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Abstract
To extend the metamaterial applications on simultaneously regulating multiple fields with transformation optics, we propose a class of manipulative cell here to manipulate thermal and DC fields simultaneously in non-conformal angular schemes. Significant behaviors of thermal cloaking, electrical concentration, and related switched functions are numerically demonstrated with appropriate media. The findings not only present an efficient method for simultaneously manipulating various energy, but also break the limitation of structural profiles in the designs of bi-functional meta-device. Moreover, it may also provide references for efficient energy manipulation and management in conventional energy techniques.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Owing to the theory of transformation optics (TO) [1] and related metamaterials, a new class of manipulative function without perturbing the surrounding ambient has been introduced into varied energy fields. The core of TO is to map material properties onto tailored spatial transformations under the premise of form invariant of corresponding governing function. In recent years, the research focus of electromagnetic metamaterials gradually changed from fundamental works to potential applications. For example, a class of tunable wideband microwave filters have been proposed based on ferrite metamaterials [2]. Upon the extensive studies of ferrite metamaterials, the random materials with negative electromagnetic parameters have been further fabricated by tailoring microstructures [3]. Based on these fundamental works, a class of paper-based metasurfaces [4] have been demonstrated accordingly, which would offer both economic and energy benefits on electronic printing. Owing to the validation of invariant governing function after spatial transformations, such manipulative method has been extended to a wide range of areas, including optics [5], acoustics [6], elastics [7], static magnetics [8], DC electric current [9], mass wave [10], thermodynamics [11], mass diffusion [12], and hydromechanics [13]. Since thermal energy is the basis and driving force for some industrial applications, the manipulation of thermal field based on metamaterial has aroused considerable research interests. The first attempt of transformation optics in thermal field is to theoretically propose a class of thermal cloak [11]. Further investigations reveal that such cloaking behavior can be obtained by engineering thermal materials [14], and a motivated method of scattering and cancel [15] is proposed to design the ultrathin cloak [16] and bilayer cloak [17]. With the extensive research on specific transformations, varied kinds of thermal meta-devices, including the typical harvester [18], rotator [11], and illusion [19], have also been demonstrated.
Such manipulative method is also employed to DC field, and both of the cloaking [20,21] and concentrating [22] behaviors are also exhibited in DC field. Furthermore, a potential bond of simultaneously manipulating these fields is presented. Among these novel designs, a class of bi-functional cloaks [23] are proposed for thermal and DC fields. Based on the cloaking design, the invisible sensor [24] for Laplace fields is also demonstrated. Besides, the strategy of simultaneously concentrating thermal flux and DC current [25] extends the range of bi-functional meta-devices. Considering the transformed characteristics of cloaking and concentrating behaviors, a series of bi-functional metamaterials [26,27] are demonstrated to respectively realize different functions in Laplace fields. However, current works remain one non-negligible challenge. That is, the conformal and continuous (circle or cylinder) profiles restrict the potentials in practical applications with angular system. The conventional transformations employed in cylinder system cannot satisfy demands in the non-singular and irregular system, and the method of scattering and cancel extremely depends on the structural profiles [28]. Besides, the attempt of achieving bi-functional devices in angular system has not been reported in Laplace field yet, as the specific thermal and electrical conductivities should be simultaneously considered and matched on the spatial transformations only in one non-conformal domain. Considering the progress of electromagnetic metamaterials [2,3,4], the potential applications of thermal and direct current metamaterials are expectable, as the tunability of field manipulations [2] in arbitrary domains would provide varied impressive and unexplored behaviors.
In this paper, a class of bi-functional devices with independent metamaterial cells is proposed based on linear map [29,30,31,32] to simultaneously and independently achieve the thermal cloaking and current concentration in angular structures. Detailed analysis of the appropriate properties is operated to determine the satisfied thermal and electrical conductivities in a specific region. Significant manipulative behaviors are respectively observed in thermal and electrical fields in the validated schemes. The findings open up an avenue for simultaneously and independently manipulating thermal and DC fields without the limitation of structural profiles. Considering the practically complicated demands of structural properties and ambient parameters in potential applications, the findings provide the appropriate method of designing meta-devices with arbitrary profiles to cope with the varied potential demands. Besides, the simultaneously and independently tunable functions for thermal and electrical fields in arbitrary domains further enriching the previous researches on functional switch with simple profiles. Hence, both the above advantages present great potentials on pushing meta-devices to potential applications, and motivating the excited energy techniques to satisfy varied functional demands under the changing external environments.
2. Theoretical design for the bi-functional device with non-continuous profile
For a non-conformal angular system, two aspects are the priorities. The first is to create target regions for multiple functions. The second is to determine the appropriate media, which can be simultaneously satisfied the demands of varied fields. For generality, polygonal schemes are selected as the non-continuous angular profiles, while the inner regions are rotated to the outer regions to build non-conformality. The transformation processes are shown in Fig. 1, which can be obtained with (rotatory) linear maps [29,30,31].
Fig. 1. General transformations of the target manipulations and medium arrangements for each independent cell. Among these, (a) – (f) are the transformation processes of the manipulations. (a) and (b) denote the conformal structures in the original domain, and the first step of internal rotation for the cloaking behavior; (c) and (d) are the first step of rotation for the harvesting performance. Both of the rotations shown in (b) and (d) would be further transformed into the one presented in (e) through second steps of regional expansion and compression. (f) is the enlarged view of the functional regions after the final transformations; In addition, (g) – (i) denote the medium arrangements of the functional regions shown in (e) and (f). (g) Schematic cell for simultaneously and independently concentrating electric current and cloaking thermal flux; (h) Independent cell employed in Types I/III, in which the extension cords of layers AD and BC intersect at one vertex of the internal region; (i) Independent cell employed in Types II/IV, in which the extension cords of layers AD and BC intersect at one vertex of the external region.
Based on the findings of designing single-function devices, the employment of alternative layered media with passive and positive conductive parameters is the favorite means to satisfy the conductive demands. However, the method is not feasible in achieving the conductivities for each field of bi-functional schemes, as the satisfied parameters for different fields are difficult to match in one selected medium. To achieve bi-functional manipulations with appropriate media in non-conformal angular schemes, the typical example of simultaneously concentrating electric current and cloaking thermal flux is selected. The schematic of general medium arrangement is also shown in Fig. 1. Figure 1(a) presents the regional configurations of satisfied media. The medium domains divided by dash lines are alternatively filled with electrically positive and inactive media to achieve the current concentration, while the parts separated by the dot lines are alternatively composited with thermally positive and passive media to realize the heat flux avoidance. Obviously, the medium configurations are arranged as continuous lattices (divided by the dash and dot lines). Besides, each medium employed in one functional lattice should be equipped with multiple conductivities for varied fields. Here, four adjacent lattices with one connected node are defined as one independent cell.
The medium configurations in Types III/I of one independent cell are demonstrated in Fig. 1(b). The height of medium A (B) is set as Δl1, and the total height for the entire cell is ΔlIII(I). The distance between the initial point (one vertex of the inner region) and the bottom of the selected cell is r5. Here, we make ${\textrm{m}_{\textrm{C},\textrm{III}(\textrm{I} )}} = \left( {{r_{3(2 )}}\cdot \cos\; \frac{{({2n - 1} )\pi }}{N} - {r_{4(1 )}}} \right)/\Delta {l_{IV({II} )}}$ to simplify the configurations. Note that, mC,III(I) is an integer i.e., the entire height of Type III/I is an integral multiple of ΔlIII(I). In addition, r5 is also an integral multiple of ΔlIII(I) ranging from 0 to ${\textrm{m}_{\textrm{C},\textrm{III}(\textrm{I} )}} - 1$. For the azimuthal ranges, the azimuthal angle for composited layer AD is defined as α1, and the entire azimuthal angle for the cell is α2. The skew angle of the cell to the central line of Type III/I is α3. Similar to the height setups, the entire azimuthal angle of Type III/I can be expressed as: $2\cdot \textrm{arccot}\left( {\textrm{cot}\; \frac{{({2n - 1} )\pi }}{N} - {r_{4(1 )}}/\left( {{r_{3(2 )}}\cdot \textrm{sin}\; \frac{{({2n - 1} )\pi }}{N}} \right)} \right)$, which is equal to α2 · mP,III(I), here mP,III(I) is also an integer. The skew angle α3 is also an integral multiple of α2, which ranges from 0 to ${\textrm{m}_{\textrm{P},\textrm{III}(\textrm{I} )}} - 1$. Based on the effective medium theory, the perpendicular and parallel electrical conductivities presented in Fig. 1(b) can be observed.
The medium configurations in one independent cell of Types IV/II are demonstrated in Fig. 1(c). the total height of the independent cell is ΔlIV(II), and the height of medium A (B) is Δl2. The distance between the initial point (one vertex of the outer region) and the top of the independent cell is set as r6. The value of total height, i.e., $\left( {{r_{3(2 )}} - {r_{4(1 )}}\cdot \cos \; \frac{{({2n - 1} )\pi }}{N}} \right)/\Delta {l_{IV({II} )}}$, is equal to mC,IV(II) · ΔlIV(II), in which mC,IV(I) is also an integer, and r6 is an integral multiple of ΔlIV(II) ranging from 0 to ${\textrm{m}_{\textrm{C},\textrm{II}({\textrm{IV}} )}} - 1$. For the azimuthal components, α4, α5, and α6 respectively denote the azimuthal angles for layer AD, the entire cell, and the skew angle to the central line. It’s noted that ${\textrm{m}_{\textrm{P},\textrm{IV}({\textrm{II}} )}}\; {\alpha _5} = 2\cdot \textrm{arccot}\left( {\textrm{cot}\; \frac{{({2n - 1} )\pi }}{N} - {r_{4(1 )}}/\left( {{r_{3(2 )}}\cdot \textrm{sin}\; \frac{{({2n - 1} )\pi }}{N}} \right)} \right) - \frac{{2\pi }}{N}$, and mP,IV(II) is also an integer. That is, the entire azimuthal angle of Type IV/II is an integral multiple of α5. In addition, the skew angle α6 is also an integral multiple of α5, which is ranging from 0 to ${\textrm{m}_{\textrm{P},\textrm{IV}({\textrm{II}} )}} - 1$. Based on the above definitions and effective medium theory, conductive components inside Type IV/II for DC and thermal fields can be expressed as follows:
Table 1. Satisfied electrical and thermal conductivities of media B, C, and D for hexagonal scheme
Table 2. Satisfied electrical and thermal conductivities of media B, C, and D for heptagonal scheme
The next step is to search the most approximate media for the independent cells. Owing to the approximate conductivities of the functional media in each part, only four kinds of natural media are employed in the independent cells for each proposed scheme. For the hexagonal scheme, ASTM A1011 carbon steel with an electric conductivity of 0.704 MS/m and a thermal conductivity of 93 W/m·K is selected as medium A, due to its best fitting conductivities to the calculated ones. The electrical insulation of polydimethylsiloxane (PDMS) with a thermal conductivity of 0.15 W/m·K is employed as medium C, as the calculated electrical conductivities were several orders of magnitude lower than the background medium. As the typical electrical insulation and thermal conduction medium, COORSTEK hot pressed aluminum nitride (AlN) with a thermal conductivity of 81 W/m·K is employed as medium B. Considering the positive and passive conductions in DC and thermal fields, silver epoxy adhesive (AA-DUCT 916) with an electric conductivity of 1 MS/m and a thermal conductivity of 1.5 W/m·K is selected as medium D. Furthermore, the abovementioned media are also employed in the heptagonal scheme, as their approximately conductive parameters to the theoretical ones indicated in Table 2. The physical models are presented in Fig. 2.
Fig. 2. Geometry simples of the proposed schemes. (a) Hexagonal scheme; (b) Heptagonal scheme; (c) and (d) are the enlarged view of functional regions I (III) and II (IV).
Numerical simulations based on the finite element method are employed with COMSOL Multiphysics 5.3a to demonstrate the bi-functional behaviors. For the general setups, the left and right sides are respectively set as high and low temperature (normalized potential) boundaries, i.e., 373 K (1 V) and 293 K (0 V). In addition, the top and bottom boundaries are electrically and thermally insulated.
3. Results and discussion
According to the above derivations, both of the electrical and thermal distributions for each proposed scheme are presented in Fig. 3. The contrast bare plate is also examined to implement a fair comparison. Obviously, all the proposed schemes significantly and simultaneously performed tailored behaviors compared with those pure carbon fiber plates. Here, we take the hexagonal scheme shown in Figs. 3(b) to illustrate the bi-functional performances. Both of the isopotential and isothermal lines in the background distributed smoothly with few perturbations. As illustrated in Fig. 3(b1), DC currents inside the functional regions were greatly restored by the manipulative cells. That is, the surrounding currents were rapidly concentrated in Types I, due to the compressive arrangements of electrical conductions and insulations (layers AD and BC). Moreover, currents inside functional Types II were expanded onto the boundaries of inner regions. Hence, potentials of the symmetry points on the internal boundaries were uniform. With such current manipulations, DC current transferred along the parallel direction with enhanced potential gradients inside the inner regions. That is, electrical concentrating behavior was observed in such a device. The temperature distribution is presented in Fig. 3(b2).
Fig. 3. Field distributions of the bare plate and proposed schemes. Among the subgraphs, (a1) – (c1) illustrate the DC fields, and (a2) – (c2) present the thermal fields. (a) Contrast bare plate; (b) Hexagonal scheme; (c) Heptagonal scheme. (The field distributions under arbitrary field flow directions can be found in the Visualization 1, Visualization 2, Visualization 3, Visualization 4)
It’s seen that the isothermal lines bypassed the surroundings of the functional regions, and distributed smoothly in the background with few perturbations. Furthermore, isothermal lines inside types III (I) were parallel to the outer boundaries, which were contributed by the alternative arrangements of layer AB and CD. Similar to the manipulative effects of types III (I), isothermal lines inside types IV (II) were parallel to the inner boundaries of function region, as the thermal flux were spread and homogenized by the pre-designed expansive effects of types IV (II). Under the combined effects, the temperatures on the internal boundaries were uniform. It led to homogeneous thermal fields (no temperature gradients) inside the inner region. Consequently, significant thermal cloaking and electrical concentrating behaviors were simultaneously obtained. Owing to the similar configurations of manipulative cells, such bi-functions were also observed in the heptagonal scheme as shown in Fig. 3(c).
To further investigate the bi-behaviors, the symmetry lines (y = 0) were selected as shown in Fig. 4. For contrasts, the values of the bare plate scheme were also presented. Here, the hexagonal scheme is illustrated in detail. It can be seen that currents propagated smoothly in the background region, which led to the homogeneous potential distributions. Owing to the electrically active conductions in orthogonal directions, current flows smoothly transferred inside Types II with tiny loss. Under the above influences, currents were concentrating in the inner regions with enhanced potential gradients. Similar to the performances in DC field, thermal flux transferred smoothly in the background and contributed to the uniform thermal profiles. Moreover, temperature drops were also observed, once the thermal flux diffused into functional types. Different from those in electrical concentrating behavior, the changing trends were gradual and terraced, due to the thermally passive conductions in orthogonal directions. When the flux transferred to the inner regions, the temperature distributions approached stabilization. With the expansive effects of types IV (II), no temperature gradients were inside the inner regions. Hence, such configurations of the manipulative cells can simultaneously and independently behave electrical concentrating and thermal cloaking performances. Similar manipulations are also observed in the heptagonal scheme as illustrated in Fig. 4(b). Moreover, better bi-functional behaviors were observed in the heptagonal scheme due to the larger anisotropies.
Fig. 4. Temperature and current distributions along the central lines of y = 0 of the proposed schemes and related contrasts. (a) Hexagonal scheme; (b) Heptagonal scheme.
Furthermore, additional functions of independent electrical cloaking and thermal harvesting, can be also achieved by rotating each schematic cell. As shown in Fig. 5, directional functions can be adjusted with the rotation of the employed media inside each cell. That is, the inverse functions, i.e., thermal harvesting and electrical cloaking behaviors, can be obtained with the rotated schematic cells, as the thermally conductive layer AB and insulative layer DC are along the vertical direction in each subcell, and electrical conductive (AD) and insulative (CB) layers are along the parallel direction. Considering the entire effective parameter of each schematic cell, the thermal and electrical conductivities in the functional regions of the switched schemes can be also approximated matched on the ones of background. The field distributions of the switched schemes are shown in Figs. 5(b) and 5(c). It can be seen that significantly inverse functions of thermal harvesting and electrical cloaking are observed in the hexagonal and heptagonal schemes. Furthermore, few perturbations are also observed in the background domains. Hence, the switchable functions can be obtained with rotated schematic cells in the same proposed schemes. Furthermore, both of the original (Fig. 3) and switched (Fig. 5) schemes can demonstrate omnidirectional field manipulations in corresponding fields, whatever the directions of incoming field flows. That is, all the proposed devices are available to provide corresponding behaviors under arbitrary directions of incoming field flows. The detailed field distributions of the schemes under arbitrary field flow directions can be found in the Visualization 1–Visualization 4.
Fig. 5. Rotated schematic cell and field distributions of the switched functions. Among these, (a1) and (a2) respectively denote the original and rotated schematic cells; (b1) and (b2) are the thermal cloaking behaviors of the proposed schemes with rotated schematic cells; (c1) and (c2) present the electrical harvesting performances of the proposed schemes with rotated schematic cells.
In general, the appropriate conductivities for arbitrary meta-devices can be achieved with the proposed expressions. That is, arbitrary structural demands (including regular and complicated structures) in potential applications can be easily satisfied with the generally conductive components. Hence, the findings would provide references on flexibly manipulating fields in arbitrary domains. Moreover, the significant demonstrations of the field manipulations exhibit potentials on efficiently programming field distributions and motivating the excited energy systems, such as the energy management in solar cells, thermoelectric devices with complicated structures and hybrid energy transports and allocations.
4. Summary
In summary, a class of manipulative cells is proposed to simultaneously and independently manipulate currents and heat flux in non-conformal angular profiles. Typically simultaneous bi-behaviors of electrical concentrating and thermal cloaking are numerically demonstrated in hexagonal and heptagonal schemes. Such manipulative cell is also available to demonstrate other multiple bi-behaviors of current avoidance and heat flux concentration through adjusting the orders of functional media. Such proposed manipulative cells could open up an avenue for simultaneously and independently tailoring field conductions, achieving bi-energy manipulations, and flexible switching functions. Furthermore, the findings also provide references on designing multiple functionalized devices for more energy types.
Funding
China Scholarship Council (CSC No. 201806120169).
Disclosures
The authors declare no conflicts of interest.
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