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Research Papers: Conduction

Conduction in Jammed Systems of Tetrahedra

[+] Author and Article Information
Kyle C. Smith

Mem. ASME
e-mail: kyle.c.smith@gmail.com

Timothy S. Fisher

Mem. ASME
e-mail: tsfisher@purdue.edu
Birck Nanotechnology Center and School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received June 13, 2012; final manuscript received April 1, 2013; published online June 27, 2013. Assoc. Editor: Jose L. Lage.

J. Heat Transfer 135(8), 081301 (Jun 27, 2013) (7 pages) Paper No: HT-12-1285; doi: 10.1115/1.4024276 History: Received June 13, 2012; Revised April 01, 2013

Control of transport processes in composite microstructures is critical to the development of high-performance functional materials for a variety of energy storage applications. The fundamental process of conduction and its control through the manipulation of granular composite attributes (e.g., grain shape) are the subject of this work. We show that athermally jammed packings of tetrahedra with ultrashort range order exhibit fundamentally different pathways for conduction than those in dense sphere packings. Highly resistive granular constrictions and few face–face contacts between grains result in short-range distortions from the mean temperature field. As a consequence, ‘granular’ or differential effective medium theory predicts the conductivity of this media within 10% at the jamming point; in contrast, strong enhancement of transport near interparticle contacts in packed-sphere composites results in conductivity divergence at the jamming onset. The results are expected to be particularly relevant to the development of nanomaterials, where nanoparticle building blocks can exhibit a variety of faceted shapes.

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Figures

Grahic Jump Location
Fig. 1

Consolidation of 400 tetrahedra. Density is indicated below each image and color indicates the number of particles belonging to the face–face cluster of a given particle at the jamming threshold density of ϕJ = 0.634 for this system. Boundaries of the periodic supercell are indicated by black edges of the surrounding cube.

Grahic Jump Location
Fig. 2

Reconstructed interface geometry. (a) Phase of a given cell is determined by that of its centroid and is indicated by the centroid's color. The true and reconstructed interfaces are depicted in black and gray, respectively. (b) Sample cross-section of the three-level refined mesh for a system of 400 tetrahedra at ϕ = 0.611. The pore phase is indicated in white, while distinct grains are shaded.

Grahic Jump Location
Fig. 3

Octree refinement of κ¯ as a function of density for 400 tetrahedra with κg = 103. For each data point, the coarsest level of the mesh has a resolution of Δx=0.025(NVg/φ)1/3, where Δx is the length of coarse cube-shaped finite volume cells, N is the number of particles, and Vg is the volume of an individual grain. The position of the jamming point is marked by the red dashed line.

Grahic Jump Location
Fig. 4

Stencil employed for discretization of flux at faces between cells of dissimilar size

Grahic Jump Location
Fig. 5

Radial distribution function of the present systems for various system sizes. Experimental [27,37] and simulated [18,38] values of other contemporary works are shown as well. The system of 25 tetrahedra is excluded because of insufficient sampling. In brackets are listed the jamming threshold densities. Rmin is the minimum possible centroidal separation between contacting tetrahedra. The referenced prior simulation results are offset by an amount of two.

Grahic Jump Location
Fig. 6

Cluster number distribution for jammed systems of tetrahedra of varying system size

Grahic Jump Location
Fig. 7

(a) Normalized temperature field T˜ for 400 tetrahedra with κg = 103 at ϕ = 0.611. The field is induced by a homogenized temperature gradient (∇T)h applied to the system along the indicated direction. The length scale of a tetrahedron edge is indicated as a reference for the size of an individual grain. (b) Histogram of the deviation δT˜ from the mean normalized temperature field. The fitted normal distribution is shown along with its standard deviation σ and mean value μ.

Grahic Jump Location
Fig. 8

Mean effective conductivity for tetrahedra with κg = 103 as a function of (a) density and (b) density deviation, ϕJ − ϕ, where ϕJ is the jamming density for the respective system. The open triangular symbol represents a cubic polynomial extrapolation to the jamming point. 50:50 denotes the bidisperse mixture of tetrahedra described in the text. For simple cubic-packed spheres ϕJ = π/6, the density at which spheres just touch. GEMA refers to the granular effective medium approximation presented in Refs. [10,11], and MG-EMA refers to the Maxwell-Garnett effective medium approximation (see Ref. [39]). Data for simple cubic-packed spheres are from [30] at low density and [28] at high density.

Grahic Jump Location
Fig. 9

Deviatoric conductivity Δκ as a function of ϕ for κg = 103. The open triangular symbol represents a cubic polynomial extrapolation to the jamming point, and 50:50 denotes the bidisperse mixture of tetrahedra described in the text.

Grahic Jump Location
Fig. 10

Mean effective conductivity as a function of grain conductivity κg for (a) 100 tetrahedra at ϕ = 0.626 and (b) 400 tetrahedra at ϕ = 0.611. Error bars represent κ1 and κ3. Theoretical data are presented based on the GEMA in Refs. [10,11] and the Zehner–Bauer–Schlünder (ZBS) model [15,16].

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