Principles of thermal design with nematic liquid crystals
aa r X i v : . [ c ond - m a t . s o f t ] S e p Principles of thermal design with nematic liquid crystals
S. Fumeron ∗ Laboratoire d’ ´Energ´etique et de M´ecanique Th´eorique et Appliqu´ee,CNRS UMR 7563, Nancy Universit´e, 54506, Vandœuvre Cedex, France.
E. Pereira
Instituto de F´ısica, Universidade Federal de Alagoas,Campus A.C. Sim˜oes, 57072-900, Macei´o, AL, Brazil.
F. Moraes
Departamento de F´ısica, CCEN, Universidade Federal da Para´ıba,58051-970, Caixa Postal 5008, Jo˜ao Pessoa, PB, Brazil.
Highly engineered materials are arousing great interest because of their ability to manipulateheat, as described by coordinate transformation approach. Based on the recently developed ana-log gravity models, this paper presents how a simple device based on nematic liquid crystal canachieve in principle either thermal concentration or expulsion. These outcomings are shown to stemfrom topological properties of a disclination-like structure, induced in the nematic by anchoringconditions.
PACS numbers: 61.30.Jf, 44.10.+i
Manipulation of heat flux is currently stimulating in-tensive research efforts because of its abundant wealthof potential applications. Most challenging stakes arerelated to thermal shielding/stealth of objects, concen-trated photovoltaics (higher conversion efficiency, reduc-tion in the cell area [ ? ]) or thermal information process-ing (heat-flux modulators [1], thermal diodes [2–5], ther-mal transistors [6, 7], thermal memories[8, 9]). Theseprospects come from the possibility to design energypaths in a similar fashion to light in transformation op-tics [10]: the use of nanostructured devices is indeed ex-pected to allow concentrating, shielding and inverting ofconductive heat flux [11–13]. Recent works brought ex-perimental evidences of their efficiency [11]. However,besides the technical issues in building such compositematerials, they also suffer from a lack of flexibility: forexample, once the device has been made, it can shield aregion from heat flux, but that function cannot be switchoff (which is of prime interest for thermal logic gates orheat flux modulators).A promising way to overcome this difficulty relies inthe use of nematic liquid crystals (NLC) devices insteadof multilayered devices. NLC are generally organic com-pounds formed by an assembly of rod-like molecules.These latter often consist in a rigid core made of twobenzene rings (responsible for crystal-like properties) as-sociated with flexible exterior chains (responsible for flu-idity). In other words, molecules in the nematic phaseshow no positional order (as in ordinary liquids), but ori-entational order (anisotropy of dielectric permittivities,elastic constants [14]..). Thus, NLC are invariant underthe symmetry group D ∞ = SO (2) × Z , that consists in ∗ Electronic address: [email protected] rotations about the molecular axis (called the director)and rotations through 180 ◦ about axes in the orthogonalplane. Generally, the alignment of the director exhibitsdiscontinuities: among them, disclinations (also calledwedge dislocations) are elastic defects belonging to thefirst homotopy group π and they appear as a conse-quence of SO (2) symmetry-breaking. An intuitive wayto understand it is to perform a ”cut and glue” Volterraprocess [15]: the disclination is generated by removinga wedge of material of dihedral angle φ and gluing theloose faces together (see Fig. 1 for the two-dimensionalcase). This generates a positive curvature disclination.The inverse procedure, inserting a wedge of material, pro-duces a negative curvature disclination (not shown). N FIG. 1: Volterra process: an angular sector φ is removedfrom a flat surface, the remaining edges are glued together,which results in a conical surface. Topological defects are the core of fruitful analogies be-tween condensed matter physics and cosmology[16, 17].They are elegantly described in terms of Riemann-Cartangeometry [18–20] and therefore they can be studied byusing techniques borrowed from General Relativity ( U theory of gravitation [ ? ]): this is the core of so-calledanalog gravity. Compared to ordinary elasticity theory(OET), this approach has two major assets : first, itsaccuracy (OET only reproduces the first-order approxi-mation of the geometric theory of defects [22]) and sec-ond its versatility (changing the kind of defect only re-quires to change the metric, instead of a complicated setof boundary conditions in OET). In this framework, adisclination in NLC acts analogously to a gravitationalline source and the medium surrounding is described bythe metric of a global cosmic string [23 ? ]: ds = dr + α r dφ + dz , (1)which describes a conical geometry. For both cosmicstrings and disclinations in elastic solids, the parameter α is related to the sector removed/added by φ = 2 π (1 − α ),where α < α > g ij = x + α y r xy (1 − α ) r xy (1 − α ) r α x + y r
00 0 1 (2)In the diffuse regime, heat conduction in the nematicphase is modified by the presence of topological defects.For a given metric g ij , heat equations governing conduc-tive flux and temperature fields are given by [26]: q i = − λ ij ∂ j T, (3) div q = p, (4)with λ ij = λ g ij , (5) p = − λ g ij ∂ j T ∂ i ln g, (6)where g = det g ij and λ is the thermal conductivity ofthe isotropic liquid phase. Generally speaking, it appearsthat a non-trivial metric modifies the thermal conductiv-ity. Moreover, it can also introduce an effective internalheat source ( p >
0) or sink ( p <
0) that is directly cou-pled to the temperature fields. In the case of (2), heatconduction locally occurs as in a monoclinic crystal [26]and as g = α = const , the disclination does not intro-duce any internal source term. Similarly to matter incosmic strings wakes [27], heat flux lines stay in planes z = const. but feel the curvature of the background con-ical geometry generated by the disclination (see Fig. 2).As the Riemann tensor is null everywhere, except at thedefect core where it has a δ -function singularity given by[28] R = 2 π − αα δ ( r ) , (7)the distortion of flux lines when approaching the defectcore also testifies that disclinations are responsible foran elastic analog of Aharonov-Bohm effect (as alreadynoticed in [29, 30]).Now that the specific features of heat conduction neara disclination have been reviewed, we are investigatingthe possibility to implement them for thermal design. A a b FIG. 2: Director field (dotted lines) and geodesics (plain lines)in the presence of a disclination line. (a) Case α >
1. (b) Case α ′ <
1. If it is the same material, α ′ = 1 /α . qualitative understanding of it can be obtained from aparametric study of the following configuration: a hol-low cylinder, inside which there is the core region whereone aims at controlling the conductive heat flux, is in-serted inside a conducting solid sandwiched between twoheated vertical plates (see figure 3). In normalized units,cold temperature is set to T c = 0, whereas hot temper-ature is set to T h = 1. The host material consists inan homogenous isotropic medium with unit thermal con-ductivity, whereas the thick cylinder consists in a NLCfor which the spatial configuration of the director ˆ n isthat of a disclination. We can think of a nematic direc-tor field configuration like the one in Fig. 2a withoutthe singularity on its axis, by isolating the axis with acylindrical wall. In order to have the same radial orien-tation of the director field, both inner and outer bound-aries can be prepared to provide homeotropic anchoringto the nematic molecules. As there is no disclinationcore, such configuration is topologically stable. That is,no escape in the third dimension can occur. The stabil-ity of the mesophase is more problematic. Indeed, liquidcrystals made from a single kind of organic moleculesare thermotropic and in practice, they exhibit a nematicphase in a very narrow range of temperatures (typically afew tens of degrees). But for thermal management, NLCwith low melting and high clearing temperatures are re-quired. This can be achieved by using eutectic liquidcrystals mixtures (or “guest-host systems”): by adjustingthe proportions of each compound (Schroeder-Van Laarlaw), the nematic range can be larger than 100 K[31, 32].The thermal conductivity of the isotropic fluid phaseis taken equal to that of the host medium, to avoid ad-ditional thermal effects (such as Kapitza resistance). Wewill consider three cases: 1) the reference case consistingin a NLC without disclination ( α = 1); 2) a strength +1disclination with the director aligned radially, for which α = p C /C = 2 (Fig. 2a); 3) again a strength +1disclination but now with the director circling the de-fect axis, for which α = p C /C = 0 . FIG. 3: Principle of the thermal control device the cylinder. Based on COMSOL Multiphysics finiteelement-based simulations, we first present results relatedto α = 1. Temperature fields and heat flux lines in Fig. 4 a b FIG. 4: Reference case without the device ( α = 1). (a) Tem-perature field. (b) Heat flux lines. are the regular solutions of steady-state heat conductionequations inside isotropic solids: heat flux lines follow thedirection of the temperature gradient isothermal lines ( x -axis), whereas isothermal lines correspond to the planes y = const in agreement with boundary conditions.Now, let us examine on Figure 5 the case worrespond-ing to α = 2. Compared to previous case, isothermal a b FIG. 5: Concentrator configuration with α = 2. (a) Temper-ature field. (b) Heat flux lines. lines bend in order to enter the core of the device. Be-sides, heat flux lines are also converging inside the innerregion, such that the norm of heat flux vectors is globallyincreasing in this area. Therefore, the device is respon-sible for focusing conductive heat inside the core, whichcan be thought of as a heat concentration phenomenon.Now, we consider on Figure 6 the case α = 0 . a b FIG. 6: Repeller configuration with α = 0 .
5. (a) Temperaturefield. (b) Heat flux lines.
It must be emphasized that these two behaviours (re-pelling and concentrating) cannot be understood in termsof geodesics of the effective metric (2), as it is usuallythe case in analog gravity models. In general relativity,geodesics are the paths followed by free-falling particlesin a curved spacetime. They are the solutions of Euler-Lagrange equations, obtained by varying the Einstein-Hilbert action, and they are a generalization of Newton’ssecond law m a = F in the case where F = 0 (uniform mo-tion). Therefore, considering the presence of a tempera-ture gradient is equivalent to add a thermal constraint toEuler-Lagrange equations. Another way of understand-ing it is to take a mechanical standpoint. Heat conduc-tion can be thought of as a phonon gas, and as such, itobeys fluid mechanics. In this frame, the temperaturegradient acts as a driving force for the heat flow (see[ ? ] for temperature gradient as an effective pressure inphonon hydrodynamics), so that the heat flux cannot beconsidered as free-falling.As previously mentioned, the main asset of our de-vice over systems based on composite materials relies inits versatility. As a matter of fact, the concentrator isachieved for a NLC such that C > C ( α > n = ˆ r ). Conversely,the cloak configuration may be achieved for the sameNLC, but with the director parallel to the boundaries FIG. 7: Principle of concentrator-repeller switching based onbistable anchoring transition. (a) Homeotropic anchoring inthe absence of external field. (b) Transition to planar anchor-ing due to electric field. This latter is shut off after transitionis performed. (ˆ n = ˆ φ ) such that α >
1. Therefore, rotating the rod-likemolecules by 90 degrees enables to switch at will fromthe concentrator to the cloak. This can be achieved byelectric-field-driven bistable anchoring (see figure 7). An-choring yields a constant orientation of the thermotropicNLC molecules due to particular surface conditions. Inthe presence of a surfactant-treated surface (for examplesilane), rods align normally to the surface and anchor-ing is said homeotropic. On the contrary, in the pres-ence of a polymer coating such as PVA (polyvinyl alco-hol), anchoring is planar and calamitic molecules alignparallel to the surface. One technical issue to developthis device is to anchor NLC molecules on curved sub-strates. Usual photopolymer-based techniques require awell-defined angle of incidence for light, which make themill-adapted for non-planar surfaces. Gupta et al. [ ? ] per-formed NLC anchoring on curved surfaces by using self-assembled monolayers (SAM) formed from alkanethiols.Two additional assets for SAM-based anchoring are that1) they are stable upon application of the electric field,and 2) polymerizable SAM are expected to be chemicallystable over years (no oxidative degradation). Above theFrederiks transition, application of an electric field leadsto unstable new states of orientation (depending on thesign of its dielectric anisotropy)[14]: as soon as the field isremoved, molecules relax back to the original orientationfixed by anchoring conditions. However, for dye-dopedNLC, sufficiently high values of the field were shown toinduce stable anchoring transitions between homeotropicand planar states: back switching does not occur evenwhen the field is off[33]. In principle, such effect can beused to switch between the concentrator and the repellerby applying during a short time an electrical potential difference between both sides of the hollow cylinder. Asthe electrodes are turned on only during the time re-quired for the anchoring transition to occur, electric-fieldinduced instabilities do not appear when the device isworking.Both configurations (concentrator and repeller) havebeen considered with horizontal temperature gradients,so that no Rayleigh-B´ernard convection can occur. How-ever, as the thermal gradient is not parallel to the gravityfield g , weak thermoconvective instabilities can developin the annulus domain. These instabilities vanish pro-vided that the device is thin enough. Although an ac-curate justification requires solving the equations of ne-matohydrodynamics, this effect can be understood qual-itatively from the usual viscous dissipation power [ ? ].For an annulus region of thickness d , with internal ra-dius r i and outer radius r e , consider a thermoconvectiveinstability with mean velocity U . Therefore, the dissi-pated power can roughly be expressed as a scaling lawaccording to : P dis = 2 η Z Z Z ¯¯ D : ¯¯ D d x ∼ η U d π ( r e − r i ) , (8)where η is the mean shear viscosity of the nematic and¯¯ D is the strain rate tensor. Therefore, inner viscous dis-sipation is maximized in a disc-like device (low thickness d and small core regions r i ) and it is also expected to beenhanced by dissipation at boundaries (anchoring condi-tions).In this work, we examined the possibility of thermalmanagement from a liquid crystal-based device. It relieson reproducing inside a hollow cylinder the topology of adisclination, which can imprint curvature to conductiveflux lines. Numerical results confirm the possibility ofheat guiding phenomena: in particular, we showed thatthe device can be tuned to either concentrate or expulseheat flux. The switching frequency is limited by relax-ation times governing the anchoring transition (severalmilliseconds for dye-doped nematics [33]). In principle,it can be designed to work in a relatively large rangeof temperatures. Concentrator or repeller configurationsare both mechanically stable and thermal instabilites canbe neglected by adequate choice of geometry (disc-like).However, it is mandatory for practical developments ofsuch device to quantitatively study these aspects throughoptimal choice of materials. This will be considered inour next works.E. Pereira thanks financial support from Brazilianagencies FAPEAL and CNPq. F. Moraes thanks finan-cial support from INCT-FCx, CNPq and CAPES. [1] P.J. van Zwol and K. Joulain and P. Ben Abdallah andJ.J. Greffet and J. Chevrier, Phys. Rev. B , 201404 (2011).[2] M. Terrano and M. Peyrard and G. Casati, Phys. Rev. Lett. , 094302 (2002).[3] B. Li and L. Wang and G. Casati, Phys. Rev. Lett. ,184301(2004).[4] B. Li and J. Lan and L. Wang, Phys. Rev. Lett. ,104302 (2005).[5] C. W. Chang and D. Okawa and A. Majumdar and A.Zettl, Science , 1121 (2006).[6] B. Li and L. Wang and G. Casati, Appl. Phys. Lett. ,143501 (2006).[7] O. P. Saira and M. Meschke and F. Giazotto and A. M.Savin and M. Mottonen and J. P. Pekola, Phys. Rev.Lett. , 027203 (2007).[8] L. Wang and B. Li, Phys. Rev. Lett. , 177208 (2007).[9] L. Wang and B. Li, Phys. Rev. Lett. , 267203 (2008).[10] J.B. Pendry and D. Schurig and D.R. Smith, Science ,1780 (2006).[11] S. Narayana and Y. Sato, Phys. Rev. Lett. , 214303(2012).[12] S. Guenneau and C. Amra and D. Veynante, Opt. Exp. , 8207 (2012).[13] M. Maldovan , Phys. Rev. Lett. , 025902 (2013).[14] P. G. de Gennes and J. Prost, The Physics of LiquidCrystals (Claredon Press, Oxford, 1992), 2nd ed.[15] M. Kleman and J. Friedel , Rev. Mod. Phys. , 61(2008).[16] I. Chuang and B. Yurke and R. Durrer and N. Turok,Science , 1336 (1991).[17] C. Furtado and F. Moraes and A. de M. Carvalho, Phys.Lett. A , 5368 (2008).[18] K. Kondo, Jpn. Nat. Congr. Appl. Mech.: Proc. (Tokyo) , 41 (1952).[19] B. A. Bilby and R. Bullough and E. Smith, Proc. R. Soc. A , 263 (1955).[20] M. O. Katanaev and I. V. Volovich, Ann. Phys. , 1(1992).[21] F.W. Hehl and P. Von Der Heyde and G.D. Kerlick andJ.N. Nester, Rev. Mod. Phys. , 393 (1976).[22] M. O. Katanaev, Phys.-Usp , 675 (2005).[23] E. Pereira and S. Fumeron and F. Moraes, Phys. Rev. E , 022506, 049904 (2013).[24] A. Vilenkin, Phys. Rev. D , 852 (1981).[25] C S´atiro and F. Moraes, Eur. Phys. J. E , 425 (2008).[26] S. Fumeron and E. Pereira and F. Moraes , Int. J. Therm.Sci. , 64 (2013).[27] M.B. Hindmarsh and T.W.B. Kibble , Rep. Prog. Phys. , 477 (1995).[28] D. Sokolov and A. Starobinskii , Sov. Phys. Dokl. , 312(1977).[29] S. Azevedo and F. Moraes , Phys. Lett. A , 374(1998).[30] C. Furtado and A. M. de Carvalho and C. A. de LimaRibeiro , Mod. Phys. Lett. A , 1393 (2006).[31] S.T. Wu and C.S. Hsu and K.F. Shyul , Appl. Phys. Lett. , 344 (1999).[32] P. Oswald and P. Pieranski, Nematic and CholestericLiquid Crystals: Concepts and Physical Properties Illus-trated by Experiments (CRC Press, 2005), 1st ed.[33] J.K. Kim and K.V. Le and S. Dhara and F. Araoka andK. Ishikawa and H. Takezoe , J. Appl. Phys. , 123108(2010).[34] L.D. Landau and E. Lifschitz,