic (piezoelectric) constants increase.

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Effective properties of spherically anisotropic piezoelectric composites

Anisotropic dielectric response occurs naturally due to the presence of gradation, like in functionally graded materials or graded biological cells. However, these materials with locally anisotropic dielectric responses can have macroscopically isotropic responses. In treating graded particles of anisotropic materials, traditional isotropic gradation methods need to be modified. In this work, we developed a first-principles approach, as well as an anisotropic differential effective dipole approximation, for calculating the dipole moment of these particles. To this end, the two methods are shown in excellent agreement. As a result, these approaches offer convenient and effective ways to investigate the dielectric properties and optical responses of graded spherical particles of anisotropic materials, as well as the electrokinetic phenomena of biological cells.

http://www.physics.fudan.edu.cn/tps/people/jphuang/Mypapers/JAP-2.pdf

Dielectric response of graded spherical particles of anisotropic materials

The perturbation expansion method is employed to compute the effective nonlinear conductivity of a random composite material characterized by a weakly nonlinear relation between the current density J and the electric field E of the form J=σE+χ|E| 2 E, where a and χ take on different values in the inclusion and in the host. We develop perturbation expansions to obtain analytic formulas of the potential to arbitrary order in X. As an example, we apply the method to deal with a spherical inclusion in a host, both of either linear or nonlinear J-E relations, and obtain the potential to second order in X

Effective conductivity of nonlinear composites of spherical particles: A perturbation approach.

A perturbative approach, which has previously been applied to study the effective nonlinear response in random nonlinear composites consisting of Kerr materials, is extended to treat random composites with components having nonlinear response at finite frequencies. For a sinusoidal applied field, the field in the composite generally includes components with frequencies at the higher harmonics. Using the potential in the absence of nonlinearity as the unperturbed potential, nonlinear response can be studied perturbatively. Expression for the effective nonlinear susceptibility at the third harmonic is derived in the dilute limit of one of the nonlinear components.

A theory of nonlinear AC response in nonlinear composites

Periodic composite materials with contact resistance on the inclusion surface are studied. A generalized Rayleigh identity, which is applicable to such systems, is derived. Calculations are made for composites containing a simple cubic array of spherical inclusions. These solutions are derived by applying the generalized Rayleigh identity. A definition of the effective conductivity suitable for systems with a discontinuity of the temperature field on surface of inclusions is suggested, and it produced reasonable effective conductivities over all the possible range of Biot’s number. It turned out that the contact resistance can change the effective conductivity dramatically and is rich in its effects on the thermal properties of composites.

Guo-Qing Gu

Papers

Effective properties of spherically anisotropic piezoelectric composites

Dielectric response of graded spherical particles of anisotropic materials

Effective conductivity of nonlinear composites of spherical particles: A perturbation approach.

A theory of nonlinear AC response in nonlinear composites

Effective thermal conductivity of a periodic composite with contact resistance