Pierre Cocheteux

Alexandre Voisin

Eric Levrat

Benoît Iung

Éric Levrat

PROGNOSTIC DESIGN: REQUIREMENTS AND TOOLS

Keywords:

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION AND PROBLEM STATEMENT

Even if maintenance is a necessity, maintenance has a negative image and suffers from a deficiency of understanding and respect. It is usually recognised as a cost, a necessary evil, not as a contributor. Moreover traditionally the scope of maintenance activities has been limited to the production vs. operation phase. But as the paradigm of manufacturing shift towards realizing a sustainable society, the role of maintenance has to change to take into account a life-cycle management oriented approach [START_REF] Takata | Maintenance: Changing Role in Life Cycle Management[END_REF] for enhancing the eco-efficiency of the product life [START_REF] Desimone | Eco-Efficiency[END_REF]. In that way, maintenance has to be considered not only in production vs. operation phase but also in product design, product disassembly, and product recycling … [START_REF] Takata | Maintenance: Changing Role in Life Cycle Management[END_REF] and has to be lead the implementation of a set of means and processes for acting on the operational requirements to carry out the global system performances [START_REF] Al-Najjar | Selecting the most efficient maintenance approach using fuzzy multiple criteria decision making[END_REF].

To support this new role in operation, the maintenance concept undergone through several major developments to lead to proactive considerations which require changes in transforming traditional "fail and fix" maintenance practices to "condition-based vs. predictive" strategies performed only when a certain level of component degradation (impacting profit/product conditions) occurs [START_REF] Djurdjanovic | Watchdog Agent --an infotronics-based prognostics approach for product performance degradation assessment and prediction[END_REF]. The business processes the most important to develop anticipation action are monitoring, diagnosis, prognosis and decision-making processes. Among these modules, the prognosis process is often considered as the main one while its goal is fundamental for implementing anticipation capabilities.

Indeed, prognosis has to predict the future health of the system and generate the different RULs (Remaining Useful Life of the system, part of the system or component) for each detected (current) or potential degradation/failure mode, by taking into account the knowledge of the system (functional and dysfunctional), past information (background), current information (current state) and future information (scenario with manufacturing and maintenance data) [START_REF] Voisin | Generic prognosis model for proactive maintenance decision support: application to pre-industrial e-maintenance test bed[END_REF].

Thus prognostic is a process dealing with a lot of knowledge (more or less formalized) which is issued from different sources. The identification and the use of this knowledge needed all along the prognostic life cycle, and particularly in the engineering phase, require a consistent methodology for guarantying and validating, at the end, requirements about prognostic results quality. About methodology, [START_REF] Lee | Intelligent prognostics tools and e-maintenance[END_REF] notice that most of the developed prognostics approaches are application or equipment specific and that a generic and scalable prognostic methodology does not exist. Moreover [START_REF] Uckun | Standardizing Research Methods for Prognostics[END_REF] shows that a huge number of paper tackling the prognostic issues present a lack of common methodology. Among the methodologies presented, any one allows to obtain a global reasoning for the prognostic design phase. The authors propose only some items to be followed in the design phase but without real consistency between the items:



Establish the requirement of the system  Perform a failure mode analysis  Determine the most critical failure mode  Determine the available model in order to prognosticate these failure modes taking into account sensor capabilities  Assess the obtained model with metrics about accuracy, precision and robustness.

Nevertheless [START_REF] Muller | Formalisation of a new prognosis model for supporting proactive maintenance implementation on industrial system[END_REF] propose a more complete methodology. It uses the functional and dysfunctional analyses of the system to define its structure and health. Stochastic processes are also formalized for supporting the degradation dynamic. The use of common analyses is relevant but the a priori choice of the technology/tool (Bayesian network) limits the application cases.

Therefore a generic methodology for prognostic design is still missing leading to answer to the following questions:

 What are the failures modes to be anticipated for the system?  What are the technologies/tools/models to be used to support this anticipation?

These questions are generic but their answers depend on the context in relation to the life phase of the system and the available knowledge. Indeed the system could already exist or could currently be designed. The amount of system knowledge differs if system components are specific or not to the application or if their model is new or well-known. This issue request the inventory of the prerequisites needed for prognostic engineering.

To face this challenge, this paper proposes a generic methodology for selecting the failure modes (e.g. the most critical) needing prognostic and its technology/tool associated to reach the prognostic requirements in terms of result quality. To reach its goal, the methodology is supported by an analysis gathering the necessary knowledge for prognostic engineering.

The relation between the system life phase and the prognostic design phase is addressed in section 2. Then section 3 describes the proposed methodology for selecting failure modes and associated prognostic tools. Section 4 delivers some conclusion and perspectives.

PROGNOSTIC DESIGN AND SYSTEME LIFE CYCLE

For the system engineering point of view (in relation with INCOSE initiative0 F1 ) the principal system is the system performing the global function and needs contributor processes for supporting it in operation phase. These last processes represent the support system whose supply chain and maintenance are the main processes. Maintenance enables system of interest to fulfill its finality by deploying strategies which could evolve during the principal system life.

The strategic decision to move to proactive maintenance can be made at different moment of the system life. This decision implies to design and build an integrated system of proactive maintenance as described in [START_REF] Voisin | Generic prognosis model for proactive maintenance decision support: application to pre-industrial e-maintenance test bed[END_REF] and thus its key process named prognostic. The context for the prognostic engineering depends on the system life phase when prognostic is engineered. During the system design phase, the physical part of the system does not exist. New paradigms promote the evaluation and simulation of the system into its different future environments in order to assess their impacts on the future system. This impact assessment could lead to (re-)design the system and in this case, maintenance issues and in particular prognostic issues and requirements could directly influence the system design. This way improve the global economical performance of system by decreasing the number of failures or their impacts on system finality (better reliability) and also by make easier the monitoring of the degradation modes and their forecasting and thus improve prognostic performances and decrease prognostic engineering time (and therefore its cost).

The decision to evolve could be made also when the system exists and operates. In this case the system is already built, thus modifications are difficult and expensive and the proactive maintenance aims at replacing a conventional strategy (corrective, timed-based, condition-based…) already performed on the system. The potential physical changes in the system are the integration of additional sensors. Thus the context is locked and the only lever to improve the global economic performance is the proactive maintenance design and above all the prognostic engineering. But the advantage, in comparison with the previous case, is the potential available knowledge which can be used to build prognostic models. The experience defined through expertise or data brings lots of information about the most frequent failure modes and is able to focus the prognostic engineering on them. This paper is interested by this case which does not authorize modification on the system it-self. Nevertheless some of proposal of this paper can be considered for both cases.

The design phase is the most important one because the other phase depends on it. Thus the prognostic engineering phase needs attention and has to follow a systems development model as the V-Model [START_REF]Systems Engineering Fundamentals[END_REF]. Therefore requirements and specifications about expected prognostic results have to be defined in first and represent the "system level design requirements" (figure 1). This phase is performed in two steps: (1) the choice of metrics for evaluating performance of the prognostic process and (2) the quantification of the expected level of these metrics. Lots of metrics are already proposed in the scientific contribution and address different points of view. But no methodology has been proposed for selecting the appropriate metrics. [START_REF] Saxena | Metrics for Evaluating Performance of Prognostic Techniques[END_REF] give however a classification of the metrics and reviews the main ones. Thus the use of several metrics seems necessary to assess the different points of view (e.g. accuracy and precision). When the decision on metrics is planned out, the quantification of them is given and become requirements for the prognostic designer. A welldone requirement establishment phase allows to make easier integration and validation phases. Then the "Items level design requirements" can be started. It allows to determine the system structure and the component requirements. In the prognostic case, failure mode can be seen as "prognostic component". Then prognostic "Items level design requirements" phase aims at answering to the question addressing the choice of the failure modes according the system performances and of their supported technologies/tools/models needed to fulfill requirements.

PROGNOSTIC DESIGN: METHODOLOGY FOR SELECTING FAILURE MODES AND ASSOCIATED PROGNOSTIC TOOLS

An industrial system has lots of ways of dysfunction with a lot of degradation and failure modes. But the frequency of occurrence, the impact on the performances and the generated corrective or preventive maintenance cost could be very different from a failure mode to another one. Thus a method for selecting them and their associated prognostic tool is necessary as shown in the figure 2.

Normal behavior analysis

Designing contributor processes like prognostic need to know the principal system and its functioning way. A system is described by different points of view with specific models:





Management point of view: description of the economical data and of the policies and strategies deployed on the system and its parts for managing it along its all life.

From this last point come requirements for choosing the criterion used to classify the most important failure modes to prognosticate. On the figure 2 this criterion is called criticality criterion and characterizes the failure occurrence (e.g. cost, ecological impact, frequency, risk of no detection…).

Dysfunctional behavior analysis

Maintenance deals with the dysfunctional behaviors which need also methods and tools to represent and formalize the knowledge. Two kinds of knowledge could be distinguished: This dysfunctional knowledge is used for two reasons: to select failure modes and indicators and to build the prognostic model. The first one needs a global view of the available knowledge. Knowledge formalized with the form of table is easy to use but no method proposes to gather all the knowledge. [START_REF] Mathew | Failure Mechanisms Based Prognostics[END_REF] propose to extend the FMECA analysis with the degradation mechanism causes of failures. In that way we propose to use FMECA as an engineering tool for formalizing the knowledge (not as a method for analysing risk) and to complete it with information needed for selecting failure modes. Therefore the FMAP (Failure Modes Analysis for Prognostic) is formalized into the table 1 composed of seven fields. Failure mode, causes and consequences are the same field that their corresponding FMECA ones. They represent failure modes, their causes and their consequences. Criticality corresponds to the criterion defined by the management point of view in the normal behaviour analysis. Influential variables field allows to identify variables which affect the failure and/or the dynamic evolution of the degradation modes which cause it. These variables describe material/technological properties, operational mode and the environment variables which impact degradation/failure modes.

Observable indicators are raw or built signals on which degradation/failure modes can be observed. This or these indicator(s) allow(s) to identify if the degradation/failure mode has occurred (diagnostic) and the degradation level (health assessment). In the prognostic context, these indicators are projected and then used to compute RULs. Finally the last field is named observable property. This field has to be carry out for each observable indicators. This field answers at the question "what can we observe on the observable indicator evolution?". Sometime only the failure or degradation is observable and sometime both are observable.

In the case of observable degradation, the dynamic between the degradation mode beginning and the failure occurrence could be more or less fast. Prognostic enables the anticipation of the failure for the purpose of (re-)scheduling a maintenance action. But a maintenance action needs a preparation time and if the RUL is too short, then the information for scheduling the preventive maintenance action happens too late and the action cannot be performed before the failure. Then the observable dynamic of the degradation mode (fast or slow) is important an information given in this field.

Degradation/Failure modes selection

After its formalization, the knowledge is used in first for selecting degradation/failure modes in order to focus on the most critical ones. Of course the first criterion is the criticality defined into the FMAP. But this criterion is not the only one because some critical degradation/failure modes could be not observable or could have no observable indicators able to be forecasted. In this case degradation/failure mode causes are considered and prognosticated. They need a function to pass from causes to subject mode (e.g. aggregation function). If one or more causes cannot also be prognosticated, the same way can be performed. But tools performing prognostic and aggregation are only models and thus insert incertitude and error sources in the final result. Therefore the choice of indicators which are representative of high-level degradation/failure modes (next of the critical mode) has to be preferred as often as possible.

Representative indicators selection

The prognostic of degradation/failure modes on component is supported by indicators forecasting. They could be representative for one or several modes and their evolutions characterize the degradation dynamic or the failure occurrence. These indicators can be measured directly on the system or built with aggregated data from other signals.

Prognostic model/tool selection

Prognostic models/tools operate the selected representative indicators forecasting by projecting it into the future. Their selections depend on the available knowledge about the evolution of degradation and failure phenomena that means about the indicators selected previously. Two sources provide knowledge: expertise and data. People who work sometime during several years in relation with a system collect a lot of information. This knowledge is most of time imprecise but brings a global view of the degradation phenomena. They enable also to fill in the lack of knowledge given by data. Well-known methods are used for formalizing this knowledge (FMECA, HAZOP…) and scientific contributions use also expertise like [START_REF] Zuashkiani | Incorporating expert knowledge when estimating parameters of the proportional hazards model[END_REF] for estimating parameters of proportional hazards model.

Knowledge from data brings more precision but most of time only on concise phenomena.

Moreover the question about data quality can be asked: how is the measure precision? , are they complete or censured? , are data with noise? and so on. For using this knowledge lots of algorithms already exist to adjust model to the data (e.g. training algorithm for neural network, regression algorithm for linear, exponential or logarithm function…). The complementarity between both sources is important: the global knowledge but imprecise supported by expertise is able to be adjusted with a better precision using data and algorithm (for example the degradation law is given by the expert and its parameters adjusted by data).

The available knowledge is one of the main factors for selecting prognostic tool/model: either the more appropriate model is directly defined by the expert or the content and the format guide the selection. This link is highlighted by the classification of data based prognostic in [START_REF] Heng | Rotating machinery prognostics: State of the art, challenges and opportunities[END_REF]. This classification criterion used in this paper is the type of available information (traditional reliability approaches -event data based prediction, prognostics approaches -condition data based prediction and integrated approaches combining both approaches).

Requirements validation

The last phase of the methodology is the validation or not of the prognostic process in comparison with its expected requirements. When the prognostic quality is insufficient, i.e. when the required levels of the metrics chosen during the phase "system level design requirements" (figure 1) are not respected, a new selection of the prognostic model/tool has to be performed. This could concern a part or the totality of the representative indicators.

The loop has to be repeated until reaching a sufficient result. Otherwise a new representative indicators selection could be performed in the same way. Then if the available knowledge is not sufficient to reach the prognostic design requirements, knowledge (data) has to be created by degradation test (run-to-failure test) or the integration of additional sensors.

CONCLUSION

This paper has proposed guidelines for prognostic design issues for a system currently in service. The main contribution is the proposal of a methodology for selecting degradation/failure modes and prognostic tools. This methodqology is supported by the functional and dysfunctional formalized knowledge enabling to classify each failure mode in relation to a specific criticality. This selection is related with a list of associated indicators to forecast and the knowledge about them leads to a choice of prognostic tool. If prognostic design requirements are not sufficient, selection and choice have to be reconsidered or knowledge has to be created by test or addition of sensors.

Method to choose more precisely the prognostic tool, adjust them and link together for reaching a global prognostic model is needed. The literature has lots of scientific contribution about component prognostic, but methodology and associated support tools for assessing future performance still missing. A first proposition has already made in [START_REF] Cocheteux | Methodology for assessing system performance loss within a proactive maintenance framework[END_REF] and could be continued and improved.

  hal-00429297, version 1 -2 Nov 2009 Author manuscript, published in "11th International Conference on The Modern Information Technology in the Innovation Processes of the Industrial Enterprises, Bergame : Italy (2009)"

Figure 1 :

 1 Figure1: systems development model: V-Model[START_REF]Systems Engineering Fundamentals[END_REF]

Figure 2 :

 2 Figure 2: Methodology for selecting failure modes and prognostic tools



  Environmental point of view: description of the system environment and their interactions, Functional point of view: functional analysis (e.g. IDEF-0, FAST…), system model (e.g. transfer function),  Operational point of view: control mode description (e.g. UML use case diagram),  Organic point of view: description of the physical structure decomposed into subsystem until components), structure function, 3D representation (e.g. CAD software models),  Technical point of view: technological description of the components, physical construction of the components,

( 1 )

 1 the knowledge about causes and consequences and (2) knowledge about the evolution of degradation and failure phenomena. The first one is most of time described in a qualitative way. It is supported by a lot of tools whose the most popular are FMECA (Failure Mode, Effects, and Criticality Analysis), HAZOP (Hazard and operability study) or cause and consequence trees. The second kind of knowledge can be classified into several criteria:  nature (probabilistic model as reliability model, based on physic law or on specific indicator evolution),  quantitative or qualitative,  the state space nature (discrete or continuous),  static model description (probabilistic law (Weibull, exponential…), state definition of a discrete model as Markov chain…)  dynamic model description (transition functions for discrete model, value of parameters…).

Table 1 :

 1 FMAP (Failure Modes Analysis for Prognostic)

http://www.incose.org/


Frank Vega

Criterion for the Riemann Hypothesis

Keywords: Mathematics Subject Classification (2010). Primary 11M26; Secondary 11A41. Keywords. Riemann hypothesis, prime numbers, Riemann zeta function, Chebyshev function

published or not. The documents may come

Criterion for the Riemann Hypothesis

Introduction

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2 . It was proposed by Bernhard Riemann (1859). The Riemann hypothesis belongs to the Hilbert's eighth problem on David Hilbert's list of twenty-three unsolved problems. This is one of the Clay Mathematics Institute's Millennium Prize Problems. In mathematics, the Chebyshev function θ(x) is given by θ(x) = q≤x log q with the sum extending over all prime numbers q that are less than or equal to x, where log is the natural logarithm.

Proposition 1.1. For every x ≥ 19035709163 [1, Theorem 1 pp. 2]:

1 -0.15 log 3 x • x < θ(x) < 1 + 0.15 log 3 x • x.

The following property is based on natural logarithms:

Proposition 1.2. For x > -1 [8, pp. 1]: x x + 1 ≤ log(1 + x) ≤ x.
Leonhard Euler studied the following value of the Riemann zeta function (1734) [START_REF] Ayoub | Euler and the Zeta Function[END_REF].

Proposition 1.3. We define [2, (1) pp. 1070]:

ζ(2) = ∞ k=1 q 2 k q 2 k -1 = π 2 6 ,
where q k is the kth prime number. By definition, we have

ζ(2) = ∞ n=1 1 n 2 ,
where n denotes a natural number. Leonhard Euler proved in his solution to the Basel problem that

∞ n=1 1 n 2 = ∞ k=1 q 2 k q 2 k -1 = π 2 6 ,
where π ≈ 3.14159 is a well-known constant linked to several areas in mathematics such as number theory, geometry, etc.

The number γ ≈ 0.57721 is the Euler-Mascheroni constant which is defined as

γ = lim n→∞ -log n + n k=1 1 k = ∞ 1 - 1 x + 1 ⌊x⌋ dx.
Here, ⌊. . .⌋ represents the floor function. Franz Mertens discovered some important results about the constants B and H (1874) [START_REF] Mertens | Ein Beitrag zur analytischen Zahlentheorie[END_REF]. The number B ≈ 0.26149 is the Meissel-Mertens constant where γ = B + H [START_REF] Mertens | Ein Beitrag zur analytischen Zahlentheorie[END_REF].

Proposition 1.4. We have [4, Lemma 2.1 (1) pp. 359]:

∞ k=1 log q k q k -1 - 1 q k = γ -B = H.
For x ≥ 2, the function u(x) is defined as follows [10, pp. 379]:

u(x) = q>x log q q -1 - 1 q .
On the sum of the reciprocals of all prime numbers not exceeding x, we have:

- 0.2 log 3 x ≤ q≤x 1 q -B -log log x ≤ 0.2 log 3 x .
In number theory, Ψ(n) = n • q|n 1 + 1 q is called the Dedekind Ψ function where q | n means the prime q divides n. For x ≥ 2, a natural number M x is defined as

M x = q≤x q.
We define R(n) = Ψ(n) n•log log n for n ≥ 3. We say that Dedekind(x) holds provided that

R(M x ) ≥ e γ ζ (2)
.

The well-known asymptotic notation Ω was introduced by Godfrey Harold Hardy and John Edensor Littlewood [START_REF] Hardy | Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic ϑ-functions[END_REF]. In 1916, they also introduced the two symbols Ω R and Ω L defined as [START_REF] Hardy | Contributions to the theory of the Riemann zetafunction and the theory of the distribution of primes[END_REF]:

f (x) = Ω R (g(x)) as x → ∞ if lim sup x→∞ f (x) g(x) > 0; f (x) = Ω L (g(x)) as x → ∞ if lim inf x→∞ f (x) g(x) < 0.
After that, many mathematicians started using these notations in their works.

From the last century, these notations Ω R and Ω L changed as Ω + and Ω -, respectively. There is another notation:

f (x) = Ω ± (g(x)) (meaning that f (x) = Ω + (g(x)) and f (x) = Ω -(g(x)) are both satisfied). Nowadays, the notation f (x) = Ω + (g(x)
) has survived and it is still used in analytic number theory as [START_REF] Tenenbaum | Introduction to Analytic and Probabilistic Number Theory[END_REF]:

f (x) = Ω + (g(x)) if ∃k > 0 ∀x 0 ∃x > x 0 : f (x) ≥ k • g(x)
which has the same meaning to the Hardy and Littlewood older notation. Putting all together yields a proof for the Riemann hypothesis.

Central Lemma

Several analogues of the Riemann hypothesis have already been proved. Many authors expect (or at least hope) that it is true. However, there are some implications in case of the Riemann hypothesis could be false. The following is a key Lemma.

Lemma 2.1. If the Riemann hypothesis is false, then there exist infinitely natural numbers x for which Dedekind(x) fails (i.e. Dedekind(x) does not hold).

Proof. The function g is defined as [11, Theorem 4.2 pp. 5]:

g(x) = e γ ζ(2) • log θ(x) • q≤x 1 + 1 q -1
.

The Riemann hypothesis is false whenever there exists some natural number x 0 ≥ 5 such that g(x 0 ) > 1 or equivalent log g(x 0 ) > 0 [11, Theorem 4.2 pp. 5].

It was proven the following bound [11, Theorem 4.2 pp. 5]:

log g(x) ≥ log f (x) - 2 x .
For x ≥ 2, the function f was by Nicolas in his seminal paper as [10, Theorem 3 pp. 376], [3, (5.5) pp. 111]:

f (x) = e γ • log θ(x) • q≤x 1 - 1 q .
If the Riemann hypothesis is false then there exists a real number b with 0 < b <

That inequality is equal to log

f (y) ≥ k • y -b • √ y • 1 √ y , but we notice that lim y→∞ k • y -b • √ y = ∞
for every possible values of k > 0 and 0 < b < 1 2 . Now, this implies that

∀y 0 ∈ N, ∃y ∈ N (y > y 0 ) : log f (y) ≥ 1 √ y .
Note that, the variable k disappears in our previous expression when we do not need it anymore. In this way, if the Riemann hypothesis is false, then there exist infinitely many natural numbers x such that log f (x) ≥ 1

√

x . Since

1 √ x0 > 2 x0
for x 0 ≥ 5, then it would be infinitely many natural numbers x 0 such that log g(x 0 ) > 0. □

Main Insight

This is the main insight.

Lemma 3.1. The inequality q≤x ( q q-1 )

log θ(x) ≥ e γ ζ(2)
7 holds for large enough x ∈ N.

Proof. By Proposition 1.4, the inequality

q≤x q q-1 log θ(x) ≥ e γ ζ( 2
)
7
is the same as

q≤x log q q -1 -B -log log θ(x) ≥ H + 6 • γ -7 • log(ζ(2))
after of applying the logarithm to the both sides and distributing the terms.

In addition, log log θ(x) < log log 1 + 0.15 log

3 x • x = log log 1 + 0.15 log 3 x + log x = log   (log x) •   1 + log 1 + 0.
q≤x log q q -1 -B -log log x - 0.15 log 4 x ≥ H + 6 • γ -7 • log(ζ(2)).
That is,

q≤x log q q -1 -B -log log x- 0.15 log 4 x -u(x) ≥ H -u(x)+6•γ -7•log(ζ(2)).
after subtracting u(x) to the both sides of the inequality. By Proposition 1.4, we can see that

q≤x 1 q -B -log log x - 0.15 log 4 x -u(x) ≥ 6 • γ -7 • log(ζ(2)).
Frank Vega By Proposition 1.5, we deduce that

- 0.2 log 3 x - 0.15 log 4 x -u(x) ≥ 6 • γ -7 • log(ζ(2)).
It is a fact that the inequality

- 0.2 log 3 x - 0.15 log 4 x -u(x) ≥ 6 • γ -7 • log(ζ(2)) holds for large enough x ∈ N due to 6 • γ -7 • log(ζ(2)) < 0 is a negative real constant and lim x→∞ - 0.2 log 3 x - 0.15 log 4 x -u(x) = 0. □

Main Theorem

This is the main theorem.

Theorem 4.1. Dedekind(x) always holds for large enough x ∈ N.

Proof. By Lemma 3.1, the inequality

q≤x q q-1 log θ(x) ≥ e γ ζ (2)
7 holds for large enough x ∈ N. By Propositions 1.2 and 1.4, the inequality

q≤x q q-1 log θ(x) ≥ e γ ζ(2) 7 is equivalent to e H-u(x) • R(M x ) ≥ e γ ζ (2) 7
.

Certainly, we have

q≤x q q-1 log θ(x) ≥   q≤x q q-1 e 1 q   • q≤x 1 + 1 q log θ(x) = e H-u(x) • q≤x 1 + 1 q log θ(x) = e H-u(x) • M x • q|Mx 1 + 1 q M x • log log M x = e H-u(x) • Ψ(M x ) M x • log log M x = e H-u(x) • R(M x )
using the Propositions 1.2 and 1.4 such that e 1 q ≥ 1 + 1 q for every prime q. Consequently, we would have [START_REF] Ayoub | Euler and the Zeta Function[END_REF] .

e H-u(x) e γ ζ(2) 6 • R(M x ) ≥ e γ ζ
We only need to prove that Using a simple numerical calculation, we can check that

γ + B 5 log(ζ(2)) > 1.26483 > 1.2 = 6 5
and therefore, the proof is done. □

Main Result

This is the main result.

Corollary 5.1. The Riemann hypothesis is true.

Proof. By Lemma 2.1, if the Riemann hypothesis is false, then there exists an infinite sequence of natural numbers x i such that Dedekind(x i ) fails. This contradicts the fact that Dedekind(x) always holds for large enough x ∈ N according to the Theorem 4.1. By Reductio ad absurdum, the Riemann hypothesis must be true as a direct consequence of Lemma 2.1 and Theorem 4.1. □

Conclusions

Practical uses of the Riemann hypothesis include many propositions that are considered to be true under the assumption of the Riemann hypothesis and some of them that can be shown to be equivalent to the Riemann hypothesis. Indeed, the Riemann hypothesis is closely related to various mathematical topics such as the distribution of primes, the growth of arithmetic functions, the Lindelöf hypothesis, the Large Prime Gap Conjecture, etc. A proof of the Riemann hypothesis could spur considerable advances in many mathematical areas, such as number theory and pure mathematics in general.

Frank Vega NataSquad 10 rue de la Paix FR 75002 Paris France e-mail: vega.frank@gmail.com

Proposition 1 . 5 .

 15 For x ≥ 2278383 [5, Theorem 5.6 (1) pp. 243]:

1 2

 1 such that, as x → ∞ [10, Theorem 3 (c) pp. 376], [3, Theorem 5.29 pp. 131], log f (x) = Ω ± (x -b ). Actually Nicolas proved that log f (x) = Ω ± (x -b ), but we only need to use the notation Ω + in this proof under the domain of natural numbers. According to the Hardy and Littlewood definition, this would mean that ∃k > 0, ∀y 0 ∈ N, ∃y ∈ N (y > y 0 ) : log f (y) ≥ k • y -b .

> 1 for x ≥ 2 [

 12 enough x ∈ N to confirm that Dedekind(x) also holds. Hence, it is enough to show that e 10, (11) pp. 379]. Next,-B + 6 • log(ζ(2)) -5 • γ < 0after applying the logarithm to the both sides of the inequality. Finally,

Acknowledgment

Many thanks to Patrick Solé and Michel Planat for their support.


Xiaodong Liu

Jincheng Yu

Bing Wang

Krishnendu Maji

Diana T Alvarez-Ruiz

Emmanuel Guilmeau

Robert Freer
email: robert.freer@manchester.ac.uk

Enhancing the thermoelectric properties of Nb-doped TiO2-based ceramics through in-situ synthesis of β-Sn inclusions at grain boundaries

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

873K. Our approach based on grain boundary engineering provides a simple and eco-friendly route to enhance the thermoelectric performance of low-cost, oxide-based ceramics. KEYWORDS: Thermoelectric, rutile, β-Sn, grain boundary inclusions, twin boundaries.

INTRODUCTION

The developing energy crisis is placing greater pressure on the urgent development of efficient and eco-friendly energy sources. Thermoelectrics are able to directly convert heat into electricity, and thereby represent a green technology which can be of great importance in contributing to the goal of carbon neutrality. Traditional high-performance thermoelectric materials usually contain rare, toxic and heavy elements, and involve complex and energy-consuming fabrication procedures [START_REF] Snyder | Complex thermoelectric materials[END_REF][START_REF] He | Advances in thermoelectric materials research: Looking back and moving forward[END_REF]. However, in spite of their lower thermoelectric performance, transition metal-oxide based thermoelectrics have considerable potential for a new generation of thermoelectric materials, because of their low cost, simple processing and the abundance of their starting materials in the earth [START_REF] Backhaus-Ricoult | Semiconducting large bandgap oxides as potential thermoelectric materials for high-temperature power generation?[END_REF][START_REF] Koumoto | Thermoelectric ceramics for energy harvesting[END_REF].

The thermoelectric performance of a typical material can be evaluated by the dimensionless thermoelectric figure of merit (ZT), which is characterized by: 𝑍𝑇 = !" ! # 𝑇, where S is the Seebeck coefficient, σ is the electrical conductivity, 𝑆𝜎 $ is the power factor, κ is the thermal conductivity and T is the absolute temperature. Hence, a high-performance thermoelectric material typically requires enhanced electrical transport properties (power factor) and/or reduced thermal transport properties (thermal conductivity). Unfortunately, it is difficult to control these two factors simultaneously because of coupling effects. Therefore, the strategy of improving one factor at expense of the other is commonly adopted to obtain high ZT values.

Among the transition metal oxides, rutile structured TiO2 based oxides are of specific interest and are candidates for future high-temperature thermoelectric devices because of their very good thermal stability. However, the key obstacle to realizing wide application is the modest thermoelectric performance of pure rutile [START_REF] Bayerl | Theoretical limits of thermoelectric figure of merit in n-type TiO2 polymorphs[END_REF][START_REF] Nam | A gigantically increased ratio of electrical to thermal conductivity and synergistically enhanced thermoelectric properties in interface-controlled TiO2-RGO nanocomposites[END_REF][START_REF] Touloukian | Thermophysical properties of matter -The TPRC data series[END_REF] (both relatively low electrical conductivity and high thermal conductivity) which results in low ZT values. Several approaches have proved to be effective in enhancing the thermoelectric performance of TiO2-based oxides. Donor doping with Nb or Ta at cation sites [START_REF] Itakura | Hall Effect and Thermoelectric Power in Semiconductive TiO2[END_REF][START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF][START_REF] Verchère | Optimum in the thermoelectric efficiency of nanostructured Nb-doped TiO2 ceramics: from polarons to Nb-Nb dimers[END_REF][START_REF] Yuan | Investigation of the thermoelectric properties of reduced Nb-doped TiO2-δ ceramics[END_REF][START_REF] Yuan | Thermoelectric properties of Nb-doped TiO2δ ceramics reduced at elevated temperature[END_REF][START_REF] Zaitsev | Preparation and characterization of nanocrystalline doped TiO2[END_REF][START_REF] Zhou | Preparation of titanium-tantalum-oxygen composite thermoelectric ceramics through highpressure and high-temperature method[END_REF], and doping with N at anion sites [START_REF] Mikami | Thermoelectric properties of nitrogen-doped TiO2-x compounds[END_REF][START_REF] Liu | Chemical tuning of TiO2 nanoparticles and sintered compacts for enhanced thermoelectric properties[END_REF] can augment the carrier concentration and help increase electrical conductivity. By substituting for matrix ions, donors introduce a high density of point defects, which can act as scattering centers for phonons and thus help to reduce thermal conductivity. Similarly, nano-structuring [START_REF] Zaitsev | Preparation and characterization of nanocrystalline doped TiO2[END_REF][START_REF] Liu | Chemical tuning of TiO2 nanoparticles and sintered compacts for enhanced thermoelectric properties[END_REF] (nano-grains and nanoparticles) can produce nano-sized boundaries or interfaces, which can greatly enhance phonon scattering and reduce the thermal conductivity. Magnéli phases [START_REF] Fan | Preparation of monophasic titanium sub-oxides of Magnéli phase with enhanced thermoelectric performance[END_REF][START_REF] Harada | Thermoelectric properties and crystallographic shear structures in titanium oxides of the Magnéli phases[END_REF][START_REF] Liu | Manipulation of planar oxygen defect arrangements in multifunctional Magnéli titanium oxide hybrid systems: From energy conversion to water treatment[END_REF][START_REF] Liu | High-thermoelectric performance of TiO2-x fabricated under high pressure at high temperatures[END_REF][START_REF] Backhaus-Ricoult | Levers for thermoelectric properties in titania-based ceramics[END_REF][START_REF] Portehault | Facile general route toward tunable Magnéli nanostructures and their use as thermoelectric metal oxide/carbon nanocomposites[END_REF] (TinO2n-1, 4 ≤ n ≤ 10) are highly oxygen-deficient, near to rutile structured TiO2-x and have been extensively investigated.

Such phases are associated with high electrical conductivity; the high density of crystallographic shear planes [START_REF] Bui~ill | Crystal Structures in the {132} CS Family of Higher Titanium Oxides TinO2n-1[END_REF][START_REF] Zhang | Andersson-Magnéli Phases TinO2n-1: Recent Progress Inspired by Swedish Scientists[END_REF] (with intervals of approximately 0.5 -1 nm) inside the Magnéli phases give rise to significant phonon scattering with negligible carrier scattering, thereby leading to improved thermoelectric performance.

The introduction of highly conductive secondary phases can also be very beneficial to the thermoelectric performance. Successful approaches including the addition of graphene, carbon nanotubes (CNTs) or reduced graphene oxide (rGO) into SrTiO3 and TiO2 oxides have been reported [START_REF] Nam | A gigantically increased ratio of electrical to thermal conductivity and synergistically enhanced thermoelectric properties in interface-controlled TiO2-RGO nanocomposites[END_REF][START_REF] Lin | Thermoelectric power generation from lanthanum strontium titanium oxide at room temperature through the addition of graphene[END_REF][START_REF] Okhay | Thermoelectric performance of Nbdoped SrTiO3 enhanced by reduced graphene oxide and Sr deficiency cooperation[END_REF][START_REF] Feng | Graphene promoted oxygen vacancies in perovskite for enhanced thermoelectric properties[END_REF][START_REF] Srivastava | Anisotropy and enhancement of thermoelectric performance of Sr0.8La0.067Ti0.8Nb0.2O3-δ ceramics by graphene additions[END_REF]. The highly conductive carbon-based secondary phases which segregate at the grain boundaries can help to reduce grain boundary electrical resistance and to increase charge transport. In spite of a small rise in the thermal conductivity, the significant improvement in charge transport leads to enhanced thermoelectric performance. Mesoscale metallic particles (Cu, Fe, Ti, W and Ta) [START_REF] Zhou | Preparation of titanium-tantalum-oxygen composite thermoelectric ceramics through highpressure and high-temperature method[END_REF][START_REF] Hao | Fabrication and Thermoelectric Properties of Magnéli Phases by Adding Ti into TiO2[END_REF][START_REF] Lu | Effect of Cu powder addition on thermoelectric properties of Cu/TiO2-x composites[END_REF][START_REF] Srivastava | Improving the thermoelectric properties of SrTiO3-based ceramics with metallic inclusions[END_REF] actively contribute to boosting thermoelectric performance through strong enhancement of carrier concentration and hence electrical conductivity. By contrast, nano-sized metallic particles (Ag) can further lower the thermal conductivity [START_REF] Liu | Bottom-up assembly to Ag nanoparticles embedded Nb-doped TiO2 nanobulks with improved n-type thermoelectric properties[END_REF] due to phase boundary phonon scattering. However, such nano-sized metallic particles usually show poor dispersion, which results in severe aggregation, creating unwanted challenges in the fabrication process.

In this work, we report a novel, eco-friendly route to fabricate high thermoelectric performance rutile-structured Nb-doped TiO2-based oxides with metallic Sn inclusions at the grain boundaries.

Assisted by advanced electron microscopy, we provide detailed microstructural information about the metallic Sn inclusions and the matrix rutile at the atomic level. The β-Sn phase exhibits exceptionally high electrical conductivity [START_REF] Plevachuk | Electrical conductivity, thermoelectric power and viscosity of liquid Sn-based alloys[END_REF][START_REF] Crum | Electrical conductivity of Sn at high pressure and temperature[END_REF], approximately three orders of magnitude higher than that of the matrix rutile. The uniformly distributed Sn metallic inclusions help to enhance the electrical conductivity by approximately 37% and the power factor by nearly 30%, together with a slight reduction of the thermal conductivity. Consequently, the ZT value is increased to 0.20 at 873K, i.e., about ~ 18% enhancement compared to the previously reported single Nb-doped TiO2 based oxides [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF].

MATERIAL AND METHODS

Polycrystalline samples were prepared from controlled mixtures of high purity powders of TiO2 (Sigma-Aldrich, >99.99%) and Nb2O5 (Sigma-Aldrich, >99.99%); (sample compositions and codes are listed in Table 1). Carbon flake, used only for sintering environment was obtained from Sigma-Aldrich, (>99.99%). The starting powders were mixed with propan-2-ol (Fisher Scientific) and YSZ balls (Pi-Kem), vibration-milled for 24 hours, then dried for 24 hours at 373 K. The dried powders were loaded into an alumina crucible (covered with a thin alumina lid) and calcined in a box furnace at 1473 K for 4 hours in the air. In order to reduce sintering temperature, 0.81wt% manganese (II) carbonate (Sigma Aldrich, >99.99%) powder was added to the calcined powders [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF], which were re-milled and dried once more. To selected batches, 1wt% SnO2 (Sigma-Aldrich, >99.99%) powder was added (See Table 1) and vibration milled once more, then dried at 373 K for 24 hours. The resulting powders were compacted in a 20 mm diameter cylindrical, stainlesssteel die, and pressed under a constant pressure of 50 MPa. For sintering, the pellets were packed in carbon flakes in a 30 mm diameter alumina crucible (covered with a thick alumina lid) and heated in air in a box furnace at 1473 K for 4 hours; the carbon flakes create a strongly reducing environment during sintering [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF]. The phase structure was identified by X-ray diffraction (XRD), using a Philips X'Pert X-ray diffractometer®) (Cu Kα radiation, λ = 1.540598 Å). Data were collected for 2 theta angles from 10° to 100°, with 0.033° step size. Rietveld refinement was conducted using Topas [START_REF] Coelho | Whole-profile structure solution from powder diffraction data using simulated annealing[END_REF].

Microstructural observation and chemical investigations were performed on a TESCAN MIRA LC FEG scanning electron microscope (SEM), equipped with an Oxford Instrument SDD energy dispersive X-ray (EDX) detector. Electron backscattered diffraction (EBSD) analysis was conducted using an Oxford Instrument Symmetry S2 ® EBSD detector. The nano-sized microstructural features were examined by transmission electron microscopy (TEM). Samples for TEM analysis were prepared by both standard crushing method and focused ion beam (FIB)

techniques. The FIB employed a FEI Helios 660 Gallium (Ga) ion source and a FEI Helios plasma dual beam system, incorporating a standard FEG SEM column and a xenon (Xe) ion beam column.

Selected area electron diffraction (SAED) patterns were obtained using a FEI Tecnai 20 analytical LaB6 TEM at 200 kV. High resolution transmission electron microscopy (HRTEM), scanning transmission electron microscopy (STEM) and energy-dispersive x-ray spectroscopy (EDXS)

were conducted using a Thermo Fisher Talos F200A FEG TEM (equipped with a HAADF/BF STEM detectors and four silicon drift Super-X EDXS detectors) at 200 kV. Atomically-resolved STEM investigations were carried out using an aberration-corrected scanning transmission electron microscope (FEI Titan G2 ChemiSTEM operated at 200 kV) equipped with HAADF/BF/ADF STEM detectors and with a convergence semi-angle of 21 mrad. The electron energy loss spectroscopy (EELS) data were acquired by a Gatan Quantum EEL spectrometer, with collection convergence semi-angle of 65 mrad. The TEM data processing was performed on GMS 3 Digital Micrograph commercial software. Ball-and stick-models and calculated SAED patterns were simulated using CrystalMaker software [START_REF] Yin | CrystalMaker 6.3[END_REF]. HAADF/ABF STEM image simulations were performed by Dr. Probe software [START_REF] Barthel | Probe: A software for high-resolution STEM image simulation[END_REF].

X-ray photoelectron spectroscopy and determination of element binding states was carried out by a High-Throughput X-ray photoelectron spectrometer (XPS) system (ESCA2SR spectrometer, Scienta Omicron), equipped with a monochromatic Al Kα source (photon energy 1486.69 eV).

Survey scans were carried out from 0 to 1200 eV (binding energy). High-resolution XPS spectra were obtained for the Ti 2p, Nb 3d, O 1s and C 1s regions. The C 1s peaks were used to standardise and calibrate the high-resolution spectra. The XPS data processing was conducted by CasaXPS software [START_REF] Walton | Peak fitting with CasaXPS : a Casa pocketbook[END_REF].

The transport properties, electrical conductivity and Seebeck coefficients were measured using a Ulvac-Riko ZEM-3 system. The temperature range was from room temperature to 873 K, with 50 K per steps. The measurements were carried out under a low-pressure He atmosphere. The carrier concentration and carrier mobility were determined at room temperature by Hall effect measurements using a physical properties measurement system (PPMS; Quantum Design) under an applied magnetic field from -7 to 7 T. Thermal conductivity was determined from room temperature to 873 K from values of bulk density (measured by Archimedes' method), thermal diffusivity (obtained by a NETZCH LFA-427 in flowing Ar atmosphere) and heat capacity (collected by a NETZCH STA 449C in an Ar atmosphere).

RESULTS AND DISCUSSION

Bulk materials and crystal structure.

All ceramic TNS samples are dark grey in color and of at least 96% theoretical density. There is a slight decrease in density with increasing Nb content, indicating that Nb doping can hinder densification, consistent with our earlier work [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF]. The relative density of TNS samples is typically 4% higher than that of the corresponding TNO samples (Table S1). This difference is attributed to the generation of an additional low-melting temperature phase; further details are given below in the XRD section.

The XRD patterns (Figure 1) reveal that the rutile phase (Tetragonal, P42/mnm) is dominant in the TNO and TNS samples. With the addition of SnO2, small amounts of β-Sn can also be detected in the TNS samples (Figure 1a). As SnO2 can be easily converted into metallic Sn under reducing conditions [START_REF] Kim | Reduction of SnO2 with hydrogen[END_REF][START_REF] Mitchell | The reduction of SnO2 and Fe2O3 by solid carbon[END_REF], the β-Sn phase in the samples should originate from the strong reducing atmosphere during sintering. Rietveld refined lattice parameters of the matrix rutile phase and secondary β-Sn phase are presented in Figure S1, Table 2 and Table S2. The refined lattice parameters for the rutile phase in TNS samples are slightly lower than those of TNO samples, but both are larger than those reported for pure rutile [START_REF] Restori | Charge Density in Rutile, TiO2[END_REF] (Table 2), and increase with increasing Nb doping; this is consistent with the deviation of the low angle rutile peaks (Figure 1b). The lattice expansion of the rutile phase in all samples can be ascribed to the high-level of Nb doping and the incorporation of cations of larger ionic radius [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF] (RNb 5+ = 0.64 Å; RTi 4+ = 0.605 Å).

Microstructural analysis

The microstructures of the TNO samples (Figure 2a) contain regular polygonal grains with a large number of pores and cracks. With increasing Nb content, the amount of porosity increases significantly (Figure S2). The SnO2-containing TNS samples (Figure 2b), exhibit similar grain morphology, but in addition, a high density of white particles (sizes approximately 1 -4 µm) are uniformly distributed within the matrix. These particles are predominantly located at the grain boundaries (Figure 2b), especially at triple grain junctions (Figure 2c). Voids (dark in color and marked by blue circles) and cracks (marked by red arrows in Figure 2b) can be observed inside the grains. The amount of porosity increases slightly with increasing Nb content, although less evident than in the TNO samples, as a result of the higher levels of densification (Figure S3, Table S1).

The crystal structure and compositions of the white particles in the TNS samples were investigated simultaneously by SEM EBSD (Figure 2d-g Low magnification (LM) SE and BSE images of TNS samples (Figure S3) show that Nb doping has minimal impact on the grain morphology: all TNS samples exhibit similar polygonal grain structure and the presence of a high density of white inclusions located at grain boundaries. EBSD band contrast and phase maps for TNS samples (Figure S4) and the band contrast map for TNO samples (Figure S5) show the grain sizes and their distribution. The average grain sizes, determined by EBSD grain size analysis, do not change in any consistent way with increasing Nb doping level (Table S3). However, compared to the TNO samples (~3.96 µm), the average grain sizes of TNS samples are slightly larger (~4.89 µm) (Table S3). On the basis of the Sn-O phase diagram [START_REF] Okamoto | O-Sn (Oxygen-Tin)[END_REF][START_REF] Cahen | Thermodynamic modelling of the O-Sn system[END_REF], metallic β-Sn transforms to liquid Sn at 504.9 K. As the highest temperature of the sintering process is 1473 K, the resulting liquid Sn will promote liquid phase sintering [START_REF] German | Review: liquid phase sintering[END_REF] and assist densification. The fact that the grain sizes in the TNS samples are only marginally larger than those in the TNO samples, in spite of liquid phase sintering, suggests that the high concentration of voids helps to inhibit grain growth. that these nano-twins belong to the (01-1) twins (Figure 3g). The blue and red circles highlight the diffraction spots for both twin domains, while the green circles denote the twin plane. The diffraction patterns match well with the simulated (011) twin SAED patterns (Figure 3g), and hence confirm the presence of (01-1) twin structures. HRTEM images (Figure 3h and 3i) show the atomic structure of (01-1) twin boundaries (denoted by the yellow and green dash boxes in

Figure 3f

). The cation columns in both twin domains have mirror symmetry to the (01-1) twin boundaries (the twin boundaries are highlighted by the yellow dash line with arrows at each end).

STEM EDX maps reveal that the constitutional elements are uniformly distributed at twin boundaries (Figure S6), indicating that there is no elemental segregation during the formation of these twin boundaries. Therefore, these TBs belong to coherent twin boundaries (CTBs) due to the uniformity and high symmetry of the structure. The measured rotation angle between the neighbouring cation columns is approximately 136°, which is in good agreement with the theoretical value (θ = 134.97°) and the reported values (θ ≈ 137°) [START_REF] Hrkac | Atomic structure and crystallography of joints in SnO2 nanowire networks[END_REF][START_REF] Zheng | Growth twins in nanocrystalline SnO2 thin films by high-resolution transmission electron microscopy[END_REF]. In addition, the HRTEM image of the region denoted by the red dash box in Figure 3f presents multiple defect structures (Figure 3j), which are located between two twin domains with widths of approximately 40 nm.

The corresponding FFT patterns confirm that these defects are typical multiple stacking faults (SFs) (Figure 3k); the presence of SFs might be due to the high degree of lattice distortion caused by higher levels of Nb doping or high concentration of oxygen vacancies [START_REF] Li | Nanoscale stacking fault-assisted room temperature plasticity in flashsintered TiO2[END_REF]. From the low magnification BF STEM image of a TNS60 sample (Figure S7b), nano-regions with darker contrast can be seen within the β-Sn matrix. In the presence of a strong reducing atmosphere, most of the SnO2 is transformed into β-Sn.

However, small amounts of nano-sized SnO2 are not sufficiently reduced, and mix with β-Sn phases leading to the formation of complex NRs. Since SnO2 has a larger lattice spacing than metallic Sn, the Sn/O intensity ratio is lower at the NR for EDX maps, which corresponds well with the FFT results. Atomically-resolved HAADF and BF STEM images of the interface between β-Sn and NR are shown in Figure 5d and 5e. The atomic arrangement of the β-Sn region is the same as in the previous STEM analysis (Figure 4g and4h). However, there exists a complex and highly distorted atomic arrangement in the NR area. A high density of nano-twin-like arrays can be observed in the BF STEM image. The highly distorted lattice and high-density of nano-sized defects in the NR structure could be thermoelectrically-beneficial by enhancing phonon scattering, which could effectively reduce thermal conductivity.

Element binding states

XPS survey scans confirm the presence of all elemental components in TNS samples (Figure S12). The high-resolution scans (Figure 7) show the Ti 2p transition and Nb 3d transitions for the TNS samples. The doublet peaks (Ti 2p3/2 peak/Ti 2p1/2 peak and Nb 3d5/2 peak/Nb 3d3/2 peak) in Ti 2p and Nb 3d transitions (Figure7 a, c, e) reveal the spin-orbital splitting of approximately 5.6 eV and 2.6 eV, respectively. Notable shoulders can be observed for all peaks, which is ascribed to the presence of cations (Ti and Nb) with different valence states. Peak fitting results indicate the presence of Ti 4+ /Ti 3+ and Nb 5+ /Nb 4+ for all TNS samples and the content of Ti 3+ and Nb 4+

increases with increasing Nb content (Table 3), in good agreement with the previous investigations [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF]. However, compared with studies of similar compositions [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF], the much lower Ti 3+ and Nb 4+ content (~ 4% -7% for Ti 3+ and ~ 10% -12% for Nb 4+ ) in the TNS samples is attributed to the lower sintering temperature and time under reducing conditions. In addition, the content of lowoxidation state Nb ions (~ 10% -12% for Nb 4+ ) is higher than that of the low-oxidation state Ti ions (~ 4% -7% for Ti 3+ ), indicating that free electrons are more likely to be trapped at Nb cation sites to form lower state Nb ions.

Thermoelectric properties.

The electrical transport properties of the TNO and TNS samples are presented in Figure 8;

values of individual parameters at room temperature are listed in Table 4. The electrical conductivity (σ) of all samples increases with increasing temperature (Figure 8a). In contrast with TNO samples, the σ value of TNS samples are significantly higher by up to ~ 40% at 873K, indicating that the reduction of SnO2 particles, forming β-Sn structured secondary phases, is highly beneficial to enhance σ. The pure β-Sn metallic phase exhibits a very high σ value of approximately 9.17 ×10 4 S cm -1 [START_REF] Plevachuk | Electrical conductivity, thermoelectric power and viscosity of liquid Sn-based alloys[END_REF], which is orders of magnitude higher than that of rutile-based ceramics [START_REF] Verchère | Optimum in the thermoelectric efficiency of nanostructured Nb-doped TiO2 ceramics: from polarons to Nb-Nb dimers[END_REF][START_REF] Yuan | Thermoelectric properties of Nb-doped TiO2δ ceramics reduced at elevated temperature[END_REF][START_REF] Zaitsev | Preparation and characterization of nanocrystalline doped TiO2[END_REF]. Therefore, the uniformly dispersed, highly conductive inclusions strongly enhance σ.

In addition, compared with earlier studies of strongly reduced TNO samples with same Nb doping content [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF], the σ value of the current TNS samples is also significantly improved by approximately 35%. At room temperature (298 K), the σ values (Table 4) of TNO samples decrease with increasing Nb content, which is similar to the σ trend reported by Liu et al [START_REF] Liu | Enhanced asymmetrical transport of carriers induced by local structural distortion in chemically tuned titania: A possible mechanism for enhancing thermoelectric properties[END_REF]. In contrast to the TNO samples, the electrical conductivity of TNS samples at room temperature shows only slight variation (Table 4). As the β-Sn phase provides a high density of carriers, the conduction activation energies (Table S4) for TNS samples (represented by purple bars in Figure 8b) are much lower than for TNO samples; consequently, the carrier concentration in TNS samples is typically double that in TNO samples (

𝑆 = % " & + ' ! ( ! ) *+ ' , -. /(1)
,1 (

134 # Equation (1)
where S is the Seebeck coefficient, 𝐶 + is the electron specific heat, n is the carrier concentration, k is the Boltzmann constant, T is the absolute temperature, e is the electron charge and 𝜇(𝜀) is the energy correlated carrier mobility. In general, the first term of the formula is dominant, and thus Equation 1 can be simplified to 𝑆 ≈

% "
& . At elevated temperature, as carrier concentration can be significantly enhanced by Nb doping, any increase in Nb content helps to reduce the S values. In addition, the TNS samples exhibit much lower absolute S values (approximately 10% -20%) than the TNO samples. This can be ascribed to the formation of metallic β-Sn structured inclusions that significantly increase the carrier concentration; consistent with Equation 1.

It is found that the absolute S values for TNS60 samples are marginally higher than those of TN75 samples, although the carrier concentration of TNS60 samples is higher than that for TN75.

This suggests the contribution from the carrier scattering (described by the second term of Equation 1) is higher in TNS60 samples. The uniformly distributed β-Sn structured metallic inclusions generate a high density of interfaces between the rutile matrix and the β-Sn cores, which can act as carrier scattering walls. In addition, the widely distributed nano-regions (NRs) with pronounced lattice distortion can also enhance carrier scattering. Furthermore, the high density of twin boundaries, stacking faults and slip bands inside the rutile matrix (Figure S13) can produce potential barriers; high-energy carriers will be able to cross the barriers, whilst the low-energy carriers are strongly scattered, thereby enhancing Seebeck coefficients. The Seebeck coefficients for TNS samples are approximately equal to those of the strongly reduced TNO samples [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF], which further confirms the enhancement in carrier scattering.

The power factor for TNS samples increases with increasing Nb content and temperature (Figure 8d). Due to the significantly improved electrical conductivity and the modified Seebeck coefficients, the power factor values for TNS samples are strongly enhanced compared to those for TNO samples. TNS60 exhibits the highest power factor value of approximately 670 µW m -1

K -2 at 873 K, showing approximately 27% enhancement compared to the TNO samples with same base composition, and about ~30% higher than for strongly reduced TNO samples with the same Nb content [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF]. Figure S14 show the reproducibility of the power factor data. These high power factor values confirm the positive effects of SnO2 addition on thermoelectric transport properties of rutile-based ceramics.

The total thermal conductivity (κ) of the samples (Figure 9a) varies from ~3 to ~4.5 W m -1 K -1

for TNS and ~ 2.6 to ~3.4 W m -1 K -1 for TNO, significantly lower than published data for single crystal rutile (measured by the comparative method) [START_REF] Touloukian | Thermophysical properties of matter -The TPRC data series[END_REF] (~6 to ~8 W m -1 K -1 ) or polycrystalline rutile (measured by the laser flash method) [START_REF] Nam | A gigantically increased ratio of electrical to thermal conductivity and synergistically enhanced thermoelectric properties in interface-controlled TiO2-RGO nanocomposites[END_REF] (~4 to ~5 W m -1 K -1 ). This supports the concept that Nb doping helps to reduce the thermal conductivity, although the total thermal conductivity is increased by approximately 10% to 30% through the addition of SnO2.

The lattice thermal conductively (κL) (Figure 9b) and electronic thermal conductivity (κE) (Figure 9b) for TNO and TNS samples were calculated using Wiedemann-Franz approximation (𝜅 = 𝜅 5 + 𝜅 4 = 𝜅 5 + 𝐿𝜎𝑇), where L is the Lorenz number (estimated using the single parabolic band model [START_REF] Chang | Enhanced thermoelectric properties of Bi2S3 polycrystals through an electroless nickel plating process[END_REF]), σ is the electrical conductivity and T is the absolute temperature. The higher κE values for TNS samples compared to TNO samples, are explained by the larger electrical conductivity in TNS samples, due to the higher carrier concentration and the presence of metallic inclusions. The highest κE value represents approximately 8% of the κ values, which indicates that κL dominates.

In the high density TNS samples, point defects (Nb substitution and oxygen vacancies) and planar defects (twins and slip bands) contribute to lowering the κL (Figure S13). However, Nb doping can effectively suppress the formation of oxygen vacancies, and thus the lowest κL for TNS60 might originate from the balanced concentration of point defects (Nb doping and oxygen vacancies). In contrast, in TNO samples, as a result of the lower density, the large number of pores in TN75 samples might be dominant in reducing the κL. The significantly improved power factor enabled by both the metallic β-Sn structured inclusions and Nb doping compensates for the slightly increased thermal conductivity, and thereby the effectively enhances thermoelectric figure of merit ZT values for TNS samples with higher Nb content (Figure 10). For example, ZT values for TNS60 samples are approximately 20% higher than for TN60 samples, with a maximum ZT value of 0.20 for TNS60 at 873 K. This peak ZT value also shows ~18% improvement over previously reported ZT for strongly-reduced, Nb-doped TiO2 with the same composition [START_REF] Liu | Controlling the Thermoelectric Properties of Nb-Doped TiO2 Ceramics through Engineering Defect Structures[END_REF]. Our investigations provide a competitive approach to enhancing the thermoelectric performance of oxide-based TE materials with advantages of low sintering temperature, easy processing and eco-friendly raw materials. This investigation presents a simple and eco-friendly method to enhance the transport properties of oxide-based ceramics and highlights the important detailed relationships between atomic structures and enhanced thermoelectric performance. The strategy employed provides a new approach to grain boundary engineering and could offer solutions to enhance the thermoelectric performance of other materials through incorporation of metallic grain boundary inclusions.

Figure 1 .

 1 Figure 1. (a) XRD patterns for the TNS and TNO samples; (b) Enlarged view of the region denoted

  ) and EDX (Figure2h-k). The forward scatter detector (FSD) mixed scanning area image shows a single particle located at triple grain junction (Figure2d); notably, parallel slip bands (marked by green arrows in Figure2d) are revealed inside neighbouring rutile grains. Due to the limited resolution, the accurate orientation of these linear feature is less evident in the EBSD IPF image (Figure2f). The EBSD phase map (Figure2g) confirms the β-Sn structure for the white particles at grain boundaries (denoted by the yellow colour) and rutile structure for the neighbouring matrix grains (denoted by the blue colour); the corresponding EDX maps (Figure2h -2k) also indicate that the particles are enriched in Sn but depleted in Ti, Nb and O. Therefore, the numerous white particles are identified as β-Snstructured inclusions, consistent with the XRD phase analysis.

Figure 2 .

 2 Figure 2. SEM backscattered electron images of the TN60 and TNS60 samples ((a) TN60, (b)

Figure 3 .

 3 Figure 3. TEM analysis of the rutile matrix in TNS60 samples. (a) (b) (c) (d) SAED patterns and

Figure 4 .

 4 Figure 4. (a) (b) (c) (d) SAED patterns of the beta-Tin phase along [001], [111], [-133] and [113]

  Figure5ashows a BF STEM image of the high concentration of nano-regions (NRs) with darker contrast; these NRs are rhombus or irregular polygonal in shape, with lengths and widths of approximately 10 -50 nm. A high magnification BF STEM image (Figure5b) presents the morphology of a single NR along the [111]T (T represents the β-Tin phase) zone axis; the corresponding FFT patterns reveal several sets of complex diffraction patterns (Figure5c). The dominant diffraction spots marked by white arrows can be indexed by [111]T β-Sn structure, while the red arrows highlight the (very low intensity) diffraction spots that belong to the [001]R (R represents the rutile) rutile structure. Apart from these two groups, there are still several diffraction spots that cannot be indexed on the basis of existing crystal structures, including all Sn -O oxides. It is inferred that these are superlattice spots resulting from the formation of a complex Sn -O superlattice. The STEM EDX maps of a single nano-region (FigureS11) show that the Sn content is slightly lower than in the surrounding regions; this can be attributed to the presence of the rutile-structured SnO2 added to the samples.

Figure 5 .

 5 Figure 5. STEM analysis of nano-regions in the beta-Tin phase. (a) BF STEM image shows the

Figure 6a (

 6a Figure 6a (HRTEM image) shows the interface between the matrix rutile and a β-Sn structured

Figure 6 .

 6 Figure 6. TEM analysis of the interface between the rutile and β-Sn phase in TNS60 samples. (a)

Figure 7 .

 7 Figure 7. High resolution XPS spectra for the Ti 2p and Nb 3d transitions of TNS samples: (a) (b)

Figure 8 .

 8 Figure 8. Electrical transport properties of TNO and TNS samples as a function of temperature.

Figure 9 .

 9 Figure 9. Thermal conductivity as a function of temperature for TNO and TNS samples: (a) total

Figure 10 .

 10 Figure 10. Dimensionless figure of merit (ZT) as a function of temperature.

Table 1 .

 1 Sample formulations and codes.

Table 2 .

 2 Lattice parameters for the rutile phase in TNO and TNS samples

	Codes	rutile	TN45	TN60	TN75	TNS45	TNS60	TNS75

a, b (Å) 4.5937 4.6150(1) 4.6226(1) 4.6315(3) 4.6128(0) 4.6216(2) 4.6279(2) c (Å) 2.9581 2.9721(0) 2.9760(1) 2.9798(2) 2.9697(0) 2.9740(0) 2.9773(1)

Table 3 .

 3 Component analysis of the XPS spectra for the TNS samples

	Sample	Ti2p3/2%		Nb3d5/2%
	Code	Ti 4+	Ti 3+	Nb 5+	Nb 4+
	TNS45	95.7	4.3	89.9	10.1
	TNS60	94.7	5.3	89.8	10.2
	TNS75	93.0	7.0	88.4	11.6

Table 4 )

 4 . The similar n values for TNS45 and TNS60 samples suggests a balance between localized carriers and carriers provided by β-Sn phase. As expected, the electrical conductivity of all samples increase with temperature and with Nb content. It is noted that changes in carrier concentrations and mobilities are not strictly monotonic with Nb doping; this may be a result of multiple counteracting mechanisms operating in the samples: (i) the high-density of

	defects such as structural distortions, oxygen interstitials and nano-sized inclusions, are believed
	to not only scatter phonons, but may also scatter electrons, thus affecting mobility in a non-
	monotonic way; (ii) the carrier concentrations are determined not only by Nb doping content, but
	also by (uncontrolled) oxygen stoichiometry.

Table 4 .

 4 Electrical conductivity (σ), Seebeck coefficient (S), carrier concentration (n) and carrier mobility (μ) of the TNO and TNS samples at room temperature.

	Sample	TN45	TN60	TN75	TNS45	TNS60	TNS75
	σ (S cm -1 )	33.86	23.57	16.50	50.21	50.95	51.88
	S (μV K -1 )	-335.27 -321.71	-350	-286.43 -308.59 -252.63
	n (10 20 cm -3 )	2.18	3.84	2.17	8.27	7.94	12.4
	μ (cm 2 V -1 s -1 )	0.971	0.384	0.475	0.379	0.401	0.261
	The negative Seebeck coefficients (S) of the TNO and TNS samples (Figure 8c) confirm n-type
	semiconducting behavior. It is noted that the absolute S values decrease with increasing Nb content
	for both the TNO and TNS samples; this can be explained by the Mott formula [55] (Equation 1):

ACKNOWLEDGMENTS

The authors are grateful to the EPSRC for the provision of funding for this work (EP/H043462, EP/I036230/1, EP/L014068/1, and EP/L017695/1 acknowledged by R.F.). The work was also supported by the Henry Royce Institute for Advanced Materials, funded through EPSRC Grants EP/R00661X/1, EP/S019367/1, EP/P025021/1, and EP/P025498/1. We gratefully acknowledge the support from the Electron Microscopy Centre in Faculty of Science and Engineering, the Xray facilities in the Department of Materials in the University of Manchester. X.L thanks China

DECLARATION OF COMPETING INTEREST

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Scholarship Council for their financial support during his Ph.D. program. All research data supporting this work are directly available within this publication.

Appendix A. SUPPORTING INFORMATION

The following supporting information is available.

Density data for TNO and TNS bulk samples; Lattice parameters and Rietveld refinements for TNS samples; Average grain size of all bulk samples; SEM SE and BSE images of the TNO and TNS samples; SEM EBSD band contrast and phases maps of the TNO and TNS samples; STEM EDX maps of the twin structure in rutile phases; STEM EDX maps of the tin structure; Simulated STEM images of the tin structure; EDX spectrum of the tin structure; EELS maps of the interface between rutile and tin; STEM EDX analysis of the nano region in tin phases; XPS survey spectra of the TNS samples; Schematic map of TNS samples; Reproducibility of the power factor data.