J Grandjean
email: julien.grandjean@univ-savoie.fr

G Le Goic

H Favreliere

Y Ledoux

S Samper

F Formosa

L Devun

T Gradel

Multi-scalar analysis of hip implant components using modal decomposition

Keywords: surface analysis, surface roughness, total hip arthroplasty, modal decomposition, surface metrology

This paper presents a metrological analysis of hip prosthesis components. When changing ceramic prostheses, the surgeon sometimes finds traces of alloy lying in the insert or the femoral head. These traces can be thin and accurate or as a wide band. From the measurements made on the contact areas of hip prosthesis components, we analyse these phenomena by highlighting the defects of form, waviness and roughness of these surfaces using a novel geometric parameterization (namely modal parameterization). The aim of this work is to isolate these defects to characterize the prostheses components. We show that this parameterization allows a multi-scale analysis of surfaces regardless of the type of wear of the prosthesis, and that the results offer some relevant explanations to the analysis of visible damage on the prostheses. In a later study, we are going to analyse the defects influence on the alteration of the performance of hip prostheses.

Introduction

Total hip arthroplasty is one of the most frequently performed procedures in orthopaedic surgery (150 000 procedures in France annually). Today, it allows numerous people suffering from degenerative disease or acute inflammation to walk without pain. From a mechanical point of view, this prosthesis is a spherical connection that functions in the same manner as the replaced joint. While replacing ceramic prostheses, a surgeon will sometimes note wear marks or metal alloy incrustations on the spherical part of the explants. These incrustations can be thin and very localized or in the form of a wide band. They can cause the prostheses to deteriorate prematurely or result in other minor malfunctions (such as noise).

This paper presents a metrological study of these phenomena to provide some principles for geometrically analysing spherical surface defects.

Geometric measurements are typically decomposed into six orders of defects that are defined according to technological and mathematical criteria (International Organization for Standardization 1998). The first orders, size and position, are processed according to mathematical models based on the theory of mechanisms. In this paper, we concentrate on form, waviness and surface roughness defects. The surfaces studied are the contact surfaces between joint components associated with the ball-and-socket connection, which has a nominally spherical form. Using scatter plots that are derived from measurements of the examined surface and obtained by extended-field confocal microscopy, modal parameterization is used to exhaustively decompose the examined surface into a vector basis without a priori knowledge of the surface's form. This decomposition sorts the defects by order of frequency and increasing complexity (Le [START_REF] Goïc | Multi scale modal decomposition of primary form, waviness and roughness of surfaces[END_REF]. This paper shows that this technique can be used to filter the form, waviness and surface roughness components, regardless of the type or amount of wear of the examined part.

A subsequent study will analyse the assemblies as a function of their scale and conditions of use and the influence of geometric defects observed on prosthetic joint components on the function and performance impairment of these assemblies.

Analysis of a hip prosthesis using the modal method

2.1. Total hip implant (THI) 2.1.1. Introduction. The hip joint connects the joint cavity of the pelvis (the acetabulum) to the femoral head (hemispherical end of the thigh bone). A THI (figure 1) consists of a cup, femoral stem and prosthetic head. The cup is the upper component of the THI and consists of a metal hemisphere (titanium or cobalt-chrome) set into the acetabulum. A polyethylene, metal or ceramic lining, referred to as the insert, is lodged into the cup. The femoral stem is a metal component (titanium, steel, or cobalt-chrome) that is inserted into the upper end of the femur. This stem supports the prosthetic head (ceramic or cobalt-chrome), which is attached to the neck of the femoral stem and fits into the insert.

The materials most commonly used to reduce the friction between the prosthetic head and insert are combined in a metal-on-polyethylene wear couple. However, polyethylene degradation is significant and provokes the phenomenon of osteolysis, wherein the organism seeks to remove foreign particles. This reaction weakens the bones and can cause the prosthesis to loosen. Polyethylene improvements, such as cross-linked polyethylene, increase its resistance to wear and thus limit the occurrence of osteolysis. However, technological improvements have led to the development of new friction-reducing material combinations, such as ceramicon-ceramic wear couples, which today compete with metalon-polyethylene wear couples. The ceramic-on-ceramic wear couple was developed in 1970 but its use was limited due to the fragility of the ceramic, which can lead to fractures in the implants. This problem was partly solved by the development of new ceramics in 1995. The risk of fracture is very limited with these new materials, with an estimated risk between 0.005% and 0.02% for alumina ceramic and an even lower risk for composite alumina-zirconium ceramics [START_REF] Chevillotte | Etude expérimentale des phénomènes de grincement ou 'squeaking' avec les couples de frottement céramique/céramique dans les prothèses totales de hanche Journées Lyonnaises de Chirurgie de la Hanche[END_REF][START_REF] Hannouche | Fractures of ceramic bearings: history and present status[END_REF].

Squeaking.

In some cases, implanted ceramic THIs emit a noise, often referred to as squeaking, when the patient moves. According to [START_REF] Swanson | Influence of prosthetic design on squeaking after ceramic-on-ceramic total hip arthroplasty[END_REF] this painless phenomenon affects 1% of THI and constitutes a real inconvenience to patients in 25% of those cases. For more severe cases, a surgical procedure known as revision is necessary to replace the implant.

The literature shows that numerous factors can provoke squeaking, such as poor implantation of a prosthetic component, the off-centred loading of the prosthesis (edge loading), the presence of foreign particles (also referred to as third-body particles) between the two spherical parts of the joint, or poor lubrication related, for example, to the patient's pathology [START_REF] Walter | A review of squeaking hips[END_REF]. [START_REF] Restrepo | The effect of stem design on the prevalence of squeaking following ceramic-on-ceramic bearing total hip arthroplasty[END_REF] and [START_REF] Jarrett | The squeaking hip: a phenomenon of ceramic-on-ceramic total hip arthroplasty[END_REF] showed that an overhanging acetabular lip can be responsible for the presence of metal particles between the spherical parts, resulting in squeaking. A second type of wear, referred to as stripe wear, is often found on the explants of squeaky prostheses and is characterized by rough areas on the head and insert [START_REF] Matar | Revision hip arthroplasty for ceramic-on-ceramic squeaking hips does not compromise the results[END_REF][START_REF] Restrepo | The noisy ceramic hip: is component malpositioning the cause?[END_REF]. [START_REF] Taylor | The role of stripe wear in causing acoustic emissions from alumina ceramic-on-ceramic bearings[END_REF] and [START_REF] Chevillotte | Etude expérimentale des phénomènes de grincement ou 'squeaking' avec les couples de frottement céramique/céramique dans les prothèses totales de hanche Journées Lyonnaises de Chirurgie de la Hanche[END_REF] also confirmed that this phenomenon could cause squeaking through testing under in vitro conditions.

Stripe wear.

Since ceramic prostheses were originally developed, rough areas have been found on the head and insert of many explants, disclosing intra-granular fractures and the erosion of rubbing surfaces [START_REF] Nevelos | Microseparation of the centers of alumina-alumina artificial hip joints during simulator testing produces clinically relevant wear rates and patterns[END_REF][START_REF] Sariali | Three-dimensional modeling of in vitro hip kinematics under micro-separation regime for ceramic on ceramic total hip prosthesis. An analysis of vibration and noise[END_REF]. This specific type of wear is called stripe wear due to the resulting band of deterioration.

2.1.4. Metallic incrustations. On certain ceramic explants, alloy incrustation marks can be found on the insert or the femoral head. These third-body incrustations may be thin and very localized or in the form of bands that are wider than those caused by stripe wear. These incrustations are likely due to impingement or friction between the ceramic and metal parts of the prosthesis or, if metal deposits are left by the surgeon during the implanting of the prosthesis, due to friction between the tools and ceramic parts.

Analysis method

To provide quantitative information on these different phenomena, which could cause the onset of squeaking, we propose to study two prosthetic heads that show deterioration by metal deposit inclusions. The wear of the first prosthetic head is superficial, whereas the second presents a more advanced stage of deterioration. We believe that these phenomena could potentially modify the state of the prosthesis surface and change the pressure and lubrication conditions of the prosthesis. The conjunction of these phenomena may cause squeaking to occur.

This study is challenging because all of the defects measured from the THIs must be decomposed to characterize and quantify them. This decomposition would allow, for example, the impact of each of these defects on the performance of the prosthesis to be measured (i.e. the coefficient of friction or onset of squeaking).

Thus, the proposed study consists of measuring the evolution of the spherical surface (form, waviness and roughness) as a function of the two aforementioned phenomena (stripe wear and metal incrustation) through confocal microscopy and a modal analysis of the measurement results.

Characterization of spherical surface defects.

The characterization of surface defects is a really important topic for researchers, but the literature provides little geometric information to quantify them on prosthetic components. Some work has already been published considering different methodologies, like multi-axis non-contact measurement method based on chromatically encoded confocal [START_REF] Tuke | 3D linear and volumetric wear measurement on artificial hip joints-validation of a new methodology Precis[END_REF] or coordinate measuring machines (CMM) with uncertainty analysis and quantification [START_REF] Carmignato | Uncertainty evaluation of volumetric wear assessment from coordinate measurements of ceramic hip joint prostheses[END_REF]. However, few methods of post-processing allow a multi-scalar analysis. The method selection criteria are mainly based on the type of defects that are to be characterized (International Organization for Standardization 1998) (position, orientation, form, waviness, roughness or micro-roughness), the multi-scalar character of the method (the ability to analyse different orders of defects simultaneously) and the method's ease of implementation and use. Based on the scale of the defects that we wish to characterize, we can consider parameterizations dedicated to defect analysis on the elementary spherical geometries or more general parameterizations dedicated to a geometric analysis in which the reference element is a plane.

To describe elementary spherical geometry defects, [START_REF] Kanada | Evaluation of spherical form errors-computation of sphericity by means of minimum zone method and some examinations with using simulated data Precis[END_REF] proposes a method based on spherical harmonics to describe the defect of sphericity (form defect). This method is also used in the field of imaging and threedimensional reconstruction. [START_REF] Kautz | Fast, arbitrary BRDF shading for low-frequency lighting using spherical harmonics[END_REF] use this approach to reduce the number of parameters corresponding to the light source in the calculation of the bidirectional reflectance distribution function. In the optics field, surface defects recurring on optical elements can be defined as a linear combination of Zernike polynomials (International Organization for Standardization 2007). Closer to the work presented in this paper, [START_REF] Favrelière | Caractérisation des défauts d'une surface sphérique par décomposition modale[END_REF] characterize the form defect of a spherical surface to describe the modal deformations of spherical elements.

To characterize the surface beyond the form defect (waviness, roughness or micro-roughness), many authors use more generic methods that are not specifically dedicated to the defects of spherical geometry elements. [START_REF] Huang | Mode-based decomposition of part form error by discrete-cosine-transform with implementation to assembly and stamping system with compliant parts[END_REF] propose to use a database of descriptors based on the discrete cosine transform (DCT) to analyse the different forms of a part. Another popular method is the discrete Fourier transform (DFT); in particular, the standard ISO-12781-2 (International Organization for Standardization 2011) refers to parameters involving the harmonic components of the DFT to evaluate the flatness of a surface.

The multi-scale analysis of a machined surface can be particularly useful for describing the quality of the manufactured surface, its anomalies (metallic incrustations for this study) and the resulting functional aspects [START_REF] Chen | Multi-scale analysis of engineering surfaces[END_REF]. In this context, different parameterizations can be used. [START_REF] Rosenboom | Surface description and defect detection by wavelet analysis[END_REF] show the manner in which a surface can be characterized and defects of very small amplitude can be detected using wavelet analysis, which allows one-dimensional (linear) or multidimensional (surface) signals to be characterized at different scales. Le [START_REF] Goïc | Multi scale modal decomposition of primary form, waviness and roughness of surfaces[END_REF] proposes to characterize the form, waviness and roughness components of a surface using discrete modal decomposition (DMD) in the context of analysing the quality of the appearance of a surface and detecting visual anomalies thereon. DMD allows a surface to be described at different scales using a very small number of parameters. In the remainder of this paper, we develop this approach through the analysis of the contact surfaces of a THI's joint components.

Modal decomposition.

DMD is adaptable to any type of geometry (profile or surface) and allows the defects of form, waviness and surface roughness to be characterized simultaneously [START_REF] Formosa | Modal expression of form defects Models for Computer Aided Tolerancing in Design and Manufacturing[END_REF]. For this study, this method is used to quantify and compare the wear of different prostheses, regardless of the nature and extent of the wear. DMD decomposes a signal into a family of discrete functions defined a priori in a similar manner as the DFT or discrete wavelet transform. In this decomposition, the signal corresponds to the measured surface and can be a flat, cylindrical or spherical surface. More precisely, the DMD allows the surface to be decomposed into a family of descriptors (discrete functions) called modes or modal deformations. This family of functions is determined from the reference geometric element's vibration modes, which come from the solution of equation (3). For example, these functions correspond to the vibration modes of a plane if the measured surface is flat; this ability of the method is denoted by the name chosen for this parameterization (modal parameterization) [START_REF] Favrelière | Modal tolerancing: from metrology to specifications PhD Thesis Université de[END_REF].

The decomposition operation consists of creating a projection vector (1) of measure V (in vector form) in the vector space (Q) formed by the vibration modes (vectors Q i ), as shown below:

Q * T • V = ((Q T • Q) -1 • Q T ) • V = λ, ( 1
)
where the resulting vector λ corresponds to the contributions λ i of each mode, referred to as the modal coordinates or modal coefficients. Because the projection is performed in the non-orthonormal basis vector (Q), the dual basis Qi * must be used. The vectors (Qi) are normalized with the infinity norm (||Qi|| ∞ = 1) to allow the contributions λ i to be obtained in metric values that are homogeneous with the measured surface. We can then express the measured surface as follows:

V = N q i=1 λ i Q i + (N q ), ( 2
)
where N q represents the number of modes selected for the decomposition and is the residual of decomposition. Modes Q i are obtained from solving the following classical problem of mechanical vibration:

K • q + M • q = 0 with ⎧ ⎨
⎩ M : generalized mass matrix K : generalized stiffness matrix q : displacement vector.

(3)

This dynamic problem (3) can be solved either analytically (see [START_REF] Leissa | Vibration of Plates[END_REF] or numerically by the finite element method (see [START_REF] Zienkiewicz | The Finite Element Method for Solid and Structural Mechanics 6th edn[END_REF]. The problem is usually solved numerically, which provides a DMD.

Compared with the Fourier transform, the modal parameterization operates similarly but it enables any type of surface to be analysed, without a priori knowledge. Moreover, the proposed method can be adapted to more complex decompositions by inserting pre-defined forms (namely the 'technological modes') in the vector basis.

Multi-scalar analysis of a spherical surface: the prosthetic head

Samples

This study only uses samples of prosthetic heads that exhibit abnormalities (i.e. stripe wear or metal incrustations, see 2.1.3 and 2.1.4). Identical processes are carried out for both parts of the ball-and-socket joint (the prosthetic head and insert). To apply this method, two samples with different levels of metal incrustations were chosen (figure 2):

• prosthetic head A (Pa), which is composed of strong ceramic BIOLOX R forte: thin metal incrustations; • prosthetic head B (Pb), which is composed of strong ceramic BIOLOX R forte: larger metal incrustations with a zone in which deterioration appears to be more advanced.

Measurement protocol

Many optical methods, such as techniques based on interferometry, laser triangulation or confocal microscopy, permit surface topography to be precisely measured. This study uses the Altisurf 520 instrument (figure 3(a)) to obtain the measurements; this instrument is based on extended-field confocal microscopy (figure 3 (b)) (Udupa et al 2000a(Udupa et al , 2000b)).

It is equipped with a chromatic confocal probe with a vertical measurement range of 300 μm, a vertical accuracy in Z of 0.06 μm and a lateral resolution of 1.55 μm . This non-contact measurement has already been used to measure prosthetic implants such as in [START_REF] Tuke | 3D linear and volumetric wear measurement on artificial hip joints-validation of a new methodology Precis[END_REF]. This study performs measurements on areas with abnormalities (figure 4) to evaluate the geometric variations of form, waviness and roughness corresponding to incrustations (figure 2). The measured surfaces are 1 mm squares composed of 250 000 measurement points (one point every 2 μm).

Multi-scalar analysis by the modal method

This section describes a multi-scalar surface analysis using the modal method on the measured surfaces. The modal method algorithms are developed on Matlab R , and Mountains R software is used to plot the surface topography.

Preamble.

The objective of this study is to characterize the geometric variations of the measured surface on the joint components of hip prostheses. The nominal (reference) surfaces of prosthetic heads are spherical. The first step of this analysis is thus to filter the spherical component of the surface by calculating the associated sphere using non-deteriorated areas. The calculation of the radius and position of the centre of the best-fitted sphere minimizes the distance between the surface and a sphere according to the criterion of least-squares.

Figure 5 shows the resulting surfaces. The form, waviness and roughness defects will be subsequently characterized from this initial step.

Implementation of the modal method

. DMD is the central step of the modal method (section 2.2). As a reminder, this step decomposes the surface meas V into a family of descriptors derived from the mechanical vibration. These descriptors correspond to the natural vibration modes of the reference geometry, which here is a plane (figure 6) after the sphere is filtered through the least-squares method. This family of descriptors forms a vector space, referred to as a modal basis. After projecting the measured surface in the modal basis, the decomposition is obtained (4). For this case, we set the number of modes (N q ) equal to 500 for the decomposition:

meas V = N q =500 i=1 λ i Q i + (N q = 500), ( 4
)
where the λ corresponds to the modal contribution λ i of each mode and Q i to the vectors of the plane modal basis.

The DMD can be visualized in the form of a modal amplitude spectrum. Figure 7 shows the spectrum with the 500 modes of the decomposition. Note that the modal spectrum of the decomposition of this surface is contained in a very rapidly decreasing double envelope. A negative amplitude on the modal spectrum represents the corresponding opposite mode. Modes (Q i ) are sorted in order of frequency, but the form of the envelope shows that they are simultaneously sorted in order of amplitude. This feature allows the DMD results to be used to reconstruct the form, waviness and roughness defects of the decomposed surfaces. Furthermore, the shape of the spectrum (rapid decrease) indicates that a small number of parameters can effectively represent the surface.

(N w ) = meas V - N w i=1 λ i Q i . ( 5
)
Figure 8 shows the residual of the modal decomposition for sample Pa.

Note that this residual is of very low amplitude (< 1.8 μm) and contains high amplitudes from the threshold N w + 1 (i.e. very low periodicity) but also aperiodic components and measurement noise. This measurement noise has already been evaluated and decreased in a previous work with a calibration protocol [START_REF] Favrelière | Modal tolerancing: from metrology to specifications PhD Thesis Université de[END_REF]. With this calibration protocol, the systematic geometrical errors related to the mechanical guiding system are corrected and the final measurement noise amplitude is decreased to 0.5 μm for all the measurement range (200 mm × 200 mm). For the two studied samples (1 mm × 1 mm), we assume that the measurement noise is lower and significantly equal to the optical sensor noise (0.06 μm). Modal decomposition therefore provides a very good approximation of the surface from a set of measured data points. (b) Form, waviness and roughness defects. The modal method can reconstruct the different components of a surface by reconstructing parts of the DMD. The form, waviness and roughness defects of a surface can then be obtained by the simple reconstruction of a part of the modal amplitude spectrum (figure 9). The form, waviness and roughness components are obtained by reconstructing part of the modal spectrum according to the following equations ( 6):

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ form V = N f i=1 λ i Q i waviness V = N w i=N f +1 λ i Q i roughness V = meas V -N w i=1 λ i Q i . ( 6
)
The choice of the two thresholds, form/waviness (N f ) and waviness/roughness (N w ), influences the way in which the defect types are classified (figure 9).The standards do not clearly define how to set these thresholds, and our studies do not provide additional information about the thresholds. However, DMD users can easily vary these thresholds, according to the visualization of reconstructed surfaces, these thresholds can be defined. To illustrate the method, we can see in figure 10 some examples of the reconstruction of the surface Pb for several selections of number of modes (variation of N f ). Thus, for each analysed object, users can define what they see as being in the order of a defect for the form, waviness or roughness. Figures 11 and12 show the reconstructions of the form, waviness and roughness components obtained by the modal method for both surfaces studied.

Influence of sphere filtering

The same studies of modal filtering are here conducted on measured surfaces without a previous best fitting of the sphere. Figures 13 and14 show the reconstructions of the form, waviness and roughness components obtained by the modal method for both the studied surfaces without the best fitting of the sphere.

The presented results show that form and waviness components are affected by the initial spherical form. However, the roughness component seems unchanged. In order to compare the two approaches, we propose to compute the modal decomposition of a nominal sphere, with radius equal to 14 mm, to determine the impact of the initial spherical form in the modal decomposition. The reconstruction results (see figure 15) showed that the nominal sphere is well reconstructed in the form component (N f = 40). Part of the reconstruction on the waviness component (N w = 150), with an amplitude of 0.2 μm, remains to be done. But we can observe that the amplitude of the roughness component is very low, 0.05 μm or 0.3% of the initial amplitude. We can therefore conclude that the two filtering approaches of the initial spherical form do not affect the roughness component.

Using the results

The multi-scale analysis allows a surface to be decomposed in terms of three main parameters: form, waviness and roughness defects. From the two prosthetic heads measured in this study, we have purposefully chosen two explants with wears that are not of the same type to demonstrate the usefulness of the proposed method. The wear on the head of Pa is thin and 'superficial' (surface deterioration), while that on the head of Pb has greater deterioration, with pronounced roughness and chipping that is visible to the naked eye (figure 2). Thus, of the chosen samples, the identification of different surface components will help to quantify the form, waviness and roughness components of both surfaces despite their very different typology.

The proposed method allows the simultaneous analysis and identification of different orders of defects on a surface without a priori knowledge of the surface's form or other characteristics. For example, there is a ten-fold difference between the form defects of samples Pa and Pb due to the chipping of prosthesis Pb (figures 11 and 12), confirming the difference already visible to the naked eye. This result is also visible in the spectrum (figure 7) when comparing the contributions of the form modes.

Similarly, the method allows the roughness defects between the two chosen samples to be identified and compared. The method illustrates the advantage of the modal method for identifying the various orders of defects of a surface in a single analysis without knowing a priori on which surface component the deterioration exists.

We show in particular that the modal method can extract roughness defects of the order of the roughness (high frequencies) with amplitudes greater than those of the form and waviness defects. This finding is visible in figure 11, in which the roughness has an amplitude of approximately 1.6 μm and the waviness has an amplitude of approximately 0.7 μm. This finding also applies to prosthesis Pb (figure 12), for which the amplitudes of the roughness and waviness are approximately 17 μm and 4 μm, respectively. Thus, the modal method allows the different surface components to be extracted and compared regardless of the surface types.

Finally, the filtering quality for form and waviness performed on the surfaces allows the stripes' amplitude to be precisely measured and enables us to see that on the surface of Pa, the stripes have a primary direction of propagation that follows the direction of the incrustation mark. It can also be noted that these stripes have a thickness that exceeds the nominal surface (figure 8), confirming the metallic deposit hypothesis. However, some of the stripes are below the nominal diameter. We can thus assume that third-body particles have already scratched the surface of the head before becoming incrusted. However, the particles can also originate from a small deposit left by the surgeon during the implantation of the prosthesis due to friction between the tools and ceramic parts that has amplified over time. The advanced incrustation of the Pb prosthesis has a roughness of a completely different type because roughness is more homogeneous and has an amplitude of approximately 17 μm (figure 12). The width of the incrustation bands is similar to that of the strip wear bands, which suggests that the roughness due to stripe wear allowed the incrustation of third-body particles in the prosthesis.

Conclusion and perspectives

The purpose of this study is to provide elements of analysis to help better understand the phenomena of metal incrustations on hip joint components. With this in mind, we develop a metrological approach to this phenomenon to determine the consequences that these incrustations have on the surface condition (roughness) and the geometric variations in terms of form and waviness defects.

We develop an innovative surface analysis and filtering method referred to as the modal method. From measurements made on the contact areas of the joint components of hip prostheses, the modal method allows different orders of geometric defects to be identified from the surfaces studied. We show that the DMD allows an accurate multi-scale analysis of surfaces with different types of defects. We show that it is possible that the metal marks visible on the prosthetic head may have provoked either more or less advanced deterioration of the prosthesis surface. While the marks on the sample Pa did not degrade the surface in more than a superficial manner (scratches on the order of the roughness or modification of the surface condition), the more 'marked' prosthesis sample Pb affected all of the geometrical surface components, resulting in a major form defect.

The distinction and characterization of the different types of defects by the DMD can be useful for classifying different prostheses or establishing a defect database. Thus, depending on the types of prostheses and conditions of use, a defect database can be established based on the distribution of the modal amplitude spectrum coefficients.

A subsequent study will consist of using this method on in vitro tests to measure the influence of geometric variations on the dysfunction of hip prostheses with these abnormalities (coefficient of friction, onset of squeaking, among others). More specifically, this future study will compare the components of deteriorated prostheses with those of new prostheses, as well as prostheses before and after a wear test. Discrete mode decomposition can also generate these defects on the numerical modelling of the prosthesis based on the modal amplitude spectrum. This approach also allows the selection of the types of defects to be modelled. Thus, one can consider coupling the in vitro tests with numerical simulations to develop analytical models of behaviour.

Figure 1 .

 1 Figure 1. Total hip implant (THI).

Figure 2 .

 2 Figure 2. Sample study: explants of prosthetic heads (provided by the Tural company) .

Figure 4 .

 4 Figure 3. Measurement method.

Figure 5 .

 5 Figure 5. Reconstruction of the surfaces obtained after the removal of the original form (sphere) by the least-squares criterion.

Figure 6 .

 6 Figure 6. First six natural vibration modes of a flat surface .

Figure 7 .

 7 Figure 7. Representation of the modal amplitude spectrum.

  3.3.3. Modal filtering.(a) Residual. The residual, denoted by , is used to compare the measured surface and the surface reconstructed from all of the decomposition modes. This step allows us to evaluate the relevance of the modal parameters and the quality of the approximation made for the surface considered. The residual of the DMD is obtained by

Figure 8 .

 8 Figure 8. Representation of the residual of the DMD calculated on the surface (Pa) .

Figure 9 .

 9 Figure 9. Steps of the modal method.

Figure 10 .

 10 Figure 10. Surface Pb and the form reconstruction with N f = 5, 10, 15, 20 and 40.

Figure 11 .

 11 Figure 11. Multi-scale analysis of the prosthetic head Pa, with N f = 40 and N w = 150.

Figure 12 .

 12 Figure 12. Multi-scale analysis of the prosthetic head Pb, with N f = 40 and N w = 150.

Figure 13 .

 13 Figure 13. Multi-scale analysis of the prosthetic head Pa without sphere fitting, with N f = 40 and N w = 150.

Figure 15 .

 15 Figure 15. Multi-scale analysis of a theoretical sphere, with N f = 40 and N w = 150.

Acknowledgments

This research effort is supported by the research program IDEFORA. These works have been done in collaboration between the Symme Lab (Annecy) and I2M Lab (Bordeaux), and applied to samples of the Tural Company, partner of the program.


Clio Azina

Iñaki Cornu

Jean-François Silvain

Yongfeng Lu

Jean-Luc Battaglia

Effect of titanium and zirconium carbide interphases on the thermal conductivity and interfacial heat transfers in copper/diamond composite materials

Keywords: metal-matrix composites (MMCs), interface/interphase, thermal properties, transport phenomena analysis

Thermal properties of metal matrix composite materials are becoming ever more relevant with the increasing demand for thermally efficient materials. In this work, the thermal conductivity and heat transfers at the interfaces of copper matrix composite materials reinforced with diamond particles (Cu/D) are discussed. The composite materials contain either ZrC or TiC interphases and exhibit, respectively, higher and lower thermal conductivities with respect to their pure Cu/D counterparts. These thermal conductivities are accounted to the presence of strong covalent bonds and increased relative densities. The role of these interphases is also discussed regarding the phonon transmission at the interfaces.

Introduction

Thermally efficient materials are constantly being studied for several engineering purposes such as heat-sinks in microelectronic components. Such materials allow proper heat dissipation and ensure a longer lifespan of the microelectronic assembly. In a constant effort to find highly conductive materials, numerous results have been reported regarding the copper(Cu)/diamond(D) system [1][2][3] , and related interphases [4][5][6][7][8][9][10][11][12][13] . Indeed, composite materials, such as Cu/D, are attractive because of their ability to combine the properties of both constituting phases. However, in the absence of an appropriate interface between Cu and D, the thermal conductance hc at the interface is low, resulting in a lower effective thermal conductivity, Keff, with respect to the conductivity of pure Cu.

Several values of Keff have been reported in the literature for the Cu/D system that vary according to the fabrication process, as well as to the properties of the reinforcement in terms of volume fraction fd, characteristic dimension d, shape, and thermal conductivity kd. Indeed, for a composite without interfacial layers, Schubert et al. reported Keff = 215 W.m -1 .K -1 for fd = 42 % and d = 120 μm 1 . On the other hand, Tao et al. reported that Keff varied from 150 down to 42 W.m -1 .K -1 when fd varied respectively from 50 to 70 % for d = 40 μm 3 . More recently, He et al. reported a value of Keff = 414 Wm -1 K -1 for fd = 90 % and d = 220-245 μm, meaning that a slightly higher value of the equivalent thermal conductivity than that of copper can be reached without the presence of an interfacial layer 9 . Yoshida and Morigami reported the highest thermal conductivity for a pure Cu/D system as Keff = 742 W.m -1 .K -1 for fd = 70 % and d = 90 -100 μm, the thermal conductance of the interface being about hc ≈ 3×10 7 W.m - 2 .K -1 2 . Their remarkable result comes from the elevated sintering temperature and pressure of Ts = 1180 °C and Ps = 4.5 GPa, respectively, during 15 min using a belt-type high-pressure apparatus 14 . However, performing sintering at very high temperatures and pressures during short durations is not available at the industrial level, although progress is continuously reported.

All these results showed that despite the weak interfacial bonding between Cu and D, an increase of the composite's thermal conductivity is expected when increasing the size of the diamond particles and performing fastest hot-pressing processes. Obviously, the dependence of the thermal conductivity on the diamond particle size indicates that the interfacial thermal barrier must be considered. However, it has also been shown that the increase in the thermal conductance at the interface between metals and diamond is still weak, and that it tends to saturate at very high pressures 15 . In a non-reactive system such as Cu-D, the creation of an interphase, for example a carbide layer or submicronic Cu particles attached through oxygen bonds to the matrix and reinforcements 16 , will allow a better transfer of thermal and thermomechanical load from one phase to the other. In this work we focus on the insertion of carbide interphases.

As reported by Schmidt-Brüken et al. 5 and Dewar et al. 6 , alloying of copper with a strong carbide forming element promotes wetting and bonding with diamond and leads to a significant increase of the thermal conductivity. It has been well established that the thermal conductivity is higher as the chemical bond between matrix and reinforcement is stronger 17 .

Bond strength enhancement has been observed with addition of Ti, Cr, Zr, and B in the Cu powder prior to the sintering process. Two main physical phenomena can explain the increase in thermal conductivity. The first one is related to the chemical bond between the carbide layer and both the diamond and Cu, and the second one is related to the role of the interphase regarding the heat transfer process at the nanoscale between two very different materials 15 . Indeed, electrons are responsible for the heat transfer in copper whereas phonons dominate in diamond. Regarding the Cu-D system, the electron-phonon coupling model is insufficient to explain the conductance at the metal-dielectric interface. Due to the low cut-off frequency of phonons in metals and the very high one for diamond, only inelastic processes, such as the three-phonon scattering process, are able to explain the measured interfacial extra conductance beyond the radiation limit, resulting from the elastic two-phonon process. The necessary coupling between phonons is then largely improved by inserting an interphase between the two materials. However, this layer must be chosen carefully in terms of its intrinsic thermal conductivity, Debye temperature and thickness. As revealed by Weber and Tavangar, the concentration of the active element, forming the carbide interphase, must not exceed the limit required for carbide formation 10 .

We report in Table 1 several thermal conductivities obtained from the literature which consider different carbide interphases, and different sintering parameters. Although some inconsistencies occur, remarkable trends can easily be identified to reach the highest thermal conductivity of the composite material. Some conditions which stand out are: (i) high heating rates dT/dt (°C.min -1 ), (ii) reinforcement contents which are close or exceed the percolation threshold, (iii) large reinforcement sizes (d > 120 μm), and (iv) interphase thicknesses, eX, that are higher than the mean free path of hot carriers. Condition (iv) allows maximizing the elastic scattering of phonons at the interface between the interphase and both adjacent materials (the matrix and the particle). The results obtained by Li et al. 12,18 that reported a thermal conductivity exceeding 900 W.m -1 .K -1 for a CuZr0.5 composite with d = 230 μm, and fd = 0.61, and a thermal conductivity exceeding 750 W.m -1 .K -1 for a CuTi0.5 with same values for d and fd, should be pointed out. The authors attribute these extremely high values of TCs to the optimized ZrC and TiC interphases formed using the gas pressure infiltration method.

Therefore, the upper limit for eX must be chosen in order to respect condition (iv) but also to lead to the best bonding at both the diamond/carbide and carbide-copper interfaces. However, the choice of the active element remains questionable regarding the values reported in Table 1 where similar values for Keff are found using either Cr, Ti, Zr or W with optimal values of d, fd and sintering parameters. In addition, it is not clear that diamond particles coated with the active element would lead to significant improvement with respect to uncoated particles. The experimental data are well predicted, according to the authors of the mentioned works, from the theoretical models of the effective thermal conductivity derived, for instance, by

Hasselman and Johnson 23 (H-J model). These models show that increasing the volume fraction of D in the Cu matrix, when hc does not reach the minimal value at a given value of particle diameter, d, leads to a significant decrease of the effective thermal conductivity of the composite. The H-J model rests on the Maxwell approach that assumes the dilution of the diamond particles within the matrix. Indeed, in the derivation by Maxwell, the assumption has been made that the average distance between the particles of the discontinuous phase is large enough so that the fields around individual particles are undisturbed by the presence of the other particles. However, as reported in the work of S. Whitaker, based on the volume averaging method [START_REF] Whitaker | The Method of Volume Averaging[END_REF] , the analytical model of Maxwell is retrieved by considering a unit cell as that described by Chang 25,26 . This configuration assumes a periodic arrangement of the unit cells and a temperature gradient for two parallel faces of the cube whereas the other perpendicular faces to the temperature gradient are insulated.

In the present paper we report the Cu(Cu-Ti)/D and Cu(Cu-Zr)/D systems obtained through the solid-liquid coexistent phase process 27 , and their corresponding thermal conductivities, as well as the heat transfers at the interphases before and after the percolation threshold. We discuss the role of the carbide interphase in the elastic and inelastic phonon-phonon processes and the contribution of chemical bonding at the involved interfaces within the measured enhancement of interfacial thermal conductance between copper and diamond. Theoretical values of Keff are calculated using both the H-J model and finite element obtained based on simulations of the heat transfer within the representative elementary volume of the composite considering randomly distributed particles. Both approaches are investigated. The average thermal resistance Rc = 1/hc, at the particle-matrix interface, is then identified by comparing the experimental values and the theoretical ones for each type of interphase. Finally, these values of Rc are compared to the theoretical expectations calculated using the Diffuse Mismatch Model (DMM).

Materials and methods

2.1.Interphase creation

The copper-based matrix is composed of a mixture of dendritic copper powder (Eckart No squeezing of the liquid was observed during the sintering process, which we explain by the very small amount of liquid generated. The heating is ensured using an induction system, while the pressure is applied through the means of a hydraulic press. The temperature is controlled with the use of a thermocouple (type K) which is inserted into the mould. The chamber is put under vacuum (10 -2 mbar range) to prevent the oxidation of the copper matrix during heating and/or cooling.

In the following sections we will be referring to the composite materials using the following nomenclature: Cu(Cu-X)y/Dz, where X is either Ti or Zr, y is the volume percentage of Cu-X alloy with respect to the total volume, and z is the volume percentage of diamond particles.

2.2.Characterization

Cryofractures were carried out in liquid nitrogen. The microstructures of each composite were analyzed using scanning electron microscopy (SEM) (TESCAN VEGA 2 SBH) in secondary and back-scattered electrons mode, to deduce chemical information through apparent chemical contrast.

X-ray diffraction patterns were collected on a PANalytical X'pert PRO MPD diffractometer.

The patterns were analyzed using EVA software (Bruker).

Due to the difficulty of machining diamond-based materials, preliminary studies were carried out on carbon fiber reinforced composite materials 27 . For the purpose of this article, the interphase thicknesses were considered to be the same for both carbon fiber and diamond particle-reinforced composites.

2.3.Thermal measurements

The thermal diffusivity of the composite materials was measured using a specific photothermal technique based on a periodic pulse (duration of 20 ns) waveform of the photothermal source 29 . This technique has been developed to deal with highly conductive materials without the need of absorbing/emitting coatings on both faces of the sample. The standard deviation of the measurement is less than 2 %. The average specific heat for each sample is calculated using the volume-based rule of mixture based on the densities and specific heats of Cu and diamond as well as on the volume fraction of diamond particles. The effective thermal conductivity is thereby calculated from the measured thermal diffusivity and the average specific heat. ) and tends to decrease with increasing fd (> 0.3). This result is not surprising since the thermal contact at the diamond-copper interface is very low due to the purely mechanical bonds and very different phonon properties, as revealed by the Debye temperature of the two materials. The effective thermal conductivity of Cu(Cu-Ti)/D with particle diameter equal to 60 μm is higher, although it remains very comparable to the thermal conductivity of copper, when fd = 0.4 (purple spheres, Figure 2). This disappointing result is explained by the fact that some Ti remains within the copper matrix and drastically decreases the thermal conductivity of the matrix and therefore that of Keff 27 .

Results and discussion

Experimental thermal conductivities

3.2.Analytical vs numerical models

The classical model of the effective thermal conductivity derived by Hasselman and Johnson (H-J, Equation 1) is valid when the volume fraction fd is less than the value at percolation fd,per = 0.52 considering the particles of diameter d 23 . It must be noted however that the percolation threshold for randomly packed spheres is about 0.31 whereas the threshold for randomly packed diamond is expected to be around 0.42 (because of the faceted shape of the particles). As presented in Figure 5(a), the dispersion process of diamond particles within the copper powder is not uniformly distributed. Therefore, we calculated also the effective thermal conductivity using the finite element (FE) method applied to a simulated medium, that is a cube with side a in which the spheres are randomly distributed, as presented in Figure 5(b). The face of the simulation box is at a temperature of 0 K whereas the parallel face is at 1 K. All the other faces are insulated. The calculation of the effective thermal conductivity has been made considering, for a given value of fd, three different random distributions of the particle. Therefore, it was verified that the calculated value does not change more than 2% between each simulation. In addition, the side a of the simulation box is chosen to be large enough in order to avoid fluctuations of the calculated values when changing the particle diameter. We found that using a = 20×d allows to fulfil this requirement. The effective thermal conductivity Keff is calculated according to fd and Rc for d = 60 μm and reported in Figure 5(c). Keff is also calculated according to Rc and d for fd = 0.42 and reported in Figure 5(d). The H-J model and the FE lead to comparable results. A small discrepancy occurs for large values of d and small values of Rc.

These results lead to conclude about the non-significant influence of the particles distribution within the matrix when fd < fd,per. As shown, the effective thermal conductivity is slightly sensitive to the particles distribution when the resistance of the interphase is very low (see the results for Rc = 4×10 -9 m 2 .K.W -1 , for instance). However, such low values cannot be reached in practice since it is already lower than the intrinsic thermal conductivity ei/ki of the interphase without considering the resistances at both the Cu-interphase and interphase-diamond interfaces. On the other hand, Figure 5(c) shows that Keff is equal or lower than that of copper when 𝑅 𝑐 ≥ 5.5 × 10 -8 𝑚 2 . 𝐾. 𝑊 -1 whatever the volume fraction fd. At this threshold value of Rc, the effective thermal conductivity Keff is equal to that of copper km. In that case, relation (1) becomes

𝑘 𝑑 = 𝑘 𝑚 (1 -𝑘 𝑚 𝑎ℎ 𝑐 ⁄ ) ⁄
. As shown in Porosity is not an issue in these composites, as the solid-liquid coexistent phase process allows full densification of high-volume fraction composites, thanks to the liquid phase which flows through the pores. K.m 2 .W -1 and kd = (1038 ± 80) W.m -1 .K -1 (we used the Levenberg-Marquardt minimization algorithm with 2×10 -8 m 2 .K.W -1 and 1500 W.m -1 .K -1 as initial values for Rc and kd respectively).

The comparison between theoretical data and measurements for the Cu(Cu-Ti)/D system is represented in Figure 7(d). The remaining quantity of Ti within the Cu matrix, at the end of the sintering process, depends on the initial quantity inserted in the composite mixture. Indeed, large volume fractions of reinforcements, for the same Ti amount will lead to less Ti within the matrix. This trend can be observed for the composites containing 20, 30 and 40 vol.% of diamond particles and the fixed amount of 4 vol.% Cu-Ti alloy. The corresponding values of the metal matrix thermal conductivity is derived from the measurements of Lloyd et al 13 . We

(b) (a)
found that the value Rc = (1.05 ± 0.2)×10 -8 m 2 .K.W -1 and kd = (1040 ± 90) W.m -1 .K -1 allows the best fit for the experimental data, using the same procedure as previously. The identified value of kd is consistent with the one we found for the Cu(Cu-Zr)/D system as well as the identified value of Rc for the Cu(Cu-Ti)/D system which is very close to that found for the Cu(Cu-Zr)/D system. Phonon density of states g(ω) measurements from neutron scattering experiments for diamond, Cu, ZrC and TiC have been made by several authors and are reported in Figure 8.

3.3.Nanoscale analysis of interfacial thermal resistances

The phonon DOS for diamond has a prominent peak at 37 THz, thus very close to the cutoff frequency (about 40 THz), which results in optical modes associated to sp 3 bonds. The phonon DOS for Cu is located at about 7 THz, which is also the cutoff frequency. It is thus clear that the high difference in cutoff frequency for Cu and diamond limits the heat transfer through elastic phonon scattering, making therefore the inelastic scattering mandatory to enhance the thermal conductance of the interface. It was well demonstrated that anharmonic inelastic scattering (one diamond phonon splits into two equal frequency metal phonons) are mainly responsible for the conduction at the interface between diamond and Cu 17,32 As reported by Liang and Tsai, using molecular dynamics (MD) simulations, inserting an interlayer whose Debye temperature is about the square root of the product of the Debye temperatures of the two solids, will enhance the thermal conductance between these two solids 33 . The optimal Debye temperature for an interfacial material between Cu and diamond should be 𝛩 𝐷,𝑖,𝑜𝑝𝑡 = √310 × 1860 ≈ 760 𝐾. Therefore, TiC and ZrC could be potential candidates regarding this criterium, although the Debye temperature of TiC is closer to the optimal (Table 2). Therefore, the DOS spectra overlap spans a much larger range of frequencies than the one obtained from the bulk DOS. However, this change cannot be easily predicted in our configuration, therefore, the bulk DOS is used to calculate the theoretical thermal conductance. Hence, the obtained thermal conductance value must be viewed as the minimal one. On the other hand, all the simulations by MD presented in previous papers assumed Lennard-Jones (LJ) potentials, even within the case of a disordered interface. This cannot be retained as a credible configuration in the present study since, as shown by Losego et al. 17 , transition from van der Walls (Cu-diamond system) to covalent bonding (Cu-TiN and TiNdiamond systems) increases strongly the interfacial thermal conductance. The thermal boundary conductance was found to be most influenced by the interfacial chemical bonding as both the phonon flux and the vibrational mismatch between the materials are each subject to the interfacial bond strength. The phonon mean free path and thermal conductivity, ki, of the interfacial material strongly influence the optimal thickness at which the interfacial thermal conductance hc increases. The thermal resistance Rc of the interphase is therefore given by:

𝑅 𝑐 = 𝑅 𝐶𝑢-𝑖 + 𝑒 𝑖 𝑘 𝑖 + 𝑅 𝑖-𝐷 (2)
In this relation ki is the thermal conductivity of the material which constitutes the interphase (TiC or ZrC), ei is the thickness of the interphase, Rcu-i is the thermal resistance of the interface between the interphase layer and copper and Ri-D is the resistance of the interface between the interphase layer and the diamond. The thicknesses used in this calculation were eTiC = 200 nm and eZrC = 125 nm. The interfacial resistance between two materials, a and b, is calculated using the asymptotic expression of the DMM at high temperature given by: 𝑅 𝑎-𝑏 =

where τa-b is the transmission coefficient of phonons at the interface between materials a and b, that is defined as 𝜏 𝑎-𝑏 = ∑ 𝑣 𝑔,𝑗,𝑏 -2 /(∑ 𝑣 𝑔,𝑗,𝑏 the Cu/D system, where scattering is mainly inelastic, the calculated resistance using the DMM has to be viewed only as qualitative. All the required properties for these calculations are reported in Table 2. Using this data, the values for τ and Ra-b reported in Table 3, are calculated. Finally, the thermal resistance for the two interphases are: Rc,TiC = (1.17 ± 0.15)×10 -8 K.m 2 .W -1 and Rc,ZrC = (1.11 ± 0.14)×10 -8 K.m 2 .W -1 . It is not surprising to find that both interfacial materials lead to comparable values of thermal resistance. Also, one can notice that these values are reasonable for our composite materials; therefore, the thickness approximation made is acceptable. On the other hand, it is clear that the electronic contribution is one order of magnitude lower that that of phonons in Rc.

Conclusions

In this work the effective thermal conductivities and heat transfers at the interphases of Cu(Cu-Ti)-D and Cu(Cu-Zr)-D systems were investigated. The composite materials were obtained using the solid-liquid coexistent phase process, which enhances the diffusion and reactivity of the carbide forming elements with the carbon reinforcements and allows full densification of composite materials with high contents of reinforcements. Cryofractures have confirmed the presence of a strong bonding between the Cu matrix and the diamond reinforcement by exhibiting the fragile rupture of the diamond reinforcements. The presence of ZrC interphases was shown to increase the thermal conductivities of the composite materials with respect to Cu/D composite materials with no interphases, which also confirms the strong bonding brought by the interphase. The highest thermal conductivities were obtained for the composite materials containing ZrC interphases rather than TiC, as Ti tends to drastically decrease the intrinsic thermal conductivity of the Cu matrix, even in very small concentrations. The effect of the diamond particle size was also discussed, and we confirmed that larger diamond particle sizes lead to higher effective thermal conductivities. The comparison of the simulations with the experimental results led to determine the thermal resistances of Rc,TiC = (1.05 ± 0.2)×10 -8 K.m 2 .W -1 for the TiC interphases and Rc,ZrC = (1.12 ± 0.2)×10 -8 K.m 2 .W -1 for the ZrC interphases. Finally, the heat transfer through the interphases was analyzed at the nanoscale. Thermal resistances at the interfaces between the metalcarbide interphase and carbide interphase-diamond were calculated using the Diffuse Mismatch Model. The values found for Rc,TiC and Rc,ZrC using this approach are very close to each other and are also very close to those extracted from bulk measurements.

Acknowledgement

  Granulate Velden GmbH) (d50 close to 35 μm) and either Cu-Ti alloyed powder with a composition of 21.79 wt.% in Ti (d50 close to 15 μm) or Cu-Zr alloyed powder with a composition of 8.6 wt.% in Zr (d50 close to 9 μm), both produced by atomization and supplied by Nanoval GmbH. The Cu-Ti content was varied between 2 and 6 vol.% and Cu-Zr was maintained at 14 vol.%, with respect to the total composite volume. The volume fractions used correspond to those which have resulted in the highest thermal conductivities in preliminary investigations. Grade MBD6 diamond particles of d50 ranging from 40 to 220 μm (Henan Zhongxing Corporation), were used as reinforcements (kd = 1000-1500 W.m -1 .K -1 thermal conductivity). The copper powders and reinforcements are mixed together using a 3D mixing device during 90 min at 20 RPM. Composite materials reinforced with diamond particles were sintered using a TermolabTM press. The powders were pre-compacted in a graphite mold before sintering. The pure Cu/D composites were sintered at 650 °C under 40 MPa of pressure for 30 min, the powders containing Ti were hot-pressed at 950 ºC under 40 MPa for 30 min, while those containing Zr were hot-pressed at 1050 °C under 40 MPa for 45 min.

Figure 1

 1 Figure 1 gathers the SEM micrographs obtained after cryofracture of the Cu/D40, Cu(Cu-Ti)4/D40, and Cu(Cu-Zr)14/D40 composite materials. One can observe the lack of cohesion at the interface between the Cu matrix and the diamond reinforcements in the case of the Cu/D40 composite material. This is due to the lack of chemical affinity between the constituents. The shock induced by the cryofracture caused a certain amount of diamond particles to detach from the matrix, leaving traces of their initial positions. The few diamond particles, which remained attached, are surrounded by voids which can clearly be seen in Figure 1(a). However, the composite materials synthesized with either Cu-Ti or Cu-Zr

Figure 1 :

 1 Figure 1: SEM micrographs of fractured surfaces of (a) Cu/D40, (b) Cu(Cu-Ti)4/D40, and (c) Cu(Cu-Zr)14/D40 composite materials after cryofracture in liquid nitrogen. The measured thermal conductivity values reported in Figure 2 correspond to those of the Cu/D and Cu(Cu-Ti)/D composite materials. As expected, the effective thermal conductivity of the Cu/D (gray spheres, Figure 2) composite material, without interphase, never exceedsthat of copper (390 W.m -1 .K -1 ) and tends to decrease with increasing fd (> 0.3). This result is

Figure 2 :

 2 Figure 2: Thermal conductivities with respect to the diamond reinforcement content of the Cu/D (gray spheres) and Cu(Cu-Ti)/D (purple spheres) composite materials. The initial volume fraction of Cu-Ti in the composites are reported for each Cu(Cu-Ti)/D sample. The XRD patterns presented in Figure 3 (a) and (b) show the contributions of TiC and ZrCrespectively within the composite materials. While the presence of impurities within the Cu matrix is not necessarily detrimental to the properties, Ti has been shown to significantly decrease the electrical conductivity of pure Cu31 . Therefore, one can expect that a small amount of Ti impurities within the composite could decrease the overall conductivity of the composite. To confirm this hypothesis a CuTi alloy (1 vol.% of Cu-Ti in Cu) has been synthesized. The value of the thermal conductivity of this alloy was deduced from the thermal diffusivity and is equal to 211 W.m -1 .K -1 . Here we show that the presence of Ti impurities can decrease the thermal conductivity of pure copper by almost half. Furthermore, SEM analyses on the alloy after chemical etching, revealed that the grain size of Cu corresponds to the initial dendrite sizes (Figure3 (c)). When compared with an equivalent CuZr alloy (kCu-Zr = 343 W.m -1 .K -1 ), the grain size of the CuTi alloy is smaller.

Figure 3 :

 3 Figure 3: XRD patterns of (a) Cu(Cu-Ti)/D and (b) Cu(Cu-Zr)/D composite materials. (c) and (d) are SEM micrographs of chemically etched CuTi and CuZr alloys respectively. Assuming the maximum of Keff is reached close to the percolation limit, we measured the thermal conductivity of Cu(Cu-Zr)/D composites with fd = 40 vol.% and we considered several values of particle diameter d, as shown in Figure 4. Results are more encouraging than those obtained with the TiC interphases even with the smallest diameter. The highest thermal conductivity Keff = 584 W.m -1 .K -1 was obtained with particles of 220 μm in diameter.

Figure 4 :

 4 Figure 4: Thermal conductivities of Cu(Cu-Zr)/D composites with fd = 40 vol.% for different particle sizes (dpart).

Figure 5 (

 5 d) this critical value increases significantly as the particle diameter increases. Finally, the uncertainty on the value of the thermal conductivity of the diamond particles must be considered since both materials constitutive of the composite are very conductive. Since measuring an accurate value on small grains is a very challenging task, the identification of both Rc and kd, starting from experimental values of Keff, is carried out and discussed later.

Figure 5 :

 5 Figure 5: (a) Diamond particles distribution measured on a cross-section view by SEM at different volume fractions, (b) random normal distribution of diamond particles within the copper matrix used for finite element calculations of Keff at different fd. Calculated effective thermal conductivity as a function of (c) the thermal resistance Rc of the interface with fd varying from 0.09 to 0.49, and (d) the particle diameter d with the interfacial thermal resistance Rc varying from 1×10 -9 to 1×10 -7 K.m 2 .W -1 with fd = 0.42. Spheres represent the realistic model and dashed lines represent the FE model.

Figure 6 :

 6 Figure 6: (a) Measured (pink spheres) and calculated (dashed line) thermal conductivities of Cu/D composite materials reinforced with diamond particles of dpart = 60 μm and an interfacial thermal resistance of (1.62 ± 0.15)×10 -7 K.m 2 .W -1 . (b) Relative density of composite materials with respect to the diamond volume fraction with and without Cu-Ti addition.

Figure 7 :

 7 Figure 7: (a) ratio of the sensitivity functions of Keff for {Rc and kd} and for {Rc and km} according to fd with km = 400 W.m -1 .K -1 , kd = 1200 W.m -1 .K -1 , Rc = 10 -7 K.m 2 .W -1 , d = 60 µm. (b) ratio of the sensitivity functions of Keff for {Rc and kd} and for {Rc and km},according to d with km = 400 W.m -1 .K -1 , kd = 1200 W.m -1 .K -1 , Rc = 10 -7 K.m 2 .W -1 , fd = 0.4. Comparison of measured effective thermal conductivity of (c) the Cu(Cu-Zr)/D composite materials with the calculated values obtained using the H-J model from identified Rc and kd parameters, and (d) the Cu(Cu-Ti)/D composite material with the calculated values (dashed line) obtained using the H-J model from identified Rc and kd parameters. Initial Ti contents (vol.%) are indicated in blue for each value of fd. Remaining contents (wt.%) of Ti within the composite materials are given by light blue spheres (right Y-axis).

  40 vol.%, R c = 1.4×10 -8 K.m 2 .W -1

, 3 ⁄

 3 with subscript g denoting either the transverse (T) or longitudinal (L) velocity, and va is the speed of sound in material a that is defined as 3 . Since the DMM assumes elastic phonon scattering (a phonon of frequency w will only transfer energy across an interface by scattering with another phonon of frequency w), it is relevant for the Cu/ZrC/D and Cu/TiC/D systems. For

Table 1 :

 1 Measured effective thermal conductivities Keff of the Cu/X/Diamond composites, X

	being the carbide interphases (B4C, Cr3C2, TiC, ZrC, WC), and hc. IHP: indirectly heated hot
	pressing, DHP: directly heated hot pressing, SPS: spark plasma sintering, PPS: pulse plasma
	sintering, GPI: gas pressure infiltration route.
				ex (nm) / hc a	Keff	Ts (°C)/ Ps (GPa)/ dT/dt
		D (µm)	fd			(°C/min) (heating
				(MW.m -2 .K -1 )	(W.m -1 .K -1 )	duration, process)
		120	0.42	-/17	482	950/-/-(1h, IHP)
	CuCr0.8 4
		120	0.42	-/94	639	950/-/-(30 min, DHP)
	CuCr0.8 1	177-210	0.5	-/50	640	-/-/-
	CuCr d 7	70	0.65 b 1000/34	562	1150/0.02/-(10 min)
	CuCr d8	100	0.5	1000/-	284	920/0.03/110 (SPS)
	CuCr d19	200	0.5	200/21	589	900/0.08/-(10 min, PPS)
	CuB0.3 4	120	0.42		490	950/-/-(1h)
	CuTi1 4	120	0.42		290	950/-/-(1h)
	CuTi0.6 11	300	0.5	272/-	620	1646/0.7/-(30 min) c
	CuTi0.5 e20	180	0.5	-/48	630	1000/0.05/50 (10 min, SPS)
	CuTi0.5 18	230	0.61 b 200 -300/-	752	1150/0.01/-(10 min, GPI)
	CuZr1 9	220 -	0.9 b	-/5.9 b	677	1500/5/-(10 min) c
		245
	CuZr0.7 4	120	0.42		285	950/-/-(1h)

  . In addition, electron-phonon interactions within the metal, lead to a thermal resistance 𝑅 𝑒𝑝 = 1 √𝐺𝑘 𝐶𝑢,𝐿 ⁄ in series with the phonon-phonon resistance, where 𝑔 = (5.73 ± 0.15) × 10 16 𝑊. 𝑚 -3 . 𝐾 -1 is the volume electron-phonon coupling factor in copper and 𝑘 𝐶𝑢,𝐿 = 8.45 𝑊. 𝑚 -1 . 𝐾 -1 is the lattice thermal conductivity of Cu. Using these values, this comes to 𝑅 𝑒𝑝 = (1.437 ± 0.19) × 10 -9 𝐾. 𝑚 2 . 𝑊 -1 .

Table 2 :

 2 Density, thermal conductivity, specific heat, Debye temperature, longitudinal and transverse velocities, and melting temperature for Cu, TiC, ZrC, and diamond, found in the literature.On the other hand, as shown in Figure8, the phonon DOS for TiC and ZrC are very similar in terms of both spectral distribution and cutoff frequency, that is about 20 THz. As stated by English et al. and Shin et al., phonon DOS for materials located at both sides of an interface differs from bulk phonon DOS for each material. In fact, the presence of interfacial atomic restructuring is expected to reduce the phonon boundary resistance by providing additional transport channels15,34 .Figure8: Phonon density of states g(ω) measured from neutron scattering experiments for diamond35 , Cu36,37 , ZrC38 , and TiC 39 .

	Material	Density (kg.m 3 )	Thermal conductivity (W.m -1 .K -1 )	Specific heat capacity (J.kg -1 .K -1 )	Debye Temperature ΘD (K)	vT ; vL (ms -1 )	Melting point (K)	Cut-off frequency ωc (THz)
	Cu	8960	400	380	310	4760; 2300	1358	7

Table 3 :

 3 Phonon transmission coefficients and thermal resistances of the interfaces involved in the studied systems.

	Interface	τ	Ra-b (K.m².W -1 )
	Cu-D	0.035	1.29×10 -8
	Cu-TiC	0.124	3.67×10 -9
	Cu-ZrC	0.171	2.65×10 -9
	D-TiC	0.205	3.03×10 -9
	D-ZrC	0.150	3.45×10 -9

The authors thank Mr. Eric Lebraud of the Institut de Chimie de la Matière Condensée de Bordeaux for performing the XRD analyses. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.


Simon-Frédéric Désage
email: simon-frederic.desage@univ-savoie.fr

Gilles Pitard
email: gilles.pitard@univ-savoie.fr

Maurice Pillet
email: maurice.pillet@univ-savoie.fr

Hugues Favrelière
email: hugues.favreliere@univ-savoie.fr

Jean-Luc Maire
email: jean-luc.maire@univ-savoie.fr

Fabrice Frelin
email: fabrice.frelin@univ-savoie.fr

Serge Samper
email: serge.samper@univ-rennes1.fr

Gaëtan Le Goïc
email: gaetan.le-goic@u-bourgogne.fr

Syntactic texture and perception for a new generic visual anomalies classification

Keywords: visual inspection, classication, Surface imperfection, geometrical description, visual eect, visual texture

The research purpose is to improve aesthetic anomalies detection and evaluation based on what is perceived by human eye and on the 2006 CIE report. 1 It is therefore important to dene parameters able to discriminate surfaces, in accordance with the perception of human eye. Our starting point in assessing aesthetic anomalies is geometric description such as dened by ISO standard, 2 i.e. traduce anomalies description with perception words about texture divergence impact. However, human controllers observe (detect) the aesthetic anomaly by its visual eect and interpreter for its geometric description. The research question is how dene generic parameters for discriminating aesthetic anomalies, from enhanced information of visual texture such as recent surface visual rendering approach. We propose to use an approach from visual texture processing that quantify spatial variations of pixel for translating changes in color, material and relief. From a set of images from dierent angles of light which gives us access to the surface appearance, we propose an approach from visual eect to geometrical specications as the current standards have identied the aesthetic anomalies.

INTRODUCTION 1.1 Motivation

Nowadays, perceived quality of visual appearance is [still] an industrial matter. The main diculty is to link objective measurements and subjective human aesthetic indicators. The CIE (International Commission on Illumination) provided a technical report CIE 175:2006 A framework for the measurement of visual appearance .

This report denes four headings under which possible measures might be made: color, gloss, translucency and texture. However, when a customer, and thus imitating his behavior, a human controller, inspect the visual appearance of a product, they can not dierentiate the four parties. Considering color and translucency as known and controlled by the industry, we have chosen to work on the gloss and texture. The goal is to identify visual aesthetic anomalies and traduce some geometric considerations from ISO standard 8785:1998 2 to visual eects considerations. A known diculty is the interdependence of these measurements because translucency can inuence color, which may inuence gloss, and texture is probably a combination of all three. perception notions in Evaluation step. There is some anomaly elements strongly detected but weakly perceived, and conversely, some anomaly elements weakly detected and strongly perceived. We have chosen to present in this paper a proposal for improvement of the syntactic description of surface imperfections to prepare the evaluation. This description must be more perceptual and must take into account the visual impact of anomalies.

Our starting technical point is the exploitation of the most advanced visual surface representation and image processing approach of texture classication.

7 Our starting theorical determination is the study and the imitation of human behavior in the surface quality inspection (Figure 1).

Methodology

There are already many research and industrial solutions exploiting anomaly appearance descriptions, but these are case by case solutions with suitable devices (limited lighting and viewing conditions). We nd examples in various elds of applications, such as for controlling the road conditions 8 or the appearance of the fruits, 9 or to control the textile manufacturing quality. 10

This work proposes a generic [perception-based] visual anomalies classication, like a bridge between geometrical specications from dimensional metrology and surface appearance specications of aesthetic anomalies. The research purpose is double. Firstly, following quality inspection formalization based on visual human behavior and providing complementary indicators of detection and perception, there is a hope to provide a better understanding of visual human mechanisms. Secondly, there is a desire to reduce the surface anomaly classication to the perceptual classication and to provide a robust imitation of controller for detection and evaluation of aesthetic anomalies. The syntactic appearance classication is based on texture attributes, such as the tone (light, dark), the roughness (rough, smooth, rippled), the light aspect (matte, speckled), the nesse (ne, coarse) and the regularity (regular or irregular). By example, scratch is viewed like a white line, i.e. a ne anomaly with a regular, smooth, light tone and speckled aspect. 11

In short, there are two steps to construct a new visual anomalies classication. Firstly, we need to provide an exhaustive restricted classication, based on geometrical considerations of the standards for surface imperfections.

Secondly, we need to describe all anomalies by their visual eects. We propose then some considerations from new visual texture processing to identify the presence or not of an anomaly.

CLASSIFICATION OF AESTHETIC ANOMALIES

Then the work is composed of two parts, similar to the human visual system. The rst part is to describe the visual impression (or stimulus) of appearance anomalies with a pattern recognition mechanism. Indeed, the rst part of the human visual system is associated with the eye and its ability to detect and pattern recognition learned by individual culture. This work is based on the standards of geometrical product specications 2 as the human capacity for interpretation allows visual recognition from the causal description of an anomaly. These norms are used to dene the set of search anomalies and associated criteria. The second part is to put into perspective the shape or patterns recognized with an ambient context to perceptual evaluation interdependent with the surrounding visual texture (object context).

Geometrical classication and standard

A nding of our previous work is that the vocabulary designating surface imperfections among industrial companies, whether or not in the same domain is rarely the same. This diculty leads to potential misunderstandings because how do we know if we talk about the same thing. We have chosen to focus the rst part of work from the ISO 8785 1998 2 dealing with surface imperfections. This standard provides anomalies terms, denitions and parameters from causal descriptions. We establish an initial classication of visual anomalies following their morphological and appearance features. Our starting point is the 31 cases outlined in the standard 2 summa- rized in a table (Figure 3) to have a global view of all surface imperfections, considering that the standard 2 is comprehensive.

Figure 3: Table of ISO 8785 Anomalies Classication from their causal descriptions

We propose to use the following ve simple scales to classify these imperfections. These scales are illustrated by the gure 4, respectively.

1. First rule : planar shape -From linear to circular. This rule is calculated based on the circularity of the bounding ellipse of the anomaly. Indeed, from this ellipse, we can extract the eccentricity, which is the ratio of the two axes of the ellipse. If it tends to 1 then the anomaly tends to look like a circle, while conversely, if the eccentricity tends to 0, then the anomaly will be linear.

2. Second rule : relief shape -From inwardly to outwardly. This rule measures the average topographic impact of the anomaly, ie if it has more removed or added to the material from the model plan. It allows to dierentiate mainly hollow compared to the bumps.

3. Third rule : density -From one to innity (a number -many). This rule measures the amount of small anomalies that constitutes a group such as we consider only a single visual anomaly. 4. Fourth rule : relative size scale -From microscopic to object scale. We study small parts size of ten centimeters maximum. Our size scale varies from mesoscopic to macroscopic, ie from the smallest visible by a human eye to a size consistent with the size of the observed part. The parts considered to be of good quality, we do not observe the anomalies consistent with the size of the part, but only at lower scales and mostly microscopic. 5. Fifth rule : appearance contrast scale -From dark to light via contrast. It is sometimes dicult to judge whether a part of the room is darker or lighter than its surroundings, considering the changes in his appearance for dierent lighting congurations / surface / viewpoint. In our approach, we consider appearance characteristic of the anomaly when is highlighted with respect to its neighborhood. For example, a scratch can vary from light to dark as the relative positioning illumination / view, but it is primarily highlighted when it shines compared to its neighborhood. In this case, we consider its appearance as light, in contrast to its neighbors. We can provide some clarications to the scale on the relative size of the anomaly, from considerations from the dimensional metrology. 12 Indeed, as shown in the gure 5, it is usual to consider four orders of composition as a macroscopic object, corresponding to the shape, waviness, roughness and microroughness. In our approach, we merge the two categories of roughness, especially because it can be dicult to discern both visually for a human without additional tools. and the pattern show the solutions set for groove-type anomalies. The advantage of this representation is that it allows you to associate a pattern to an anomaly, which is easier perceptually than a sentence to distinguish and compare, because it can be independent of the language used. If we consider at least three levels by previous classication rules, there are at least 3 5 cases, namely 243 cases. The standard allows to summarize all the case with just over 10% of cases (13%). Nevertheless we still want to reduce the classication of aesthetic anomalies relative to the geometric perception, to establish a generic and comprehensive automatic detection.

Restricted classication by geometrical perception

Previous work

13,14 has shown the possibility of reducing and optimizing only 4 types (perceptual descriptions) of anomalies, and virtually no loss in detection of surface appearance anomalies. These four categories are:

1. Mark: Any trace left on the surface of the part representing a lack of timely matter.

2. Deformation: Any change in shape, altering the initial shape.

3. Heterogeneity: Any change in spot color on the surface, shadow or cloud.

Pollution: A foreign body.

There is also a higher order than these four groups to distinguish between progessive or non anomalies. In our approach, we consider only those that are not. However, if it does not aect the anomaly detection method, it has an increased importance in the perception of scalable anomalies. Indeed, scalable anomalies are at risk of growing degradation medium or long terms, while non-scalable anomalies have a direct impact on the appearance or product specications.

Classication by visual eects

For recall, a photography, as image texture, is a visual texture slice and an image is a local result of color, material and relief combination. There is surface texture such as only relief texture, and the surface visual texture is described by a BRDF eld.

We consider a new decomposition 7 of the BRDF associated (Figure 10 Regardless of what eld of possibilities, we propose a classication (Figure 12) of the possible inuence of defects on the surface condition. That majority is observed anomalies inuence the relief, which correlates with the fact that mainly the techniques used for their detection are the responsibility of the topography. Nevertheless, one can recall the major diculty of the techniques of topography or dimensional metrology which need to be at the right scale to measure because if sampling of the surface is not good, it can not detect the anomaly . This is the advantage and disadvantage of the main visual detection because the visual eect is "visible" at multiple scales. Indeed, the fact that an aesthetic anomaly is visible at dierent scales is the fact that it is disturbing.

Our classication also provides that certain types of defects can be perceived by dierent visual eects. The name "Possible" here means "Yes" or "No". To illustrate this limit, we have taken the example of a mean scratch (average depth and width) whose visual impact is measured on dierent scales (Figure 13 (a)). Of course, in the real case, we must take into account the optical properties of the material and its color to perform the measurement in good conditions. In our example, the material considered is purely specular, as can be a metal, and there is no chromatic eect.

Measurement of changes and impacts on perceived visual texture

To measure and identify a deviation of the visual texture, we propose to use a method of analysis texture, similar to the work of Alice Porebski 16 (Application to aw identication on decorated glasses printing) which shows the possibility of classication and learning signatures of visual appearance and defects associated from the analysis of a co-occurrence matrix. For this, we rely on the extension of this method to visual texture analysis. 7

By cons, by the approach of visual texture analysis, it is appropriate to use ve new scales related to the texture.

1 We propose to use the following ve simple scales to classify these imperfections :

1. First rule : Tone -From light to dark.

2. Second rule : Roughness -From rough to smooth, via rippled.

3. Third rule : Light aspect -From matte to speckled.

4. Fourth rule : Finesse -From ne to coarse.

5. Fifth rule : Regularity -From regular to irregular.

The trend of anomalie inuence on visual texture is currently more light, more rough, more speckled, more coarse and more irregular.

CONCLUSION AND PERSPECTIVES

We present a new generic and reduced classication of aesthetic anomalies from syntactic surface texture analysis and human perception. We show a comparison between geometrical dimensional considerations like an analysis of anomaly origin and appearance and perceptual analysis like an analysis of anomaly visual impact.

Perspectives in the short or medium terms are to identify the dierent scales in visual perception visual texture indicators and integrate the human perception of values described by the laws of Gelstat, through perception indicators, to improve automatic assessment of aesthetic anomalies according to criteria of human perception.

Figure 1 : 1 Figure 2 :

 112 Figure 1: The vision scheme from CIE 1

Figure 4 :

 4 Figure 4: The ve appearance rules to describe surfaces imperfections from ISO 8785 -1999 (a) The scale of circularity criterion for planar shape, i.e. the eccentricity of ellipse bounding (b) The scale of relief criterion, i.e. the mean relief of anomaly (c) The scale of density, i.e. the number of labels (d) the relative scale of anomaly compared to object (e) the contrast scale compared to the neighborhood of anomaly

Figure 5 :

 5 Figure 5: The 4 orders scale of surface imperfections considered in dimensional metrology 12

Figure 6 :Figure 7 :

 67 Figure 6: Primary Kiviat evaluation diagram for anomaly

Figure 8 :

 8 Figure 8: Restricted classication according to 4 groups 13

Figure 9 :

 9 Figure 9: Correspondence between the 4 groups and initial descriptions. (a) Kiviat diagram for Mark-type anomalies (b) Kiviat diagram for Deviation-type anomalies (c) Kiviat diagram for Heterogeneity-type anomalies (d) Kiviat diagram for Pollution-type anomalies

15

 15

Figure 10 :

 10 Figure 10: Decomposition of BRDF (a) Phong model-like BRDF decomposition (b) BRDF decomposition following surface state

Figure 11 :

 11 Figure 11: Possible combinations of surfaces states change

Figure 12 : 13 3. 2 Figure 13 :

 1213213 Figure 12: Classication by visual impact -Restricted classication according to the change of surfaces states 13

ACKNOWLEDGMENTS

We thank our SYMME partners in MESURA project as well as Savoie Mont Blanc Industries and Conseil General 74 to enable us to carry out this research by giving us resources.