Many-Particle Thermal Invisibility and Diode From ... · We confirm it in simulation and experiment. As an application, the concept of many-particle thermal invisibility helps us

Jin Shang1

Department of Physics,

State Key Laboratory of Surface Physics,

Key Laboratory of Micro and Nano

Photonic Structures (MOE),

Fudan University,

Shanghai 200433, China;

Collaborative Innovation Center

of Advanced Microstructures,

Nanjing 210093, China

Chaoran Jiang1

Department of Physics,

State Key Laboratory of Surface Physics,

Key Laboratory of Micro and Nano

Photonic Structures (MOE),

Fudan University,

Shanghai 200433, China;

Collaborative Innovation Center

of Advanced Microstructures,

Nanjing 210093, China

Liujun XuDepartment of Physics,

State Key Laboratory of Surface Physics,

Key Laboratory of Micro and Nano

Photonic Structures (MOE),

Fudan University,

Shanghai 200433, China;

Collaborative Innovation Center

of Advanced Microstructures,

Nanjing 210093, China

Jiping HuangDepartment of Physics,

State Key Laboratory of Surface Physics,

Key Laboratory of Micro and Nano

Photonic Structures (MOE),

Fudan University,

Shanghai 200433, China;

Collaborative Innovation Center

of Advanced Microstructures,

Nanjing 210093, China

e-mail: [email protected]

Many-Particle ThermalInvisibility and Diode FromEffective MediaInvisibility has recently been achieved in optics, electromagnetics, acoustics, thermotics,fluid mechanics, and quantum mechanics; it was realized through a properly designedcloak structure with unconventional (anisotropic, inhomogeneous, and singular) materialparameters, which limit practical applications. Here, we show, directly from the solutionof Laplace’s equation, that two or more conventional (isotropic, homogeneous, and non-singular) materials can be made thermally invisible by tailoring the many-particle local-field effects. Our many-particle thermal invisibility essentially serves as a new class ofinvisibility with a mechanism fundamentally differing from that of the prevailingcloaking-type invisibility. We confirm it in simulation and experiment. As an application,the concept of many-particle thermal invisibility helps us propose a class of many-particle thermal diodes: the diodes allow heat conduction from one direction with invisi-bility, but prohibit the heat conduction from the inverse direction with visibility. Thiswork reveals a different mechanism for thermal camouflage and thermal rectification byusing composites, and it also suggests that besides thermotics, many-particle local-fieldeffects can be a convenient and effective mechanism for achieving similar controls inother fields, e.g., optics, electromagnetics, acoustics, and fluid mechanics.[DOI: 10.1115/1.4039910]

1 Introduction

Invisibility means an object cannot be seen or detected, and it isalways a fascinating topic in fantasy or science fiction. Since thelast decade, it has genuinely entered the realm of scientificresearch [1,2]. So far, invisibility has attracted enormous researchinterests in optics [1,4–8], electromagnetics [2,9–14], acoustics[11,15–17,19], thermotics [18,20–29], fluid mechanics [30,31],and quantum mechanics (matter waves) [11,32,33]. In the dura-tion, cloak structures (say, designed with conformal mapping [1]or coordinate transformation [2]) dominate the existing invisibilityresearch [2,4–9,11–17,19,21–33], which are convenient for theo-retical settings. However, such cloaks for achieving invisibility

require unconventional material parameters that need to satisfyone or more of the following three properties: (i) anisotropy (e.g.,the parameter like a dielectric constant or thermal conductivity isa tensor, possessing different values of radial and tangentialcomponents), (ii) inhomogeneity (say, the parameter has a spatialgradation profile in the cloak region), and (iii) singularity (e.g.,the parameter should be zero or infinite at certain positions). Suchrequirements (i)–(iii) usually could not be satisfied by naturallyoccurring materials or chemical compounds, thus significantlylimiting the practical applications. To be free from or to compro-mise with (i)–(iii), researchers have proposed various schemes,see Refs. [1,3–9,11–17,19,21–33]. For example, anisotropy indesign invisibility could be overcome by conformal transforma-tion optics using isotropic inhomogeneous medium [1,3]. G€om€oryet al. [12] designed a special cylindrical superconductor-ferromagnetic bilayer that can exactly cloak uniform static mag-netic fields [12], hence yielding an exact magnetic cloak. Exactbilayer thermal cloaks have been similarly proposed [25–27].

1J. Shang and C. Jiang contributed equally to this work.Contributed by the Heat Transfer Division of ASME for publication in the

JOURNAL OF HEAT TRANSFER. Manuscript received October 14, 2017; final manuscriptreceived March 19, 2018; published online May 25, 2018. Assoc. Editor:Alan McGaughey.

Journal of Heat Transfer SEPTEMBER 2018, Vol. 140 / 092004-1Copyright VC 2018 by ASME

Nevertheless, even though such exact bilayer cloaks [12,25–27]are independent of condition (i) and (ii), they still require condi-tion (iii): the magnetic permeability [12] or thermal conductivity[25–27] of the inner layer must be zero. In general, most schemes[4–9,11–17,19,21–33] based on cloak structures are faced with thesame problem: they could not be free from all the (i)–(iii) due tothe underlying mechanisms, thus limiting applications. These lim-itations inspire us that it is necessary to resort to another mecha-nism. Accordingly, here we propose a many-particle systeminstead, which is different from the cloak structure. By choosingmany-particle local-field effects appropriately, we shall show thatour system exhibiting thermal invisibility is free from (i)–(iii)indeed, which satisfies isotropy (the thermal conductivity is a sca-lar), homogeneity (objects have a random distribution in theregion, yielding homogeneous equivalent thermal conductivitiesof the subregions everywhere), and nonsingularity (thermal con-ductivities have a finite nonzero value). Compared to the previousrandom medium method that needs complex algorithm optimiza-tion [34,35], this mechanism is easy to conduct. The concept ofmany-particle thermal invisibility further yields a class of many-particle thermal diodes with invisibility.

2 Many-Particle Thermal Invisibility

2.1 Two-Dimensional Case

2.1.1 Theory. For a circular particle with thermal conductivityj and radius R0 embedded in an infinite uniform matrix with jm,the solution of Laplace’s equation, r2T¼ 0, in polar coordinates(r, h) is

Tinnerðr; hÞ ¼ A0 þ B0lnðrÞ þX1

m¼1

½Am cosðmhÞ þ Bm sinðmhÞ�rm

þX1

m¼1

½Cm cosðmhÞ þ Dm sinðmhÞ�r�m

Touterðr; hÞ ¼ A00 þ B00lnðrÞ þX1

m¼1

½A0m cosðmhÞ þ B0m sinðmhÞ�rm

þX1

m¼1

½C0m cosðmhÞ þ D0m sinðmhÞ�r�m

where Tinner (r, h) and Touter (r, h) denote the distribution of tem-perature in the particle and matrix, respectively. Here, A0, B0, Am,Bm, Cm, Dm, A00; B00; A0m; B0m; C0m, and D0m are undetermined coeffi-cients. The accompanying boundary conditions are

Tinner r ¼ 0ð Þ <1

Touter r !1; hð Þ ! T0 þ c0 r cos hð Þ

Tinner r ¼ R0ð Þ ¼ Touter r ¼ R0ð Þ

jm@Touter

@r

����r¼R0

¼ j@Tinner

@r

����r¼R0

Here, T0 denotes the temperature at r¼ 0 and c0 is a constantdepending on the applied temperature gradient. Then, we obtain

Touter r; hð Þ ¼ T0 þ c0 r cos hð Þ � j� jm

jþ jmc0 R2

0

cos hð Þr

(1)

So, we are allowed to write the Clausius–Mossotti factor a as

a ¼ j� jm

jþ jm(2)

which describes the degree of thermal contrast between the circu-lar particle and matrix. Clearly, a¼ 0 corresponds to zero thermalcontrast, i.e., the particle and matrix share the same material.

Accordingly, a larger a indicates a larger contrast between theparticle and matrix.

To proceed, we consider a two-dimensional (2D) square sys-tem, which contains a central square area and an environmentoccupied by a material with thermal conductivity jm surroundingthe central square area. n kinds of circular particles, each withthermal conductivity ji and area fraction pi (i ¼ 1; 2; 3;…; n),occupy the whole central square area with a random distribution,thus yielding a many-particle system. In the presence of an exter-nal temperature gradient, if the existence of the central squarearea does not disturb the temperature distribution or heat flow inthe environment, the central square area is thermally invisible.For this purpose, we need to set the central square area to possessa special effective thermal conductivity that must be equal tojm. In this regard, what we need is to let the thermal contrasts(Clausius–Mossotti factor) of all the particles within the centralsquare area be canceled out. Thus, we obtain

hClausius�Mossotti factori2D ¼ 0 (3)

where h� � �i2D denotes the area average of � � � over the centralsquare area. Equation (2) owns at least two features. On the onehand, when the thermal conductivities of the particles, ji, satisfyEq. (2), the effective thermal conductivity of the central squarearea equals jm. This is because the many-particle local-fieldeffects lead to the overall disappearance of the thermal contrastsbetween all the particles and the environment. In other words, fora specific particle, when we apply Eq. (1) (for one particle in aninfinite uniform matrix) to Eq. (2) (for many particles in a randomdistribution), we have assumed the specific particle to be embed-ded in an effectively uniform matrix (which is composed of manyother particles). Accordingly, the Clausius–Mossotti relationEq. (1) can be applied to Eq. (2). On the other hand, if the centralsquare area is divided into plenty of subareas, each includingmany particles distributed randomly, the equivalent thermal con-ductivities of the subareas are still equal to jm according toEq. (2). That is, these subareas become homogeneous indeed.

Without loss of generality, we first consider the n¼ 2 case thatthe central square area includes two materials with j1 and j2.According to Eq. (2), we set j1> jm> j2. The material of j1

tends to expel isotherms and attract heat flux lines (Fig. 1(c))while the other material of j2 has an opposite effect (Fig. 1(d)).Thus, the distortions of heat flux lines contributed by the twomaterials can be canceled out at certain area fractions accordingto Eq. (2). Figure 1(a), together with Fig. 1(j), shows the result offinite element simulations based on the commercial software COM-

SOL MULTIPHYSICS. When compared with Fig. 1(b) for the case ofpure environmental material, the temperature distribution in theenvironment shown in Fig. 1(a) has almost not been disturbed bythe presence of the central square area containing two differentmaterials. Thus, the thermal invisibility is achieved (Fig. 1(a)). Ifwe extend the central square area from two materials (Fig. 1(a)) tothree materials (Figs. 2(a) and 2(b)) whose thermal conductivitiesare set to satisfy Eq. (2) as well, the same thermal invisibilitybehavior appears as well (see Fig. 2(a)). Similarly, more kinds ofmaterials (n> 2) can be added into the central square area, andsame thermal invisibility behaviors can always come to appear aslong as the thermal conductivities and area fractions of thesematerials satisfy Eq. (2).

2.1.2 Experiment. We further perform an experiment to con-firm the above-mentioned simulation results. The experimentalsetup is depicted in Fig. 1(e); detailed experimental methods canbe found in experimental details. We use an infrared camera todetect the distribution of temperature in the experimental sample.Figure 1(f), together with Fig. 1(j), depicts the case of the mixtureof coalesced polydimethylsiloxane and copper (T2) in an environ-ment of brass (H62), and it shows a satisfactory thermal invisibil-ity indeed, which echoes with the simulation result shown inFig. 1(a) (a quantitative comparison is shown in Fig. 1(j)). For

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Fig. 1 Two-dimensional results of (a)–(d) finite element simulations under the boundary condition of heat insulation and (f)–(i)experiments based on (e) the experimental setup (where HOT and COLD represent the hot and cold source, respectively): the colorsurface in (a)–(d), (f)–(i) denotes the distribution of temperature; the white lines in (a)–(d) represent the isothermal lines. (a) and (f)show a 20 cm 3 20 cm system (the experimental sample of (f) is depicted in the upper right corner of (e)), which owns a centralsquare area (6.7 cm 3 6.7 cm) containing the first material (as shadowed in the central square area of (a)) of thermal conductivity0.15 W/(m K) and area fraction 36.4% randomly embedded in the second material of 400 W/(m K) and 63.6%; the first (second) mate-rial can be seen as an assembly of a kind of circular particles with different sizes; outside the central square area is the environ-ment occupied by the material of jm 5 109 W/(m K). (b) and (g) show the case of the pure environmental material jm 5 109 W/(m K),and their central squares (6.7 cm 3 6.7 cm) denote only the position for the sake of comparison. (c) and (h) are same as (a) and (f),respectively, but the central square area only includes the material of 400 W/(m K). (d) and (i) are also same as (a) and (f), respec-tively, but the central square area only contains the material of 0.15 W/(m K). (j) shows the quantitative comparison of temperaturedistribution between simulation and experiment, for the structure of (a) and (f). The “I–V” represent the five positions as depictedin (e).

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understanding the experimental thermal invisibility (Fig. 1(f)), wealso plot more relevant experimental comparisons (seeFigs. 1(g)–1(i), which agree well with Figs. 1(b)–1(d) (simulationresults), respectively).

In view of the good agreement between our simulation(Figs. 1(a)–1(d)) and experiment (Figs. 1(f)–1(i)) (see alsoFig. 1(j)), now we may conclude that Eq. (2) is indeed useful forachieving many-particle thermal invisibility.

2.2 Three-Dimensional Case. Figures 1, 2(a), and 2(b)show only two-dimensional cases according to Eq. (2). For com-pleteness, we can obtain three-dimensional Clausius–Mossottifactor a as

a ¼ j� jm

jþ 2jm(4)

and three-dimensional invisibility can be similarly achieved using

hClausius�Mossotti factori3D ¼ 0 (5)

where h� � �i3D denotes the volume average of � � � over the wholecentral cubic volume. In Eq. (4), the Clausius–Mossotti factor isfor three dimensions. As a model demonstration, we perform finiteelement simulations for a three-dimensional case, (see Figs. 2(c)and 2(d)). The central cubic volume contains two different materi-als, whose thermal conductivities satisfy Eq. (2). Consequently, agood thermal invisibility behavior appears indeed (see Fig. 2(c)).

2.3 Discussion. A feature of our design is that heat canflow cross the invisible region, namely, the central square area(Figs. 1(a), 1(f), and 2(a)) or the central cubic volume (Fig. 2(c)).Although the isotherms and heat flux lines in the invisible region

Fig. 2 (a) Two-dimensional and (c) three-dimensional finite element simulation results of the(b) two-dimensional and (d) three-dimensional structure, respectively: the color surface in (a)and (c) represents the distribution of temperature; the direction and length of the arrows in (a)and(c) denote the pathway and strength of heat flux, respectively. In (a) and (c), the boundarycondition is also set at heat insulation. (a) depicts a 20 cm 3 20 cm system; its central squarearea (6.7 cm 3 6.7 cm) includes three different materials (randomly distributed with each other,as is shown in (b)) with thermal conductivities 0.15 W/(m K), 270.5 W/(m K) and 400 W/(m K) andarea fractions 33.3%, 33.3%, and 33.4%; each of the three materials is also occupied by a kindof circular particles with different sizes; outside the central square area is the environmentoccupied by the material of jm 5 109 W/(m K). (c) displays three orthogonal planes of a20 cm 3 20 cm 3 20 cm system; its central cubic volume (6.7 cm 3 6.7 cm 3 6.7 cm) contains twomaterials with thermal conductivities 0.15 W/(m K) and 155.6 W/(m K) and volume fractions 20%and 80% (as is shown in (d)), which are randomly embedded in each other; either of the twomaterials is also an assembly of a kind of spherical particles with different sizes; outside thecentral cubic volume is also the environment of jm 5 109 W/(m K).

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may be distorted due to the random distribution of materials, thetemperature field within the environment is not disturbed, thusbeing called invisibility. Note that the small distortion at the closeproximity of the invisible region is caused by the nonuniform dis-tribution of the circular or spherical particles, which can be over-come if one adopts particles as more as possible. Now we can saythat the interior localized distortions in such invisible regions can-cel out each other as a result of many-particle local-field effects,and then they do not affect the exterior temperature field in theenvironment. This is the reason why the thermal invisibility canbe achieved (Figs. 1(a), 1(f), 2(a), and 2(c)). So, similar invisiblefunctions can be realized via other designs, such as with periodicstructures and particles having different shapes/sizes. Previousworks about thermal cloaks (e.g., Ref. [36]) determine thermalconductivities by using coordinate transformation. In this work,the effective medium method is applied to approximately realizethe desired thermal conductivity because one cannot find it in

Fig. 3 Blinding sensors: two-dimensional finite elementsimulation results: (a) same as Fig. 1(a), but four small squares(sensors) are symmetrically added, each with 3 cm 3 3 cm, 0.026W/(m K), and a center-to-center separation of 6.7 cm between itand the central square area, (b) same as Fig. 1(b), but the samefour small squares as those in (a) are added, (c) and (d) aresame as (a) and (b), respectively, but the thermal conductivityof each small square has a conductivity of 500 W/(m K) instead.The distribution of temperatures in the four small squares in (a)(or (c)) is the same as that in (b) (or (d)); in this sense, a thermalcomposite (in (a) or (c)) and a pure environmental material (in(b) or (d)) inside the central square area cannot be distin-guished by the sensors located in the small squares, that is, themany-particle invisibility helps to hide this central square areafrom the sensors. Same results hold also for (e)–(h), which arethe same as (a)–(d), respectively, but showing the case of pointheat sources. In (e)–(h), the left/lower (right/upper) boundariesare set at heat insulation (a constant temperature).

Fig. 4 Many-particle thermal diode: simulation results. (a) sche-matic graph showing the diode structure, where the left part has acentral square area containing material A (jA 5 0.15 W/(m K), say,polydimethylsiloxane) randomly embedded in material B(jB 5 400 W/(m K), say, red copper) with area fraction 36.4%. Mate-rials C and D occupy the other areas with temperature-dependentthermal conductivities following the second expression of Eq. (3)in Ref. [29], jC 5 jl 1(js2jl )/f exp ½(T 2Tc)/1:0�11g and jD 5 jl 1(js2jl )/fexp ½2(T 2Tc)/1:0�11g with jl 5 0.026 W/(m K) (e.g., air),js 5 109 W/(m K) (e.g., brass) and Tc 5 298 K, which may be exper-imentally realized with the aid of shape-memory alloy accordingto the design depicted in Fig. 2 of Ref. [29]. (b) the distribution oftemperature (color surface) and isotherms (white lines) in thethermal diode for high flux JH 5 1.31 3 104 W/m2. (c) same as (b),but the positions of the heat source and cold source areexchanged, thus showing the case of low flux JL 5 3.88 3 102 W/m2.The corresponding rectification ratio (JH 2 JL)/(JH 1 JL) 5 94%.

Journal of Heat Transfer SEPTEMBER 2018, Vol. 140 / 092004-5

natural materials. Our work is different from the existingresearches since we theoretically design the device via the effec-tive medium method from the very beginning. In addition, it isalso worth mentioning that our thermal invisibility corresponds tothe fact that the hidden objects are able to feel the temperaturegradient or heat flow, thus yielding a one-way invisibility.

Our many-particle thermal invisibility is robust against bothout-of-plane infrared cameras (Figs. 1(f) and 1(g)) and in-planesensors (Fig. 3). Regarding these sensors, they may output electric(or other) signals transformed from temperature signaturesaccording to thermoelectric or pyroelectric effects. Differing fromthe out-of-plane infrared camera, the existence of in-plane sensorscan disturb the distribution of temperature outside the centralsquare area. But, as shown in Fig. 3, this distortion caused by thesensors does not affect the validity of thermal invisibility eventhough the thermal environment changes from line heat sources(Figs. 3(a)–3(d)) to point heat sources (Figs. 3(e)–3(h)). In otherwords, the sensors outside become blind because they cannot beused to distinguish a thermal composite and a pure environmentalmaterial located in the central square area.

3 Thermal Diode With Invisibility

As a model application, our thermal invisibility can help to pro-pose a class of thermal diodes; the research of such a diode wasstarted in 2004 [37] and has received much attention due to poten-tial applications [38]. Here, we propose a different kind of many-particle-system-based thermal diodes (Fig. 4(a)): the diodes allowheat flow from one direction with invisibility (Fig. 4(b)), but pro-hibit the heat flow from the inverse direction with visibility(Fig. 4(c)). Such a diode with the invisibility or visibility providesthe heat rectification with an additional freedom of control, whichmay be useful in the areas where both thermal camouflage andthermal rectification are needed. Imagine one wants to hide adetector in the thermal diode (such as medium A). This can berealized by adding a compensation material (such as medium B).Then, one can detect thermal signals when the thermal diodeworks on the ON-state (as we have mentioned in Sec. 2.3 that “thehidden objects are able to feel the temperature gradient or heatflow, thus yielding a one-way invisibility.”). However, when thethermal diode works on the OFF-state and the thermal signals dis-appear, the invisible function will be invalid and it shows a warn-ing that the thermal diode is turned off.

4 Conclusion

In summary, we have experimentally demonstrated that tailor-ing many-particle local-field effects can cause granular compo-sites to be thermally invisible in an environment. This invisibilityhas a mechanism essentially differing from that of the well-knowncloaking-type invisibility. Our concept of many-particle thermalinvisibility has helped to propose a class of many-particle thermaldiodes with invisibility. This work provides a new design strategyfor thermal camouflage and thermal rectification by using compo-sites. And it also implies that, besides thermotics, many-particlelocal-field effects could be a useful mechanism for achieving sim-ilar controls in other fields like optics/electromagnetics, acoustics,and fluid mechanics, where the electric permittivity and magneticpermeability, mass density and modulus, and diffusion coefficient,respectively, play the same role as the thermal conductivity inthermotics. What one needs to do is to apply effective mediummethods to different fields. It may also be useful in implementa-tion of conformal thermal devices in future.

Acknowledgment

We appreciate the useful discussions with Professor H. Zhao,Professor H. P. Fang, Professor Y. L. Han, Professor Y. J. Deng,Mr. Q. Ji, and Mr. G. L. Dai.

Funding Data

� Science and Technology Commission of Shanghai Munici-pality (16ZR1445100).

� National Fund for Talent Training in Basic Science (No.J1103204).

� Fudan’s Undergraduate Research Opportunities Program(L.X.).

Appendix: Experimental Details

The four experimental samples (Figs. 1(f)–1(i)) are manufac-tured by chemically etching, and they are covered by 0.01 mmthick polydimethylsiloxane films in order to eliminate the reflec-tion. The size of the holes in the copper with conductivity400 W/(m K) is in the millimeter scale. Since polydimethylsi-loxane is a kind of soft matter, the holes are filled with polydi-methylsiloxane (with 0.15 W/(m K)) and the defects made byair bubbles can be neglected. Additionally, even though thereexists a little air, the resulting contact resistance has almost noeffect on the overall result because air’s conductivity is close topolydimethylsiloxane’s conductivity. Outside the central squareis occupied by brass with 109 W/(m K). All the samples areplaced on 2.0 cm thick plates of expanded polystyrene. A stableheat source is served by a tank filled with hot water (313 K),whose heat capacitance is much higher than the samples. Thecold source corresponds to a same tank filled with ice-watermixture (273 K). The room temperature is about 293 K. Thesamples are detected by using the infrared camera (Flir E60)with a resolution of 0.1 K.

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Journal of Heat Transfer SEPTEMBER 2018, Vol. 140 / 092004-7