# Directed Thermal Diffusions through Metamaterial Source Illusion with Homogeneous Natural Media

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Derivations of Conductivities Tensors and Geometrical Models

#### 2.1. Theoretical Derivation of Conductivities Tensor

_{n}and OC

_{n}into OA

_{n}′ and OC

_{n}′. The linear mapping can be carried out by validating the relations of the original and transformational domains along the directions of the principle axes. That is, the relevant expansive/compressive coefficients should be solved through the following expressions:

_{n}C

_{n}to segment A

_{n}′C

_{n}′, it can be directly realized in a simplified way by approximating the length of arc A

_{n}C

_{n}to segment A

_{n}C

_{n}. However, such a simplification would introduce major errors into the transformations, once the inscribed polygon with fewer edge numbers (N), i.e., the length of arc A

_{n}C

_{n}is not small enough compared with the radius of the inner regions (r) [14]. Hence, we further refine the configurations inside the part A

_{n}A

_{n}′C

_{n}′ C

_{n}of Type I into m parts through the bisecting angle of A

_{n}′OC

_{n}′, in order to reduce the errors due to the simplifications. Each refined configuration is assigned a serial number p

_{(A~D)}, ranging from 1 to m. That is, the general vertices of an arbitrary element inside Type I can be obtained considering the positions of azimuths:

_{1}are the radii of the inner and outer circles. n denotes the serial number of each adjacent element inside Type I considering the positions of azimuth, ranging from 0 to N − 1. m are the multiples of equal dividing for the central angle and, p

_{A}and p

_{C}are the serial number of each characteristic component with equal central angle which is counting from the side of OA. According to the linear mapping function, the approximate inner triangle $O{A}_{\mathrm{n},{\mathrm{p}}_{\mathrm{A}}}{C}_{\mathrm{n},{\mathrm{p}}_{\mathrm{C}}}$ can be mapped into the outer $O{A}_{\mathrm{n},{\mathrm{p}}_{\mathrm{A}}}^{\prime}{C}_{\mathrm{n},{\mathrm{p}}_{\mathrm{C}}}^{\prime}$ with the following transformation processes upon the linearly expansive processes.

_{I,n}, b

_{I,n}, d

_{I,n}, and e

_{I,n}are the elements of ∂x′/∂x, ∂x′/∂y, ∂y′/∂x, and ∂y′/∂y in the related Jacobi matrix used for combining such transformation processes. Owing to the 2D domain, the elements of ∂z′/∂x and ∂z′/∂y are zero, i.e., c

_{I,n,p}, and f

_{I,n,p}are independent of the changes in the z direction. Hence, the related transformed Jacobi matrix can be derived upon solving the coefficients of a

_{I,n}, b

_{I,n}, d

_{I,n}, and e

_{I,n}. The above equations can be transformed into matrix forms through using the correspondences on the both sides of the simultaneous equations in Equation (3).

_{I,n}, b

_{I,n}, d

_{I,n}, and e

_{I,n}can be obtained through multiplying the inverse matrices of the corresponding parameters in the original region at the both sides of Equation (4). Considering the positions of characteristic points, the general expansive components in the refined elements of Type I can be achieved as follows.

_{B}and p

_{D}are the corresponding serial number of each characteristic component counting from the side of OB. Considering the mapping relations and the characteristic points, the expansive coefficients of the elements inside Type II can be derived by taking Equation (7) into Equation (1). Hence, the general expansive coefficients of Type II can be obtained.

_{i}

_{,n}·e

_{i}

_{,n}− b

_{i}

_{,n}·f

_{i}

_{,n}, in which κ denotes the original conductivity between the inner and outer circles. Hence, the conductivity components can be expressed as follows:

#### 2.2. Selections of Materials and Geometric Profiles of Thermal Source Illusions

_{P}and κ

_{N}are the relative conductivities for the positive and negative layers. γ

_{P,l}and γ

_{N,l}denote the composited fractions of the two layers in certain regions and the sum of γ

_{P,l}and γ

_{N,l}is 1.

_{1}), the radii of the outer (inscribed polygons) and inner circles are set as 0.1 m and 0.02 m, i.e., r

_{1}= 0.1 m and r = 0.02 m. The entire systems are placed at the center of the square plates with dimensions of 400 mm × 400 mm, filled with nickel steel (50% Ni) with a thermal conductivity of κ = 19.6 W∙m

^{−1}∙K

^{−1}. In addition, the inner circles are used for diffusing the thermal flux generated by the sources. Hence, the nickel steel (50% Ni) is also employed to fabricate the inner circular regions as it has small effect on the restoring function. For the parts between the outer circle and corresponding inscribed polygons, nickel steel is also employed, as the related spaces are unchanged during the transformation process. Considering the characteristics of the designed distribution of thermal fields under related polygonal profiles, each area inside Types I and II of all the schemes is divided into two parts: 10% portions of the area at the positions of the geometric deformations are considered as one part, while the remaining 90% are another. The 10% separations are beneficial to restrict the regions with extreme conductivities in relatively small areas. In addition, this also helps by reducing the limit parameters at the geometric deformations, resulting in better approximate mapping. The following considerations are to select the appropriate positive and negative layers for creating the 10% and 90% parts of the functional regions. For the 10% part in each type, p

_{A(B)}and p

_{C(D)}should be 1, and the values of m should be 22.4, 15, 12.8, and 11.8 for the triangle, square, pentagon, and hexagon schemes, respectively. In the 10% part of Type I, the relative conductivities for the triangle and square schemes are far larger than natural positive mediums, and the relative conductivities for the pentagon and hexagon are 291.52 and 199.75 W∙m

^{−1}∙K

^{−1}with nearly 100% fractions. In order to satisfy the requirements of the conductivities for the 10% parts in Type I, the pure copper with a conductivity of 398 W∙m

^{−1}∙K

^{−1}is selected for the triangle, square, and pentagon schemes. Aluminum alloy (6063) with a conductivity of 201 W∙m

^{−1}∙K

^{−1}is employed in the hexagon scheme. Furthermore, the adjacent 10% parts of the Types II are all filled with polydimethylsiloxane (PDMS) to create essential anisotropies coupled with those former 10% parts at the geometric deformations (near the vertices).

_{A(B)}is 2, and the values of m and p

_{C(D)}should also be 22.4, 15, 12.8 and 11.8 for the triangle, square, pentagon, and hexagon schemes, respectively. Taking the above newly defined points into Equations (5), (6) and (8), (9), the corresponding conductivity components and related fractions of the positive and negative layers can be achieved by Equations (14)–(16). Here, the negative and positive layers are respectively defined as layers A and B. Owing to the symmetries of the distribution of the functional regions, the configurations of the conductivities in the positive and negative layers are uniform. The desired conductivities of positive layers B and related fractions for the 90% part of Types I and II are determined as a function of the conductivities of the negative layers A, as shown in Figure 2.

^{−1}∙K

^{−1}is employed to completely fill in the layers A, owing to its low conductivity and preference of industry. For the conductivities of layers B, the values for the proposed triangle, square, pentagon, and hexagon schemes should be 44.015, 32.172, 28.233, and 26.232 W∙m

^{−1}∙K

^{−1}, respectively. Considering the previous studies [29,32,34], the conductivities inside layers B could be achieved by alternately arranging negative and positive mediums, owing to the approximate values calculated by Equation (16) and ${\kappa}_{\mathrm{P},\mathrm{l}}={\alpha}_{\mathrm{P},\text{}\mathrm{m}}{\kappa}_{\mathrm{P},\mathrm{m}}+{\alpha}_{\mathrm{N},\mathrm{m}}{\kappa}_{\mathrm{N},\mathrm{m}}$ (where α

_{P,m}and α

_{N,m}denote the area fractions of the positive and negative media composited in layers B, and κ

_{P,m}and κ

_{N,m}are the corresponding conductivities for the positive and negative media). In order to design the schemes with fewer kinds of media, the negative media composited in layers B are also PDMS. Hence, the total fractions of employed mediums inside this part (90% part) can be obtained.

_{P,m}and α

_{N,m}are the fractions of the positive and PDMS composited in the each positive layer. β

_{P,T}and β

_{N,T}denote the total fractions of the positive and negative media (PDMS) employed in each 90% part of all types, respectively. Furthermore, the more balanced configurations of the positive media and PDMS, the optimal expected performances would occur, owing to the higher anisotropies inside the functional regions.

^{−1}∙K

^{−1}, ductile iron with a conductivity of 75 W∙m

^{−1}∙K

^{−1}, and steel SAE 1010 with a conductivity of 59 W∙m

^{−1}∙K

^{−1}are selected as the potential candidates for the proposed schemes. The total contents of the candidates in each scheme are illustrated in Figure 3. In order to employ fewer kinds of media in each scheme, only one positive medium, which approximately and simultaneously satisfied the fractional demands in Types I and II. Hence, die-casting aluminum A380 is selected for the triangle scheme with total fractions of 40.54% and 45.05% for Types I and II. Ductile iron is used for Types I and II of the square scheme with corresponding total fractions of 41.78% and 49.42%. Moreover, steel SAE is employed both in the pentagon and hexagon schemes, due to its most approximate fractions to the demands, i.e., 42.70% and 50.83% for Types I and II of the pentagon scheme, and 39.71% and 44.12% for Types I and II of the hexagon scheme.

_{s}= 373 K. Furthermore, a scheme of pure nickel steel (50% Ni) plate with dimensions of 400 mm × 400 mm is established to make fair contrasts with the proposed schemes.

## 3. Demonstration of the Proposed Schemes and Discussions

#### 3.1. Temperature Distributions of the Proposed Thermal Source Illusions

#### 3.2. Characteristics of Temperature Distribution

_{l}denotes the radius on the boundary of the inscribed triangle. That is, the nature of thermal diffusions led to the local expansion of energy distributions with pure nickel steel inside the domains between the inscribed triangle and the outer circle. Similar to the other schemes, the maximums occurred, where the azimuths approached the vertices of the inscribed polygons, and the minimums were observed at the azimuths where the symmetry axes were located, i.e., θ = 45°+nπ/4, 36° + nπ/5, and 30° + nπ/6 for the square, pentagon, and hexagon schemes (n ranges from 0 to N − 1). Furthermore, the inner regions inside the isothermal lines were enlarged with the increasing side counts of the inscribed polygons, which forced boundaries to approach the outer circles caused by the decreasing central angles. Meanwhile, the above phenomena also meant that the thermal energy distributions were expanded by the functional elements compared with the contrast scheme shown in Figure 5b, i.e., the minimum radii of the isothermal lines of T = 302 K were larger than those in Figure 5b.

_{D}= |T

_{sc}− T

_{pl}|, where T

_{sc}and T

_{pl}are the temperatures on the measured lines of the proposed schemes and bare plate, respectively. The standard deviations of the temperature deformations on the measured lines, used to express the effectiveness, can be written as:

_{m}is the numbers of the measured points on related measured lines. The standard deviations of the temperature deformations on the measured lines are shown in Figure 9. Note that the larger standard deviations indicate higher discreteness of the temperature deformations between observed values and their averages. Consequently, larger deformations would appear on corresponding azimuths of the proposed schemes, owing to the symmetries illustrated in Figure 5. Hence, more significant performances of illusions would be observed. As Figure 9 illustrates, the standard deviation on each measured line was 0 for the contrast bare plate, which contributed to shape the profiles of the temperature distributions as circles. For the proposed schemes, the standard deviations on each measured line increased with decreasing side numbers, which meant that the more significant illusions were achieved in the schemes with fewer diffusion elements inside the functional regions. For the independent schemes, the trends of the standard deviations as a function of the radii of the measured lines were similar, i.e., first increased in the functional regions and then decreased in the outer regions. In addition, the more approximate deformations would be observed in the schemes with more diffusion elements, i.e., more side counts. Hence, the more approximate temperature distributions were observed on the related azimuths with fewer side counts, due to the reduced central angles, which were in accordance with those shown in Figure 5 and validated in the electromagnetic field [14]. For the thermal diffusions in the outer spaces, the standard deviations decreased as the radii increased, due to the natural characteristics of the diffusions inside the isotropic medium of the outer spaces with thermal losses. Furthermore, the thermal fluxes were able to transfer efficiently, which could be orthogonal to the pre-designed boundaries of the inscribed polygons in the outer domains, once the media inside the outer spaces were compiled following certain conditions.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Structural Parameters (m) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Scheme | Triangle | Square | Pentagon | Hexagon | Bare plate | ||||||||

Outer radii | 0.1 | 0.1 | 0.1 | 0.1 | / | ||||||||

Inner radii | 0.02 | 0.02 | 0.02 | 0.02 | / | ||||||||

Background | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | ||||||||

Employed mediums for each proposed scheme | |||||||||||||

Type | I | II | Center and background | ||||||||||

Triangle | 10% copper, 40.54% die-casting aluminum A380, 49.46% PDMS | 45.05% die-casting aluminum A380, 54.95% PDMS | Nickel Steel (50% Ni) | ||||||||||

Square | 10% copper, 41.78% ductile iron, 48.22% PDMS | 49.42% ductile iron, 50.58% PDMS | Nickel Steel (50% Ni) | ||||||||||

Pentagon | 10% copper, 42.70% steel SAE, 47.30% PDMS | 50.83% steel SAE, 49.17% PDMS | Nickel Steel (50% Ni) | ||||||||||

Hexagon | 10% aluminum alloy (6063), 39.71% steel SAE, 50.29% PDMS | 44.12% steel SAE, 55.88% PDMS | Nickel Steel (50% Ni) | ||||||||||

Bare plate | Nickel Steel (50% Ni) | ||||||||||||

Material parameters | |||||||||||||

Material | Copper | Aluminum alloy (6063) | PDMS | Nickel steel (50% Ni) | Aluminum A380 | Ductile iron | Steel SAE | ||||||

Conductivity (W∙m^{−1}∙K^{−1}) | 398 | 201 | 0.15 | 19.6 | 96.2 | 75 | 59 | ||||||

Density (kg∙m^{−3}) | 8930 | 2690 | 1000 | 8260 | 2710 | 7100 | 7830 | ||||||

Specific heat (J∙kg^{−1}∙K^{−1}) | 386 | 900 | 1450 | 460 | 963 | 450 | 460 | ||||||

Boundary conditions for all the schemes | |||||||||||||

Left wall | Right wall | Top wall | Bottom wall | Ambient | Central point source | ||||||||

Thermal isolation | Thermal isolation | Thermal isolation | Thermal isolation | 293 K | 373 K |

## Appendix B

#### Appendix B.1. Comparison between the Proposed Methodology and Well-Established Methods

**Figure A1.**Comparison of the outside isothermal profile distributions among the ideal values of the assumed EMA in the functional regions, simulated values of the classically alternative layer configuration schemes without the reserved 10% area portions and the observed values of the proposed schemes. (

**a**) The triangle scheme; (

**b**) the square scheme; (

**c**) the pentagon scheme; and (

**d**) the hexagon scheme.

#### Appendix B.2. Independence Analysis of the Presented Schemes

Illusive Schemes (Figure 6) | Grid Number | ||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

Triangle | 39,214 | 48,961 | 59,865 | 69,023 | 78,165 |

Square | 46,650 | 57,732 | 67,836 | 76,962 | 86,142 |

Pentagon | 54,685 | 65,985 | 74,986 | 86,316 | 97,645 |

Hexagon | 62,947 | 73,546 | 82,643 | 91,687 | 102,567 |

**Figure A2.**Measured temperatures of the proposed schemes with varied meshes: (

**a**) the triangle scheme; (

**b**) the square scheme; (

**c**) the pentagon scheme; and (

**d**) the hexagon scheme.

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**Figure 1.**Schematic of the illusive performance and the design of the source illusion device. The blue lines in (

**a**,

**b**) denote the general heat flux. (

**a**) Original thermal fields; (

**b**) illusive thermal fields; (

**c**) transformation process for the proposed source illusion device; and (

**d**) enlarged view of the functional elements labelled by Types I and II.

**Figure 2.**(

**a**) The thermal conductivities of the positive layers B; and (

**b**) the related fractions of the positive layers B.

**Figure 3.**Total contents of the contents of the potentially positive candidates for the proposed schemes. (

**a**) Triangle scheme; (

**b**) square scheme; (

**c**) pentagon scheme; and (

**d**) hexagon scheme.

**Figure 4.**Geometrical models for the proposed thermal source illusive schemes with varying inscribed polygons. (

**a**) The triangle scheme; (

**b**) the square scheme; (

**c**) the pentagon scheme; and (

**d**) the hexagon scheme (see the section in the Appendix for the summary for the employed mediums and related fractions).

**Figure 5.**Distribution characteristics of the thermal fields for the contrast bare plate scheme. (

**a**) Temperature distributions of the entire bare plate scheme; (

**b**) distribution characteristics on the isothermal line of T = 302 K; and (

**c**) temperature distributions on the measured line of r = 0.1 m.

**Figure 6.**Temperature distributions for proposed schemes at t = 1000 s. (

**a**) The triangle scheme; (

**b**) the square scheme; (

**c**) the pentagon scheme; and (

**d**) the hexagon scheme.

**Figure 7.**Distribution characteristics on the isothermal lines of T = 302 K for proposed schemes. (

**a**) The triangle scheme; (

**b**) the square scheme; (

**c**) the pentagon scheme; and (

**d**) the hexagon scheme.

**Figure 8.**Distribution characteristics on the measured lines of r = 0.1 m for proposed schemes. (

**a**) The triangle scheme; (

**b**) the square scheme; (

**c**) the pentagon scheme; and (

**d**) the hexagon scheme.

**Figure 9.**Standard deviations of the temperature deformations for the proposed schemes as a function of the radii of the additional measured lines.

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## Share and Cite

**MDPI and ACS Style**

Xu, G.; Zhang, H.; Jin, L.; Jin, Y.
Directed Thermal Diffusions through Metamaterial Source Illusion with Homogeneous Natural Media. *Materials* **2018**, *11*, 629.
https://doi.org/10.3390/ma11040629

**AMA Style**

Xu G, Zhang H, Jin L, Jin Y.
Directed Thermal Diffusions through Metamaterial Source Illusion with Homogeneous Natural Media. *Materials*. 2018; 11(4):629.
https://doi.org/10.3390/ma11040629

**Chicago/Turabian Style**

Xu, Guoqiang, Haochun Zhang, Liang Jin, and Yan Jin.
2018. "Directed Thermal Diffusions through Metamaterial Source Illusion with Homogeneous Natural Media" *Materials* 11, no. 4: 629.
https://doi.org/10.3390/ma11040629