Liquid Crystals as Phase Change Materials for Thermal...

Research ArticleLiquid Crystals as Phase Change Materials forThermal Stabilization

Eva KlemenIiI 1 andMitja Slavinec 12

1Faculty of Natural Sciences and Mathematics University of Maribor Maribor Slovenia2Academic Scientific Union of Pomurje (PAZU) Murska Sobota Slovenia

Correspondence should be addressed to Mitja Slavinec mitjaslavinecumsi

Received 19 February 2018 Accepted 12 April 2018 Published 15 May 2018

Academic Editor Charles Rosenblatt

Copyright copy 2018 Eva Klemencic and Mitja Slavinec This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Thermal stabilization exploiting phase change materials (PCMs) is studied theoretically and numerically Using the heat sourceapproach in numerical simulations we focus on phase change temperature as a key factor in improving thermal stabilization Ourfocus is to analyze possible mechanisms to tune the phase change temperature We use thermotropic liquid crystals (LCs) as PCMsin a demonstrative systemUsing the Landau-deGennesmesoscopic approach we show that an external electric field or appropriatenanoparticles (NPs) dispersed in LCs can be exploited to manipulate the phase change temperature

1 Introduction

In order to achieve and maintain an optimum ambient tem-perature one must regulate heat flow using different heatingor cooling systems which are usually large energy consumersRecently several studies [1ndash5] focused on the development ofinnovative materials to passively reduce temperature fluctua-tions Phase changematerials (PCMs) are suitable candidatesas they undergo a phase change during which they exchangelatent heat with the surroundings

The main function of PCMs is saving excess heat that isaccumulated at higher temperature conditions and releasingit at a lower temperature Therefore the efficiency of asystem using PCMs is strongly correlated with phase changetemperature For example if one wants to decrease roomtemperature oscillation it would be optimal to have a PCMwith phase change around the desired room temperature Ifthe room temperature exceeds the temperature of the PCMcomposite and reaches the phase change temperature thePCM undergoes a phase change During the phase changethe PCM accumulates the excess heat in form of latent heatThe opposite happens as the temperature lowers towards thephase change temperature and the latent heat is released tothe surroundings

In this paper we focus on analyzing possible mechanismsto tune the phase change temperature in order to obtainoptimal thermal stabilizationWe used a nematic liquid crys-tal (LC) system as an experimental testing ground LCs areexperimentally suitable due to their liquid character softnesslarge susceptibility to relatively weak external perturbationsand optical anisotropy [6 7]We considered thermotropic LCin which by decreasing the temperature from the isotropic(I) phase the nematic (N) phase is obtained via first-ordercontinuous symmetry breaking phase transition [8ndash10] Inthe uniaxial bulk NLC phase rod-like molecules exhibitlong-range orientational ordering commonly presented bythe nematic director 997888119899 Fluctuations about 997888119899 describe theuniaxial order parameter 119878 Note that |997888119899 | = 1 and states plusmn997888119899are equivalent due to the so-called head-to-tail invarianceAt the mesoscopic scale we described nematic orientationalordering by the nematic tensor order parameter Q [11] Ourmain goal was to identify different mechanisms to controlandmanipulate the phase change temperature In general thepresence of nanoparticles (NPs) external electric fields andother perturbations gives rise to elastic distortions [12ndash16]that can result in the temperature shift of the phase transition

First we introduce a numerical model to analyze thermalstabilization using PCMs The model is based on the heat

HindawiAdvances in Condensed Matter PhysicsVolume 2018 Article ID 1878232 8 pageshttpsdoiorg10115520181878232

2 Advances in Condensed Matter Physics

T1 T2

휆PCMcPCM휌PCM

Figure 1 Schematic representation of two layers one integrating PCMs

source approach and finite difference method Next wepresent a theoretical Landau-de Gennes model to examinethe impact of the external electric field and NPs on the phasechange temperature of NLC system We present and discussresults in Section 3 In the final section we conclude ourfindings

2 Modeling

21 Thermal Stabilization with PCM In this section wedescribe the use of PCMs for the improvement of thermalstabilization We considered one-dimensional heat transferthrough the planar surface of a two-layered composite (Fig-ure 1) with one layer integrating homogeneously distributedPCMs Our numerical model is based on the heat sourceapproach [17ndash20] in which latent heat is modeled as anadditional heat source In terms of enthalpy (ℎ) the transientheat transfer equation in one dimension is expressed as

120588120597ℎ120597119905 = 12058212059711987921205971199092 (1)

where 120588 is density and 120582 thermal conductivity of each layerAssuming high porosity of the composite layer the

expansion of PCMs during the phase transition does notchange the total volume Additionally we considered differ-ent physical properties of the solid (119904) and liquid (119897) phase ofPCMs

In order to simplify their numerical simulations otherstudies [17 19 20] determined the temperature range wherephase transition occurs In this study we assumed that phasetransition is instant at one specific critical temperature119879119888Weintroduced the factor 119903 that equals zero when the PCM is inthe solid phase (119879 lt 119879119888) and one when the PCM is in theliquid phase (119879 gt 119879119888)

Using a model of effective physical quantities [18] wedefined the effective specific heat capacity (119888119890) thermalconductivity (120582119890) and density (120588119890) as follows

119888119890 = 120572 (119888(119897)PCM119903 + 119888(119904)PCM (1 minus 119903)) + (1 minus 120572) 119888119860 (2a)

120582119890 = 120572 (120582(119897)PCM119903 + 120582(119904)PCM (1 minus 119903)) + (1 minus 120572) 120582119860 (2b)

120588119890 = (1 minus 120572) 120588119860 + 120572 (120588(119897)PCM119903 + 120588(119904)PCM (1 minus 119903)) (2c)

where 120572 represents the relative amount of PCM integrated inthe composite layer and subscripts119860 and119861 correspond to twodifferent layers of studied composite (see Figure 1)

We derived the heat transfer equation for the compositeincorporating PCMs in terms of the enthalpy ℎ119890 = 119888119890119879+120572119903119876119871as follows

120588119890119888119890 120597119879120597119905 + 120588119890120572119876119871 120597119903120597119905 = 120582119890 12059711987921205971199092 (3)

where 119876119871 is the latent heat of the integrated PCM Fornumerical simulations we used the finite difference method

22 Phase Change Temperature Shift Next we theoreticallyanalyzed possible mechanisms to tune the phase changetemperature We considered the NLC rectangular cell wherelateral confinement walls impose strong homogeneous tan-gential anchoring (Figure 2) The characteristic confinementsize 119877 is assumed to be larger than the cell thickness ℎ

Using the nematic tensor order parameter [6] wedescribed LC orientational ordering as follows

Q = 3sum119894=1

119904119894997888119890 119894 otimes 997888119890 119894 (4)

where 997888119890 119894 are eigenvectors 119904119894 are eigenvalues of Q andthe symbol otimes represents tensor product In case of uniaxialordering it holds as follows [12]

Q = 119878 (997888119899 otimes 997888119899 minus 13 I) (5)

where I is the identity matrix

Advances in Condensed Matter Physics 3

y

x

z

R

R

h

Figure 2 NLC cell of thickness ℎ along the 119911-Cartesian coordinate with conflicting strong homogeneous tangential anchoring

Next we expressed the free energy density as a sum of thecondensation nematic term 119891119899 elastic term 119891119890 external fieldterm 119891119891 and surface term 119891119904 [12 13]

119891119899 = 1198600 (119879 minus 119879lowast)2 TrQ2 minus 1198613TrQ3 + 1198624 (TrQ2)2 (6a)

119891119890 = 1198712 |nablaQ|2 (6b)

119891119891 = 1205760Δ1205762 997888119864 sdot Q997888119864 (6c)

119891119904 = 1199082 Tr (Q minus Qs) (6d)

where 1198600 119861 119862 are material-dependent constants and 119879lowast isthe isotropic supercooling temperatureThe elastic term eval-uates deviations from a spatially homogeneous orientationalordering in terms of the elastic constant LThe external fieldterm describes the contribution of an external electric field997888119864 where Δ120576 stands for the electric field anisotropy constantand 1205760 represents the electrical permittivity constant Sincewe imposed homogeneous tangential anchoring at the lateralsurfaces with the anchoring strength 119908 Q119904 describes thedegree of orientational ordering enforced by the confiningsurface

The I-N phase change temperature is determined bymaterial-dependent constants

119879IN = 119879lowast + 1198612(41198600119862) (7)

Next assuming uniaxial ordering described by (5) weimposed a distortion to orientational ordering on a character-istic length scale 119877 On average it holds as follows (see (6b))[21 22]

119891119890 sim 1198712 11987821198772 (8)

where the overbar (sdot sdot sdot ) indicates spatial averaging

Then we expressed the average contributions of thecondensation nematic term and elastic term to the averagefree energy density 119891 as follows [23ndash25]

119891 sim 1198600 (119879 minus 119879lowast)3 1198782 minus 1198619 1198783 + 1198629 1198784 + 1198712 11987821198772equiv 1198600 (119879 minus 119879eff

lowast )3 1198782 minus 1198619 1198783 + 1198629 1198784

(9)

where

119879efflowast = 119879lowast minus 32 11987111986001198772 (10)

Considering 119871 1198600 and 119879lowast are material-dependent coeffi-cients one could vary the effective phase change temperatureof the NLC system by varying the typical distortion length119877 We obtained the phase change temperature shift Δ119879 =119879IN minus 119879IN(119877) expressed as follows (see (7)) [26 27]

Δ119879IN (119877) = minus32 11987111986001198772 (11)

Since the temperature shift is connected to 119877 we focusedon imposing different distortions that can be described by alinear deformation imposed characteristic length 119877

Below we present two experimentally attainable possi-bilities influencing the phase change temperature of NLCthe presence of an external electric field and NPs Since thepresence of NPs and the external electric field could inducebiaxiality we introduce the degree of biaxiality as follows [28]

1205732 = 1 minus 6 (TrQ3)2(TrQ2)3 (12)

which is zero for uniaxial states and one when the stateexhibits a maximum degree of biaxiality


For further convenience we introduce three charac-teristic lengths the external field coherence length 120585119891 =radicLS(1205760Δ1205761198642) the surface anchoring length 119889119890 = 119871119908and the bare biaxial order parameter correlation length120585119887 = radicLC119861 In addition we introduce parametrization andscaling as described in the following [12]

Q (119909 119910) = (1199023 + 1199021) 997888119890 119909 otimes 997888119890 119909 + (1199023 minus 1199021) 997888119890 119910 otimes 997888119890 119910+ 1199022 (997888119890 119909 otimes 997888119890 119910 + 997888119890 119910 otimes 997888119890 119909) minus 21199023997888119890 119911otimes 997888119890 119911

(13)

where 1199021 1199022 and 1199023 are 119909 and 119910 functions and 997888119890 3 = 997888119890 119911 is aneigenvector ofQ

In this scaling we define the scaled temperature as 119905 =(119879 minus 119879lowast)(119879lowastlowast minus 119879lowast) Here the superheating temperature ofthe nematic phase is 119879lowastlowast = 119879lowast + 1198612241198600119862 For numericalconvenience we introduce 120591 = 1 + radic1 minus 119905With this in mindwe express the equilibrium uniaxial order parameter as 119878eq =119878lowastlowast120591 where 119878lowastlowast = 1198614119862 = 119878eq(119879lowastlowast)

Finally we obtain bulk Euler-Lagrange equations forvariation parameters 1199021 1199022 and 1199023 by minimization of thefree energy density [29]

Δperp1199021(120585(0)119887119877 )2

minus 12059161199021 + 211990221199021 minus 11990212 (311990222 + 11990221 + 11990223)

+ 14 (120585(0)119887120585119891 )2

((997888119890 119909 sdot 997888119890 )2 minus (997888119890 119910 sdot 997888119890 )2) = 0(14a)

Δperp1199022(120585(0)119887119877 )2

minus 12059161199022 + 13 (11990221 + 11990223 minus 311990222)

minus 11990222 (311990222 + 11990221 + 11990223) + 112 (120585(0)119887120585119891 )2

= 0(14b)

Δperp1199023(120585(0)119887ℎ )2

minus 12059161199023 + 211990221199023 minus 11990232 (311990222 + 11990221 + 11990223)

+ 12 (120585(0)119887120585119891 )2

(997888119890 119909 sdot 997888119890 ) (997888119890 119910 sdot 997888119890 ) = 0(14c)

where Δperp = 12059721205971199092 + 12059721205971199102 Euler-Lagrange equations aresolved using the standard relaxation method numerically

3 Results and Discussion

In this section we first analyze the numerical results of aone-dimensional heat transfer through composite wall with afocus on varying the phase change temperature In our simu-lations the composite wall (labeled GPCM) consisted of twolayers 200 cm thick panel at the outside and 30 CM thickgypsum wallboard with PCMs at the inside The latter is alsotrue in real building constructions as the purpose of PCM is

to stabilize the interior temperatureThe outside temperaturefluctuated according to typical daynight temperatures in thesummertime (from 15∘C to 35∘C) and the room temperature(119879room) was set to a constant value in our case to 21∘C Weexamined cases for three different phase change temperatures(119879119888) of PCM and obtained different scenarios for the optimalthermal stabilization depending on the initial temperature(119879119894) of the system

Figure 3 depicts the incoming heat flux (119902in) timedependency for three different cases First we set the initialtemperature of the composite wall to the average value ofthe outside temperature (119879119894 gt 119879room) It is evident that thepresence of PCMs affects 119902in as soon as the phase changetemperature T119888 is reached At this point 119902119899 is close to zeroor zero for 119879119888 = 119879room (black solid curve) until the latentheat storage capacity becomes full By varying the phasechange temperature we show that the time period of a steadyincoming heat flux is the longest when 119879119888 is slightly above theroom temperature (red dotted curve) On the other hand inorder tomaintain the constant room temperature in this casethe incoming heat flux should be compensated using differentcooling devicesThe zero incoming heat flux is reachable onlyby setting the phase change temperature equal to the roomtemperature Negative values of 119902in for the case when 119879119888 lt119879room (blue dashed curve) correspond to a decrease of a roomtemperature below the desired value Next we analyzed casesfor 119879119894 lt 119879room (Figure 3(b)) and 119879119894 = 119879room (Figure 3(c))Figures 3(b) and 3(c) confirm that the incoming heat flux canbe entirely or partly stored in the form of the latent heat atthe 119879119888 It is also evident that temperature oscillations in allthree cases are reduced noticeably especially for 119879119888 = 119879roomand 119879119888 gt 119879room The numerical obtained results show thatthermal stabilization could be more efficient by using a smallvariation of the phase change temperature Nevertheless it isevident that the incoming heat flux fluctuates themost for thecomposite with no PCMs (black dotted curve)

To better understand the efficiency of PCM compositeswe numerically analyzed thermal stabilization for threedifferent composites used in real-life applications Figure 4shows a comparison of an alternative composite GPCM(black solid curve) with composites labeled BS1 (blue dashedcurve) and BS2 (red dotted curve) BS1 and BS2 bothconsist of classic building materials brick wall of thickness500 cm for BS1 and 200 cm for BS2 and 30 cm thickinsulation layer of Styrofoam placed at the outside of awall We simulated typical outside temperature fluctuationsfor the summertime as described above In this case thephase change temperature of GPCM is set to the roomtemperature It is noticeable that thicker wall of BS1 lowerstemperature oscillations in comparison with BS2 This isexpected due to larger thermal capacity Furthermore it isalso evident that the alternative composite GPCM of equalthickness as BS2 improves thermal stability even more Theincoming heat flux after 48 hours of simulations is closeto 0 for GPCM and around 5 for BS1 Therefore wecan conclude that both systems BS1 and GPCM optimizethe thermal stability but by using alternative compositeswith PCMs construction walls can be approximately 2 timesthinner


0 6 12 18 24 30 36 42 48

t (h)

Tc lt TroomTc = Troom

Tc gt Troom0 PCM

minus06

minus04

minus02

00

02

04

06

qin

(rel

ativ

e uni

ts)

08

10

(a)

10

08

06

04

02

00

minus02

minus04

minus06

0 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(b)10

08

06

04

02

00

minus02

minus04

minus060 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(c)

Figure 3The time dependency of the incoming heat flux (relative units) for three values of phase change temperature 119879119888 = 119879room (black solidcurve) 119879119888 gt 119879room (red dotted curve) and 119879119888 lt 119879room (blue dashed curve) For comparison the composite without PCM (black dotted curve)is included Initial temperature is set to (a) 119879119894 gt 119879room (b) 119879119894 lt 119879room and (c) 119879119894 = 119879room

Note that in all simulations we considered temperatureoscillations between 15∘C and 35∘C for total time of 48hours In order to obtain more realistic results additionalsimulations using different temperature ranges and longersimulation time are welcomed According to our numericaloutcomes (Figures 3 and 4) we assume similar thermalstabilization with the main difference in the direction of theheat flow when simulating low temperatures In this casewe expect that the lowest temperature oscillations would beobtained for 119879119888 = 119879room and 119879119888 lt 119879room Since even smallvariation in 119879119888 affects the thermal stability it is reasonableto explore possible mechanisms to develop PCM compositeswith tuneable phase change temperature

We analyzed the impact of external electric fields andNPson the phase change temperature 119879119888 of NLC We considered

square-shaped NP that enforces strong tangential boundaryconditions at its surface in the presence and absence ofan external electric field In our study we set the lateralconfinement size119877 below 1micron comparable to 120585119887Figure 5demonstrates typical nematic configurations in the absenceand presence of an external electric field Figure 5(a) depictsthe diagonal structure which is stable for cases when 119877 ≫ 120585119887and 119864 = 0 At the corners of the cell opposing conditionsgive rise to defects In the center of each defect a negativeuniaxial order along 997888119890 119911 is established and is surroundedby a rim of a maximal degree of biaxiality By increasingthe external electric field strength we obtained qualitativelydifferent configurations for 119877120585119891 = 10 (Figure 5(b)) and for119877120585119891 = 100 (Figure 5(c)) In the latter case the externalelectric field triggered a surface order-reconstruction type


0 6 12 18 24 30 36 42 48

t (h)

01

03

05

07

qin

(rel

ativ

e uni

ts)

09

10

00

02

04

06

08

BS1BS2GPCM

Figure 4 The time dependency of the incoming heat flux (relative units) for three real-life composites BS1 (blue dashed curve) BS2 (reddotted curve) and GPCM (black solid curve)

1

09

08

07

06

05

04

03

02

01

(a)

1

09

08

07

06

05

04

03

02

01

(b)

1

09

08

07

06

05

04

03

02

01

(c)

Figure 5The degree of biaxiality 1205732 in the absence of NPs for119877120585119887 = 7 Uniaxial state is presented in black a maximum degree of biaxiality ispresented in white The diagonal structure is formed (a) in the absence of an external electric field (b) for 119877120585119891 = 10 and (c) for 119877120585119891 = 100

of structural transition Therefore by application of strongenough external field one obtains the qualitatively differentconfiguration of NLC which would affect the phase transi-tion temperature according to (11)

Next we studied the impact of square-shaped NP in theabsence of an external electric field (Figure 6) We placed NPat four different positions of the NLC cell (i) at the center(ii) at the bottom boundary (iii) at the left boundary and(iv) at the left bottom edge The NP acts as a source of elasticdistortions that result in locally induced biaxiality

We obtained an equal configuration (but rotated) for twopositions of NP close to the left boundary and close to thebottom boundary (Figures 6(b) and 6(c))This shows that thepresence of NPs in an NLC cell affects the typical distortionlength R Regarding (10) and (11) we conclude that NPs affecteffective phase change temperatures Note that in generalseveral NPs could be introduced which increases complexityand richness of phenomena

Additionally we studied the combined impact of NPs andexternal electric fields The external electric field broke the

symmetry of the system (see Figure 7) and enabled an order-reconstruction type transition at the bottom plate

4 Conclusions

In this study we numerically assessed the impact of PCMson thermal stabilization Numerical simulations based onthe heat source method confirm that PCMs integrated inthe composite material reduced temperature oscillationsand therefore improved thermal stabilization We focusedon different cases by varying the phase change tempera-ture around room temperature and showed that the effi-ciency of the thermal stabilization depends on both thephase change temperature and the desired room tem-perature Therefore it is reasonable to develop tuneablePCM composites Our main goal was to find and ana-lyze possible mechanisms to control and manipulate phasechange temperature We considered NLC cells as PCMs asthey are commonly used in theoretical and experimentaltesting


(a) (b) (c) (d)

Figure 6 Absence of external electric field 119877120585119887 = 7 Qualitatively different configurations are obtained for NP placed at the (a) center (b)bottom boundary (c) left boundary and (d) left bottom edge The color scheme is the same as in Figure 5

(a) (b) (c) (d)

Figure 7 Configurations for combined impact of the external electric field 119877120585119891 = 50 and four different positions of NPs (a) at the center(b) bottom boundary (c) left boundary and (d) left bottom edge for 119877120585119887 = 7 The color scheme is the same as in Figure 5

In our theoretical section we showed that phase changetemperature can be shifted by varying the typical distortionlength 119877 We then examined the impact of external electricfields and NPs on the NLC configuration and consequentlyon the phase change temperature We demonstrated thatwe could shift the phase change temperature by changingthe external electric field in the absence of NPs Whenan external electric field is absent an NP can also effec-tively change the typical characteristic length of the NLCsystem

LCs can be used as PCMs to improve thermal stabi-lization Furthermore one can even tune the phase changetemperature with relatively simple mechanisms One of themain disadvantages of using LCs for thermal stabilizationis relatively high cost corresponding to the relatively smallamount of latent heat Nevertheless LCs have potentialfor future applications as PCMs especially in the spaceindustry

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] D Zhou C Y Zhao and Y Tian ldquoReview on thermal energystorage with phase change materials (PCMs) in building appli-cationsrdquo Applied Energy vol 92 pp 593ndash605 2012

[2] V V Tyagi and D Buddhi ldquoPCM thermal storage in buildingsa state of artrdquo Renewable amp Sustainable Energy Reviews vol 11no 6 pp 1146ndash1166 2007

[3] D Li L Yang and J Lam ldquoZero energy buildings andsustainable development implications mdashA reviewrdquo Energy vol51 pp 1ndash10 2013

[4] N Bhikhoo A Hashemi and H Cruickshank ldquoImprovingthermal comfort of low-income housing in Thailand throughpassive design strategiesrdquo Sustainability vol 9 no 8 article no1440 2017

[5] M Casini Smart Buildings Advanced Materials and Nan-otechnology to Improve Energy-Efficiency and EnvironmentalPerformance Woodhead Publishing (Elsevier) AmsterdamThe Netherlands 2016

[6] M Kleman and O D Lavrentovich Soft Matter PhysicsSpringer-Verlag Berlin Germany 2002

[7] H de J Wim Liquid crystal elastomers Materials and Applica-tions Springer-Verlag Berlin Germany 2012

[8] V Popa-Nita ldquoStatics and kinetics at the nematic-isotropicinterface in porousmediardquoTheEuropean Physical Journal B vol12 no 83 1999

[9] D E Feldman ldquoQuasi-long-range order in nematics confinedin random porous mediardquo Physical Review Letters vol 84 no21 pp 4886ndash4889 2000


[10] A Aharony and E Pytte ldquoInfinite susceptibility phase inrandom uniaxial anisotropy magnetsrdquo Physical Review Lettersvol 45 no 19 pp 1583ndash1586 1980

[11] N J Mottram and C Newton ldquoIntroduction to Q-tensortheoryrdquo Research Report no 10 University of StrathclydeMathematics Glasgow UK 2004

[12] S Kralj and A Majumdar ldquoOrder reconstruction patterns innematic liquid crystal wellsrdquo Proceedings of the Royal Society ofLondon vol 470 no 2169 2014

[13] M Ambrozic S Kralj and E G Virga ldquoDefect-enhancednematic surface order reconstructionrdquo Physical Review E Sta-tistical Nonlinear and Soft Matter Physics vol 75 no 3 ArticleID 031708 2007

[14] N Schopohl and T J Sluckin ldquoDefect core structure in nematicliquid crystalsrdquo Physical Review Letters vol 59 no 22 pp 2582ndash2584 1987

[15] N D Mermin ldquoThe topological theory of defects in orderedmediardquo Reviews of Modern Physics vol 51 no 3 pp 591ndash6481979

[16] S Kralj Z Bradac andV Popa-Nita ldquoThe influence of nanopar-ticles on the phase and structural ordering for nematic liquidcrystalsrdquo Journal of Physics Condensed Matter vol 20 no 24Article ID 244112 2008

[17] Y Dutil D R Rouse N B Salah S Lassue and L Zalewski ldquoAreview on phase-change materials Mathematical modeling andsimulationsrdquo Renewable Sustainable Energy Reviews vol 15 pp112ndash130 2011

[18] A M Borreguero M Luz Sanchez J L Valverde M Carmonaand J F Rodrıguez ldquoThermal testing and numerical simulationof gypsum wallboards incorporated with different PCMs con-tentrdquo Applied Energy vol 88 no 3 pp 930ndash937 2011

[19] S N AL-Saadi and Z Zhai ldquoModelling phase change materialsembedded in building enclosure a reviewrdquo Renewable andSustainable Energy Reviews vol 21 pp 659ndash673 2013

[20] A Guiavarch D Bruneau and B Peuportier ldquoEvaluationof thermal effect of pcm wallboards by coupling simplifiedphase change model with design toolrdquo Journal of BuildingConstruction and Planning Research vol 02 no 01 pp 12ndash292014

[21] G Cordoyiannis A Zidansek G Lahajnar et al ldquoInfluenceof confinement in controlled-pore glass on the layer spacingof smectic- A liquid crystalsrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 79 no 5 Article ID051703 2009

[22] Z Bradac S Kralj and S Zumer ldquoEarly stage domain coars-ening of the isotropic-nematic phase transitionrdquoThe Journal ofChemical Physics vol 135 no 2 p 024506 2011

[23] V Popa-Nita and S Kralj ldquoLiquid crystal-carbon nanotubesmixturesrdquoThe Journal of Chemical Physics vol 132 no 2 ArticleID 024902 2010

[24] R Repnik A Ranjkesh V Simonka M Ambrozic Z Bradacand S Kralj ldquoSymmetry breaking in nematic liquid crystalsAnalogy with cosmology and magnetismrdquo Journal of PhysicsCondensed Matter vol 25 no 40 Article ID 404201 2013

[25] A Ranjkesh M Ambrozic S Kralj and T J Sluckin ldquoCom-putational studies of history dependence in nematic liquidcrystals in random environmentsrdquoPhysical Review E StatisticalNonlinear and Soft Matter Physics vol 89 no 2 Article ID022504 2014

[26] S Kralj G Cordoyiannis A Zidansek et al ldquoPresmec-tic wetting and supercritical-like phase behavior of octyl-cyanobiphenyl liquid crystal confined to controlled-pore glass

matricesrdquo The Journal of Chemical Physics vol 127 no 15Article ID 154905 2007

[27] V Popa-Nita and S Kralj ldquoRandom anisotropy nematic modelNematic-non-nematic mixturerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 73 no 4 Article ID041705 2006

[28] M Ambrozic F Bisi and E G Virga ldquoDirector reorientationand order reconstruction competing mechanisms in a nematiccellrdquoContinuumMechanics andThermodynamics vol 20 no 4pp 193ndash218 2008

[29] M Slavinec E Klemencic M Ambrozic and M KrasnaldquoImpact of nanoparticles on nematic ordering in square wellsrdquoAdvances in Condensed Matter Physics vol 2015 Article ID532745 11 pages 2015

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T1 T2

휆PCMcPCM휌PCM

Figure 1 Schematic representation of two layers one integrating PCMs

source approach and finite difference method Next wepresent a theoretical Landau-de Gennes model to examinethe impact of the external electric field and NPs on the phasechange temperature of NLC system We present and discussresults in Section 3 In the final section we conclude ourfindings

2 Modeling

21 Thermal Stabilization with PCM In this section wedescribe the use of PCMs for the improvement of thermalstabilization We considered one-dimensional heat transferthrough the planar surface of a two-layered composite (Fig-ure 1) with one layer integrating homogeneously distributedPCMs Our numerical model is based on the heat sourceapproach [17ndash20] in which latent heat is modeled as anadditional heat source In terms of enthalpy (ℎ) the transientheat transfer equation in one dimension is expressed as

120588120597ℎ120597119905 = 12058212059711987921205971199092 (1)

where 120588 is density and 120582 thermal conductivity of each layerAssuming high porosity of the composite layer the

expansion of PCMs during the phase transition does notchange the total volume Additionally we considered differ-ent physical properties of the solid (119904) and liquid (119897) phase ofPCMs

In order to simplify their numerical simulations otherstudies [17 19 20] determined the temperature range wherephase transition occurs In this study we assumed that phasetransition is instant at one specific critical temperature119879119888Weintroduced the factor 119903 that equals zero when the PCM is inthe solid phase (119879 lt 119879119888) and one when the PCM is in theliquid phase (119879 gt 119879119888)

Using a model of effective physical quantities [18] wedefined the effective specific heat capacity (119888119890) thermalconductivity (120582119890) and density (120588119890) as follows

119888119890 = 120572 (119888(119897)PCM119903 + 119888(119904)PCM (1 minus 119903)) + (1 minus 120572) 119888119860 (2a)

120582119890 = 120572 (120582(119897)PCM119903 + 120582(119904)PCM (1 minus 119903)) + (1 minus 120572) 120582119860 (2b)

120588119890 = (1 minus 120572) 120588119860 + 120572 (120588(119897)PCM119903 + 120588(119904)PCM (1 minus 119903)) (2c)

where 120572 represents the relative amount of PCM integrated inthe composite layer and subscripts119860 and119861 correspond to twodifferent layers of studied composite (see Figure 1)

We derived the heat transfer equation for the compositeincorporating PCMs in terms of the enthalpy ℎ119890 = 119888119890119879+120572119903119876119871as follows

120588119890119888119890 120597119879120597119905 + 120588119890120572119876119871 120597119903120597119905 = 120582119890 12059711987921205971199092 (3)

where 119876119871 is the latent heat of the integrated PCM Fornumerical simulations we used the finite difference method

22 Phase Change Temperature Shift Next we theoreticallyanalyzed possible mechanisms to tune the phase changetemperature We considered the NLC rectangular cell wherelateral confinement walls impose strong homogeneous tan-gential anchoring (Figure 2) The characteristic confinementsize 119877 is assumed to be larger than the cell thickness ℎ

Using the nematic tensor order parameter [6] wedescribed LC orientational ordering as follows

Q = 3sum119894=1

119904119894997888119890 119894 otimes 997888119890 119894 (4)

where 997888119890 119894 are eigenvectors 119904119894 are eigenvalues of Q andthe symbol otimes represents tensor product In case of uniaxialordering it holds as follows [12]

Q = 119878 (997888119899 otimes 997888119899 minus 13 I) (5)

where I is the identity matrix


y

x

z

R

R

h




119891119890 = 1198712 |nablaQ|2 (6b)

119891119891 = 1205760Δ1205762 997888119864 sdot Q997888119864 (6c)

119891119904 = 1199082 Tr (Q minus Qs) (6d)



119879IN = 119879lowast + 1198612(41198600119862) (7)


119891119890 sim 1198712 11987821198772 (8)




lowast )3 1198782 minus 1198619 1198783 + 1198629 1198784

(9)

where



Δ119879IN (119877) = minus32 11987111986001198772 (11)



1205732 = 1 minus 6 (TrQ3)2(TrQ2)3 (12)





(13)




Δperp1199021(120585(0)119887119877 )2

minus 12059161199021 + 211990221199021 minus 11990212 (311990222 + 11990221 + 11990223)

+ 14 (120585(0)119887120585119891 )2

((997888119890 119909 sdot 997888119890 )2 minus (997888119890 119910 sdot 997888119890 )2) = 0(14a)

Δperp1199022(120585(0)119887119877 )2

minus 12059161199022 + 13 (11990221 + 11990223 minus 311990222)

minus 11990222 (311990222 + 11990221 + 11990223) + 112 (120585(0)119887120585119891 )2

= 0(14b)

Δperp1199023(120585(0)119887ℎ )2

minus 12059161199023 + 211990221199023 minus 11990232 (311990222 + 11990221 + 11990223)

+ 12 (120585(0)119887120585119891 )2

(997888119890 119909 sdot 997888119890 ) (997888119890 119910 sdot 997888119890 ) = 0(14c)








0 6 12 18 24 30 36 42 48

t (h)


Tc gt Troom0 PCM

minus06

minus04

minus02

00

02

04

06

qin

(rel

ativ

e uni

ts)

08

10

(a)

10

08

06

04

02

00

minus02

minus04

minus06

0 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(b)10

08

06

04

02

00

minus02

minus04

minus060 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(c)






0 6 12 18 24 30 36 42 48

t (h)

01

03

05

07

qin

(rel

ativ

e uni

ts)

09

10

00

02

04

06

08

BS1BS2GPCM


1

09

08

07

06

05

04

03

02

01

(a)

1

09

08

07

06

05

04

03

02

01

(b)

1

09

08

07

06

05

04

03

02

01

(c)







4 Conclusions



(a) (b) (c) (d)


(a) (b) (c) (d)




Data Availability




References



































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






y

x

z

R

R

h




119891119890 = 1198712 |nablaQ|2 (6b)

119891119891 = 1205760Δ1205762 997888119864 sdot Q997888119864 (6c)

119891119904 = 1199082 Tr (Q minus Qs) (6d)



119879IN = 119879lowast + 1198612(41198600119862) (7)


119891119890 sim 1198712 11987821198772 (8)




lowast )3 1198782 minus 1198619 1198783 + 1198629 1198784

(9)

where



Δ119879IN (119877) = minus32 11987111986001198772 (11)



1205732 = 1 minus 6 (TrQ3)2(TrQ2)3 (12)





(13)




Δperp1199021(120585(0)119887119877 )2

minus 12059161199021 + 211990221199021 minus 11990212 (311990222 + 11990221 + 11990223)

+ 14 (120585(0)119887120585119891 )2

((997888119890 119909 sdot 997888119890 )2 minus (997888119890 119910 sdot 997888119890 )2) = 0(14a)

Δperp1199022(120585(0)119887119877 )2

minus 12059161199022 + 13 (11990221 + 11990223 minus 311990222)

minus 11990222 (311990222 + 11990221 + 11990223) + 112 (120585(0)119887120585119891 )2

= 0(14b)

Δperp1199023(120585(0)119887ℎ )2

minus 12059161199023 + 211990221199023 minus 11990232 (311990222 + 11990221 + 11990223)

+ 12 (120585(0)119887120585119891 )2

(997888119890 119909 sdot 997888119890 ) (997888119890 119910 sdot 997888119890 ) = 0(14c)








0 6 12 18 24 30 36 42 48

t (h)


Tc gt Troom0 PCM

minus06

minus04

minus02

00

02

04

06

qin

(rel

ativ

e uni

ts)

08

10

(a)

10

08

06

04

02

00

minus02

minus04

minus06

0 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(b)10

08

06

04

02

00

minus02

minus04

minus060 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(c)






0 6 12 18 24 30 36 42 48

t (h)

01

03

05

07

qin

(rel

ativ

e uni

ts)

09

10

00

02

04

06

08

BS1BS2GPCM


1

09

08

07

06

05

04

03

02

01

(a)

1

09

08

07

06

05

04

03

02

01

(b)

1

09

08

07

06

05

04

03

02

01

(c)







4 Conclusions



(a) (b) (c) (d)


(a) (b) (c) (d)




Data Availability




References



































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








(13)




Δperp1199021(120585(0)119887119877 )2

minus 12059161199021 + 211990221199021 minus 11990212 (311990222 + 11990221 + 11990223)

+ 14 (120585(0)119887120585119891 )2

((997888119890 119909 sdot 997888119890 )2 minus (997888119890 119910 sdot 997888119890 )2) = 0(14a)

Δperp1199022(120585(0)119887119877 )2

minus 12059161199022 + 13 (11990221 + 11990223 minus 311990222)

minus 11990222 (311990222 + 11990221 + 11990223) + 112 (120585(0)119887120585119891 )2

= 0(14b)

Δperp1199023(120585(0)119887ℎ )2

minus 12059161199023 + 211990221199023 minus 11990232 (311990222 + 11990221 + 11990223)

+ 12 (120585(0)119887120585119891 )2

(997888119890 119909 sdot 997888119890 ) (997888119890 119910 sdot 997888119890 ) = 0(14c)








0 6 12 18 24 30 36 42 48

t (h)


Tc gt Troom0 PCM

minus06

minus04

minus02

00

02

04

06

qin

(rel

ativ

e uni

ts)

08

10

(a)

10

08

06

04

02

00

minus02

minus04

minus06

0 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(b)10

08

06

04

02

00

minus02

minus04

minus060 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(c)






0 6 12 18 24 30 36 42 48

t (h)

01

03

05

07

qin

(rel

ativ

e uni

ts)

09

10

00

02

04

06

08

BS1BS2GPCM


1

09

08

07

06

05

04

03

02

01

(a)

1

09

08

07

06

05

04

03

02

01

(b)

1

09

08

07

06

05

04

03

02

01

(c)







4 Conclusions



(a) (b) (c) (d)


(a) (b) (c) (d)




Data Availability




References



































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






0 6 12 18 24 30 36 42 48

t (h)


Tc gt Troom0 PCM

minus06

minus04

minus02

00

02

04

06

qin

(rel

ativ

e uni

ts)

08

10

(a)

10

08

06

04

02

00

minus02

minus04

minus06

0 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(b)10

08

06

04

02

00

minus02

minus04

minus060 6 12 18 24 30 36 42 48

t (h)

qin

(rel

ativ

e uni

ts)


Tc gt Troom0 PCM

(c)






0 6 12 18 24 30 36 42 48

t (h)

01

03

05

07

qin

(rel

ativ

e uni

ts)

09

10

00

02

04

06

08

BS1BS2GPCM


1

09

08

07

06

05

04

03

02

01

(a)

1

09

08

07

06

05

04

03

02

01

(b)

1

09

08

07

06

05

04

03

02

01

(c)







4 Conclusions



(a) (b) (c) (d)


(a) (b) (c) (d)




Data Availability




References



































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






0 6 12 18 24 30 36 42 48

t (h)

01

03

05

07

qin

(rel

ativ

e uni

ts)

09

10

00

02

04

06

08

BS1BS2GPCM


1

09

08

07

06

05

04

03

02

01

(a)

1

09

08

07

06

05

04

03

02

01

(b)

1

09

08

07

06

05

04

03

02

01

(c)







4 Conclusions



(a) (b) (c) (d)


(a) (b) (c) (d)




Data Availability




References



































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






(a) (b) (c) (d)


(a) (b) (c) (d)




Data Availability




References



































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





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