Concurrent guiding of light and heat by transformation optics and transformation thermodynamics via soft matter

Controlling light and heat via metamaterials has presented interesting technological applications using transformation optics (TO) and transformation thermodynamics (TT). However, such devices are commonly mono-physics and mono-purpose, because the used metamaterial is designed to deal with one type of physical mechanisms. Here we demonstrate, for the first time, how to connect TO and TT via the liquid crystal 4-Cyano-4’-pentylbiphenyl (5CB) and, to exemplify such link, we present a multiphysics, multi-purpose device that simultaneously controls light and heat using such material. The anisotropic multiphysics properties of 5CB bond TO and TT, expanding the usage of these theories. The device, composed by 5CB confined between two right circular concentric cylinders, concentrates light (as a converging lens) and simultaneously repels heat from the inner cylinder when the molecules are along the direction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{{\boldsymbol{\rho }}}$$\end{document}ρˆ and it disperses light (as a diverging lens) and concurrently concentrates heat to the inner cylinder, without disturbing the external temperature field, when the molecules are along the direction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{{\boldsymbol{\theta }}}$$\end{document}θˆ, contributing for saving materials and designing miniaturized multiphysics systems.

Liquid crystals. Liquid crystals are materials composed by anisotropic molecules that has stable phases between the isotropic liquid and the solid crystal 18,19 and they have good application for display 26 and non-display usages 2,27-29 of these phases is the nematic one, where the molecules (rod-like, in this work) are randomly positioned in the space, but they have an average molecular orientation along a local molecular field n called director. One possibility on disturbing such molecular field is the presence of topological defects of the nematic phase, that, in fact, they can be present in different physical systems [30][31][32] . Depending on the direction of n, one can enhance the anisotropy of different macroscopic properties: thermal 24,27,28 , optics 2,33-35 , acoustics 19,36 , etc. Many anisotropic properties of the nematic phase of liquid crystals are related to the local molecular director n 18,36,37 . For a given n, the components λ ij of the thermal conductivity tensor are given, in Cartesian coordinates, by 37 ij iso ij a i j ij being λ = λ λ + ⊥ iso 2 3 , λ λ λ = − ⊥ a and the parallel and perpendicular molecular thermal conductivities are λ and λ ⊥ . It is reasonable to regard the temperature dependence of the molecular thermal conductivities. Thus, for the liquid crystal 5CB, we have 38 where λ 0 , λ 1 , λ 1, , α and α ⊥ are material constants and T NI and T C are, respectively, the nematic-to-isotropic temperature and the clearing-point temperature of the regarded liquid crystal. On the electromagnetic phenomena and disregarding absorption, whether such absorption is weak for 5CB 39 or it doesn't rule the deflection of light, the components of the permittivity tensor for the nematic phase of liquid crystals are given by 37 and the parallel and perpendicular molecular permittivity constants are ε and ε ⊥ . For simplicity and based on the comparison between our obtained results shown in the next section and the published results using ray optics 5 , we regard μ μ δ iso ij a i j ij 0 for the components of the permeability tensor. For the liquid crystal 5CB, the temperature dependences of the permittivity tensor can be related to the ones of the molecular refractive indexes by 40 c 0 where A, B and γ are material constants, (Δn) 0 is the birefringence when in crystal phase and T C is the clearing-point temperature of the regarded liquid crystal. Due to the birefringence, the refractive index of the extraordinary ray of the group velocity of an electromagnetic wave inside the nematic phase is given by 41

Results and Discussion
Connection between TO and TT. Differently of solid metamaterials, that are designed to work for a specified type of energy, nematic liquid crystals are anisotropic materials for simultaneous types of energy. This happens because the similarity between the thermal conductivity tensor, Eq. (2), and the permittivity tensor, Eq. (5), producing the same effective metric tensor by Eq. (1), resulting in Such metric tensor, that TO and TT require do modify the material properties, can be achieved by a unique molecular director configuration → n r ( ) of nematic liquid crystals, connecting both theories. For example, the molecular directors ρ =n and θ =n in cylindrical coordinates arise the following metric tensor 24 : . We substitute the metric (10) in (9) to obtain the components of the material tensors or the components of the molecular field n, where the thermal conductivity tensor is explicitly The thermal conductivities λ ρρ and λ θθ are, respectively, the ones of the radial and polar directions.
Observe that, from now on, any nematic liquid crystal described by Eq. (9) can compose devices that are naturally multiphysics ones. To give only one example of connection between TT and TO, we present in the next subsection a device that displays a simultaneous omnidirectional controlling of the propagation of heat and electromagnetic waves using the nematic liquid crystal 5CB. To highlight such connection, we regard the temperature dependence of the molecular thermal conductivities using the Eqs (3) and (4) and the refractive indexes by Eqs (6) and (7).
Simulations. For the proposed device, the Laplace and Fourier equations for the propagation of heat and the Maxwell equations for the propagation of electromagnetic waves will be solved using a finite element software (COMSOL Multiphysics). The artifact is composed of concentric cylinders, with the 5CB liquid crystal 38,40,42 filling the region between them and the same material (here it was water, where we are disregarding absorption) filling the regions in the inner cylinder and out the outer one, as in Fig. 1. The structure of concentric cylinders keeping liquid crystal has been experimentally known for a long time 43,44 and studied exhaustively [45][46][47] . This device was analyzed with the molecular director n in two different directions: in the radial direction ρ =n and in the azimuthal direction θ =n , with such directors switched by, for example, an external electric field via the Freedericks transition 18,37 . One can prepare these directors, for instance 48 , coating the exterior surface of the inner cylinder with indium tin oxide (ITO), setting a homeotropic strong anchoring energy, and the interior surface of the outer cylinder with Polymethylmethacrylate, producing a weak anchoring energy. Because the device has symmetry for each plane orthogonal to the axis of the concentric cylinders, we model cross-section views of these cylinders in a right rectangular prism. Such model consists on a rectangle with dimensions 200 × 120 μm containing on its center two concentric circles with radii 42.25 μm and 20 μm. These geometric dimensions can avoid convection in the liquid crystal for the applied temperature gradient, as can be seen in details in 37 . The region between the circles is filled with the 5CB liquid crystal, while water fills the region in the inner cylinder and between the outer cylinder and the rectangle. For the rectangle, we set fixed temperatures at 296 K and 306 K on opposite 120 μm sides, while we define insulating walls for the other ones. We use a finite element software to simulate the propagation of heat and radio-frequency (RF) electromagnetic wave in such device, with both molecular thermal -Eqs (3) and (4) -and dielectric -Eqs (6) and (7) -properties depending on the temperature.
The thermal case. A better understand of the action of the device on the thermal phenomena is acquired determining the isothermal surfaces (solving the Laplace equation) and the heat flux vectors (solving the Fourier equation) for the two work modes of our controller. We present a reference case with water everywhere in Fig. 2 Fig. 3. In Fig. 3(a), we have the molecular director θ =n and the isothermal surfaces between the cylinders are deformed to increase the temperature gradient inside the inner cylinder. Another important aspect on this Fig. 3(a) is that the temperature field outside of the device is not deformed, meaning that an outside observer measuring only its local temperature field is unable to detect the presence of the device, characterizing a cloaked thermal concentrator. In Fig. 3(b), we have the director ρ =n and the isothermal surfaces between the cylinders and outside of the controller are deformed to decrease the temperature gradient inside the inner cylinder, representing a thermal repeller. One can comprehend the thermal concentration and repelling of the controller from the ideas of transformation thermodynamics 3,49 . Among the needed conditions that the conductivity tensor must satisfy to produce a cloaked (i.e., without disturbing the exterior temperature field) concentrator and a cloaked repeller, we have, respectively, the following ones: ρρ θθ (13) The Eq. (11) shows that the condition (12) for the concentrator is met in our device when θ =n , while the condition (13) for the repeller is not met in our device when ρ =n . In the works 3,27,50 , the thermal conductivities were considered constants. In our simulations, we have used the temperature nonlinear equations for the molecular thermal conductivities (12) and (4) and we found similar qualitative results. Thus, Eqs (12) and (13) are approximate conditions for the cloaked effects, justifying the distortions in Figs 3 and 4, with the real ones can be found in 3 . This means that the controller concentrates or repels the heat flux even when the thermal conductivities depend on the temperature. The action of the controller on the temperature field was calculated and it is shown in Fig. 4. We observe the concentration of heat through the inner cylinder in Fig. 4(a) without disturbing the external temperature field, and in Fig. 4(b) we observe the repelling of heat from the inner cylinder, perturbing the external temperature. Again, we have results similar to the ones in 3,27,50 . Having the value of the temperature for each spatial position, we link the action of this device on the thermal propagation to the propagation of RF wave by temperature dependent refractive indexes, Eqs (6) and (7).
The electromagnetic case. How the controller regulates the propagation of the RF waves due to the temperature dependence of the material properties is a critical matter on the designing of the device. And since the molecular director n determines the refractive index N g by Eq. (8) and, consequently, the dielectric tensor ε ↔ , one can also change at will the propagation of the RF waves by changing the orientation of n applying an electric and/or magnetic field. Solving the electromagnetic wave equation for ρ =n and θ =n , we found the intensity of the electric field. For As a reference case of the simulation, we have Fig. 5 with water everywhere. Since the planar wave comes from all the left boundary, we observe a scattering of the wave and an attenuation of the electric field intensity along the propagation of the wave, where we also have the electromagnetic absorption from the water.
In Fig. 6, the director is ρ =n and we noticed a concentration of RF waves after its passage by the device. For Fig. 6(a), the perfect absorbers horizontal wall creates two symmetrical "paths" of high intensity light, while in Fig. 6(b) such two regions appear after the passage of the light through the device. The creation of these regions of high intensity of light has two equivalent interpretation: by Fermat principle 26,51 or by the effective curved space where the liquid crystal lies 2,5,33 . On the latter geometrical approach, the light path represents a geodesic of such . Axial transversal view of the temperature field created by the liquid-crystalline controller. In (a), the molecular director n between the cylinders is in the azimuthal direction θˆ. One notices a high gradient of temperature is observed inside the inner cylinder, not disturbing the external thermal field. In (b), with the molecular director ρ =n between the cylinders, we have a low gradient of temperature inside the inner cylinder, disturbing the external thermal field. Colors represent temperatures in kelvins.
Scientific REPORTS | (2018) 8:11453 | DOI:10.1038/s41598-018-29866-w effective space, with a distorted trajectory in the region with the liquid crystal. Thus, the light in the liquid crystal describes paths between two spatial points A and B that minimizes the integral where l is a parameter along the path and g ij are the components of the metric tensor 23 . Although this description is more suitable for ray optics 52 , our wave optics simulations, with the simplification μ ij ≈ Diag(μ 0 ), produced results similar to those in 5 , where the ray optics was used-being the approximation μ ij ≈ Diag(μ 0 ) reasonable 41and the temperature dependence for the molecular refractive indexes were not considered. Therefore, this indicates the device sustains its purpose even under such thermal link. The absence of the two regions of high intensity light before the device for the perfect conductor horizontal wall in Fig. 6(b) is an indication of destructive interference phenomena due to the reflection of the waves. Thus, when the molecular director is ρ =n , the device behaves like a converging lens, concentrating light that emerges from it.
In Fig. 7, the director is θ =n and we noted a relevant reducing of the emerging RF waves from the device for both kinds of horizontal walls, when comparing to Fig. 5. Again this event comes from the deflection of light by the liquid crystal between the cylinders with molecular director θ =n . Such deflections can be interpreted, once more, by the effective curvature felt by light 2,52 . The decreasing on the intensity of the wave is according to results published in 5 . As a last comment, the annular regions around the boundaries of the concentric cylinders are consequence of the spatial meshing of the finite element simulation. Thus, when the molecular director is θ =n , the device behaves like a diverging lens, dispersing light that emerges from it.
Observe that, when ρ =n , heat is repelled from the inner cylinder and the RF waves is concentrated after emerging from the device. The opposite situation takes place when θ =n , where heat is concentrated on the inner cylinder, without perturbing the exterior temperature field, and RF waves is dispersed after emerging from the device. Such opposite pictures come from the different physics equations that were solved: diffusion equation for heat and wave equation for light.

Summary and Conclusions
In this article, we studied how coexisting heat flux and electromagnetic waves are controlled by liquid crystal in its nematic phase, confined between concentric cylinders. Such thermal coupling with the propagation of the electromagnetic wave was made through the temperature dependence of the liquid crystal refractive indexes n and n ⊥ , using Eqs (6) and (7) 40 . For the molecular field ρ =n , the controller repels heat from the inner cylinder and it focuses the electromagnetic radiation to two strips of very high intensity emerging from the device. For the molecular field ρ =n , the controller concentrates heat to the inner cylinder without perturbing the exterior temperature field and decreases the intensity of the emerging electromagnetic radiation from the device. This is due to the fact that we are dealing with two energy transport regimes: thermal diffusion and wave propagation. An approach to understand such results is that the molecular fields ρ =n and θ =n creates an effective curved space between the concentric cylinders, disturbing the propagation of heat and light in such region 5,24,27,33 . Although we considered the temperature dependence on the refractive indexes 40 and on the molecular thermal conductivities 38 , our reported results are similar to the ones without such considerations 5,27 . This indicates a kind of thermal robustness of the proposed multiphysics controller and it can be a manifestation of the topological invariance of the physical properties of the liquid crystal phase. We say "topological invariance" because the temperature dependencies does not change the topological charge of the directors 41 ρ =n and θ =n , keeping consequently 48 the working modes of our proposed controller unchanged.
Controlling the amount of light and heat in a given region presents good possibilities for technological applications, for example, the energy saving by reuse of the heat produced secondarily by main physical systems, for use in other systems of interest. Another advantage of the proposed multiphysics controller is the spatial miniaturization on the development of energy manipulation projects, since a single artifact will act as a driver for different types of energy (light and heat). . The upper and lower walls are perfect absorbers and perfect conductors, respectively, in the pictures (a) and (b). Colors represent the intensity of the electric field in volts per meter.