Olaf Groh-Samberg

Ingrid Tucci

Qualitative Interviewing of Respondents in Large Representative Surveys

Keywords: mixed methods, qualitative data, longitudinal data, life course JEL Code: C81, C83, Z13

Large representative surveys are using mixed methods to an ever-increasing degree. Biomarkers, register data, and experiments, for example, provide different types of data that can be linked with survey data. The use of qualitative interviewing of participants in longitudinal surveys is still rare, however, in the social sciences. Yet qualitative methods have proven just as valuable as quantitative methods in providing insights into social reality by reflecting the multidimensionality of individual life courses and lived realities. Furthermore, indepth interviews can provide a better understanding of individual decision-making processes and behavior resulting from more or less unconscious strategies. They also provide insights into decisive turning points in people's lives. Finally, by linking quantitative and qualitative data, the reliability of longitudinal information can be analyzed thoroughly in terms of accuracy as well as meaningfulness.

Introduction

In the social and behavioral sciences, the use of mixed methods to address a particular research question typically involves a combination of quantitative and qualitative methodologies (see [START_REF] Brannen | Mixing methods: the entry of qualitative and quantitative approaches into the research process[END_REF][START_REF] Bryman | Integrating quantitative and qualitative research: how is it done?[END_REF]Tashakkori and Teddlie 2003). As an increasing range of data becomes available for scientific research -as documented throughout this volume and in the Discussion Paper Series of the German Data Council -the possibilities for mixed method approaches are growing. However, the use of mixed methods to link data from large representative surveys to qualitative data is still rare. A recent trend in longitudinal surveys worldwide consists in the linkage of survey data with data from different sources using diverse methodologies. For example, birth cohort studies or household panels like BHPS, HILDA, PSID, and SOEP, are collecting biomarkers, objective health measures, data from experiments, daily experience sampling or register and institutional context data to survey respondents (see the respective chapters in this volume and, e.g., the new British household panel Understanding Society). In this context of methodological innovations of longitudinal surveys, conducting in-depth qualitative interviews with sub-samples of respondents is one important and promising, yet only recently developing issue.

Up to now, qualitative methods have been used primarily with quantitative data to "embellish" analyses (Mason 2006a). However, mixed methods approaches in the sense of a triangulation of quantitative and qualitative data collected from the same respondents might help to understand the mechanisms underlying human behavior and individual life courses (see, e.g., Giele and Elder 1998). This is particularly true with respect to individual decisionmaking processes, coping strategies, and biographical "turning points," i.e., events or experiences that play a decisive role an individual's life course by correcting trajectories (see Abbot 1997). The importance of decision-making is not only central to the so-called rational actor model that has become a common reference model in the economic and social sciences and is typically associated with the large-scale quantitative data analysis (see [START_REF] Goldthorpe | The Quantitative Analysis of Large-scale Data Sets and Rational Action Theory: For a Sociological Alliance[END_REF]; it is even considered a broader "unifying framework" for the behavioral sciences [START_REF] Gintis | A framework for the unification of the behavioral sciences[END_REF].

However, as quantitative research along these lines only observes the contexts, determinants, and outcomes of individual decisions -which are measured at least indirectly by means of proxy information -the decision-making process itself can only be modeled in a "reduced form" due to the lack of information on what is really going on in the individual's mind. This is exactly where qualitative in-depth interviews with sub-samples of survey respondents offer possibilities for new research prospects. Qualitative interviews may provide insights into how people select relevant information, what relevance they assign to them, and how their values, attitudes, perceptions, states of knowledge, and conscious as well as unconscious strategies are shaped by and shape their behavior. Thus, qualitative methods can provide insights into something that still remains a "black box" for quantitative methods that aim to connect measured "inputs" with measured "outcomes" of human decisions and behavior and strive to establish a "causal link" by testing the theoretically derived hypotheses. From a qualitative perspective, this causal link appears to present itself as a dynamic and recursive system of "meanings." This does not mean, however, that the two methods are incompatible (see [START_REF] Brannen | Mixing methods: the entry of qualitative and quantitative approaches into the research process[END_REF][START_REF] Kelle | Sociological Explanations Between Micro and Macro and the Integration of Qualitative and Quantitative Methods[END_REF]). Rather, by developing explanations of human behavior -for example, regarding educational decisionsthe assumptions of quantitative research typically derived from the rational actor model, or any other theory, can be more directly tested, specified, and enriched or even rejected by means of qualitative methods that allow a deeper understanding of how choices come about.

The State of the Art

Although still rather rare, the linkage of survey data with qualitative interviews seems to have reached scientific maturity, and is being discussed increasingly within the scientific community (see Tashakkori and Teddlie 2003). Although there are still forces at work promoting the separation of quantitative and qualitative methods -separate training courses, academic journals, funding schemes, and university chairs -efforts are also underway seeking to actively press forward with mixed method research (see, e.g., [START_REF] Bryman | Integrating quantitative and qualitative research: how is it done?[END_REF]Mason 2006b).

It has become apparent that mixed methods are not a third way, or even a third methodology in their own right, and that there exists a broad variety of means by which mixed method approaches can be rationalized and employed in empirical research. A metaanalysis by [START_REF] Bryman | Integrating quantitative and qualitative research: how is it done?[END_REF] of more than 200 research projects employing mixed methods reveals that mixed methods are mainly employed in sociology, and that they combine selfadministered questionnaire surveys with semi-structured interviews to address specific research questions. Mixed methods are typically used to produce "complementary" data or to "enhance" data, facilitating the examination of different perspectives or different aspects of a particular research question. However, there is no strict methodology that determines how different methods should be linked. Rather, there are good arguments for designing and linking mixed methods based on theoretical principles in order to produce non-redundant and non-trivial results (see [START_REF] Kelle | Sociological Explanations Between Micro and Macro and the Integration of Qualitative and Quantitative Methods[END_REF].

Mixed methods approaches used to be used primarily in larger-scale research projects aiming to explore new, uncharted research fields. The seminal work of Marie [START_REF] Jahoda | Marienthal: The Sociography of an Unemployed Community -Translated by the authors with John Reginait and Thomas Elsaesser[END_REF] "Marienthal: The Sociography of an Unemployed Community" dealt with the challenges posed by the external economic shock of mass unemployment during the 1930s. The sociology and psychology of the time was entirely incapable of predicting how modern society might respond to such a shock, so the research team attempted to collect as wide a variety of data as possible, ranging from the observation of walking speed, conventional household interviews to content analysis of school essays. Once testable concepts had been produced -such as the concept of individual stages of unemployment experiences -they could then easily be tested using standard quantitative methods or more focused qualitative interviews from predefined samples. This gave rise to Lazarsfeld's idea that qualitative methods could be used to develop hypotheses and that quantitative methods could be used to test hypotheses. Following this idea, mixed methods research designs often use qualitative interviews and ethnographic research to develop a hypothesis, and survey questionnaires to test the hypothesis. However, the strict two-stage model of sequentially combining qualitative and quantitative methods has not become widespread (see [START_REF] Leech | A typology of mixed methods research designs[END_REF][START_REF] Creswell | Advanced mixed methods research designs[END_REF]. Rather, many larger mixed methods research projects use qualitative methods to supplement quantitative surveys in order to gain a fuller understanding of the "real lives" of the individuals and households surveyed (see, e.g., [START_REF] Portes | No Margin for Error: Educational and Occupational Achievement among Disadvantaged Children of Immigrants[END_REF]Mayer andSchulze 2009a, 2009b).

The unique potential of qualitative projects based on longitudinal survey respondents

In some sense, longitudinal surveys such as household panel or birth cohort studies can be said to follow in the tradition of [START_REF] Jahoda | Marienthal: The Sociography of an Unemployed Community -Translated by the authors with John Reginait and Thomas Elsaesser[END_REF] in establishing a large survey to analyze how households adapt to social and economic changes and in turn contribute to social change.

Longitudinal surveys provide a constantly expanding body of diverse data and are therefore becoming multiple or mixed method enterprises to an increasing degree. Conducting qualitative interviews with long-term survey respondents provides a unique opportunity for a real triangulation of different types of data on people's life courses. In long-running longitudinal studies, it is possible to conduct biographical interviews with long-term respondents for whom more than a decade of prospectively collected panel data are available.

In principle, the longitudinal data can also be linked with register data from employment or social insurance agencies.

Triangulations like these would make it possible to thoroughly analyze the validity, reliability, and meaningfulness of panel data. Biographical crises or "turning points" in the life course as reported in qualitative interviews can be checked against the standardized yearly measures collected in longitudinal surveys (e.g., life satisfaction). Is it possible to detect biographical crisis through quantitative longitudinal data? Are respondents able to remember negative events like unemployment or the timing of a divorce? Does the use of combined methodologies affect non-response behavior (item non-response as well as partial unit nonresponse or panel attrition)?

Mixed method research designs are often used for validation purposes: this is the case with qualitative interviews or experiments being used to validate and/or improve measures in survey questionnaires (see, e.g., [START_REF] Dohmen | Individual Risk Attitudes: New Evidence from a Large, Representative Experimentally-Validated Survey[END_REF] for measuring risk aversion). Cognitive interviewing has been developed as a qualitative tool for this purpose (see [START_REF] Willis | Cognitive Interviewing: A Tool for Improving Questionnaire Design[END_REF].

Moreover, by drawing on a large ongoing survey, one can systematically select respondents who appeared to be particularly interesting in the quantitative analysis for qualitative interviews. A common feature of such designs is the construction of typologies by clustering survey data and then selecting "representative" respondents for each cluster, or by selecting extreme cases or even outliers for more in-depth analysis (see [START_REF] Portes | No Margin for Error: Educational and Occupational Achievement among Disadvantaged Children of Immigrants[END_REF] for an outlier analysis).

Apart from investigating the methodological effects arising from the type of data, qualitative interviewing of respondents to longitudinal surveys allows insights in a wide range of particular research questions, such as school choice, educational and occupational aspirations, and family formation. Qualitative interviews can be carried out with entire households and address issues such as family relations within and across households, social networks, perceptions of neighborhoods, schools, employers, and how these shape life goals and individual behavior. However, these rich opportunities have only recently entered the research agenda of longitudinal surveys.

Review of qualitative projects based on longitudinal survey respondents

To date, very few projects have been carried out involving qualitative interviews with respondents to longitudinal surveys, but a growing number of such projects have started recently or are currently under planning: Qualitative data was collected on members of eight households selected from this first large-scale longitudinal study of household poverty in South Africa.

Challenges

Linking qualitative in-depth interviews to quantitative surveys poses new challenges. First of all, ethical and data protection issues have to be considered and resolved (see [START_REF] Leahey | Convergence and confidentiality? Limits to the implementation of mixed methodology[END_REF].

For legal reasons, survey respondents have to declare their willingness to participate in the survey, and this declaration should explicitly include their agreement to participate in personal in-depth interviews. Moreover, respondents need to understand exactly how qualitative interviews -or the transcript, audio, or video file -will be linked with the quantitative microdata.

For longitudinal survey respondents, time-consuming in-depth interviews may negatively affect survey participation, and requests to divulge intimate biographical details could impair the respondent's relationship to the interviewer. From what we know so far about the effects of introducing new and more demanding kinds of surveying in ongoing longitudinal studies, they seem to strengthen rather than weaken respondents' personal commitment to the survey.

An important challenge in developing the social science research infrastructure in the future relates to the rules of access to qualitative data on survey respondents. Those responsible for managing longitudinal surveys need to establish working models that can provide external researchers the opportunity to interview respondents.

Recommendations

The inclusion of qualitative in-depth interviewing in the repertoire of data collection methods used in sample surveys is a highly promising innovation in terms of both methodological and substantial research. However, there is still a long way to go in laying the foundations and exploring the possibilities and limits of such an approach.

 Theory & methodology: More extensive use of qualitative methods in surveys should be based on theoretical and methodological proposals that guide the triangulation of qualitative and quantitative methods.

 Ethics, data protection, and access: Ethical and data protection issues need to be addressed. Rules for access to samples of respondents should be established.

 Exploration: The possibilities and problems of conducting semi-structured and biographical interviews should be explored with rather small test samples of long-term survey respondents, focusing on methodological issues of "triangulating" life courses.



  For the German context, about three dozen interviews were conducted with respondents from the 1971 birth cohort of the German Life History Study. Using narrative interviews, Mayer and Schulze (2009a) used a "modest mixed-methods strategy" to analyze the life courses of this generation in West and East Germany and, in another study, to study parenthood processes in order to provide evidence of mechanisms resulting in delayed family formation (Mayer and Schulze 2009b: 12).  In a project at the University of Manchester on interactions between and within generations, data from the English Longitudinal Study of Ageing were linked to qualitative interviews of between 25 and 30 respondents and approximately 20 of their descendants. 1 The goal of the study was to understand intergenerational transfers and communication and the role played by older people.  In a project conducted at the Center for Longitudinal Studies at the University of Manchester, qualitative interviews are planned with about 180 respondents (aged 50) from the 1958 British Birth Cohort Study in order to understand the driving forces and the dynamics underlying voluntary social engagement.  In the UK, qualitative interviews are planned for the new British household panel "Understanding Society" (Usoc) 2 .  In the US, Portes and Fernández-Kelly (2008) also used mixed methodologies to analyze data from the Children of Immigrants Longitudinal Study (CILS). They conducted narrative interviews with 50 second-generation youths and their families to understand how young respondents have coped with disadvantages during their childhood and teen years and to examine their educational success.  Also in the US, researchers linked data from the Women's Employment Study (WES) with qualitative data gathered from a sub-sample of the survey's respondents (approximately



  70) in order to analyze processes of union formation among low-income women and to formulate hypotheses that can be tested by the use of panel data (see[START_REF] Seefeldt | Working after welfare: how women balance jobs and family in the wake of welfare reform[END_REF].3 Researchers have used mixed methodologies on data from the South African KIDS Survey (KwaZulu-Natal Income Dynamics Study) in order to understand the factors explaining transitions into or out of poverty (seeAdato, Lund and Mhlongo 2006).

  /www.understandingsociety.org.uk/news/latest/ and http://www.esrc.ac.uk/ESRCInfoCentre/research/resources/UKHLS.aspx

	1	More details on this project are available at http://www.socialsciences.manchester.ac.uk/realities/research/generations/
	2	For	more	details,	see	http://www.understandingsociety.org.uk/design/features/qualitative.aspx	as	well	as
		http:/

More details on this project are available at http://cairo.pop.psu.edu/allen/Project.cfm?ProjectID=189.


José F Bartolomé

Isabel Montero

Marcos Díaz

Sonia López-Esteban

José S Moya

Sylvain Deville

Laurent Gremillard

Jérome Chevalier

Gilbert Fantozzi

Jérôme Chevalier

Accelerated Aging in 3-mol%-Yttria-Stabilized Tetragonal Zirconia Ceramics Sintered in Reducing Conditions

published or not. The documents may come   L'archive ouverte pluridisciplinaire

The aging degradation of yttria-stabilized tetragonal zirconia ceramics (Y-TZP) is considered today an issue of important technological interest, mainly if long-term performance is required, 1 e.g., hip and knee prostheses (>10 years) or solid oxide fuel cells (SOFCs).

Recently, alarming problems related to the aging of the 3Y-TZP (zirconia doped with 3 mol% Y2O3) femoral head in total hip replacement have been reported. [START_REF] Pecharroma´n | Percolative Mechanism of Aging in Zirconia Containing Ceramics for Medical Applications[END_REF] In particular, resistance to steam sterilization and the hydrothermal stability of yttria-containing zirconia in the body have been questioned. Aging occurs by a tetragonal-to-monoclinic (t-m) phase transformation of grains on any surface in contact with water or body fluids which involves surface roughening, grain pull-out, and the formation of microcracks on the specimen surface and consequently strength degradation. This accelerated low-temperature degradation (LTD) leads to premature failure of components.

The phase stability of YSZ ceramic electrolytes is of key importance to long-term SOFC applications. Current efforts are aimed at lowering the operating temperature of solid oxide fuel cells from above 900° to 500°C to improve the longevity and cost. of the peripheral materials and the electrical power generation efficiency. [START_REF] Haile | Fuel Cell Materials and Components[END_REF] Tetragonal zirconia is considered a promising candidate to be used as electrolyte in intermediate-temperature solid oxide fuel cells (IT-SOFC), because of its good mechanical and electrical properties in comparison with cubic zirconia at lower temperatures. [START_REF] Weppner | Tetragonal Zirconia Polycrystals-A High Performance Solid Oxygen Ion Conductor[END_REF] However, the lowtemperature degradation caused by tetragonal-to-monoclinic transformation under hydrothermal conditions leads to physical degradation; this makes its application in IT-SOFC devices challenging as water vapor is produced at the anode when the cell is in operation. Moreover, the fuel cell must be able to withstand thermal cycling and the ability to operate under pressures higher than atmospheric pressure over the lifetime of the cell (>50 000 h). [START_REF] Badwal | Stability of Solid Oxide Fuel Cell Components[END_REF] Therefore, the fuel-cell components are susceptible to slow crack growth (SCG).

Considerable efforts have been exerted to elucidate the mechanism of hydrothermal degradation in Y-TZP. The yttrium depletion by Y(OH)3 formation in the presence of water vapor was proposed by Lange et al. [START_REF] Lange | Degradation during Aging of Transformation-Toughened ZrO2 -Y2O3 Materials at 250°C[END_REF] as the cause of the aging-induced degradation. However, this mechanism cannot explain the effect of grain size. [START_REF] Munoz-Saldana | Mechanical Properties and Low-Temperature Aging of Tetragonal Zirconia Polycrystals Processed by Hot Isostatic Pressing[END_REF] Moreover, it has been demonstrated that the degradation occurs even in dry air and under vacuum, depending on the type and content of alloying elements in Y-TZP, and the loss of Y2O3 is not involved in LTD when aged in air. [START_REF] Kim | Fracture Toughness, Ionic Conductivity, and Low-Temperature Phase Stability of Tetragonal Zirconia Codoped with Yttria and Niobium Oxide[END_REF] Sato et al. [START_REF] Sato | Changes in Crystalline Phase and Microstructure on the Surface of Yttria-Doped Tetragonal Zirconia Polycrystals (Y-TZP) by Annealing in Humid Conditions[END_REF] suggested that the hydrothermal degradation of Y-TZP is controlled by the breakage of Zr-O-Zr bonds to form Zr-OH due to the reaction between the chemisorbed H2O and Zr 4+ at the specimen surface, which results in the release of strain energy that would ensue if the t-m transformation would occur. However, this model cannot rationalize the composition and grain-size dependence of the degradation rate. Yoshimura 10 proposed that the accumulated strain area resulting form the migration of OH -at the surface and in the lattice serve as a nucleus of the m-ZrO2 phases in the t-ZrO2 matrix. However, studies on oxygen diffusion in Y-TZP and the effect of oxygen on the tetragonal-to-monoclinic transformation pose problems for the Yoshimura et al. theory . 11,12 Recently, Kim et al. [START_REF] Kim | Fracture Toughness, Ionic Conductivity, and Low-Temperature Phase Stability of Tetragonal Zirconia Codoped with Yttria and Niobium Oxide[END_REF] reported that LTD of Y-TZP is governed not by the existence of H2O but the relaxation process of internally strained lattice due to a thermally activated oxygen vacancy diffusion from the surface into the interior of the specimen. There is a general agreement in literature that the degradation is accelerated as the material is exposed to water or water vapor because the residual stress facilitates the t-m phase transformation and the reaction between the Zr-O-Zr bond and H2O. Therefore, the rate of LTD is governed by both the number of oxygen vacancies and the instability of t-ZrO2.

It has also been pointed out that modifying the grain boundaries by doping or by an inhomogeneous yttria concentration [START_REF] Ramesh | Environmental Degradation of CuO-Doped Y-TZP Ceramics[END_REF] is thought to play an important role in governing the hydrothermal degradation of Y-TZP ceramics.

In the SOFCs, nickel (Ni) and Y2O3 -stabilized zirconia (YSZ) cermet is widely used as an anode material because of its high catalytic property and low cost. [START_REF] Minh | Ceramic Fuel Cells[END_REF] A general method currently used for anode manufacturing is mechanical mixing of NiO and Y2O3 -ZrO2 powders, pasting mixing composite powders on a zirconia electrolyte plate by screen printing or spraying and then sintering under reducing atmosphere to form the Ni/YSZ cermets. [START_REF] Simwonis | Properties of Ni/YSZ Porous Cermets for SOFC Anode Substrates Prepared by Tape Casting and Coat-Mix (R) Process[END_REF] The phenomenon of aging-induced tetragonal-to monoclinic phase transformations in zirconia ceramics sintered in air has been widely documented; however, a direct study of the accelerated aging process in yttria-stabilized tetragonal zirconia ceramics sintered under reducing conditions is still lacking. The aim of the present work is to explore the effect of the sintering atmosphere on the accelerating aging of yttria-stabilized tetragonal zirconia ceramics.

II. Experimental Procedure

The following commercially available tetragonal zirconia polycrystals powder (Y-TZP 3 mol.%; TZ-3YS, Tosoh Corp.) has been used as starting material with an average particle size of d50 =0.2 µm, a BET specific surface area of 6.7 m 2 /g, and the following chemical analysis (wt%): ZrO2 (95.00), Y2O3 (4.98), Al2O3 (0.005), Fe2O3 (0.005), SiO2 (0.002), and Na2O (0.003).

The powders were isostatically pressed at 200 MPa. The resulting cylindrical bars (6mm diameter and 50-mm length) were fired in a 90% Ar/10% H2 atmosphere or in air at 1430°C for 2 h. The heating and cooling rate was kept at 600°C/h. The bulk densities of all the samples were measured using the Archimedes method. SEM micrographs were obtained using a HITACHI S-4700 field emission scanning electron microscope (Japan) with an accelerating tension of 20 kV. Polished samples were preliminarily thermally etched in a 90% Ar/10% H2 or in air atmosphere at 1400°C for 30 min, with heating and cooling rates of 300°C/h. Samples were then gold-coated before SEM analysis. The grain size of the 3Y-TZP was determined using a linearintercept method on representative SEM micrographs.

X-ray photoelectron spectra (XPS) were recorded with a hemispherical electron analyzer (Leybold 10) equipment. The measurements were done with MgK a excitation (hv=253.6 eV) in vacuo at <10 -9 Pa at room temperature. The acceleration voltage and current of the non monochromatized X-ray source were 11 kV and 20 mA, respectively. The samples showed charging of <2eV in XP spectra because of their insulator nature. The binding energies were corrected for charging effect by assuming a constant binding energy of the C 1s peak position of graphitic carbon at 284.6 eV.

Aging experiments were conducted using a steam autoclave at 140°C, with a 2-bar pressure. Polished samples were located in steam autoclaves (Fisher Bioblock Scientific, France) and left in steam atmosphere for different periods (up to 20 h). Samples were located on a grid in the autoclave so that they were not soaked in water during aging, but only subjected to steam atmosphere. Before increasing the temperature to 140°C, the enclosure was left open to evacuate the air atmosphere initially present in the steam autoclave, thus ensuring a 100% humidity atmosphere during aging. X-ray diffraction data were obtained with a diffractometer using Ni-filtered CuK a radiation. The tetragonal/monoclinic zirconia ratio was determined using the integrated intensity (measuring the area under the diffractometer peaks) of the tetragonal (101) and two monoclinic (111) and (1 11) peaks as described by Toraya et al. [START_REF] Toraya | Calibration Curve for Quantitative Analysis of the Monoclinic Tetragonal ZrO2 System by X-ray Diffraction[END_REF] Diffractometer scans were obtained from 27° to 33°, at a scan speed of 0.2°/min and a step size of 0.02°.

III. Results and Discussion

The density of the sintered 3Y-TZP samples was found to be approx. 98%. Figure 1 shows SEM micrographs of the thermally etched polished surfaces of both a ZrO2 sample sintered in air and in 90%Ar/10% H2 atmosphere. In both cases the average grain size of zirconia was found to be about 0.3 µm.

The monoclinic phase volume fraction evolution during aging at 140°C in steam for the two ceramics is shown in Fig. 2. For both 3Y-TZP samples, the starting surfaces after polishing are free of monoclinic phase. The m-ZrO2 fraction increases at a quite-steady rate with aging time, reaching 35% for the sample sintered in air and 55% for the one sintered in 90%Ar/10%H2 atmosphere after 20 h of aging at 140°C. Fig. 1. Polished surfaces of (A) the ZrO2 sample sintered in air and (B) the ZrO2 sample sintered in 90%Ar/10% H2 atmosphere, after being thermally etched at 1400°C for 0.5 h in air and reducing conditions, respectively. Fig. 2. Surface monoclinic phase fraction versus exposure time at 140°C in steam for the zirconia materials sintered in air and in 90%Ar/10% H2 atmosphere.

It is known that the oxygen vacancies created either by reducing the oxygen partial pressure or by alloying zirconia with aliovalent metal oxides (Mg 2+ ,Ca 2+ , and Y 3+ ) play an important role in stabilizing the zirconia structure. [START_REF] Fabris | A Stabilization Mechanism of Zirconia Based on Oxygen Vacancies Only[END_REF] Therefore, in our 3Y-TZP samples, in addition to the oxygen vacancies produced by substitution of yttrium cations for zirconium cations to maintain electrical neutrality, which are expected to have a negative charge, uncharged oxygen vacancies are introduced by sintering in a low oxygen partial pressure in response to thermodynamic considerations. The concentration of thermodynamic oxygen vacancies appears to be linked to oxygen partial pressure with the following empirical semi-quantitative relationship:

𝑉 " ∝ 𝑂 2 -1/6
where 𝑉 " and 𝑂 2 represent the concentration of thermody namic oxygen vacancies and the oxygen partial pressure, respectively. We can roughly calculate the concentration of thermodynamic oxygen vacancies. [START_REF] Wang | Thermochemistry of the U-O and Zr-O Systems[END_REF] A pressure of about 10 -24 Pa at 1400°C, which can be readily achieved in a hydrogen atmosphere, leads to about 2000 ppm intrinsic thermodynamic oxygen vacancies. Then, in the sample obtained under reducing conditions, the concentration of oxygen vacancy created by the Y 3+ doping (-15 000 ppm) increases by the sintering in reducing atmosphere, leading to a high vacancy diffusion rate which governed the t-m phase transformation during aging at low temperatures. This statement is in agreement with the X-ray irradiation experiment in air performed by Gremillard 20 on the 3Y-TZP sample surface. After 1h in an autoclave at 134°C in water vapor, he observed that the color changes from dark to white due to the elimination of oxygen vacancies. Fig. 3. XPS of the Fe 2p core level for the zirconia samples obtained after air and H2 treatments, curves a and b, respectively. Spectra are displaced in intensity scale to allow better observation.

On the other hand, if there was significant change in grain boundary composition, it can be inferred that the grain boundary regions are active sites for monoclinic nucleation during aging. The near-surface region of the ceramics treated in 90% Ar/10% H2 atmosphere and in air was analyzed by XPS. Figure 3 shows the Fe 2p core level spectra of the ceramics treated in 90% Ar/10% H2 atmosphere and in air, curves a and b, respectively. The position of the Fe 2p 3/2 peak at a binding energy of about 710.5 eV is characteristic of the iron oxide. [START_REF]X-ray Photoelectron Spectroscopy Database[END_REF] Both peaks Fe 2p3/2 and 2p1/2 are accompanied by satellites structures on their high binding energy side. These satellite structures in the XPS spectra were caused by the Fe 3d-O 2p hybridization. In the case of the spectrum a, the satellite of the 2p3/2 main peak is clearly observed on their high binding energy side, at about 7 eV. This result indicates the presence of Fe 3+ . [START_REF] Fujii | In-Situ XPS Analysis of Various Iron Oxide Films Grown by NO2 -Assisted Molecular-Beam Epitaxy[END_REF] In contrast, the signal of the Fe 3+ satellite in spectrum b has smeared out. It can be due to the major contribution of the Fe 2+ satellite at the spectrum. Thus, the shoulder of the Fe 2p3/2 toward higher energies (spectrum b) is characteristic of the formation of Fe 2+ ions. In addition, the Fe 2p1/2 peak position shifts toward low binding energies about 1 eV. [START_REF] Fujii | In-Situ XPS Analysis of Various Iron Oxide Films Grown by NO2 -Assisted Molecular-Beam Epitaxy[END_REF] Moya et al., [START_REF] Moya | Black Color in Partially Stabilized Zirconia[END_REF] using electron spin resonance techniques, detected the exolution of iron impurities (ppm) which are present at the starting powders, from the bulk to the grain boundary of yttria-partially-stabilized zirconia (3Y-TZP) polycrystals under reducing conditions. The isolated Fe 3+ ions are removed from the ZrO2 lattice, by the reduction, forming a new aggregate phase at the grain boundaries in the polycrystalline zirconia. This process can be understood by considering that reduction of Fe 3+ to Fe 2+ increases ion size and consequently diminishes the solid solution of iron. The presence of Fe 2+ in the sample sintered in reducing atmosphere has been clearly established by XPS (Fig. 3). The segregation of Fe 2+ ions at the grain boundaries and their subsequent aggregation in a separate phase, i.e., magnetite, may produce a volume expansion, as has been reported by Chein and Ko [START_REF] Chein | Causes of Refractory Disintegration in a Carbon Monoxide Environment[END_REF] On the other hand, the formation of a second phase in the grain boundary may result in a decrease of the surface energy. A monoclinic phase at room temperature is then formed when this free-energy term is low enough. [START_REF] Winnubst | The Aging Behavior of Ultrafine Grained Y-TZP in Hot Water[END_REF] Therefore in the sample sintered in reducing atmosphere the aging process can be accelerated by both cooperating mechanisms, increasing the bulk concentrations of oxygen vacancies and at the same time modifying the grain boundary region by the exolution of Fe 2+ ions.

IV. Conclusions

It has been observed an aging rate increase of about 50% in a 3Y-TZP dense sample sintered in reducing conditions versus a sample with similar density and average grain size values but obtained by sintering in air. This fact is explained in terms of the following cooperative mechanisms: (a) the increase of the oxygen vacancy concentration that is introduced by sintering in a low oxygen partial pressure, leading a high vacancy diffusion rate which governed the t-m phase transformation during aging at low temperatures, and (b) the presence at the grain boundaries in the polycrystalline zirconia of a new aggregate phase, i.e., magnetite, formed because of the exolution of Fe 2+ ions that promote a volume expansion affecting the stability of the tetragonal phase.


Jérome Chevalier

Sylvain Deville

Etienne Münch

Romain Jullian

Frédéric Lair

Jérôme Chevalier

Critical effect of cubic phase on aging in 3mol% yttria-stabilized zirconia ceramics for hip replacement prosthesis

Keywords: Aging, Degradation, Zirconia, Hip replacement prosthesis

Introduction

The concept of stress-induced phase transformation in zirconia ceramics represents one of the most remarkable innovations in the ceramic field. Indeed, it was shown in the 1970s, firstly by Garvie et al. [1] and then by Gupta et al. [START_REF] Gupta | Effect of stress induced phase transformation on the properties of polycristalline zirconia-containing tetragonal phase[END_REF], that zirconia exhibits a transformation toughening mechanism, which increases its crack propagation resistance. The stress-induced phase transformation involves the transformation of metastable tetragonal grains to the monoclinic phase at the crack tip, which, accompanied by a volume expansion, induces compressive stresses. Yttria-stabilized zirconia ceramics, usually called Y-TZP, belong to the family of toughened materials. They can exhibit a strength of more than 1 GPa with a toughness of about 6-10 MPa.m 1/2 . One of the most successful applications of yttria-tetragonal zirconia polycrystal (Y-TZP) ceramics so far is found in orthopedics, with femoral heads for total hip replacement. The use of zirconia has opened the way towards new implant designs that were not possible with alumina, more brittle. It has been estimated that about 500 000 patients have been implanted with zirconia femoral heads since 1985, 200 000 for these past 5 years.

Nevertheless, alarming problems related to aging of zirconia have been recently reported. In particular, the resistance to steam sterilization and the hydrothermal stability of yttria-containing zirconia in the body have been questioned. The phenomenon of zirconia aging is known in the ceramic community for more than 20 years, from the work of Kobayashi et al. [START_REF] Kobayashi | Phase change and mechanical properties of ZrO2-Y2O3 solid electrolyte after ageing[END_REF] who discovered a serious limitation of Y-TZP ceramics for applications near 250°C. Aging occurs by a slow tetragonal-to-monoclinic phase transformation of grains on any surface in contact with water [START_REF] Yoshimura | Role of H2O on the degradation process of Y-TZP[END_REF], or body fluid. This transformation leads to surface roughening [START_REF] Chevalier | Low temperature aging of Y-TZP ceramics[END_REF], grain pull out and micro-cracking [START_REF] Lilley | Review of low temperature degradation of tetragonal zirconia ceramics[END_REF]. Aging was thought to be very limited and well controlled near the ambient temperature. However, the US Food and Drug Administration (FDA) have recently reported on the critical effect of the well-developed steam sterilization procedure (134°C, 2-bars pressure) on zirconia implants [7]. This procedure is now forbidden for zirconia. More important, in 2001, the FDA and the Australian Therapeutic Goods Administration (TGA) [8,9] announced that firms distributing orthopedic implants were recalling series of Y-TZP hip prostheses due to a fracture episode of zirconia ceramic femoral heads. The failures origin could be related to an accelerated tetragonalto-monoclinic phase transformation of zirconia on a limited number of batches [10]. These failures, even if limited in number, have a strong negative impact for the use of zirconia in orthopedics.

There is today a trend for certain surgeons to go back to less-competitive solutions, abandoned several years ago. Alternative approaches are also under consideration, for example zirconia-toughened alumina ceramics [START_REF] Chevalier | Extending the lifetime of orthopaedic implants[END_REF][START_REF] Rack | A new ceramic material for orthopedics[END_REF]. However, it should be kept in mind that zirconia has been implanted successfully for more than 15 years with an exceptionally low failure rate [START_REF] Cales | Fracture des têtes de prothèses de hanche en zircone[END_REF]. A pragmatic approach could consider a better understanding of the key mechanisms of zirconia aging and the relationships between processing, microstructure and aging resistance. In this respect, the aim of this paper is to investigate on the possible detrimental role of the appearance of a cubic phase for sintering 3Y-TZP (namely 3 mol % yttria-tetragonal zirconia polycrystal) ceramics. Our work on this issue started with the evidence that 100% monoclinic phase was never reached in our experiments on 3Y-TZP even after very long exposure in steam [START_REF] Chevalier | Low temperature aging of Y-TZP ceramics[END_REF]. This was in agreement with previous experimental works (for example in [START_REF] Luand | Low-temperature aging of t-ZrO2 polycrystals with 3 mol % Y2O3[END_REF]), but not with a common feeling that 3Y-TZP is fully tetragonal after sintering, which would mean that all grains would transform after aging to the monoclinic structure after long-term aging.

A careful reading of the Y2O3-ZrO2 phase diagram [START_REF] Scott | Phase relationships in the zirconia-yttria system[END_REF] shows, however, that 3Y-TZP at sintering temperature (typically in the range between 1400°C and 1550°C) should present two phases : an yttrium-reach cubic phase and a tetragonal phase with an amount of yttrium lower than the nominal 3 mol % (see Fig. 1). This was experimentally demonstrated by Ruiz and Readey [START_REF] Ruiz | Effect of heat treatment on grain size, phase assemblage, and mechanical properties of 3 mol % Y-TZP[END_REF], and more recently by Matsui et al. [START_REF] Matsui | Cubic formation and grain-growth mechanisms in tetragonal zirconia polycrystal[END_REF] who showed that 3Y-TZP heated in the range 1450-1600°C (i.e. temperatures used for processing most of zirconia parts) exhibited a dual phase microstructure, consisting of both cubic and tetragonal polymorphs. Depending on sintering temperature and duration, cubic grains exhibiting 6-7 mol % yttria could coexist with tetragonal grains containing less than 2 mol % yttria. Thermal treatments at high temperatures were sometimes proposed to obtain tough and strong yttria-stabilized compounds [START_REF] Ruiz | Effect of heat treatment on grain size, phase assemblage, and mechanical properties of 3 mol % Y-TZP[END_REF][START_REF] Casellas | On the transformation toughening of Y-ZrO2 ceramics with mixed Y-TZP/PSZ microstructures[END_REF] with a large transformation toughening capability. However, aging occurs by a slow transformation of metastable tetragonal grains at the surface in contact with body fluids, by a nucleation and growth mechanism [START_REF] Chevalier | Low temperature aging of Y-TZP ceramics[END_REF], and nucleation is likely to arise at grains less stable than the average. Therefore, the hydrothermal stability of tetragonal grains in equilibrium with a cubic phase has to be questioned. [START_REF] Scott | Phase relationships in the zirconia-yttria system[END_REF]). The dashed lines show the typical range of sintering temperatures of 3 mol % yttriastabilized zirconia.

Materials and methods

The experiments were conducted from a starting ultra-pure Y-TZP powder containing 3 mol % (5.2 wt%) yttria, produced by a co-precipitation method (Tosoh). The material was processed by cold isostatic pressing at 300 MPa, followed by pressureless sintering at a temperature of 1450°C or 1550°C, for 2 or 5 h. Small cylinders of 10 mm diameter and 5 mm thickness were ground and mirror-polished to reach a sufficiently low surface roughness to observe any small surface uplift due to t-m transformation and to begin aging experiments with a monoclinic content close to 0%. Monoclinic content was measured by an X-ray diffraction (XRD) technique (CuK α radiation, with a penetration depth in zirconia estimated to be about 5 μm from Ref. [START_REF] Emsley | The elements[END_REF]) and calculated from the modified Garvie and Nicholson equation [START_REF] Garvie | Phase analysis in Zirconia systems[END_REF]. Aging experiments were carried out in steam at 134°C under 2-bars pressure, which corresponds to a sterilization procedure. The t-m transformation being thermally activated (with an activation energy of 106 kJ/mol [START_REF] Chevalier | Low temperature aging of Y-TZP ceramics[END_REF]), it can be calculated that 1 h of this treatment corresponds roughly to 4 years in vivo [START_REF] Chevalier | Low temperature ageing of zirconia hip joint heads[END_REF]. This treatment is therefore a good indicator of the aging sensitivity in vivo of a given zirconia ceramic. Aging was followed by XRD and by atomic force microscopy (AFM) (D3100, Digital Instruments Inc.) to follow topographic changes induced by the transformation. The AFM was used in contact mode, with an oxide-sharpened silicon nitride probe exhibiting a tip radius of curvature of 20 nm. The scans were performed at the surface with an average scanning speed of 10 μm/s. Some specimens were thermally etched at 1300°C for 30 min so as to reveal the microstructure. They were observed by scanning electron microscopy (SEM) (FEI, ESEM, XL30) equipped with an energy dispersion spectroscopy (EDS) system (FEI, EDXi). The specimens were gold-coated to avoid any surface charging effect. EDS spectra were conducted in specific regions for a 30 kV voltage, for 5 min, to obtain well-resolved Y-K α and Zr-K α peaks. Grain size data were obtained from representative SEM micrographs using the ASTM E112 standard.

Results

Microstructures

All specimens exceeded 98% of theoretical density. Fig. 2 (a)-(d) show representative SEM micrographs of polished and thermally etched 3Y-TZP specimens heat treated at 1450-1550°C for 2-5 h. The corresponding grain sizes are given in Table 1. The grain size is very small and homogeneous for specimens sintered at 1450°C for 2 h. It becomes larger for specimen sintered at 1450°C for 5 h or 1550°C for 2 h but still unimodal. However, few coarser grains (∼1 μm) could be observed in some occasions. The microstructure becomes clearly bimodal when sintering is achieved at 1550°C, 5 h, with the occurrence of grains larger than 2 μm. Transmission Electron Microscopy (TEM) by Rhüle et al. [START_REF] Rühle | Microstructural studies of Y2O3 containing tetragonal ZrO2 polycrystals (Y-TZP)[END_REF] has revealed that these grains were cubic in nature. This has been confirmed recently by Ruiz and Readey [START_REF] Ruiz | Effect of heat treatment on grain size, phase assemblage, and mechanical properties of 3 mol % Y-TZP[END_REF], who showed the evidence of these cubic grains above 1500°C. According to the phase diagram [START_REF] Scott | Phase relationships in the zirconia-yttria system[END_REF] (see Fig. 1), these grains should exhibit larger amount of yttria than tetragonal grains. EDS spectra were therefore conducted on one specimen sintered at 1550°C for 5 h on specific regions of the surface, i.e. at the interior of presumed cubic grains (points denoted A and B in Fig. 2d), in presumed tetragonal regions at the periphery of these cubic grains (i.e. point denoted C in Fig. 2d) and in one region situated far from large (cubic) grains (denoted D in Fig. 2d). The results were analyzed in terms of mol % of yttria to compare with the starting powder (3 mol %), in Table 2. Cubic grains exhibit a large yttria content, while tetragonal grains at their periphery are depleted in yttria. Far from these regions, the yttria content is that of the starting powder, within the error range of the measurement method. 2. Yttria content in some selected regions of Fig. 2 Region A (center of a cubic grain) B (center of a cubic grain) D (tetragonal grains at the periphery of cubic grains) E (tetragonal grains far from cubic grains) % yttria (±0.5 mol %) 5.5 mol % 5.0 mol % 2.3 mol % 3.3 mol %

Low temperature aging kinetics

The relationship between the amount of monoclinic phase and aging time for the four materials is plotted in Fig. 3. The shape of the four kinetics is basically the same, but the transformation occurs at different rates: the higher the temperature and dwell duration, the higher the transformation rate. It could be related to the overall grain size [START_REF] Luand | Low-temperature aging of t-ZrO2 polycrystals with 3 mol % Y2O3[END_REF][START_REF] Lawson | Environmental degradation of zirconia ceramics[END_REF]: the larger the mean size of the tetragonal grains, the lower their stability. However, the grain size of tetragonal grains of the material sintered at 1450°C for 5 h and 1550°C for 2 h is about the same, the only difference lying on the possibility to form a larger portion of cubic grains in the latter material. These grains attract yttrium from neighboring tetragonal grains which in turn must be less stable. The difference in transformation rates is particularly important for the first stage of the transformation, where nucleation of the new monoclinic phase is predominant [START_REF] Chevalier | Low temperature aging of Y-TZP ceramics[END_REF].

The larger the portion of cubic phase present in the material (theoretically, from the phase diagram and experimentally from SEM and EDS results), the larger the nucleation rate of phase transformation.

AFM and SEM observations of transformation

In order to demonstrate the ability of cubic grains to destabilize neighboring tetragonal grains, AFM observations were performed in the material sintered at 1550°C for 5 h, at the beginning of the aging process (after 1 h of treatment), and then when the monoclinic content measured by XRD was almost constant (after 24 h of treatment).

As suggested before, aging obviously initiates preferentially in regions surrounding cubic grains. This is illustrated in Fig. 4, where the same zone is observed before aging (Fig. 4a) and after 1 h autoclave (Fig. 4b). The line scan conducted at the periphery of a cubic grain after 1 h aging provides confirmation of t-m transformation in this region (Fig. 4c). After a longer autoclave treatment, cubic grains appeared to be unaffected by the aging (they are in their stable structure at the ambient temperature), while all surrounding area were completely transformed (see Fig. 5, taken for the same zone before aging and after 24 h autoclave). This result is critical, since it shows that cubic grains act as nucleation sites for transformation because surrounding tetragonal grains exhibit less yttria than initially in the raw powder. By comparison, the initiation of aging occurred at a much lower rate in the material processed at 1450°C for 5 h. After 24 h of treatment, the surface appears homogeneously transformed (Fig. 6). Some isolated slightly larger grains seem to be, however, not transformed (i.e. points denoted A and B in Fig. 6 for example). These grains could be cubic in nature, since a significant fraction of cubic phase is also expected from the phase diagram. These results are confirmed by SEM observations (Fig. 7), which reveal after aging a very important surface degradation (with a large amount of grain pull-out and some micro-cracking) in the material processed at 1550°C for 5 h, especially at the periphery of cubic grains. Micro-cracking in the cubic grains occurs since large stresses are induced by the transformation of the tetragonal neighboring grains, with no option to accommodate them.

Discussion

The negative impact of cubic grains on the resistance of zirconia to steam sterilization or aging in vivo has not been considered so far. The present results confirm that 3Y-TZP zirconia ceramics sintered above 1450°C can exhibit a dual microstructure, formed with cubic and tetragonal grains. Here we show that cubic grains are enriched with yttrium, causing a depletion of yttria in the surrounding tetragonal grains. Preceding works have shown that transformation occurs by a nucleation and growth mechanism [START_REF] Chevalier | Low temperature aging of Y-TZP ceramics[END_REF]. Transformation is thought to arise first for grains less stable than the average, propagating then to the neighbors due to large stresses and micro-cracks induced by the transformation. The cubic grains can therefore act as nucleation sites for the transformation. This was detrimental for the zirconia ceramics sintered at 1550°C in this study. However, even for specimens sintered at lower temperature, i.e. 1450°C, the presence of a cubic phase is expected from the phase diagram, and evidence for some non-transformed grains after aging was provided by AFM. These grains can have a significant effect on aging. It is surprising that no attention was paid to this aspect in the literature so far, while it was suggested from a simple observation of the equilibrium diagram.

We do not claim that this effect can explain the accelerated aging of some specific batches of zirconia ceramics in the recent failure events. However, it shows that processing can play a role which was not always totally clarified in the past. Further work on the relation between processing-microstructure-aging resistance should be put forward. It should be kept in mind that tetragonal-to-monoclinic transformation occurs by a nucleation and growth mechanism and that nucleation arises at grains which are less stable than the average. This means that the presence of cubic grains, but also of residual stresses, scratches or residual pores should play a role on aging of a given zirconia sample.

Conclusion

By considering that a cubic phase should be present in yttria-stabilized zirconia ceramics after sintering, the impact of cubic grains on aging was studied. Cubic grains were identified for materials sintered at 1550°C; they were shown to be enriched by yttrium, which in turn leads to a decrease of the yttrium content in the neighboring tetragonal grains. These grains will act as the further nucleation sites for tetragonalto-monoclinic transformation. In this sense, the presence of cubic grains has a detrimental impact on aging resistance. The processing of 3Y-TZP should be conducted at sufficiently low temperatures to avoid the occurrence of a dual cubic-tetragonal microstructure, but sufficiently high temperatures to obtain fully dense materials. This means that a narrow range of sintering temperature, certainly between 1400°C and 1450°C, has to be chosen to process aging resistant 3Y-TZP. The role of post sintering treatments, such as hot isostatic pressing (HIP), conducted at higher temperatures, but smaller duration, should be studied carefully.

Fig. 1 .

 1 Fig. 1. Selected region of the ZrO2-Y2O3 equilibrium phase diagram (after Scott [15]). The dashed lines show the typical range of sintering temperatures of 3 mol % yttriastabilized zirconia.

Fig. 2 .

 2 Fig. 2. Scanning electron microscopy images of 3 mol % Y-TZP ceramics: (a) sintered at 1450°C for 2 h, (b) sintered at 1450°C for 5 h, (c) sintered at 1550°C for 2 h, (d) sintered at 1550°C for 5 h. Points denoted from A to D are regions where EDS spectra were performed.

Fig. 3 .

 3 Fig. 3. Aging kinetics, at 134°C under 2-bars pressure, of 3 mol % Y-TZP ceramics: (•) sintered at 1450°C for 2 h, (○) sintered at 1450°C for 5 h, (■) sintered at 1550°C for 2 h, ( ) sintered at 1550°C for 5 h.

Fig. 4 .

 4 Fig. 4. Atomic force microscopy images of the same zone, (a) before and (b) after 1 h aging at 134°C, for a 3Y-TZP ceramic sintered at 1550°C for 5 h. Arrows indicate surface uplifts due to the t-m transformation of tetragonal grains. (c) Corresponds to a line scan at the periphery of a cubic grains after 1 h aging.

Fig. 5 .

 5 Fig. 5. Atomic force microscopy images of the same zone, (a) before and (b) after 24 h aging at 134°C, for a 3Y-TZP ceramic sintered at 1550°C for 5 h.

Fig. 6 .

 6 Fig.6. Atomic force microscopy images of a 3Y-TZP ceramic sintered at 1450°C for 5 h, after 24 h aging at 134°C.

Fig. 7 .

 7 Fig. 7. Scanning electron micrograph of a 3Y-TZP ceramic sintered at 1550°C for 5 h, after 24 h aging at 134°C.

Table 1 .

 1 Average grain size of 3Y-TZP ceramics as a function of sintering tempera-

	ture and duration
	Sintering condition 1450°C, 2 h 1450°C, 5 h 1550°C, 2 h 1550°C, 5 h
	Average grain size 0.4 μm	0.6 μm	0.6 μm	0.9 μm (t) 4 μm (c)
	(±0.1 μm)

Acknowledgments

Part of this work was conducted during a student project in the Materials Science and Engineering department ('Science et Génie des Matériaux') at INSA. The authors are grateful to the Clyme (Consortium Lyonnais de Microscopie Electronique) for providing the microscopy facilities.


Sylvain Deville

Jérome Chevalier

Gilbert Fantozzi

Ramón Torrecillas

José F Bartolomé

José S Moya

Jérôme Chevalier

Atomic force microscopy study of the surface degradation mechanisms of zirconia based ceramics

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Among engineering ceramics, zirconia based ceramics represent a very large range of materials, for which applications are countless, from gas filters to bioceramics for orthopaedic applications. The discovery of the transformation toughening effect in the 70's was followed by an extensive period of deep investigations in order to understand the underlying mechanisms and potential implications and applications of this phenomenon This behaviour is accounted for by the ability of zirconia to transform from its metastable tetragonal phase at ambient temperature to its stable monoclinic phase, under the action of thermal or mechanical stress. The volume increase related to the structural differences is the source of increased crack resistance propagation, partially closing the crack when propagating.

Though very beneficial in terms of mechanical properties, this phase transformation has also proven to be very detrimental in terms of surface degradation. Zirconia might transform at low temperature (even at room temperature), especially under humid conditions, leading to grain pop-up and pull-out, microcracks formations and subsequent surface degradation. This phenomenon, occurring with time at the surface of zirconia is now referred as aging. There is now a regain of interest from a biomedical point of view, since degradation and rupture of hip joints prosthesis related to the transformation have been reported recently.

µm

From a fundamental point of view, the transformation is now recognised as being of martensitic nature: diffusionless, athermal, involving shape change dominated by shear and occurring at high speed. The transformation has been observed on specimens with direct methods such as interferometry and Normarski interference, or in situ on thin foils by transmission electron microscopy. Indirect measurements by thermal differential analysis or X-Ray diffraction provide with informations on the global behavior of the material.

The underlying mechanisms governing the transformation are still a matter of debate. Further progress in the understanding of the phenomenon requires local direct observation at grain-size scale and in real conditions (bulk samples) if possible.

Considering the scale at which surface modifications are occurring, the very low number of studies that might be found so far is not surprising. Quite fortunately, the development of scanning tunnelling microscopy (STM) and atomic force microscopy (AFM) in the last ten years provides the scientific community with very powerful tools to investigate the surface relief at a nanometer scale. Very few studies 1,2 using AFM have been reported so far, and since they were still preliminary, they provide little quantitative informations on the transformation. In the results reported here, AFM has been used to investigate very locally several aspects of the transformation: quantification and prediction of the long term ageing behaviour, microstructural analysis of the ageing behaviour of zirconia toughened alumina, transformation toughening mechanisms analysis, influence of the cubic phase on the ageing behaviour, crack propagation characterization and more importantly martensitic relief observation and characterization.

Experimental methods

Yttria stabilized samples were processed using yttria-stabilized zirconia powder (3Y-TZP, Tosoh TZ-3YS, Tosoh Corporation, Tokyo, Japan). Green bodies, after uniaxial cold pressing at 10 bars, were isostatically cold pressed at 1300 bars and finally sintered at 1773 K for five hours in air. Zirconia toughened alumina samples were processed by a classical powder mixing processing route. A high purity alumina powder α-Al2O3 >99.9 wt.% (Condea HPA 0.5, Ceralox division, Arizona, USA) was mixed with 25 wt.% (i.e. 16.7 vol.%) of 3Y-TZP powder. Samples were sintered at 1873 K for two hours in air. Ceria stabilized zirconia (CeO2-TZP) materials were processed by classical powder mixing processing route, using Zirconia Sales Ltd. powders, with uniaxial pressing and sintering at 1813 K for two hours.

Mirror polished samples were cleaned with ethanol and observed by atomic force microscopy (D3100, Digital Instruments Inc.) in contact mode, using oxide sharpened silicon nitride probes with an average scanning speed of 2 µm/s. Autoclave ageing tests were performed in steam autoclave at various temperatures for different times. Scanning electron microscopy observations were carried out with an ESEM FEG (FEI) in backscattered electrons imaging mode on gold coated samples.

Results and discussion

Measurements and prediction of the long-term ageing behaviour

By following the transformation of a constant zone as a function of the autoclave ageing treatment time, it was possible assessing the nucleation and growth nature of the transformation.

The size of a statistically significant number of monoclinic spots was measured and found to vary linearly with their age, even in the very first stages of growth, where only a fraction of a single grain is transformed. The measured parameters could be incorporated into the prediction models of the transformation [START_REF] Gremillard | Modeling the aging kinetics of zirconia ceramics[END_REF] .

Microstructural analysis of the ageing behaviour of zirconia toughened alumina

AFM was used to analyse the influence of microstructural parameters in zirconia toughened alumina composites, were alumina is the matrix phase. This kind of materials has been the object of several studies, and factors controlling the transformation have been underlined before [START_REF] Deville | Low temperature ageing of zirconia toughened alumina ceramics and its biomedical implications[END_REF] . The size effect has been proven to be the most significant factor in controlling the transformation, though other factors such as residual stresses and interface energy between the two phases are also to be considered. The existence of a percolation threshold of the zirconia content, above which transformation is propagating, has been demonstrated recently [START_REF] Pecharroman | Percolative mechanisms of aging in zirconia containing ceramics for medical applications[END_REF] . However, very little studies are reported concerning the local observation of the transformation and the influence of microstructural parameters. Using a combined analysis with SEM and AFM on the same zone of bulk samples surface (Fig. 1), the influence of zirconia agglomerates on the transformation was analysed. It was shown that either grains within the zirconia agglomerates and small single grains embedded within the matrix were transformable. As previously hypothesised, the transformation of small single grains might be related to the presence of large residual stresses resulting from the difference between thermal expansion coefficients of alumina and zirconia.

Crack propagation characterization

AFM was used to investigate the crack propagation path at the surface of double torsion samples and Vickers indent induced cracks. A preliminary thermal etching treatment was performed so as to reveal the grain boundary. Grain boundary grooves and the slight grains relief induced by thermal etching, related to the crystallographic orientation of the grain with respect to the surface were used to study the inter-or intragranular nature of the crack propagation. AFM observations (Fig. 2) were found to be much more precise than scanning electron microscopy. In particular, the observation of crack branching and bridging zones is very clear.

Transformation toughening mechanisms analysis

The relief in the surroundings of propagated crack was observed locally on Ce-TZP double torsion samples. It was found possible observing martensitic relief at the surface of transformed grains (Fig. 3). Depending on the stabilizing oxide content, the transformation was found to be very easily stress induced or not.

Effect of the cubic phase on the ageing behaviour

3Y-TZP samples sintered at high temperatures (i.e. above 1723 K) were characterised and found to exhibit a dual phase microstructure at ambient temperature with a significant amount of large grains of cubic phase, in agreement with the zirconia-yttria phase diagram. The autoclave ageing stability of the cubic phase was assessed by AFM (Fig. 6). More importantly, the detrimental effect of the cubic phase on the ageing behaviour has been proven, in that the yttrium depletion zones around yttria enriched cubic grains appeared as preferred nucleation sites [START_REF] Chevalier | Critical effect of cubic phase on aging in 3 mol.% yttria stabilized zirconia ceramics[END_REF] . Fig. 6: Surface transformation of a 3Y-TZP sample exhibiting a large amount of cubic phase grains, after 24 hrs of autoclave ageing at 413 K. The large cubic grains are clearly not transformed.

Martensitic relief observation and characterization

AFM has been used to investigate very locally the martensitic aspects of the transformation. The main discovery is the observation [START_REF] Deville | Martensitic relief observation by atomic force microscopy in yttria stabilized zirconia[END_REF] for the first time of very clear martensitic relief at a nanometer scale, on several types of materials (3Y-TZP, 10Ce-TZP, and Al2O3-(3Y-TZP)). It was shown that using AFM, it is possible to observe the apparition of self accommodating martensitic variants at the surface of bulks samples, after thermal ageing (Fig. 4). The progressive transformation of each grain, step by step (Fig. 5), is clearly observed. One of the great points using AFM is the possible quantitative measurements of surface relief, e.g. angle relationships between self accommodating variants, etc… Those informations could be related to the underlying crystallography, and the observations reported here fit very well the theoretical predictions of phenomenological theories.

Conclusions

The purpose of this communication is to show that AFM is an extremely powerful tool to investigate the surface degradation mechanisms of zirconia containing ceramics. AFM is a unique method to provide 3D measurements at a nanometer scale with relative ease of image interpretation. In particular, it is possible imaging nanometer scale martensitic relief, providing very valuable informations for the mechanistic understanding.

Fig. 1 :

 1 Fig. 1: Observation of the same zone of ZTA by AFM (left, relief contrast) and SEM (right, species contrast) after autoclave thermal ageing treatment at 413 K. The transformation was found to occur either in zirconia agglomerates (a) or isolated grains (b and c). The comparison of AFM and SEM images allowed characterizing the microstructural environment of transformed zirconia grains.

Fig. 2 :

 2 Fig. 2: AFM observation of a propagated crack on a (1.6 vol.% ZrO2)-alumina sample (height image and deflection image). The Vickers indent induced crack path is clearly identified, as well as the crack bridging zones. Alumina grain pull-outs resulting from the polishing process are clearly visible.

Fig. 3 :

 3 Fig. 3: Self accommodating martensitic variant pairs formed in the surroundings of a propagated crack of a double torsion test Ce-TZP sample.

Fig. 4 :

 4 Fig. 4: Observation of thermal martensite self-accommodating variant pairs at the surface of a Ce-TZP sample. The characteristic shape of the relief could be explained by the phenomenological theory of the martensite crystallography.

Fig. 5 :

 5 Fig. 5: Progressive transformation of a single grain in 3Y-TZP at 383 K (0, 5 and 10 min). The apparition and growth of the variant pairs is clearly observed. (Vertical scale 35 nm)

nm

Acknowledgments

Financial support of the European Union under the GROWTH 2000 program, project BIOKER, reference GRD2-2000-25039.


Pascale Charpin
email: pascale.charpin@inria.fr

Gohar M Kyureghyan
email: gohar.kyureghyan@ovgu.de

On sets determining the differential spectrum of mappings

Keywords: Boolean function, bent function, APN mappings, monomial binomial, permutation, hyperplane, cryptographic criteria, differential uniformity

The differential uniformity of a mapping F : F 2 n → F 2 n is defined as the maximum number of solutions x for equations F (x+a)+F (x) = b when a ̸ = 0 and b run over F 2 n . In this paper we study the question whether it is possible to determine the differential uniformity of a mapping by considering not all elements a ̸ = 0, but only those from a special proper subset of F 2 n \ {0}.

We show that the answer is "yes", when F has differential uniformity 2, that is if F is APN. In this case it is enough to take a ̸ = 0 on a hyperplane in F 2 n . Further we show that also for a large family of mappings F of a special shape, it is enough to consider a from a suitable multiplicative subgroup of F 2 n .

Introduction

The differential uniformity of a mapping on F n 2 is an important parameter for cryptological applications. It is defined as follows. Let F :

F n 2 → F n 2 and
is called the difference mapping of F defined by a, or the derivative of F in direction a. Set δ(a, γ)

= |{x ∈ F 2 n , D a F (x) = γ}|, (1)
for any a ̸ = 0 and γ in F n 2 (recall that |A| denotes the cardinality of the set A). The differential spectrum of F is the multiset consisting of integers δ(a, γ) with their multiplicities. The differential uniformity of F is defined as

δ(F ) = max a̸ =0, γ∈F n 2 δ(a, γ). ( 2
)
The image set of a difference mapping D a F contains at most 2 n-1 elements, since D a F (x) = D a F (x + a) for any a ∈ F n 2 . Clearly, the image set of a difference mapping D a F is of that maximal size if and only if D a F is 2-to-1. A mapping is called almost perfect nonlinear, abbreviated APN, if all its difference mappings are 2-to-1. Note that the APN mappings can be defined also as those having differential uniformity 2. APN mappings provide the optimal resistance against the differential cryptanalysis when they are used as an S-box [START_REF] Nyberg | Perfect nonlinear S-boxes[END_REF]. Whether there exist APN permutations on F 2 n for even n is an important research problem. Only one (up to affine equivalence) such mapping is known for n = 6. It was found by a group of NSA around John Dillon in 2009, [START_REF] Browning | An APN permutation in dimension six[END_REF]. Yet, seven years later, the most famous and important open question concerning the APN functions remains: Does there exist an APN permutation on F 2 n for n even and greater than 6?

To verify the APN property of F it necessitates, a priori, to check that all difference mappings D a F are 2-to-1. Actually, it is well-known that not all D a F must be checked: It was proved in [2, Eurocrypt 93] that it is sufficient to check 2 n-1 well-chosen D a F . In Section 2, we reconsider this result and show that it is equivalent to the statement: A mapping F : F n 2 → F n 2 is APN if and only if D a F are 2-to-1 for all non-zero a on a hyperplane of F n 2 . There are only few families of APN mappings which are known. Most of the known mappings with small differential uniformity are described as univariate polynomials over finite fields. In this paper we study possibilities to reduce the complexity of computing the differential uniformity of mappings. In particular we consider checking the APN property in Theorems 3 and 4.

In Section 3 we show that the differential spectrum of a class of permutations described by Zieve [START_REF] Zieve | On some permutation polynomials over F q of the form x r h(x (q-1)/d )[END_REF] is determined by a small number of derivatives (Theorem 6). Finally we focus on binomials and give some numerical results.

Verifying the APN property 2.1 A combinatorial problem

In [START_REF] Beth | On almost perfect nonlinear permutations[END_REF], Beth and Ding introduced a so-called differential representation set of F n 2 , which is defined as follows:

Definition 1 Let S be a subset of F n 2 \ {0}. If S satisfies x ∈ F n 2 , y ∈ F n 2 with x ̸ = 0, y ̸ = 0, x ̸ = y ⇒ {x, y, x + y} ∩ S ̸ = ∅, ( 3
)
then S is called a differential representation set of F n 2 . Moreover, S is said minimal when it has minimal size.

It is proved in [START_REF] Beth | On almost perfect nonlinear permutations[END_REF] that the size of differential representation set S is equal to or greater than 2 n-1 -1. This is easy to see. Indeed, set

S ′ = F n 2 \ (S ∪ {0}) = {s 1 , s 2 , . . . , s ℓ }, ℓ = 2 n -|S| -1.
Thus, the ℓ-1 elements s

1 +s i , 2 ≤ i ≤ ℓ, belong to S so that |S| ≥ 2 n -|S|-2 providing |S| ≥ 2 n-1 -1.
In particular, a minimal differential representation set of F n 2 has cardinality 2 n-1 -1. The next theorem shows that the minimal differential representation sets are exactly the hyperplanes of F n 2 without the zero element.

Theorem 1 A subset S ⊂ F n 2 is a minimal differential representation set of F n 2 if and only if S ∪ {0} is an hyperplane of F n 2 . Proof. Let k := |S| = 2 n-1 -1. Evidently, if S ∪ {0} is an hyperplane of F n 2 then S satisfies (3). So suppose that S satisfies (3) with k = 2 n-1 -1.
Our goal is to prove that S ∪ {0} is an hyperplane. We proceed by induction. For n = 2 it is clear that the property holds. We assume that the statement is true until n -1 where n ≥ 3.

Let H be any hyperplane of F n 2 and denote by H its complement in

F n 2 . Set T = (S ∪ {0}) ∩ H and T = S ∩ H. Then |T | ≥ 2 n-2 since T satisfies (3) in H \ {0}. Therefore T ≤ 2 n-2 . Note that if |T | = 2 n-1 then T = H. So we suppose now that |T | < 2 n-1 .
Fix y ∈ H \ T . Then for all z ∈ H \ T we get y + z ∈ H. But y + z ∈ T because y ̸ ∈ S and z ̸ ∈ S. The set of elements y + z, z describing H \ T has cardinality c with c ≥ 2 n-2 . This is impossible unless

|T | = 2 n-2 .
If |T | = 2 n-2 then T is a subspace of dimension n -2, from the induction hypothesis applied to H. In this case, we have

F n 2 = T ∪ (a + T ) ∪ (b + T ) ∪ (a + b + T ), with H = (T ∪ a) + T,
for some (a, b). If T is neither equal to b + T nor equal to (a + b) + T then there are

x ∈ b + T \ T , y ∈ (a + b) + T \ T providing x + y ∈ a + T which contradicts (3). So T is a coset of T , completing the proof. ⋄ Remark 1
Theorem 1 is a special case of a result proven in [START_REF] Bose | A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes[END_REF] as mentioned in [START_REF] Pott | Vectorial quadratic bent functions as a product of two linearized polynomials[END_REF].

The APN property

In the remaining of the paper we consider mappings on finite fields F 2 n . We first state the APN property as a property on each derivative.

D α F (x) = F (x) + F (x + α) = F (y) + F (y + α) = D α F (y).
After that, one prove easily that

D x+y F (x) = D x+y F (x + α) and D x+y+α F (x) = D x+y+α F (x + α).
Thus, the functions D x+y F and D x+y+α F are not 2-to-1, which is a contradiction since either x + y or x + y + α belong to H. ⋄

Next we apply Theorem 2 to a special class of mappings. Recall that the hyperplanes (that is (n -1)-dimensional F 2 -subspaces) of F 2 n are the sets

H α := {x ∈ F 2 n | T r(αx) = 0},
defined by all nonzero α ∈ F 2 n . Further, we denote by Im(G) the image set of a mapping G.

Corollary 1 Let H = {x|T r(x) = 0} and F : F 2 n → F 2 n be defined by F (x) = G(x) + R(x)T r(x),
where G and R are arbitrary mappings on F 2 n . Then for every a ∈ H we have

D a F (x) = (G(x) + G(x + a)) + T r(x)(R(x) + R(x + a)). ( 5
)
Suppose that the mappings G(x) and P (x

) := G(x) + R(x) are APN. Set I a = Im(D a G) ∩ Im(D a P ). Then F is APN if and only if for any nonzero a ∈ H and every b ∈ I a there is x ∈ F 2 n with D a G(x) = b and T r(x) = 1, or D a P (x) = b and T r(x) = 0.
Proof. The equality ( 5) is directly obtained by replacing T r(a) = 0 in

D a F (x) = D a G(x) + T r(x)R(x) + T r(x + a)R(x + a).
Consider a ∈ H and the equation

D a F (x) = b for some b ∈ F 2 n . Note that D a F (x) = { D a G(x) if T r(x) = 0 D a P (x) if T r(x) = 1.
In particular, any solution of 5) reduces to

D a F (x) = b solves either D a G(x) = b or D a P (x)
D a F (x) = D a G(x) + T (x)R(a).
Also,

D a P (x) = b is equivalent to D a G(x) = b + R(a). Example 2 The mapping F (x) = x 3 +x 4 •T r(x) on F 2 7 is of shape discussed in Example 1.
It is easy to check that F(x) has differential uniformity 8. This example shows that for computing the differential uniformity of a generic mapping it is not enough to consider only the differential mappings defined by elements of a hyperplane, like it is the case for APN mappings by Theorem 2.

Problem 1 Suppose that δ(a, b) ≤ 4 for every nonzero a on hyperplane H α and every b ∈ F 2 n , as it is the case for the mapping F considered in Corollary 1. Under which conditions can we conclude that δ(F ) = 4?

For several classes of mappings, it is well-known that to verify the APN property it is sufficient to check (p a ) for very particular values a. The most simple case is when F (x) = x t for some fixed positive integer t. In this case it is enough to check (p 1 ) only, since

x t + (x + a) t = a t ( ( x a ) t + ( x a + 1 ) t ) implying δ(a, γ) = δ(1, γ/a t ).
Note that in this case we get the differential uniformity of F by computing the derivative in point 1 only. When F is a polynomial whose coefficients are in a subfield of F 2 n , Theorem 2 yields another general simplification. • For n = 7 we have C = 9 and it is enough to check the derivatives for

Theorem 3 Let H = { α ∈ F 2 n | T r(α) = 0 }. Set n =
D a 2 k F (y) = F (y + a 2 k ) + F (y) = (F (x + a) + F (x)) 2 k = (D a F (x)) 2 k where y = x 2 k , since F ∈ F 2 k [x]. It is clear that D a 2 k F is 2-to-1 if and only if D a F is, completing the proof. ⋄ If 2 n -1 is prime, then there are (2 n -2)/n cyclotomic cosets (all of size n) modulo 2 n -1, providing |I| = (2 n -2)/(2n
a ∈ { α i , i ∈ {1, 3,

Component functions

The components functions of the mapping F : F 2 n → F 2 n can be used to compute its differential uniformity. Using Theorem 2 we directly simplify the characterization of APN mappings given in [1, Theorem 2], which is stated in Theorem 4 below. We use the notation from [START_REF] Berger | On Almost Perfect Nonlinear functions over F n 2[END_REF]: f λ , with λ ∈ F * 2 n , are the component functions of F , i.e., the Boolean functions x → T r(λF (x)). Further, for a Boolean function f , we set

F(f ) := ∑ x∈F 2 n (-1) f (x) and D a f (x) := f (x) + f (x + a).
Theorem 4 Let H be any hyperplane in F 2 n , and

F : F 2 n → F 2 n with component functions f λ , λ ∈ F 2 n . Then, for any nonzero a ∈ F 2 n , it holds ∑ λ∈F 2 n F 2 (D a f λ ) ≥ 2 2n+1 , ( 6
)
where the equality holds if and only if

D a F is 2-to-1. In particular, F is APN if and only if every nonzero a ∈ H satisfies ∑ λ∈F 2 n F 2 (D a f λ ) = 2 2n+1 . ( 7
)
Proof. We sketch the proof, which is similar to that of [1, Theorem 2]. Set

A = ∑ λ∈F 2 n F 2 (D a f λ ) for some a. Then A = ∑ λ,x,y ∈F 2 n
(-1) T r(λ(F (x+a)+F (x)+F (y+a)+F (y))) , and hence

A = 2 n |{(x, y) ∈ F 2 n × F 2 n | D a F (x) = D a F (y)}| = 2 2n+1 +2 n |{(x, y)| D a F (x) = D a F (y), x ̸ = y ̸ = x+a}|, implying (6). Moreover A = 2 2n+1 if and only if D a F is 2-to-1, i.e., D a F (x) = D a F (y) for y ∈ {x, x + a} only.
Theorem 2 completes the proof.

⋄

Bent and semi-bent functions have been intensively studied since of their particular role in theory and applications of Boolean functions. The statement of Theorem 2 is related with Theorems V.2 and V.3 from [START_REF] Canteaut | On cryptographic properties of the cosets of R(1, m)[END_REF]. These results show that for checking that a given Boolean function is bent (resp. semi-bent) it is enough to look on the derivatives defined by nonzero elements a ∈ H, where H is a hyperplane. For clarity we discuss this fact for bent functions in more detail below.

Let n be even. A Boolean function f : F 2 n → F 2 is said to be bent when ∑

x∈F 2 n (-1) f (x)+T r(ax) = ±2 n/2 , for all a ∈ F 2 n . ( 8
)
Equivalently, bent functions are those having all derivatives D a f (x) balanced.

Recall that a Boolean function g is balanced if it takes equally often the values 0 and 1.

Theorem 5

Let n be even and H be a hyperplane of F 2 n . Then a function f : F 2 n → F 2 is bent if and only if the derivatives D a f (x) are balanced for all nonzero a ∈ H.

Proof. Assume that D a f is balanced for a ∈ H \{0}. Then for every u ∈ F 2 n the derivative of the function x → f (x) + T r(ux) in direction a is balanced as well. Further we use the classical formula

A λ := ( ∑ x∈F 2 n (-1) f (x)+T r(λx) ) 2 = ∑ a∈F 2 n (-1) T r(λa) ∑ x∈F 2 n (-1) f (x)+f (x+a) , for λ ∈ F 2 n . If H = H α = {x|T r(αx) = 0} for some nonzero α ∈ F 2 n , then A α = ∑ a∈H ∑ x∈F 2 n (-1) Daf (x) - ∑ a̸ ∈H ∑ x∈F 2 n (-1) Daf (x) = 2 n - ∑ a̸ ∈H ∑ x∈F 2 n (-1) Daf (x) ,
and then

A 0 + A α = 2 n + ∑ a̸ ∈H ∑ x∈F 2 n (-1) f (x)+f (x+a) + A α = 2 n+1 .
It is easy now to prove that the last equality implies A 0 = A α = 2 n (for instance by induction). Replacing f by x → f (x)+ T r(ux) we get A α+u = 2 n for any u and hence [START_REF] Charpin | Cubic monomial bent functions: a subclass of M[END_REF]. ⋄

Example 4

We consider the components of the function F studied in Example 1:

f λ (x) = T r(λF (x)) = T r (λ(G(x) + T r(x)R(x))) , λ ∈ F * 2 n ,
where G is APN and R is linear. Suppose that λ is such that the function

x → T r(λG(x)) is bent. Then f λ is bent when R satisfies T r(λR(a)) = 0 for all a ∈ H ( 9
)
where H = {a|T r(a) = 0}. Indeed

F(D a f λ ) = ∑ x∈F 2 n (-1) T r(λ(G(x+a)+G(x)+T r(x)R(a))) = ∑ x∈H (-1) T r(λDaG(x)) + (-1) T r(λR(a)) ∑ x̸ ∈H (-1) T r(λDaG(x)) .
The sum above equals 0 for all a ∈ H when ( 9) is satisfied.

On differential spectrum of a class of permutations

In this section we consider mappings on F 2 n of shape

F (x) = x k (h(x ρ )),
where k is a fixed positive integer and h : F 2 n → F 2 n . The subset of such bijective mappings was characterized by Zieve in [START_REF] Zieve | On some permutation polynomials over F q of the form x r h(x (q-1)/d )[END_REF]. Lemma 1 is the binary version of [START_REF] Zieve | On some permutation polynomials over F q of the form x r h(x (q-1)/d )[END_REF]Lemma 2.1]. We denote by G V the subgroup of F * 2 n of order V , where V is a divisor of 2 n -1.

Lemma 1 Let 2 n -1 = ρV , k ∈ N and h : F 2 n → F 2 n . Then the mapping F (x) = x k (h(x ρ )) is a permutation on F 2 n if and only if (i) gcd(k, ρ) = 1 and (ii) x → x k h(x) ρ permutes G V .
Using the above lemma it is easy to describe families of permutations on F 2 n , for example we have:

Proposition 1 Let 2 n -1 = ρV where gcd(ρ, V ) = 1. Then F (x) = x k (h(x ρ )), where k = iV with 1 ≤ i ≤ ρ -1, ( 10
)
is a permutation on F 2 n if and only if gcd(i, ρ) = 1 and h(G V ) ρ = G V .
Proof. To apply Lemma 1, we prove that the conditions on F correspond to (i) and (ii) respectively. Obviously, gcd(i, ρ) = 1 is here equivalent to (i).

Set P (x) = x iV h(x) ρ . For y ∈ G V we have P (y) = h(y) ρ . Thus, P permutes

G V if and only if h(G V ) ρ = G V . ⋄
Permutations described in Lemma 1 are among a large class of mappings whose differential uniformity can be computed by considering elements from a suitable subgroup of F * 2 n only:

Theorem 6 Assume that n is such that 2 n -1 = ρV with ρ > 1, V > 1 and gcd(ρ, V ) = 1. Let F : F 2 n → F 2 n be defined by F (x) = x k (h(x ρ )), where h(x) = ℓ ∑ i=0 ν i x i , 1 ≤ ℓ < V ( 11
)
and

ν i ∈ F 2 n . Then F is APN if and only if for any a ∈ G V the mapping y → ℓ ∑ i=0 ν i u iρ ( y k+iρ + (y + 1) k+iρ )
is 2-to-1. Moreover, to compute the differential spectrum of F it is sufficient to study the equations D u F (x) = b for u ∈ G V , and then

δ(F ) = max{ δ(u, b) | u ∈ G V , b ∈ Im(D u F ) }.
Proof. We compute D a F (x) for a ∈ F 2 n , a ̸ = 0:

D a F (x) = x k (h(x ρ )) + (x + a) k (h((x + a) ρ )) = ℓ ∑ i=0 ν i ( x k+iρ + (x + a) k+iρ ) = a k ( ℓ ∑ i=0 ν i a iρ ( (x/a) k+iρ + ((x/a) + 1) k+iρ ) ) . ( 12
)
Since gcd(ρ, V ) = 1, every a ∈ F * 2 n can be written as a = uv with u ∈ G V and v ∈ G ρ . Replacing in [START_REF] Pott | Vectorial quadratic bent functions as a product of two linearized polynomials[END_REF] x = ay and a = uv we get

D a F (ay) = (uv) k ( ℓ ∑ i=0 ν i u iρ ( y k+iρ + (y + 1) k+iρ ) ) . ( 13
)
Thus F is APN if and only if

y → ℓ ∑ i=0 ν i u iρ ( y k+iρ + (y + 1) k+iρ )
is 2-to-1 for any u ∈ G V . To compute δ(F ) we need to compute the number δ(a, b) of solutions x of the equations

D a F (x) = b, a ∈ F * 2 n , b ∈ F 2 n .
According to (13), the equation above reduces

u k ( ℓ ∑ i=0 ν i u iρ ( y k+iρ + (y + 1) k+iρ ) ) = b ′ , b ′ = b v k , showing that δ(a, b) = δ(u, b ′ ). Further δ(F ) is the maximal value of the δ(u, b ′ ), u ∈ G V , b ′ ∈ F 2 n , completing the proof. ⋄
The next result is derived from [1, Theorem 2] similarly to Theorem 4:

Corollary 3 Notation is as in Theorem 6. Let F : F 2 n → F 2 n be given by [START_REF] Nyberg | Perfect nonlinear S-boxes[END_REF]

with components f λ , λ ∈ F 2 n . Then F is APN if and only if for all a ∈ G V : ∑ λ∈F 2 n F 2 (D a f λ ) = 2 2n+1 . ( 14
)
If for example ρ = (2 n -1)/3 we need to do very few computations:

Example 5 Let n = 2m, 2 n -1 = 3ρ where 3 does not divide ρ. In this case, to compute the differential spectrum of F (given by ( 11)) we need to compute the image of D u F for u ∈ F * 4 only. Note that 3 divides ρ if and only if 3 divides m.

Binomials with low differential uniformity

In this section we study specific families of binomials derived from [START_REF] Nyberg | Perfect nonlinear S-boxes[END_REF]. Let 2 n -1 = ρV , we consider binomials given by:

F (x) = νx k + x k+ℓρ , ν ∈ F * 2 n , 1 ≤ k ≤ 2 n -ρ -2, 1 ≤ ℓ < V. ( 15
)
As we showed in Theorem 6 the differential spectrum of F (x) can be computed with elements of a suitable subgroup of F * 2 n , if gcd(ρ, V ) = 1. Recall that G ρ denotes the subgroup of F * 2 n of order ρ. The first natural question to be answered is whether there are APN binomials given by (15). The next example shows that such an APN mapping is known.

Example 6

Let n = 2m with m odd and consider the binomials on F 2 n given by

F (x) = x 3 (ν + x 2 m +1 ) = νx 3 + x 2 m +4
where ν ∈ ωF 2 m with ω ∈ F 2 2 \ F 2 . By Theorem 6, to compute the differential uniformity of F (x) it is enough to consider a ∈ F * 2 m . Note that x → νx + x 2 m +2 is a permutation [START_REF] Charpin | Cubic monomial bent functions: a subclass of M[END_REF]Theorem 5.1]. This class includes the first example of an APN mapping which is not equivalent to a power mapping. It was presented in [START_REF] Edel | A new APN function which is not equivalent to a power mapping[END_REF]Theorem 2], due to Edel, Kyureghyan and Pott:

x → x 3 + ux 36 , u ∈ ωF * 2 5 where ω ∈ F 4 \ {0, 1}. The latter mapping is not bijective.

It is mentioned in [START_REF] Edel | A new APN function which is not equivalent to a power mapping[END_REF] that a complete computer search for APN binomials x r + ux s on F 2 n was done for n ≤ 10.

In the remaining of this work we are interested in bijective binomials, defined by (15), of degree at least 3, which have low differential uniformity. Proposition 1 yields a large class of bijective binomials of type (15):

Corollary 4 (cf. [8, Corollary 4]) Let 2 n -1 = ρV , ρ > 1 and V > 1, where gcd(ρ, V ) = 1. Let ν ∈ F * 2 n and F (x) = x k (ν + x ρ ), k = jV where 1 ≤ j ≤ ρ -1. ( 16
)
Then F is a permutation on F 2 n if and only if gcd(j,

ρ) = 1, ν ̸ ∈ G V and (ν + G V ) ρ = G V .
Note that, concerning the differential spectrum of binomials ( 16), by Theorem 6 we need to study the derivatives y → ν ( y jV + (y + 1) jV ) + u ρ ( y jV +ρ + (y + 1) jV +ρ ) , u ∈ G V .

Numerical results

In this section, we present some interesting numerical results that we obtained for binomials discussed in Section 3.1. We did experiments for binomials F (x) = ax k + x k+ρ with the smallest values of V or for small n. Finding such bijective F having δ(F ) = 4 seems difficult; actually we obtained no example unless when F is affine equivalent to a monomial, as we explain below.

For n = 8 we obtained easily such binomials F satisfying δ(F ) = 6 (see, for instance, Table 1). This holds (not so easily) for n = 10 and V = 3. However we have no example for n = 12 and V = 5 and incline to conjecture that for n > 10 such binomials F satisfy δ(F ) > 6.

Table 1. With notation of (15): n = 8, ℓ = 1, ρ = 85 and V = 3. We give the differential spectrum of bijective differentially 6-uniform binomials of the form

F (x) = νx k + x k+85 , ν = α 3 , 1 ≤ k ≤ 169,
where α is a root of the primitive polynomial for any e ∈ F * 4 , i.e., i ∈ {0, 1, 2}. There are other values of k for which F is bijective but has a differential spectrum already in the table, corresponding to some k ′ , where k ′ < k. Note that k is not always a multiple of V (as it was assumed in Corollary 4). There are bijective differentially 6-uniform binomials of the form

x 8 + x 4 + x 3 + x 2 + 1. The subgroup G 3 is F * 4 = { α i•85 , i = 0,
F (x) = νx k + x k+341 , 1 ≤ k ≤ 681.
We give below two examples where ν = α 3 , α being a root of the primitive polynomial X 10 + X 3 + 1.

Proof. We have k + ρ = i(2 m + 1) + 1 + (2 m -1) = i(2 m + 1) + 2 m and then

F (x) = νx i(2 m +1)+1 + x i(2 m +1)+2 m = L(x i(2 m +1)+1 ) where L : x → νx + x 2 m . Clearly L is a permutation if and only if ν ̸ ∈ G V . Further, F is permutation if and only if gcd(i(2 m + 1) + 1, 2 m -1) = gcd(2i + 1, 2 m -1) = 1, completing the proof. ⋄
Claim 1 shows that some properties of F , given by ( 17), are actually properties of the monomial mapping M : x → x k , in particular the differential spectrum of F is the differential spectrum of M . For n = 10, we found 9 such binomials F which are differentially 4-uniform. More generally, it is easy to check that (33i + 1)(33j + 1) ≡ 1 (mod 2 10 -1) ⇐⇒ 2ji + i + j ≡ 0 (mod 31). since ((2 m + 1)i + 1)((2 m + 1)j + 1) ≡ (2 m + 1)(2ji + i + j) + 1 (mod 2 2m -1).

Thus, for any i we compute j to have the compositional inverse of x → x k .

Claim 3 Let F be given by

F (x) = x d , d = i(2 m + 1) + 1, i ∈ [1, 2 m -1] with gcd(2i + 1, 2 m -1) = 1.
Then F is a permutation with compositional inverse x → x ℓ where ℓ = j(2 m + 1) + 1 such that j(2i + 1) + i ≡ 0 (mod 2 m -1).

In [START_REF] Blondeau | Differential properties of power functions[END_REF]Table 1] all differentially 4-uniform bijective monomials are listed when 6 ≤ n ≤ 25. For n = 10 there are 3 sporadic differentially 4-uniform monomials appearing in this list (in fact there are 6 by including for each exponent also its inverse). In Claim 2 we identified, accidentally, the expressions for these sporadic differentially 4-uniform monomials.

Remark 2 Bracken and Leander proved in [START_REF] Braken | A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree[END_REF] that the mapping, M (x) = x 2 2r +2 r +1 , r odd, on the field F 2 n with n=4r, which is a highly nonlinear permutation, satisfies δ(M ) = 4. Note that 2 3r (2 2r + 2 r + 1) = 2 r + 1 + 2 3r = 2 r (2 2r + 1) + 1.

According to Claim 3, we can express the compositional inverse of such mapping M 2 3r . It is M ′ (x) = x j(2 2r +1)+1 , j(2 r+1 + 1) + 2 r ≡ 0 (mod 2 2r -1).

Conclusion

Many questions arise when the computational aspects of the differential uniformity of mappings are discussed. In this paper our purpose was to create restricted corpus that it is easier to explore. We are conscious that the main problem is the size of n (when considering mappings on F 2 n ). For high values of n computations become impossible; in this case finding some iterative tools seems to be a possible solution, which need to be understood. Another research direction, which need to be explored is identifying families of mappings for which the computation of differential spectrum is considerably easier than for a generic case. As Section 3 shows after identifying such families, easy computations lead already to interesting observations.

Example 1

 1 = b depending on its trace. This shows that D a F (x) = b has at most 4 solutions, since the mappings G and P are APN. Clearly, if b ̸ ∈ I a then D a F (x) = b has at most two solutions. Observe, if x is a solution for D a G(x) = b then the second solution is x + a and it holds T r(x) = T r(x + a), since a ∈ H. Similarly the solutions x and x + a of D a P (x) have the same trace, if they exist. Assume that b ∈ I a . Suppose x 1 solves D a G(x) = b and x 2 solves D a P (x) = b. Then the equation D a F (x) = b has 4 solutions if and only if T r(x 1 ) = 0 and T r(x 2 ) = 1, completing the proof.⋄ Notation is as in Corollary 1 and its proof. Assume that R is a linear mapping. Then G + R is APN as soon as G is APN and (

F 2 n

 2 expressed by a polynomial in F 2 [x]. Set C = |I| = (2 n -2)/(2n). Then we have: • If n = 5 then C = 3, and F is APN as soon as D a F satisfies (p a ) for only 3 elements a ∈ {α i , i = 1, 7, 15}, where α is a root of the primitive polynomial x 5 + x 2 + 1.

1 ,

 1 2}. The spectrum is presented as follows: {δ(e, b), b ∈ Im(D e F )}, (number of times δ(e, b) occur),

Example 7

 7 With notation of (15): n = 10, ℓ = 1, ρ = (2 10 -1)/3 = 341 and V = 3.

Claim 2

 2 Let n = 10. Consider the mapping M : x → x k over F 2 n where k takes the following values k = 33i + 1, i ∈ { (1, 10), (7, 14), (9, 24), (26, 27) } and i = 30 (18) (where we give by pairs one exponent and its compositional inverse). These mappings are bijective and satisfy δ(M ) = 4. The pair (1, 10) corresponds to the quadratic one with its inverse and i = 30 corresponds to the inverse function since (2 5 -2)(2 5 + 1) + 1 = -(2 5 + 1) + 1 = -2 5 = 2 5 (-1).

  ks where s > 1 and k ≥ 1. Let β be a primitive element of F 2 n . Let F be a mapping on F 2 n which is given by a polynomial in F 2 k [x]. Let I be a set of representatives of 2Proof. From Theorem 1, we can choose any hyperplane H to check the APN property. Here H is the hyperplane which is invariant under the Frobenius isomorphism σ : a → a 2 . Thus, taking a ∈ H we get a 2 k ∈ H and

k -cyclotomic cosets modulo 2 n -1 and I = {i ∈ I | β i ∈ H}.

Then, F is APN if and only if it satisfies (p a ) for all a ∈ I.

  By Theorem 2 if F is not APN then for any hyperplane H α there is at least one a such that D a F is not 2-to-1. Can we improve this observation in any direction? Such a belongs to 2 n-1 hyperplanes. Thus there are at least two such elements a. Is it possible to have a better lower bound on the number of such a?

5, 9, 11, 15, 23, 29, 55} }, where α is a root of x 7 + x + 1. Problem 2

Acknowledgment

In this version we correct the statements of Proposition 1 and Corollary 4 in our previous version. We thank deeply Hubert Kiechle for his careful reading of our paper and for pointing us a gap in the proof of Proposition 1

Some binomials with differential uniformity 4 appeared when n = 10. They are of the form:

where

Let F be given by ( 17). Then F is affine equivalent to the monomial of exponent k = i(2 m +1)+1. More precisely F (x) = L(x k ), L(x) = νx+x 2 m , and F is a permutation if and only if ν ̸ ∈ G 2 m +1 and gcd(2i + 1, 2 m -1) = 1.

Fortunately, Proposition 1 and Corollary 4 are quite isolated from the rest of our results, which aim to reduce computation complexity of the differential uniformity. All numerical results reported in [START_REF] Berger | On Almost Perfect Nonlinear functions over F n 2[END_REF] were doublechecked and are correct.