International Journal of Heat and Mass Transfer 79 (2014) 954–967

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Effect of interface on the thermal conductivity of thermal barriercoatings: A numerical simulation study

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.08.0880017-9310/Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Key Laboratory of Inorganic Coating Materials,Chinese Academy of Sciences, Shanghai 201899, PR China. Tel.: +86 02169906320; fax: +86 021 69906322.

E-mail addresses: L.Wang@mail.sic.ac.cn, glacier_hit@126.com (L. Wang).

L. Wang a,b,⇑, X.H. Zhong a, Y.X. Zhao a,c, J.S. Yang a,c, S.Y. Tao a, W. Zhang b, Y. Wang d, X.G. Sun d

a Key Laboratory of Inorganic Coating Materials, Chinese Academy of Sciences, Shanghai 201899, PR Chinab State Key Laboratory of High Performance Ceramics and Superfine Microstructure, Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 200050, PR Chinac University of the Chinese Academy of Sciences, Beijing 100039, PR Chinad Laboratory of Nano Surface Engineering, School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China

a r t i c l e i n f o

Article history:Received 20 December 2013Received in revised form 30 August 2014Accepted 30 August 2014

Keywords:Thermal barrier coatingsThermal insulationInterfaceFinite element simulationPhysical mechanism

a b s t r a c t

Interface is an important structure in the materials, some rather peculiar physical phenomena can occurat the interface under the application of the exterior physical field. Especially, the grain boundary, phaseinterface and layer interface are significant factors for improving the thermal insulation behavior of ther-mal barrier coatings (TBCs). In this paper, finite element method was employed to simulate the heattransfer behavior of TBCs with different interfacial characteristic based on several different interfacialthermal resistance (ITR) models. The simulation results indicate that the heat flux around the interfacehas exhibited fantastic changing characteristic, the thermal insulation effect of TBCs would be enhancedwith the area of the interface increasing. The interface roughness (amplitude) also has a very importanteffect on the effective thermal conductivity of the as-sprayed TBCs. A novel method, ComputationalMicromechanics Method (CMM), was utilized to depict the heat transfer behavior of actual coatings withirregular inner interface. The ‘‘thermal rectification’’ mechanism of heat diffusion around the interfacemake that the heat flux which passes through the interface has exhibited different flow characteristiccompared with the positions without interface. In addition, some simple experiments have furtherverified the existence of the ITR between the bond-coat and top-coat. The investigation results will alsoprovide us a powerful guide to design coating with high thermal insulation property using the physicaltheory and mechanism of the ITR.

Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Thermal barrier coatings (TBCs) are very important structuraland functional ceramic coating materials. The aspect of the thermalinsulation reflects the functional characteristic. Very complicatedphysical mechanism exists in the process of heat transfer in theTBCs which should be understood deeply. Generally, the typicalTBCs are composed of double layers (metallic layer and ceramiclayer). The Y2O3 partially stabilized ZrO2 (YSZ) layer which wasusually acted as the thermal insulation layer in the TBCs has lowthermal conductivity and plays a key and vital role in protectingthe underlying super-alloy substrates, reducing the workingtemperature and increasing working efficiency of hot sectioncomponents [1–3]. However, the durability requirements of TBCs

for these applications are increasing rapidly with the developmentof the modern power industry. A higher temperature gradient willalso be required. The most direct route to ensure that hot sectioncomponents coated with TBCs have high Thrust-Weight ratio anda long lifespan is to improve the thermal insulation property.Demand for TBCs with excellent thermal insulation performanceis becoming more and more urgent [4].

There are many factors that will affect the effective thermalconductivity of the TBCs, such as the intrinsic thermal conductivityof the ceramic layer of TBCs, the microstructure of the TBCs (archi-tecture of the pores and cracks) and so on. Pores are oftenbeneficial to decrease the thermal conductivity of the TBCs. Somenano-sized pores can also play a special role in changing the ther-mal shift or heat diffusion behavior in the coatings [5]. Cracks alsoplay an important role in changing the thermal conductivity of theTBCs. Horizontal cracks may be helpful to increase the thermalinsulation of the APS-TBCs, but segmentation cracks in the TBCsmay be harmful to decrease the thermal insulation, although theycan improve the thermal shock life as it can release some stress

L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967 955

concentration in the APS TBCs. In addition, the effective thermalconductivity of the gas trapped within the pores is influencednot only by the temperature and pressure, but also by the dimen-sion of the pores at the micro-scale level [6]. This effect dependsupon the Knudsen number, and is called Knudsen effect or rarefac-tion effect. Especially for submicron or nanoscaled pores, thedecrease in the thermal conductivity of the trapped gas appearsobvious [7,8]. Previous investigations have further shown thatpores are of consequences in decreasing the thermal conductivityof materials, and the heat flux around the pores exhibited specialflow state compared with the other positions without defects[9,10]. Golosnoy et al. [11] have reviewed the actual physicalmechanism of how heat transfer takes place in plasma-sprayed(zirconia-based) TBCs during operation of gas turbines. Theythought that the pore architecture (i.e., its morphology, connectiv-ity and scale) has a strong influence on the heat flow. The contribu-tions from convective, conductive and radiative heat transfer areconsidered under a range of operating conditions. The characteris-tics are illustrated with experimental data and modelingpredictions. But the role of interface role has seemed to be omitted.In fact, interface is a very important phase in the compositesystem. The physical field in the interface often has the fancifulphenomenon [12,13].

When heat flows across an interface between two differentmaterials (phases) or two adjacent layers, there exists a tempera-ture jump at the interface, then the interfacial thermal resistance(ITR) can be defined as follows:

R ¼ DT=J ð1Þ

where J is the heat flux density, namely, the heat flow across a unitarea in unit time. DT is the temperature difference between twosides of the corresponding interface. R is the value of the ITR. Thisproblem was recognized as early as 1941 when Kapitza [14]discovered the temperature jump at an interface between solidand liquid. Generally, as a result of the temperature discontinuityat interfaces, a multi-phase or composite material will exhibit areduced effective thermal conductivity due to the existence of theITR.

Computational simulation is an effective routine to solve theITR and the related heat conduction problems. Usually, two majorcomputational methods have been used, i.e. micro-scale simula-tion and macro-simulation. Molecular dynamics (MD) simulationis an atomic-level (nano/sub-micro scale) method which has beenused to simulate the thermal transport across micro-interfaces. Itcan describe the physical characteristic of the ITR actually. Gener-ally, in the process of the MD simulation, the positions andmomenta of a set of atoms evolve according to Newtonian equa-tions of motion, restricting the validity of the atoms’ movementto the classical limit. In this limit, MD simulation is an ideal andeffective method for predicting thermal boundary resistance dueto that the nature of the phonon scattering are not necessary beingconsidered. The only required input to an MD simulation is theatomic interaction (i.e. potential function) [15]. The computationalefficiently will improve as the classical interatomic potentials donot necessary to consider the electronic properties. The details ofthe thermal transport physics can be studied using lattice dynam-ics (LD) or non-equilibrium Green’s function (NEGF)-basedmethod. The ITR can be predicted and obtained by evaluating atheoretical expressions using the phonon properties which are cal-culated from these methods [16,17]. The LD and NEGF based meth-ods can be used to obtain phonon properties at temperaturesbelow the Debye temperature where quantum effects are impor-tant. The LD method can be also viewed as an equilibrium MDbased approach which uses fluctuation dissipation theorem orthe Green–Kubo method. It was usually used to calculate the heatflux as the function of the time in a balance system, and the

thermal conductivity can be obtained by the Green–Kubo equationeventually. As for the NEGF-based approach, the effects of inelasticscattering can be modeled using this method. And the harmonicapproximation is typically applied in order to reduce the computa-tional expense when modeling interfaces [18–20].

Tomar et al. [21] have investigated the correlations betweenthermal conduction and mechanical strength in ZrB2/SiC interface.Their analyses indicate that the strength reduction with increase intemperature is strongly correlated to phonon and electron thermaldiffusivity change. With increase in temperature, phonon thermaldiffusivity increases in the case of ZrB2 and reduces in the casesof SiC as well as the interface. Tomar et al. [22,23] have also indi-cated that the thermalmechanical behavior or mechanical strengthof the microstructural interface (including the phase interfaces inthe multicomponent materials and grain boundary) in the materi-als can be also calculated by MD, and the classical MD replaces acomprehensive quantum mechanical treatment of interatomicforces with a phenomenological description in the form of an inter-atomic potential.

Indeed, in the field of the micro-scale heat transfer, the ITR ofdifferent material system is the hot topic of the investigation inthe current, as the ITR will affect the heat transfer characteristicdirectly, and thus affect the design of the micro/nano componentsand the heat optimization.

Besides the MD simulation, finite element simulation (macro-scale simulation) combined with the experimental measurementis also an effective method to investigate the problem of the ITR.Based on the fact that the TBCs must have a certain thermal insu-lation capability, the typical structural characteristic of TBCs is fullof defects (pores, voids and micro-cracks) in the ceramic coating.Some previous work investigated the influence of the interfacialconductance on the thermal conductivity of the TBCs from theexperimental methods [24], and some macro-scale numerical(finite difference) and analytical models have been developed forthe simulation of heat flow through plasma-sprayed coatings[25]. The effective thermal conductivity of the porous air plasmasprayed TBCs was strongly dependent on its intrinsic structure,such as pores, cracks, pore/coating interface, crack/coating inter-face, layer interface. Besides the scattering effect at the defects orinterfaces, the ITR is another important factor which will affectthe eventual effective thermal conductivity of the APS-TBCs. Theradiative properties on the effective thermal conductivity of theTBCs has been investigated and extensively developed by Zhao’sgroup [26]. They have developed a Finite-Difference-Time-Domain(FDTD) method which was employed to simulate the radiative heattransfer behaviors of TBCs with different types of microstructures[27]. In this paper, the ITR will be focused on. The current work willtry to investigate the influence of the interface on the heat transferbehavior of the TBCs fabricated by APS from both the aspects offinite element simulation and the physical model of heterogeneousmaterial systematically. Especially, the methods of the averaget-matrix approximation (ATA) and coherent-potential approxima-tion (CPA) have been used to describe the effective thermalconductivity of the actual TBC model. The ‘‘thermal rectification’’of heat diffusion around the interface in the TBC has been calcu-lated and discussed in detail based on several different physicalmodels of the ITR.

2. Simulation method and procedure

2.1. Model basis for finite element analysis

Defects (pores, voids and micro-cracks) are usually distributedat random in the as-sprayed TBCs which was coated on the turbineblades (Fig. 1(a)) due to the complicated physical process of

Hot gas

A-super-alloy substrate, B-metallic bond-coatC- ceramic top-coat

ΔT

AC

B

Cooling gas

Cooling gas temperature

Gas temperature

Tem

pera

ture

TGO

TBC

20μm 2μmMicrocrack

Inter-splat interface F2

F5

F1

F3

F4

F1:Phonon conduction F2:radiation scattering at GBsF3:Knudsen conduction F4:Gas phase conduction (λ~50nm)F5: Radiation scattering at interfaces[8,11]

a) b)

c)

Fig. 1. Heat transfer mechanism in the plasma-sprayed TBCs which coated on the turbine blades (a) Turbine blades which are coated with TBCs (b) thermal gradient in theTBCs (c) schematic illustration of heat transfer mechanism in the plasma-sprayed TBCs.

956 L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967

thermal spraying. The thermal gradient across the whole coatingsystem in the through-thickness direction (spray direction) canbe seen in Fig. 1(b). In fact, in the process of the thermal spraying,the subsequent droplets impinge onto the former droplets and pro-duce plastic strain, transferring heat into the former solidifieddroplets (splats), and pores among different splats will be createdas the adjacent splats do not overlap fully. When the feedstock par-ticles fly from the spray nozzle and go through different positionsof plasma flame, their velocities and temperatures are different.Different splats have different melting conditions that result in dif-ferent shrinking volumes after their impacting on the formersplats. Therefore, there are amounts of interfaces among the splatsand between the defects and the dense coating. Typical feature ofthe as-sprayed coating is shown in Fig. 1(c). There are five factorswhich play determined role in affecting the thermal conductivityof the as-sprayed TBCs. The first one is the phonon conductionwhich is caused by the crystal lattice vibration, the second one isthe radiation scattering at grain-boundaries which will be calcu-lated and discussed in the subsequent section, the third one isthe Knudsen conduction which is induced in the inter-splat inter-face, the fourth factor is the gas phase conduction, especially thewavelength is about 50 nm, the last factor is the radiation scatter-ing at all kinds of interfaces, such as the splat interfaces, the inter-faces between the coating and defects and so on [17,24]. In thispaper, the influence of interface on the heat diffusion behavior willbe focused on.

2.2. Model description and materials parameters

The coating sample is assumed to be cylinder plate like, two-dimension (axisymmetric) mathematical perfect model withoutpores as shown in Fig. 2 is established in the current work firstly.Then, pores with different spatial and geometrical characteristicswill be introduced into the model. The temperature of the top coat-ing surface was set as 1473 K. And the simulation time is assumedto be long enough, the sample will not experience thermal cycleprocess at high temperature (usually over 1473 K), so the ther-mally grown oxide (TGO) need not be considered. The model hasthree layers: DZ125 Ni-based superalloy substrate, NiCoCrAlYbond-coating (dense, without defects), ceramic top coating. Thethickness of the substrate, bond coating and top coating are6 mm, 100 lm and 300 lm respectively, and the radius of the cyl-inder sample is 10 mm. The property parameters of the modelcame from Refs. [28,29].

2.3. Simulation method, initial and boundary conditions andsimulation process

In the current work, the thermal analysis was performed withthe ANSYS commercial available software package (afflicted withANSYS finite element code-APDL). The plane 55 element (fournodes element) and plane 35 element (6-node triangular element)were used in the model without defects and with defects. Both

I-substrate

II-bond coating

III-top coating

O

O’

radius axis

I-substrate

II-bond coating

III-top coating

symmetry axis“unit cell”

I

IIIII

dT/dn=0

Fig. 2. Model used in the finite element simulation.

L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967 957

plane elements have one degree of freedom, temperature, at eachnode [30]. According to the basic theory of Fourier’s law of heatconduction in three dimension direction, the heat conductionequation can be expressed as:

@

@xkx@T@x

� �þ @

@yky@T@y

� �þ @

@zkz@T@z

� �þ qm ¼

@

@tqcTð Þ ð2Þ

Where T is the transient temperature, t is the time, q is the density,kx, ky, kz are thermal conductivities in x, y and z direction, respec-tively. C is the heat capacity, qv is the intensity of the internal heatsource in the corresponding space.

As for the process of fabrication and cooling of plasma sprayedcoating, the intensity of the internal heat source, i.e. qv is equal tozero. And only the heat conduction in the through-thickness direc-tion (spray direction) has been considered, so the above equationcan be simplified as follows:

@T@t¼ k

qCp

@2T@y2 ð3Þ

When the left and right boundaries of the coating model are consid-ered to be adiabatic, i.e. dT/dn = 0, the effective thermal conductivityin the through thickness direction (spray direction) can becalculated from [31]:

keff ¼h

DTW

Zsbot

kðnrTÞds ð4Þ

where h and W is the average thickness and the width of the coatingmodel, respectively, DT is the temperature difference between thetop side and bottom side of the coating model, and sbot is the lowerhorizontal boundary of the coating model with normal vector n. Itcan be further seen that the effective thermal conductivity keff isinversely proportional to the temperature difference DT accordingto the formula (4). The boundary conditions can be described as fol-lows: the constant initial temperature was prescribed on the sur-face of the top-coating. The heat convection coefficient betweenthe environment and the backside of the substrate was set ashs = 150 W/(m2 K), the environmental temperature was defined asTa = 298 K. Heat transfer induced by the heat radiation wasneglected. The left and right boundaries are considered to be adia-batic, i.e. dT/dn = 0 was imposed on both the left side and right sideof the finite element model. The solution time was set as 600 swhich is long enough, when the entire coating system reach tothe thermodynamic equilibrium state, i.e. the distribution of nodetemperature, heat flux and thermal gradient kept constant as thesolution time prolong. The mesh is fine and dense enough, as thesolution of the nodal value and the element value of the entire

model have the same order of magnitude, and the relative error istiny. Then the simulation results obtained from the current finiteelement model with enough element number can be acceptable.In addition, it is well known that the radiative properties of the TBCsat high temperatures will also play an important role in changingthe heat transfer behavior of the as-sprayed TBCs [32], but it wasnot considered in the current work as the simulated temperatureis not high enough. The simulation results were extracted fromthe ANSYS visualization when the solution steps are completed.

When the left and right boundaries of the ‘‘unit cell’’ areassumed to be adiabatic, i.e. dT/dn = 0, the heat flux from the topto the bottom of the ‘‘unit cell’’ is equal to that of the exchangeof the heat flux between the bottom and the environment, so

qin ¼ �qout ð5Þkeff ðT1 � T2Þ

dc¼ hsðT2 � TaÞ ð6Þ

where T1 and T2 are temperatures imposed on the top and the bot-tom of the ‘‘unit cell’’, hs is the heat convection coefficient betweenthe bottom of the finite element model and the environment, Ta isthe environment temperature which can be defined as constant,dc is the thickness of the ‘‘unit cell’’.

Based on the description above, the effective thermal conduc-tivity keff of the ‘‘unit cell’’ can be calculated as:

keff ¼hsðT2 � TaÞdc

ðT1 � T2Þ¼ hsðT2 � TaÞdc

DTð7Þ

2.4. Computational Micromechanics Method (CMM) simulationprocedure

The Computational Micro-mechanic (CMM) Methods can bedescribed as follows: Firstly, the scanning image was treated inthe suitable image processing software, and the figure was savedas the format .DXF. Then the figure was further polished in theoperation interface of the AutoCAD software. The figure was fur-ther saved as the format of .DXF. The saved file was converted toAPDL (ANSYS Parametric Design Language) code in the GraphicUser Interface (GUI) of the ANSYS Software by the program devel-oped by us. Then an image which can be used to simulate in theANSYS can be obtained.

The porosity of the as-sprayed coating can be calculated as fol-lows: Firstly, several zones (here the number of zones is denoted asN) of a SEM image of the cross section of the as-sprayed coatingwere selected at random, then the image from one zone Xi wasconverted to a digital image with mi � ni pixel which can bedefined as a matrix G(Xi):

958 L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967

GðXiÞ ¼

gð0;0Þ gð0;1Þ . . . gð0;ni � 1Þgð1;0Þ gð1;1Þ . . . gð1;ni � 1Þ

. . . . . . . . . . . .

gðmi � 1;0Þ gðmi � 1;1Þ . . . gðmi � 1; ni � 1Þ

26664

37775 ð8Þ

As for a single layer coating, only the pore phase and the densecoating phase is considered, so the g (n,f) can be defined as:

gðn; fÞ ¼0 ðwith a poreÞ

1 ðwithout a poreÞ

�0 6 n 6 mi � 1;0 6 f 6 ni � 1ð Þ

ð9Þ

where g (n,f) indicate a pixel point. As for the image from one zoneXi, the porosity can be calculated as:

uðXiÞ ¼Ni gðn; fÞ ¼ 0½ �

mi � ni� 100% ð10Þ

where Ni [g (n,f) = 0] represents the number of points whereg (n,f) = 0.

The mean porosity of the as-sprayed coating (u) can be deter-mined as:

u ¼PN

i¼1uðXiÞN

� 100% ð11Þ

2.5. Measurement of effective thermal conductivity of the free standingcoating

Generally, the thermal conductivity k (W/m K) of the freestanding coatings can be determined by:

k ¼ a � Cp � q ð12Þ

where q is the density of the free standing coating (kg/m3) Cp

indicates the heat capacity of constant pressure (J/(kg K)), a is the

(a)

0 50 100 150 214001405141014151420142514301435144014451450145514601465147014751480

withou with in

(c)

T (K)

Tem

pera

ture

(K)

Dislocati

Top-coat

Bond-coat

A

B

Fig. 3. The temperature distribution (a) without interface thermal resistance (b) with incenter) when the interface between the bond-coat and top-coat is flat.

thermal diffusion rate (m2/s). The free standing coatings with thesuitable thickness were prepared by plasma spraying on the graph-ite substrate with the diameter of 20 mm, and the as-sprayed freestanding coatings were obtained by peeling off the coatings fromthe graphite substrate. The thermal diffusion rate and the specificheat of the corresponding freestanding coatings were measuredby the LFA 427 laser flash (Made by NETZSCH Company, Germany).The density of the free standing coatings can be measured by Archi-medes drainage method. And then the effective thermal conductiv-ity of the as-sprayed freestanding coating can be calculated byEq. (12).

3. Results and discussion

In this section, the influence of interface with different spatialand geometrical characteristics on the effective thermal conductiv-ity of TBCs will be discussed in detail. Especially, the ‘‘thermal rec-tification’’ phenomenon of heat diffusion around the interface inthe TBCs has also been analyzed and discussed based on the phys-ical models of the ITR.

3.1. Interface between the metallic layer and the ceramic layer

Fig. 3 shows the temperature distribution of TBCs when theinterface between the metallic layer and the ceramic layer is flat.It can be seen that when the ITR is considered, the insulation tem-perature has increased, which means the effective thermal conduc-tivity decrease. The insulation temperature (the temperaturedifference between the top and the bottom of the model) withoutconsidering the ITR is 35 K. The effective thermal conductivity is1.30 W/(m K). But when the ITR was considered, the insulationtemperature increases to 68.0 K, and the effective thermal conduc-tivity decreases to 0.65 W/(m K) (Fig. 3(a) and (b)). From Fig. 3(c), it

(b)

00 250 300 350 400

t interface thermal resistanceterface thermal resistance

Interface

Top-coat Bond-coat

T (K)

on (μm)

Top-coat

Bond-coat

A

B

terface thermal resistance (c) along the path AB (from the top center to the bottom

A

B

C

D

(b)T(K)

Top-coat

Bond-coat

Top-coat

Bond-coat

(a)

0 50 100 150 200 250 300 350 4001405141014151420142514301435144014451450145514601465147014751480

without thermal resistance along the path AB with thermal resistance along the path AB with thermal resistance along the path CD

(c)

T(K)

Tem

pera

ture

(K)

0 5 10 15 20 25 30

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

with interface thermal resistance without interface thermal resistance

(d)

Effe

ctiv

e th

erm

al c

ondu

ctiv

ity(K

)

dislocation (μm) amplitude (μm)

Fig. 4. The temperature distribution (a) without interface thermal resistance (b) with interface thermal resistance (c) along the path AB (from the top center to the bottomcenter) when the interface exhibit sinusoidal curve characteristic.

L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967 959

can be seen that when the interface was introduced, the tempera-ture has exhibited a jump behavior around the interface, the inter-face indeed play an important role in impeding the transfer of theheat flux at the interface. The ‘‘thermal rectification’’ of the heatdiffusion around the interface has been shown, which eventuallyresult in the temperature jump near the interface. This phenome-non has also been observed and verified in Li’s work [33–35]. The‘‘thermal rectification’’ characteristic tends to occur around theinterface in the multi-phase or composite materials.

Fig. 4 shows the temperature distribution of the TBCs when theinterface has exhibited the sinusoidal periodic morphology. It canbe seen that when the ITR is considered, the insulation temperaturehas increased, which means the effective thermal conductivitydecrease. The insulation temperature without considering the ITRis 35.1 K, the effective thermal conductivity is 1.30 W/(m K). Butwhen the ITR was considered, the insulation temperature increaseto 64.9 K, and the effective thermal conductivity decrease to0.68 W/(m K) (Fig. 4(a) and (b)). From Fig. 4(c), it can be seen thatwhen the interface was introduced, the temperature has exhibiteda jump behavior around the interface. The interface indeed plays animportant role in impeding the transfer of the heat flux. It can bealso seen that the effective thermal conductivity has exhibited theincreasing tendency with the amplitude of the sinusoidal curve(i.e. increasing the roughness of the interface). It must be speciallypointed out that the ‘‘thermal rectification’’ of heat diffusion aroundthe positions of ‘‘wave crest’’ and ‘‘wave trough’’ of the interface hasnot exhibited different characteristic obviously, but the amplitudeof the interface will have an important influence on the tempera-ture distribution and the effective thermal conductivity.

3.2. Interface in the ceramic layer

Fig. 5(a) shows the model of the TBCs with spherical poreswhen considering the ITR between the pores and the coating.

Fig. 5(b) shows the model of the TBCs with spherical pores whenconsidering the ITR between the ellipsoid pores and the coating.It can be seen that the spherical pores have not the orientation,but the direction of the ellipsoid pores can be further considered.The angle between the long axis of the ellipsoid pores and theinterface between the metallic layer and the ceramic layer (normalto the thermal spray direction) is defined as orientation angle h. Itcan be seen from Fig. 5(c), the effective thermal conductivitydecrease with the pore size increasing and the effective thermalconductivity of the TBCs considering the ITR is lower than that ofthe TBCs without considering the ITR. With the increasing of thepore sizes, the effective interface area also increase, the interfaceplays an important role in impeding the transfer of the heat flux.Fig. 5(d) shows the effective thermal conductivity of the TBCs withthe ellipsoid pores as the function of the orientation angle h. It canbe seen that the effective thermal conductivity is the lowest whenthe long axis is perpendicular to the thermal spray direction (i.e.orientation angle is equal to zero degree), and the effective thermalconductivity is the highest when the long axis is parallel to thethermal spray direction (i.e. orientation angle is equal to 90�),and when the ITR is considered, the difference between the effec-tive thermal conductivity of TBCs considering the ITR and withoutconsidering the ITR when the orientation angle is equal to 0� reachto the maximum value. The simulation results further indicate thatthe ‘‘thermal rectification’’ of heat diffusion around the coating/pore interface will take an effect when the orientation of the poreshas changed.

Fig. 6 shows the effective thermal conductivity of the TBCs withand without pore/coating ITR under the condition of different poresizes (ellipsoid pores, the size was defined as a � b, where a and bis the length of long and short axis of the ellipsoid pore, respec-tively). It can be seen that the effective thermal conductivity willdecrease with the pore size increasing, and the effective thermalconductivity of the TBCs considering ITR is lower than that of the

(a) (b)

10 12 14 16 18 20 220.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3 without pore/coating interface thermal resistance with pore/coating interface thermal resistance

(c)

θ

K*

(W/m

.K)

R(μm)-20 0 20 40 60 80 100 120 140 160 180 200

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35 without pore/coating interface thermal resistance with pore/coating interface thermal resistance

(d)

K*

(W/m

.K)

θ (deg.)

Top-coat

Bond-coat

“Unit cell”

Fig. 5. The effective thermal conductivity of TBC considering the pores with different size and orientation and the interface thermal resistance model of (a) spherical pores (b)ellipsoid pores, (c) and (d) are the corresponding finite element simulation results of effective thermal conductivity of (a) and (b), respectively.

Pore size (μm2)

K*

(W/m

.K)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3 without pore/coating phase interface thermal resistance with pore/coating phase interface thermal resistance

15.7×8 19.8×9.1 20×10.2 22.2×11.3 24.3×12.4

Fig. 6. The effective thermal conductivity of the TBCs with and without pore/coating interface thermal resistance under the condition of different pore sizes(ellipsoid pores, the size was defined as a � b, where a and b is the length of longand short axis of the ellipsoid pore).

960 L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967

TBCs without considering the ITR. Although this, if there are toomany pores in the actual coating, the strength of the coating willdecrease. Therefore, the expandable scale of the pore size is lim-ited. It is a good advice to increase the pore sizes without sacrific-ing the coating strength. The simulation results further indicatethat the ‘‘thermal rectification’’ of the heat diffusion around theinterface can be amplified when there existed amounts of coat-ing/pore interfaces or increasing the size of the interface, the ‘‘ther-mal rectification’’ has a superimposed characteristic.

As the pores existed in the coating are irregular (with differentshapes and different sizes) and distributed at random, it will bemore realistic to predict the effective thermal conductivity of theTBCs based on the actual microstructure. Gupta’s work hasreported the OOF (object originate finite element) [36] methodwhich was used to solve the effective thermal conductivity ofactual coating. But in this paper, a novel method based on actualimage, Computational Micromechanics Method (CMM) wasemployed in order to reflect the thermal conductivity of actualcoating considering the ITR more factually. The coating can beviewed as composite materials composed of pores and dense coat-ing phases. Fig. 7(a) just shows the primitive SEM image of theceramic layer of TBCs. It can be seen that there are many pores(with different size and different shape, distributed at random)

(d)

(b)(a)

20µm

T(K)

q(W/m2)

q(W/m2)

(c) (e)

(f)

T(K)

Fig. 7. Image treated by CMM (a) primitive SEM image of ceramic layer of TBC (b) finite element model, (c) temperature distribution and (d) distribution of heat flux in the Ydirection (along the spray direction) without interface thermal resistance [37] (e) temperature distribution and (f) distribution of heat flux in the Y direction (along the spraydirection) with interface thermal resistance.

L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967 961

and cracks in the ceramic layer. The influence of cracks on the ther-mal conductivity is not considered. The primitive SEM image willbe converted to a model with two typical zones by the programdeveloped by us (Fig. 7(b)). Although some places were not trans-formed well, it still reflects the characteristic of the defects approx-imately. The pore volume fraction is about 9.63%. The finiteelement mesh can be generated in the ANSYS Graphic User Inter-face (GUI) environment. The mesh reflects the given distributionof material properties since each element can be assigned a partic-ular material property which is defined by the average pixel valuebeneath it. The thermal resistances of the irregular interfaces havealso been considered. Fig. 7(c) and (d) show the temperature distri-bution in the ceramic layer for the actual coating material withoutITR and with ITR, respectively. The boundary line of temperaturefield distribution is not flat as shown before due to the presenceof irregular pores. The thermal insulation temperature is 58.7 Kconsidering the ITR. The effective thermal conductivity is 0.86 W/

(m K). If the ITR is not considered, the insulation temperature is55 K and the effective thermal conductivity is 0.94 W/(m K) whichcan be also seen in our published paper [37]. Fig. 7(e) and (f) showthe heat flux distribution in the ceramic layer without ITR and withITR, respectively. When the ITR is considered, It can be seen fromFig. 7(f) that the heat flux in the pores filled with air will havethe lowest heat flux and the direction is just opposite to that ofthe other zones, and both the maximum absolute value of the heatflux is located at the bottom of the coating and near the edge of theirregular pores. In fact, it must be specially pointed out that theheat flux in the pore is plus which means the heat flux is alongthe y direction, so the effective thermal conductivity of the poresis negative when considering the whole composite system. Thisis very interesting that the thermal conductivity is negative whichhas also been reported in these literatures [38–43]. The ‘‘thermalrectification’’ of the heat diffusion around the interfaces in the TBCsmakes the heat flux exhibit this fantastic characteristic and result

962 L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967

in the decrease of the effective thermal conductivity of the TBCssignificantly. Compared with Fig. 7(e), this interesting phenome-non seems not happen. The maximum heat flux value is higherthan that of the case without ITR. In addition, it an be further con-cluded that it will be meaningful to predict the effective thermalconductivity or insulation temperature of actual TBCs using CMMbased on the actual SEM image of the coating fabricated by APSand using the ‘‘thermal rectification’’ theory of heat diffusionaround the coating/pore interfaces.

3.3. Interface among the grains

Although pores indeed play an important role in reducing theeffective thermal conductivity of the as-sprayed TBCs, the intrin-sic thermal conductivity of material also needs to be considered,especially conventional materials and nanostructured materials.In the preceding section of this paper, only pores were consid-ered in the finite element simulation and CMM analysis. Theintrinsic thermal conductivity parameters of materials withoutdefects were imported to the finite element mode. There is ahigh probability that phonon–phonon interactions will occurwhereby the mean free path will be further reduced in nano-structured TBC compared with conventional TBC at relativelylow temperature (usually below 1200 �C), because there aremore grain boundaries and internal micro-interface, and the dis-tance between the interface decreases in nanostructured TBC.According to Debye theory about the thermal dynamics analysis,the thermal conductivity of nanostructured TBC is reduced dueto the decrease of mean free path for phonon scattering com-pared with that of conventional TBC. Particularly when the grainsize is of the same order as the mean free path for phonon scat-tering [44], the thermal conductivity of the nanostructured TBCwill be drastically reduced.

Based on the fact that the thermal conductivity of pure ZrO2

ceramic has a certain relationship with the grain size, suppose thatthe grain boundary as the function of the grain size, the grainwidth and grain size are W and d, respectively (Fig. 8).

Considering the area A, the grain boundary area Sgrain can bedetermined as follows:

Phonon wave

Phonon wave

Fig. 8. Elastic wave pass through the (a) conventional ZrO2 grain (b) nanostructuredZrO2 grain.

K�11 ¼ K�22 ¼ Km 2þ f b11ð1� L11Þ 1þ hcos2 hi� �

þ b33 1� L33ð Þ 1� hcos2��

2� f b11L11 1þ hcos2 hið Þ þ b33L33 1� hcos2 hið Þ½ �

K�33 ¼ Km 1þ f b11ð1� L11Þ 1� hcos2 hi� �

þ b33ð1� L33Þhcos2 hi�

1� f b11L11 1� hcos2 hið Þ þ b33L33hcos2 hi½ �

Sgrain ¼Affiffi

3p

4 � d2

� �2 � 6� 6� 1

2�W � ðd=2Þ ¼ 4AWffiffiffi

3p

dð13Þ

If W = dm (0 < m < 1/2), then

Sgrain ¼4Affiffiffi

3p dm�1 ð14Þ

It can be seen that the length of the transmission path of thephoton wave will increase with the grain size decreasing, and thescattering effect will also improve, so the nanostructured TBC willhave lower thermal conductivity than that of the conventional TBC.As nanostructured TBC have a large number of grain boundaries,phonon scattering will enhance significantly. In addition, the exis-tence of a nano and micron pores, oxygen vacancies and soluteatomic point defects in the nanostructured TBC can also enhancethe phonon-scattering effect, which makes the insulation effectof the nanostructured TBC improve significantly. In fact, the ‘‘ther-mal rectification’’ of the heat diffusion around the grain boundarieswill have a coupled effect with the phonon scattering, and thisresult in the decease of the effective thermal conductivity of theTBC significantly. In all, grain size and grain boundaries play animportant role in controlling the thermal transport in polycrystal-line materials, particularly when the grain size has reduced to thenano-scale.

3.3.1. 3D modelIn this section, the discussion about the influence of the inter-

face on the thermal insulation of the TBCs based on the 3D modelwill be presented in order to further clarify the effect of the inter-face. Fig. 9(a) shows the 3D model of the ‘‘unit cell’’ which includeeight ellipsoid pores in the cell, the orientation angle between thelong axis and the plus y direction is defined as h, the model size is300 � 300 � 300 lm3, the intrinsic thermal conductivity of theTBCs without pores and the pores are set to be 0.55 W/(m K) and0.02 W/(m K), respectively.

The effective thermal conductivity for the multiphase compos-ites can be obtained using the analytic solution method. Generallyspeaking, the thermal conductivity along the 1, 2, 3 direction canbe written as follows [45]:

hi�

ð15Þ

ð16Þ

where km is the thermal conductivity of the matrix, L11, L22, L33 arethe phenomenological coefficients of 1, 2 and 3 direction, respec-tively. The hcos2hi can be written as follows:

hcos2 hi ¼RqðhÞ cos2 h sin hdhR

qðhÞ sin hdhð17Þ

where q(h) is the orientation distribution function, The bii can bedefined as follows:

bii ¼Kc

ii � Km

Km þ LiiðKcii � KmÞ

ð18Þ

When the composite is uniform, the thermal conductivities of thecomposite in the 1, 2, 3 direction are the same, and they can be cal-culated as follows:

Kc11 ¼ Kc

22 ¼ Kc33 ¼

K1þ aK

Km

ð19Þ

θ

T=1473K

dT/dn=0

dT/dn=0

T(K) T(K)

q(w/m2) q(w/m2)

(b)

(d)

(c)

(e)

(a)

Fig. 9. 3D model (a) Temperature distribution in the Z direction with the orientation angle of the ellipsoid type pores equal to (b) 0�. (c) 60�, and heat flux distribution in theZ direction with the orientation angle of the ellipsoid type pores equal to (d) 0�. (e) 60�.

L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967 963

where a can be defined as follows:

a ¼ RK Km

Rð20Þ

RK is the interfacial thermal resistance and R is the radius of thepores.

The distribution of the temperature and heat flux along the Zdirection for the model with different orientation angle h can beseen in Fig. 9(b)–(e). It can be seen that the insulation tempera-tures are different for the two models in which the pores havingdifferent orientation angles. The models whose pores have the ori-entation angle 0� and 60� are defined as type I and type II model,respectively. It can be seen from Fig. 9(b) and (c), the insulation

temperature of the model I and model II is 74.2 K and 78.7 K, andthe corresponding thermal conductivity is 0.42 W/(m K) and0.41 W/(m K), respectively. The effective thermal conductivity ofthe model I is slightly higher than that of the type II model.Fig. 9(d) and (e) shows the distribution of the heat flux of the typeI and type II model, respectively. It can be seen that on the surfaceof the pores, the heat flux has the minimum value, while far fromthe surface the pores, the heat flux value became larger. As for thetype II model, it can be seen that the spatial distribution character-istic of the model II is very different from the type I model, but Itcan be also seen that on the surface of the pores, the heat fluxhas the minimum value, while far away from the surface the pores,the heat flux value became larger, i.e. the ‘‘thermal rectification’’

K*(

W/m

.K)

R(µm) 0 10 20 30 40 50 60 70 80 90

36

38

40

42

44

0.44100.44860.45620.46390.47150.47910.48670.49440.5020

Orientation angle θ (deg.)

b (µ

m)

(W/m.K)

(a) (b)

Fig. 10. Contour plot of the effective thermal conductivity calculated using the CPA method (a) and the effective thermal conductivity as the function of the radius of thespherical pores using different calculation methods (b).

964 L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967

effect of the heat diffusion around the coating/pore interface willbe stronger when the interface is far away from the surface ofthe pores.

The contour plot of the effective thermal conductivity calcu-lated using the CPA (coherent-potential approximation) methodis shown in Fig. 10(a). It can be seen that when the orientationangle is the same, the effective thermal conductivity will decreasewith increasing the pore size. Especially very interesting, the

a)

d)

M

Second

Interface

q

q

q

q

InterfaceLayer 1

Layer 2

K1

K2Ri

Second p

Fig. 11. Physical models of ‘‘local effect’’ of heat diffusion around the interface in the thmodel of (a) heat diffusion through the phase interface without evident barrier (d) heatthe phase interface with large barrier, the heat diffusion may exhibit inverse direction arimpurity in the coatings, the matrix can be viewed as the dense coating phase.)

lowest thermal conductivity is located at the middle top positionof the contour plot, which means when the orientation is near to45�. and the pore size is the largest, and then the TBCs will havethe lowest effective thermal conductivity. Fig. 10(b) shows theeffective thermal conductivity as the function of the radius of thespherical type pores using different calculation methods. It canbe seen that the relative error of the effective thermal conductivityis very small for the ATA (average t-matrix approximation) method

b)

c)

e)

atrix

phase

q

q

q

Layer 1

Layer 2Heat barrier layer

K1

K2

Ki

hase

ermal barrier coatings (a) heat diffusion through the layer interface (b) equivalentdiffusion through the phase interface with partial barrier (e) heat diffusion throughound the interface. (Note: the second phase may also be viewed as pores, cracks and

0 150 300 450 600 750 900 1050 12003456789

10111213141516

k 1(W/m

.K)

T( )0 150 300 450 600 750 900 1050 1200

1.08

1.10

1.12

1.14

1.16

1.18

1.20

k 2(W/m

.K)

T( )

0 150 300 450 600 750 900 1050 12001.22

1.24

1.26

1.28

1.30

1.32

1.34

1.36

1.38

1.40

Kef

f(W/m

.K)

T( )

Theoretical value Experimental value

Bond-coating

Top-coating

Bond-coating

Top-coating

TC

BC

TC

BC

Interface

(a) (b)

(c) (d)

Interface

Fig. 12. Comparison of the theoretical and experimental value of the effective thermal conductivity for the free standing YSZ/NiCoCrAlY coating, (a) free standing NiCoCrAlYand its effective thermal conductivity as the function of the temperature, (b) free standing YSZ and its effective thermal conductivity as the function of the temperature,(c) free standing YSZ/NiCoCrAlY coating, (d) the comparison curve of the theoretical and experimental value of the effective thermal conductivity for the free standing YSZ/NiCoCrAlY coating.

L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967 965

and analytic solution, while the effective thermal conductivity islower when using the CPA method to calculate the effective ther-mal conductivity. This is because that the CPA method has consid-ered the ITR effect which will decrease the effective thermalconductivity more or less. In fact, the ‘‘thermal rectification’’ ofheat diffusion around the interface has been considered in theCPA method. And the equivalent interface barrier layer was consid-ered simultaneously in this method.

Fig. 11 shows the physical models of ‘‘thermal rectification’’ ofheat diffusion around the interface in the TBCs. Firstly, the inter-face between the two adjacent layers has been considered(Fig. 11(a)), when the heat flux flow across the interface, the inter-face has a strong barrier effect to the heat flux, so the model can beequivalent to the Fig. 11(b), the interface can be viewed as thatthere is an additional heat barrier layer which was embedded thecenter of the two adjacent layers (layer 1 and layer 2). The effectivethermal conductivity of the whole system can be deduced asfollows:

The effective thermal conductivity in the spray direction(through-thickness direction) can be computed with Fourier’sequation, i.e.

keff ¼Qt

WDTð21Þ

Here, keff is the effective thermal conductivity of the whole system,q is the total steady-state heat flux per unit thickness through atransverse cross-section of the model, W and t are width and height

of the model, respectively. And DT is the temperature differencebetween the top and bottom boundaries. Note that q is constantfor any transverse cross-sections under steady-state conditions.

Then it can be concluded that the following expression can beobtained for the layer 1, heat barrier layer and layer 2, respectively.i.e.

k1 ¼Qt1

WDT1ð22Þ

ki ¼Qti

WDTið23Þ

k2 ¼Qt2

WDT2ð24Þ

where t1, t2 and t3 are the thickness of layer 1, heat barrier layer andlayer 2, respectively. DT1, DTi and DT2 are the temperature differ-ence between the top and bottom boundaries of layer 1, heat barrierlayer and layer 2, respectively.

While the total insulation temperature can be calculated asfollows:

DT ¼ DT1 þ DTi þ DT2 ð25Þ

And the thickness aspect satisfies the following equation, i.e.

t ¼ t1 þ ti þ t2 ð26Þ

According to the Eqs. (22)–(26), the total effective thermal conduc-tivity of the coating system considering the ITR effect can beexpressed as:

966 L. Wang et al. / International Journal of Heat and Mass Transfer 79 (2014) 954–967

keff ¼t1 þ ti þ t2t1k1þ ti

kiþ t2

k2

ð27Þ

Fig. 11(c)–(e) show the physical models of ‘‘thermal rectification’’ ofheat diffusion around the interface in the TBCs when consider thatthe second phase (impurity, micro-cracks, pores, voids, and otherreinforced phase) was embedded in the matrix (dense coating). Itcan be seen from Fig. 11(c) that the heat diffusion around the inter-face has exhibited fluent flow characteristic without evident barrier.From Fig. 11(d), it can be seen that the heat diffusion around theinterface between the matrix and the second phase has exhibitedrelatively fluent flow characteristic around the interface with par-tial barrier effect. From Fig. 11(e), it can be seen that the evidentbarrier effect has occurred around the interface, even there existedthe cases of heat diffusion with inverse direction just as the redarrow indicated in the corresponding positions in the figure. Basedon the analysis and discussion above, the following issues can beimagined. i.e., if we want to get a low thermal conductivity in onedirection of the materials, we can increase the thermal conductivityof the perpendicular direction. Generally, the effective thermal con-ductivity of the TBCs in the through-thickness direction (spray-direction) is usually demanded to be low, so we can increase theeffective thermal conductivity of the interface direction which isvertical to the spray direction of the TBCs. This may be controlledand adjusted by the microstructure of the as-sprayed coating whichcan be changed by the actual spray technique.

Fig. 12 shows the comparison of the theoretical and experimen-tal value of the effective thermal conductivity for the free standingYSZ/NiCoCrAlY coating. Fig. 12(a) shows the cross-section image ofthe free standing NiCoCrAlY bond-coating, pores and embeddedoxides can be seen in this coating, the inserted figure displaysthe thermal conductivity of the NiCoCrAlY bond-coating as thefunction of the temperature. Fig. 12(b) shows the cross-sectionimage of the free standing YSZ top-coating. Pores and cracks whichare distributed at random can be also observed. The inserted figuredisplays the relationship between the thermal conductivity of theYSZ top-coating and the temperature. Fig. 12c) shows the cross-section image of the free standing YSZ/NiCoCrAlY double layercoating. The inserted figure shows the 3D and 2D cross-sectioncoating model. Fig. 12(d) shows the comparison curve of the theo-retical and experimental value of the effective thermal conductiv-ity for the free standing YSZ/NiCoCrAlY coating. It can be seen thatthe theoretical and experimental value of the effective thermalconductivity are nearly coincident. At the most temperature points,the experimental value of the effective thermal conductivity isusually lower than that of the theoretical value, the difference orthe reduction of the experimental value may come from the contri-bution of the ITR of the splats and defects (pores, cracks)/coating.The detailed investigation about the influence of these factors onthe effective thermal conductivity of the as-sprayed TBCs will befurther reported in our future work.

The APS-TBCs often exhibit lamellar structure with a lot of ellip-soid pore structures, so the effective thermal conductivity is usu-ally lower than that of the perfect TBCs, while the TBCs preparedby electrical beam-physical vapor deposition (EB-PVD) often exhi-bit columnar grain structural characteristic, the pore channelbetween the columnar grains is often parallel to the thermal fluxdirection, so the effective thermal conductivity is often higher thanthat of APS-TBCs. So it is very important to control the pore volumefraction, spatial distribution, orientation and morphology in orderto obtain the TBCs with optimized or reduced thermal conductiv-ity. But the interface between the defects existed in the TBCs hasalso an important effect on the effective thermal conductivity ofTBCs, so it must be considered. The paper has offered usefulguidance to the design, processing and application of the corre-sponding coating system.

4. Conclusions

In this paper, the influence of interface on the effective thermalconductivity of the thermal insulation effect of TBCs was investi-gated systematically. Especially, the ‘‘thermal rectification’’ of theheat diffusion around the interface in the TBCs has been calculatedand discussed in detail. The corresponding results can be summa-rized as follows:

(1) The ‘‘thermal rectification’’ of the heat diffusion around thecoating/pore interface has an amplified role when increasingthe area of the interface. The ‘‘thermal rectification’’ of heatdiffusion around the coating/pore interface will take aneffect when the orientation of the pores has changed.

(2) The bond-coating/top-coating interface roughness also has avery important effect on the effective thermal conductivityof the as-sprayed TBCs. The effective thermal conductivitywill increase with the interface amplitude increasing withconsidering the interfacial thermal resistance, but this ten-dency is not evident without considering the interfacialthermal resistance. The ‘‘thermal rectification’’ of heat diffu-sion around the positions of ‘‘wave crest’’ and ‘‘wave trough’’of the interface has not exhibited different characteristicobviously, but the amplitude of the interface will have animportant influence on the temperature distribution andthe effective thermal conductivity.

(3) The CPA method is more accurate compared with the ATAmethod in calculating the effective thermal conductivity ofthe as-sprayed TBC, and at the same, the CPA method canbe broaden to the other multi-phase composite systemwhich is attributed to that the ‘‘thermal rectification’’ of heatdiffusion around the interface has been considered in theCPA method, and the equivalent interface barrier layer wasconsidered simultaneously in this method.

(4) The experimental verification work has been performed. Thetheoretical and experimental values of the effective thermalconductivity of the TBCs are nearly coincident. At the mosttemperature points from ambient temperature to 1150 �C,the experimental value of the effective thermal conductivityis usually lower than that of the theoretical value. The differ-ence or the reduction of the experimental value may comefrom the contribution of the ITR of the splats and defects(pores, cracks)/coating.

Conflict of interest

None declared.

Acknowledgments

We express our gratitude to the high Speed Computational Cen-ter of Harbin Institute of Technology (HIT) for providing softwaresupport. This work was jointed supported by the National NaturalScience Foundation of China (NSFC) under the Grant No. 51202277,Young Scholar Project (No.12ZR1452000) supported by the Shang-hai Science and Technology Committee and 2012 Innovation Fundof SICCAS (Y35ZC6160G). The current work was also supported bythe Major Program for Basic Research of Shanghai Science andTechnology Committee (Grant No. 12DJ1400402).

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