Models of thermal conductivity of multilayer wear resistant coatings

Surface & Coatings Technology 204 (2009) 629–634

Contents lists available at ScienceDirect

Surface & Coatings Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s u r f c o a t

Models of thermal conductivity of multilayer wear resistant coatings Leonid Braginsky a,⁎, Andrey Gusarov b, Valery Shklover a a b

Department of Materials, ETH Zürich, 8093 Zürich, Switzerland ENISE, 58 rue Jean Parot, 42023 Saint-Etienne, France

a r t i c l e

i n f o

Article history: Received 20 June 2009 Accepted in revised form 31 August 2009 Available online 12 September 2009 PACS: 65.80.+n 63.22.−m 68.65.−k 68.65.Ac 44.35.+c

a b s t r a c t The thermal conductivity of non-metallic nano- and microstructured materials is a key parameter when considering wear resistant coatings. Here, different models of phonon transport and estimation of thermal conductivity are analyzed. The hopping mechanism of phonon transport was found to be applicable for small-grain-size materials in understanding thermal conductivity, whereas large grain size materials can be studied using the correlation function approach. The Chapman–Enskog approach to the Boltzmann transport equation assuming that the transport of phonons is controlled by scattering on the grain boundaries has been analyzed. The Fourier law of thermal conductivity is obtained with the thermal conductivity inversely proportional to the specific surface of the boundaries. © 2009 Elsevier B.V. All rights reserved.

Keywords: Nanostructures Thermal conductivity Phonon propagation

1. Introduction In order to meet requirements of high wear resistance, cutting tools need to be coated with hard, high-temperature stable, ceramic coatings. Ideally, these ceramic coatings should have chemical stability, phase stability, oxidation resistance, diffusion resistance, hardness, hot hardness, low high-temperature interdiffusion, and high adhesion to a substrate. To enable these useful properties, multilayer coatings are often used. Thermal barrier properties, which provide heat deflection from the cutting tools into both the work material and chip, can be considered as one of the potential beneficial properties of multilayer coatings. A review of different models of thermal conductivity in non-perfect materials, including suggestions for criteria in selecting materials with low thermal conductivity at high temperature can be found in [1], where it was also noted that the influence of a variety of microstructural properties present in non-perfect materials increases the complexity of the selection process considerably. Early analysis [2] resulted in the estimation (1) for the relative temperature change ΔT =T0 −T1 of the tool surface with (T1) and without (T0) a thin monolithic coating (Fig. 1). This relative temperature change is expressed by: pffiffiffi ΔT Dκc π pffiffiffiffiffiffiffi ; = T0 20κt ατ ⁎ Corresponding author. E-mail address: [email protected] (L. Braginsky). 0257-8972/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2009.08.052

ð1Þ

where D(mm) is the thickness of the coating, κc and κt are the thermal conductivities (TC) of the coating and the tool, α is the thermal diffusivity of the tool, and τ is the cutting time. Expression (1) shows that the effect of the thermal barrier coating (TBC) is greater when either the relation κc/κt is larger, the coating thickness D is larger, or the cutting time τ is shorter. It was apparent that efficient thermal insulation by very thin coatings can appear in very short-acting processes [2]. On the other hand, experimental studies [3] point to the possibility of a thermal barrier effect for specific metal/coatings couples with controlled dissipation and conduction of friction energy. The effect of a particular multilayer coating(6 μm−TiC/3 μm−Al2O3/ 1 μm−TiN) on the temperature of a WC-Co tool was simulated with FEM using two models: (a) individual coating layers with intrinsic properties and (b) a single composite coating layer with effective properties [4]. It is important to stress that the coating considered in [4] contains a low-κ Al2O3 layer between two high-κ layers.1 Both models (a) and (b) show that the thermal barrier effect between the chip and the tool is more effective at the initial cutting stage and may depend on the thickness of the Al2O3 layer. It was also assumed that the observed decreased cutting temperature of the tool-chip interface compared to the uncoated tool could be assigned to the modification of the tool surface. In this regard, the reduction of cracking depth due to the application of a very thin (1 μm) external ceramic thermal 1 Thermal conductivity κ of TiC, Al2O3 and TiN change from 33 to 44 W/m K, from 28 to 5.5 W/m K and from 21 to 28 W/m K, respectively in the temperature range from 100 to 1500 °C [4].

630

L. Braginsky et al. / Surface & Coatings Technology 204 (2009) 629–634

The model considered in Section 4 supposes the grain size to be large in comparison with the phonon mean free path. This permits us to introduce the local thermal conductivity and estimate the effective thermal conductivity of the non-homogeneous media using the correlation function approach. The latter can be estimated from the optical or SEM image of the coating cross-section. The correlation function approach allows us to consider the three-dimensional problem of thermal conductivity using the two-dimensional image of the coating. In Section 5 we estimate the temperature gradients which can arise in the coating due to its imperfection. We show that layering can be used for the spreading of such gradients. Being large, such gradients can destroy the tool and the coating because of the thermal stress.

Fig. 1. A schematic illustrating the temperature change ΔT/T0 on the tool surface due to the thermal protection effect of thin monolithic coatings with thermal conductivity Kc and thickness D, see expression (1). Kt and α are thermal conductivity and thermal diffusivity of the tool.

barrier layer in (steel H-13)/Ti/TiAlN/TBC in the die casting [5] can be probably explained by a relatively short high-temperature loading of the cyclic test.2 It seems, however, that mentioned studies contain an indirect evidence of the possibility that the influence of microstructure modifications of the coating/tool couple can greatly affect the tool-chip interface temperature. The possibility of significant heatdeflection action of a coating was demonstrated by modeling studies [6] showing the increase of the heat partition coefficient for the systems TiC/Al2O3/TiN and TiC/TiCN/Al2O3/TiN. Thermophysical solutions are routinely applied in designing multilayer wear protective coatings as seen in, for example, the coating WC/MoB2/(Tix,Al1 − x)B/(MoyB1 − y)N with high thermal conductivity for high-speed cutting of high-hardness steel [7]. Similarly, the multilayer coating WC/(Ti1 − xAlxN with high thermal conductivity and very good heat dissipation was effective in high-speed machining of steel [8]. Also the coatings WC/(Ti1 − xAlxN or Ti1 − xAlxC1 − mNm)/AlN/Al2O3 with high thermal conductivity surface heat release layers of thickness 0.5–15 μm and excellent heat dissipation for cutting steels and cast iron [9] could be mentioned as examples. In this paper we propose several approaches to modeling the heat distribution in multilayer coatings for wear protection. This is consistent with the suggestion [10] of the necessity of different approaches in modifying the coating microstructure to block different regions of the phonon spectrum. We analyze three models for estimating the thermal conductivity of mono- and multilayered coatings. The first model, considered in Section 2 uses the Boltzmann equation to investigate phonon propagation in multiphase structures. This is a first-principles approach, which allows us to consider the all important mechanisms of the phonon scattering. It should be noted that the multilayer coatings can also be considered as the multiphase structures. This ensures the valid application of this approach to such structures as well. The second model considered in Section 3 is semiphenomenological. The temperature dependence of thermal conductivity of single crystalline material κi(T) is supposed to be known, and we investigate the effect of nanocrystallinity on the heat transfer using the phonon hopping model. This model holds at high temperatures when the characteristic phonon wavelength λ = 2πℏυ/kBT (here υ is the speed of sound) is less than the grain size. The mechanism is efficient, if the mean free path of the phonons is less or comparable with the grain size.

2 The conditions used in thermal cycling are 8 s of dipping in Al at 650–700 °C, 15 s in air, 7 s in water and finally 15 s in air [5].

2. Application of Boltzmann transport equation to thermal conductivity in multiphase structures Short-wavelength phonons near the boundary of the Brillouin zone with the wavelength comparable to the interatomic distance are effectively scattered by point defects and other phonons while longwavelength (about several interatomic distances or greater) phonons can skirt such scatterers. Therefore, the phonon thermal conductivity of homogeneous materials is often controlled by the long-wavelength phonons. The typical characteristics of these phonons are, for example, a wavelength of ~1 nm [11] and the mean free path of the order of several tens of nanometers [12] in GaAs at room temperature. In this case, the nanostructured multiphase materials, such as multilayers [11] or dispersions [12], can have considerably lower thermal conductivity because of scattering of the long-wavelength phonons by the phase boundaries. Introducing the phonon intensity: Iω ðr; Þ = υℏωnω ðr; ÞDω = 4π;

ð2Þ

which characterizes the energy flux at the point r and direction Ω transferred by phonons with the circular frequency ω, reduces the Boltzmann transport equation (BTE) for phonons to the form similar to the radiation transfer equation [13]. Here, υ is the phonon group velocity, ℏ the Planck's constant, Dω the density of states per unit volume, and nω(r,Ω) the distribution function of phonons, which tends to the equilibrium Bose distribution at the temperature T: nω =

1 ; eℏω = kB T −1

ð3Þ

where kB is the Boltzmann constant. A multiphase structure (see Fig. 2) is characterized by the volume fractions of phases fγ and the specific surfaces of interfaces Aγδ defined as the surface of the boundary between phases γ and δ per unit volume of the medium. Partial phonon intensity Iωγ can be defined by averaging the local intensity over the volume of phase γ by the analogy with radiation transfer [14]. Then, the multiphase transport

Fig. 2. Characteristics of a two-phase structure: volume fractions fγ and fδ and partial phonon intensities Iωγ and Iωδ .

L. Braginsky et al. / Surface & Coatings Technology 204 (2009) 629–634

model can be applied [14]. In case of two phases shown in Fig. 2, the model is described by two coupled BTE: γ

∇Iω

σωγ

=

 1 γ 0 γ 0 0 γ γ ∫ Iω ð ÞPω ð ; Þd −Iω ð Þ −βγω Iω ð Þ 4π 4π

σωγγ

ð4Þ with the internal scattering (by point defects, phonons, etc.) in phase γ characterized by the pair of the scattering coefficient σγω and the scattering phase function Pγω. Scattering by phase boundaries is given by separate terms in the right hand sides of Eq. (4). Thus, βγω is the γγ coefficient of extinction by phase boundaries. The pair of σγγ ω and Pω corresponds to reflection of phonons by phase boundaries back into phase γ. The pair of σωδγ and Pωδγ corresponds to transmission of phonons from phase δ into phase γ. In the case of statistically isotropic two-phase medium, the coefficients responsible for scattering by the phase boundaries are given as [14]: Aγδ Aγδ ργ Aγδ ð1−ρδ Þ γγ δγ = ; σω = ; σω = ; 4fγ 4fγ 4fγ

ð5Þ

   A 1 0 0 + σω ∫ Iω ð Þd −Iω ð Þ ; 4 4π 4π

ð6Þ

with the internal scattering coefficient σω and the scattering coefficient by phase boundaries A/4. The Chapman–Enskog analysis [15] is applied to Eq. (6) to derive the thermal conductivity. In the first approximation, ð0Þ

ð1Þ

ð1Þ

ð0Þ

Iω = Iω + Iω ; Iω ≪ Iω ;

ð11Þ

κ=

υCv : 3ðA = 4 + σÞ

ð12Þ

The thermal conductivity given by Eq. (12) is independent of the phonon spectrum because it is comprised in Cv. Therefore, one can think about the simplest one-frequency Einstein model when referring to Eq. (12). A more complicated stepwise approximation:

f σσ ;;

σω =

0 < ω < ω0 ; ω0 < ω < ωD

0 1

ð13Þ

containing four free parameters, σ0, σ1, ω0, and ω1 is useful to distinguish the short- and long-wavelength phonons. The Chapman–Enskog Eq. (9) can also be reduced to the Fourier's law (10) in this approximation. However the phonon spectrum is required to calculate the thermal conductivity in this case. The Debye model with Dω ∝ω2 gives: κ=

  υCv p 1−p ; + 3 A = 4 + σ0 A = 4 + σ1

ð14Þ

where

where ργ is the hemispherical reflectivity of the interface from the δγ side of phase γ. The phase functions Pγγ ω and Pω can be derived from the physics of phonon interaction with the boundaries. For example, the models of acoustic mismatch and diffuse mismatch [11] become relevant here. The applicability of the model to nanostructured materials is illustrated below by the experimentally studied influence of 1–5 nmdiameter ErAs inclusions in a In0.53 Ga0.47As matrix [12]. At small volume fraction of the dispersed phase the system of Eq. (4) is reduced to the single BTE [14]. If both the internal scattering in the matrix phase and the scattering by phase boundaries between the inclusions and the matrix are supposed to be isotropic, the transport equation becomes:

∇Iω =

4π

and the thermal conductivity is proportional to the specific heat Cv:

σωδγ

σ δδ 1 σ γδ δ 0 δδ 0 0 δ 0 γδ 0 0 ∫ Iω ð ÞPω ð ; Þd + ω ∫ Iω ð ÞPω ð ; Þd

+ ω 4π 4π 4π 4π 4π

γ βω

∞ 0

1 γ 0 γγ 0 0 γ 0 δγ 0 0 ∫ I ð ÞPω ð ; Þd ; ∫ I ð ÞPω ð ; Þd + 4π 4π 4π ω 4π 4π ω   1 δ 0 δ 0 0 δ δ δ ∫ I ð ÞPω ð ; Þd −Iω ð Þ −βδω Iω = σωδ ð Þ

∇Iω 4π 4π ω +

where the heat flux is defined as: q = ∫ dω∫ Iω ð Þd ;



631

ω0

p = ∫ωD θ 0

=

x4 ex x4 ex T 1= θ dx ∫ dx; θ = ; 2 x 0 TD ðe −1Þ ðe −1Þ2 x

ð15Þ

ωD the Debye frequency, and TD the Debye temperature. The parameter p represents the fraction of the specific heat Cv corresponding to the lowfrequency long-wavelength phonons with 0 <ω <ω0. Fig. 3 compares the model expressed by Eqs. (12) and (14) with the experimental data [12] at T = 300 K demonstrating a reduction in the thermal conductivity with an increased volume fraction of the dispersed phase. The phonon group velocity is accepted to be equal to the longitudinal sound speed in In0.53 Ga0.47As, υ = 5 km/s, the value of the specific heat used for calculations is Cv = 1.65 J/(cm3K), and TD = 330 K. The dependence of the specific surface A on the volume fraction of the ErAs inclusions is obtained assuming that the inclusions are spheres with a mean diameter of 2 nm. The only adjustable parameter of the Einstein model is chosen to be σ = 0.55 nm− 1 to fit the thermal conductivity of the matrix without inclusions (left point in Fig. 3). The resulting curve (full line in Fig. 3) does not agree with the experimental measurements for the matrix with inclusions. The conclusion is that ignoring the spectral

ð7Þ

where the zeroth-order term is the equilibrium phonon intensity: ð0Þ

Iω =

υDω ℏω ; 4π eℏω = kB T −1

ð8Þ

and the following integral equation is obtained for the first-order term: ð0Þ

∇Iω =

   A 1 ð1Þ 0 0 ð1Þ + σω ∫ Iω ð Þd −Iω ð Þ : 4 4π 4π

ð9Þ

Two model functions of σω are investigated. In the simplest approximation σω = σ = const corresponding to the grey body model for radiation transfer, Eq. (9) is reduced to Fourier's law: q = −κ∇T;

ð10Þ

Fig. 3. Thermal conductivity (TC) κ of In0.53 Ga0.47As with ErAs nano-sized inclusions versus the volume fraction of the inclusions fErAs at T = 300 K: points, experimental data [12]; full line, Einstein model, Eq. (12); broken line, Debye model, Eq. (14).

632

L. Braginsky et al. / Surface & Coatings Technology 204 (2009) 629–634

scattering properties of the matrix phase accepted in the Einstein model is not satisfactory. The three degrees of freedom of Eq. (14) following from the Debye model allow more precise fitting of the experimental data. Thus, the broken curve in Fig. 3 calculated for p = 0.01, σ0 = 0.0065 nm−1, and σ1 = 3.5 nm− 1 quantitatively follows the experimental tendency of decreasing the thermal conductivity with increasing fraction of the dispersed phase. The spectral internal scattering coefficient σω corresponding to the above parameters is shown in Fig. 4. The curve identified by the Debye model is characterized by a strong gap in the frequency range below ω0/ωD = 0.211 corresponding to the longwavelength acoustic phonons. These phonons with a mean free path estimated as 1/σ0 ~ 150 nm determine the thermal conductivity of the matrix phase. Notice that this is a small fraction of all phonons, and they are responsible for the fraction of the specific heat as small as p = 0.01. The nanostructuring aims to introduce scatterers (phase boundaries) effective in this frequency range. The presented model can be developed to distinguish the longitudinal and transversal vibrations and to take into account more realistic phase functions of scattering.

Fig. 5. Granular Al2O3 TEM image. A is the size of the neck between crystallites and t is the transparency of the intercrystallite boundary for the phonon hopping. The scale bar corresponds to 200 nm.

is the factor of disorder, and: 3. Thermal conductivity of granular materials having small grains: the hopping model Consider the phonon transport in a granular material, assuming scattering at the grain boundaries as the main mechanism restricting the heat propagation. This means the grain size is smaller or comparable with the phononic mean free path. Suppose, nevertheless, the grain size is large in comparison with the characteristic wavelength of the thermal phonons λ = 2πℏν/kBT. Fig. 5 presents the structure of one such coating. Suppose the phonons are initially localized in each nanocrystallite and consider their hopping between neighboring crystallites. We assume the crystallite size d much exceeds the wavelength of phonons. This is the case of high temperatures T ≫ 2πℏν/kBd. The equilibrium Bose distribution function (3) is supposed to validate for the phonons in each crystallite. These assumptions lead to the following expression for the thermal conductivity [16]: κph =

― 2 kB Zt AΦðηÞTD 1 = θ ∫ BðxÞdx: 0 3πℏa2 d

ð16Þ

Here Z is the coordination number (Z = 6 in the simplest case), d is the ― crystallite size, A is an average lateral area of necks between them, η is the mean deviation of the neck size, a is the lattice constant of the material,

ΦðσÞ =

1+

2η2 Z

!−1

BðxÞ =

  4 x 9 4 x e 1 2 x− : θ x 2 ðe −1Þ2 θ

The parameter t determines the transparency of the intercrystallite interface for phonon hopping. It can be considered as adjustable and found from the measurement of the thermal conductivity at a given (e.g. room) temperature. Eq. (16) predicts an increase of the thermal conductivity at T < TD, followed by a plateau: κph =

― 2 kB Zt AΦðηÞTD 20πℏa2 d

at T > TD. Phonon–phonon interaction is also important at high temperatures when it leads to κ, which is either smaller or comparable to the predictions of Eq. (16). Expression (17) takes this fact into account: 1=θ

κph = kB T∫

0

―

κi BðxÞZt AΦ ― dx: 2 3πℏk−1 κ a i d + kB TD BðxÞZt AΦ B

ð17Þ

Here κi is the thermal conductivity of the single crystalline material. Eq. (17) can be used to estimate the dependence of the thermal conductivity on the mean size of the crystallites. Comparison with the well-known Kapitza model: κph =

κi 1+

κi Gk d

(where Gk is the Kapitza conductance) yields: Gk =

Fig. 4. Spectral internal scattering coefficient σω versus phonon frequency ω identified from the experimental data.

― k2B Zt AΦðηÞTD 1 = θ BðxÞdx: ∫ 0 3πℏa2 d

Good agreement of the Kapitza model with experiments has been demonstrated previously [17]. Fig. 6 presents results of the TC measurements and its estimation using Eq. (17) for the same t=0.03 value of the adjustable parameter. We found a good agreement with experiment at intermediate (300 °C800 °C) temperatures. Disagreement at small temperature T < 300 °C is due to the long-wavelength phonons (λ >d), whose contribution has not been considered in estimation Eq. (17). Hopping model has been used to explain temperature dependence of thermal conductivity in some other micro and nanocrystalline materials: polycrystalline GaN and AlGaN films [18,19], Ge/Si quantum

L. Braginsky et al. / Surface & Coatings Technology 204 (2009) 629–634

633

Fig. 7 presents results of the TC simulation on some modeled structures. We consider them as the two-dimensional cross-section images of the three-dimensional coatings and assume the coating to be isotropic on average in the plane parallel to the substrate. This assumption allows us to investigate the three-dimensional problem of TC using the two-dimensional image of the coating. We see that TC of all the structures are inside the limiting values estimated by Eq. (18), but demonstrates sometimes non-linear behavior. The aspect ratio of the inclusions is the main parameter that determines their effect on TC. Similar results for the gas-filled pour structures have been obtained in [24]. 5. Heat pulse propagation in layered coating Irregularity of the coated surface can result in inhomogeneity of the temperature field. The cause of the irregularity can originate from roughness of the coating or a defect in it, which can result in so-called hot spots. Strong temperature gradients near this dot can destroy the coating and the tool due to the thermal stress. Assume that the instantaneous point heat source is localized at the coordinate origin. Then the temperature T obeys the heat flow equation: ∂T ∂2 T ∂2 T −λ∥ 2 −λ⊥ 2 = Qδðr; tÞ; ∂t ∂r ∂z

Fig. 6. Comparison of the measured and calculated thermal conductivities of two Al2O3 samples of different grain [270 nm (left), 315 nm (right)] and neck [250 nm (left), 280 nm (right)] sizes. Different types of dots correspond to different series of measurements. The bold curves have been obtained by estimating Eq. (17) with t = 0.03.

where r, z, λ∥ = κ∥/cρ, and λ⊥ = κ⊥/cρ are the coordinates and components of the thermal diffusivity along and across the substrate, respectively, Q ρc is the energy of the point heat origin, ρ is the density and c is the heat capacity of the material. The solution of Eq. (19) in the half-space is: Tðr; z; tÞ =

dot superlattices [20], nanocrystalline diamond [21,22]. The hopping model was proven to be in good agreement with experiments for these structures. This is the case of materials for which thermal properties are to a large degree determined by the grain boundaries. 4. Thermal conductivity of granular materials having large grains: correlation function approach Consider a coating composed of two types of grains of different materials, and assume the size of the grains to be large in comparison with the mean phonon free path. It is apparent that the thermal conductivity of such a material not only depends on the size of the different grains, but also on their distribution. The distance between the grains as well as the relative positions of the small and large grains is also important. In order to take this fact into consideration, we have developed a correlation function approach [23]. Indeed, the large size of the grains permits us to introduce the local thermal conductivity, which is equal to the thermal conductivity of the grain at a local point. The correlation function is the average value of the product of local thermal conductivities taken in ― different points of the coating: Wðr2 −r1 Þ = κðr1 Þκðr2 Þ. The black-andwhite digital image of the coating can be used to calculate this value. Perturbation theory allows us to express the effective thermal conductivity of the two-component granular media via its correlation function. We showed that the effective thermal conductivity of the two-component media lies between κmin and κmax: −1 −1 κ−1 min = f κ1 + ð1−f Þκ2

κmax = f κ1 + ð1−f Þκ2 :

ð18Þ

Here κ1 and κ2 are the thermal conductivities of the appropriate grains, and f is the relative volume density of the grains (1). We found an accurate estimation of κ using the diagrammatic technique.

ð19Þ

Q e−ðr pffiffiffiffiffiffi 2 4π λ∥ λ⊥

2

2

= λ∥ + z = λ⊥ Þ = ð4tÞ

t3 = 2

:

ð20Þ

Assuming D to be the thickness of the coating, we can estimate the maximal temperature gradient on the tool surface as:

j ∂T∂r j = 2eπ D Q −3 = 2 2

4

 3 = 2 λ⊥ : λ∥

ð21Þ

Fig. 8 presents results of modeling using Eq. (20). We found large temperature gradients in Fig. 8(a). Low-TC monolithic coating Fig. 8(b) results in retardation of the heat pulse propagation, but to the same values of the gradients. Such coating can prevent the tool damaging only in the presence of external cooling. The multilayer

Fig. 7. Relative change of thermal conductivity vs volume fraction of nanoparticles for the modeling structures composed of spherical (green circles), elongated (red triangles), and composite (blue pentagonals) inclusions. Pink rhombuses present thermal conductivity of elliptic particles, whose aspect ratio 3:5 is close to that of the composite inclusions. Black straight and dashed lines correspond to the maximal and minimal bounds found in Eq. (18), respectively. κ1/κ2 = 5 for all the structures. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

634

L. Braginsky et al. / Surface & Coatings Technology 204 (2009) 629–634

size. Thermal conductivity of the small-grain-size nanostructures can be studied using the hopping model, whereas the effect of the large grains can be estimated using the correlation function approach. Actually, the grains of both scales are present in real coatings. To estimate both effects, we can use the correlation function approach using the phonon thermal conductivity Eq. (16) or Eq. (17) as the initial values for the corresponding materials. Addition of high-TC inclusions elongated in the coating plane can appreciably increase the in-plane component of TC leaving its cross-plane component unchanged. The effect can be estimated by Eq. (18). The same effect can be reached also by addition of the high-TC layer. In the case of a textured grain coating or a multilayered material the phonon scattering becomes anisotropic. This can also be taken into account but requires a numerical analysis of the Boltzmann equation. The similar method is applicable to multilayer coatings where phonons are scattered by the interfaces between the layers. Consideration of phonon transport using Boltzmann transport equation shows that short-wavelength phonons are effectively scattered inside the matrix while long-wavelength phonons are effectively scattered by phase boundaries. The proposed models can be used for experimental identification of the frequency dependent phonon scattering properties.

References

Fig. 8. Heat pulse propagation in high-TC monolithic coating (κ∥ = κ⊥ = 8.3 W/mK) (a), low-TC monolithic coating (κ∥ = κ⊥ = 1.3 W/mK) (b), and multilayer coating (κ∥ = 8.3 W/mK, κ⊥ = 1.3 W/mK) (c).

coating Fig. 8(c) ensures the fast spreading of the heat pulse along the coating. 6. Conclusion The three models considered here investigate the grain nanostructures of different relation between phonon mean free path and the grain

[1] D.R. Clarke, Surf. & Coat. Technol. 163–164 (2003) 67. [2] A.N. Reznikov, A.N. Reznikov, Thermophysical Aspects of Metal Cutting Processes, Mashinostroenie, Moscow, 1981, p. 212, in Russian. [3] W. Grzesik, Wear 240 (2004) 9. [4] Y.-C. Yen, A. Jain, P. Chigurupati, W.-T. Wu, T. Altan, Machin. Sci. Technol. 8 (2004) 305. [5] A. Srivastava, V. Joshi, R. Shivpuri, R. Bhattacharya, S. Dixit. Surf. Coat. Technol. 163–164 (2003) 631. [6] W. Grzesik, J. Mater, Proc. Technol. 176 (2006) 102. [7] T. Ogami, Y. Tanaka, JP2006088318-A, 27 August 2004. [8] Mitsubishi Materials, JP200360603-A, 6 March 2002. [9] Shinko Kobelco Tool KK. JP2003039211-A, 2 August 2001. [10] G. Cahill, W.K. Ford, K.E. Goodson, G.D. Mahan, A. Majumdar, H.J. Maris, R. Merlin, S.R. Phillpot, J. Appl. Phys. 93 (2003) 793. [11] G. Chen, Phys. Rev. B 57 (1998) 14958. [12] W. Kim, S.L. Singer, A. Majumdar, J., M.O. Zide, D. Klenov, A.C. Gossard, S. Stemmer, Nano Letters 8 (2008) 2097. [13] A. Majumdar, J. Heat, Transfer 115 (1993) 7. [14] A.V. Gusarov, Phys. Rev. B 77 (2008) 144201. [15] J.H. Ferziger, H.G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland, Amsterdam, 1972. [16] L. Braginsky, N. Lukzen, V. Shklover, H. Hofmann, Phys. Rev. B 66 (2002) 134203. [17] L. Braginsky, V. Shklover, H. Hofmann, P. Bowen, Phys. Rev. B 70 (2004) 134201. [18] D. Kotchetkov, A.A. Balandin, Mater. Res. Soc. Symp. Proc. 764 C3.43 (2003) 215. [19] W. Liu, A.A. Balandin, Appl. Phys. Lett. 85 (2004) 5230. [20] M. Shamsa, W. Liu, A.A. Balandin, Appl. Phys. Lett. 87 (2005) 202105. [21] L. Liu, M. Shamsa, I. Calizo, A.A. Balandin, V. Ralchenko, A. Popovich, A. Saveliev, Appl. Phys. Lett. 89 (2006) 171915. [22] M. Shamsa, S. Ghosh, I. Calizo, V. Ralchenko, A. Popovich, A.A. Balandin, J. Appl. Phys. 103 (2008) 083538. [23] L. Braginsky, V. Shklover, Phys. Rev. B 78 (2008) 224205. [24] F. Cernuschi, P. Bison, A. Moscatelli, Acta Materialia 57 (2009) 3460.