Laurent Ughetto*' Didier Dubois

Henri Prade
email: prade@irit.fr

Fuzzy interpolation by convex completion of sparse rule bases

This paper proposes an approach to the interpolation be tween sparse fDZzy rules, founded on a unique prlnclple which supple ments the classical approximate reasoning machinery, The case of rules of the form A ➔ B, where A and B are lntervals 1s flrst dlscusse and then extended to fuzzy sets.

INTRODUCTION

Fuzzy rule-based systems have been either used as a con venient tool for synthesizing control laws from data, or in a knowledge representation and reasoning perspective in AI. Beside the natural way of representing imprecise and graduai pieces of knowledge that fuzzy rules provide, both applica tions take advantage of the partial overlapping of their con dition parts in the inference mechanism. Indeed, when the input demain is covered by the premises of the rules, well founded and established reasoning mechanisms associated to fuzzy rule-based systems have been proposed to generalize the Modus Ponens, as the so-called Compositional Rule of Infer ençe (CRI) introduced by Zadeh in 1973 [1]. By contrast, in case of sparse rules, a special interpolative reasoning mecha nism is needed.

Interpolation relies on the assumption of graduai and smooth variation of a variable with respect to another one. This is related to the idea of distance which underlies the natu ral way human beings fill the gap between extreme rules when they have to cope with an input "Xis A*" which does not fit any of the premises "X is A.'' of available sparse rules. Inter polative reasoning naturally subsumes some idea of similarity, where at least two rules of the form "if X is Aï then Y is B," must be used simultaneously. Indeed the result of an interpola tion depends on the reference values provided by the rules that are the closest to the input value "Xis A*". K6czy and Hirota [START_REF] K6czy | Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases[END_REF], [START_REF] K6czy | Ordering, distance and closeness of fuzzy sets[END_REF] have proposed some empirical interpolative methods for reasoning from sparse rules, i.e., rules where the Aj's do not achieve a coverage of the input universe U, but are rather disjoint and even distant from one another. The problem is then to compute some plausible conclusion "Y is B*" for an input "Xis A*", with B* in between some B, and B;,just as A* is in between the corresponding Ai and A;, using distances between fuzzy sets. Working on the a-cuts (A)"' = (9., a) of fuzzy sets A, a natural idea is to compute the a-eut of B* as [. �*, t'J = (.X•k; +(1-À) •ll.;, N •b; + (1-.X') •b;] on the basis of the solutions À and N of the equations g,_* = À•!J.i + (1-.X) •9.; and ii* = .X' • a; + (1 -.X') • ii.; where (g*, ii*] is the a-eut of A*. Unfortunately, the a-cuts of such a B* may not be nested and we may even have fl.* > b. See Yan et al. [START_REF] Yan | Reasoning conditions on K6czy's interpolative reasoning method in sparse fuzzy rule bases[END_REF], Qiao et al. [START_REF] Qiao | An improvement to K6czy and Hirota's interpolative reasoning in sparse fuzzy rule bases[END_REF], Hsiao et al. [START_REF] Hsiao | A new interpolative reasoning method in sparse rule-based systems[END_REF] for further developments along this line.

This interpolation problem can also be formalized as the computation of a convex sum of B i and B; based on a parame ter À imprecisely constrained by uo = .À •u1 + (l-.X)• u2, with 'Uo restricted by A*, u1 by A, and u 2 by A; [START_REF] Dubois | Fuzzy logic, control engineering and artifi cial intelligence[END_REF], [START_REF] Dubois | On fuzzy interpolative reasoning[END_REF]. However, while the above-mentioned drawback is avoided, the result of the inference may be very imprecise.

A general method bas been proposed by Bouchon-Meunier et al. [START_REF] Bouchon-Meunier | Analogy and fuzzy interpolation in case of sparse rules[END_REF], [START_REF] Bouchon-Meunier | Raisonnement interpolatif à partir de schéma analogi que flou[END_REF] on the basis of the concepts of location and shape of a fuzzy set and the evaluation of the distinguisha bility of two fuzzy sets with regard to these concepts and in agreement with the analogical scheme proposed in [START_REF] Bouchon-Meunier | Severa! forms of fuzzy analogical reasoning[END_REF]. More precisely, if A* is between A; and Ai+ 1 , with respect to some order on fuzzy sets, a conclusion B* regarding variable Y is constructed between B, and Bi+ l , in such a way that the lo cation and shape distinguishabilities of B* and B;, of B* and B;+i, are analogous to the location and shape distinguishabil ities of A* and A,, of A* and A,+ 1 . However, several degrees of freedom remain in this rather general scheme.

In this paper, the proposed approach relies on a unique lin ear interpolation principle which, once accepted, fully deter mines the result of the inference. The underlying idea of the approach, first briefty suggested at the end of [START_REF] Dubois | Practical computing in fuzzy logic[END_REF], and closely related to some mechanism first partially described in [START_REF] Buisson | un générateur de systèmes-experts qui prend en compte l'imprécision et l'incertitude des connaissances[END_REF] is the following. In classical reasoning, the two rules

(A n A') ➔ (B n B') and (A u A') ➔ (B u B') are di rect consequences of the set of rules { A ➔ B, A' ➔ B'}.
The basic principle of the proposed approach is to augment this machinery on the real line, by assuming that it is also al lowed to derive from the two rules any new rule A À ➔ B>., where À E (0, 1), and where AÀ = À

• A + (1 -.X) • A' and BÀ = À• B + (1 -.X) • B', obtained by interval arithmetics.
Considering two sparse rules A; ➔ B; and A;+ 1 ➔ B;+1, and an observation A* in between A; and A i + 1 , the method consists in computing a union or an intersection of several ,4 >. which best fit A*. Then, the union or intersection of the cor responding BÀ gives the corresponding output B*, according to a fuzzy linear interpolation scheme between the two initial rules, as it will be shown.

In Section II, this paper first studies linear interpolative rea soning between two non fuzzy rules, i.e., rules of the form A ➔ B, where A and B are intervals. The case of one in put and one output rules is first considered, followed by a brief discussion of interpolation with multiple input rules. Then, in Section m the results are extended to fuzzy rules, by approx imating A* with a union or an intersection of A À s (and then B* with BÀs).

Il. CASE OF NON-FUZZY RULES

This section deals with interpolative reasoning between the Iwo non-fuzzy sparse rules of the set K = { Ao ➔ Bo, A1 ➔ B 1 } . The operator ➔ is then the classical logic implication. The rules in K being sparse (or disjoint) means that A 0 nA 1 = 0. In this section, as only intervals (instead of fuzzy sets) are considered, an interval A is denoted [!!,a]. Now, consider a piece of information A* (e.g., an observa tion) on the input variable X. The set A* is also supposed to be an interval, which lies in between Ao and A 1 , on the in put universe U. It encompasses the case where A* somewhat overlaps with A 0 and/or A 1 , but still being in between (i.e., !!o $ g,* $ a* $ a 1 ), as the interpolation mechanism also has influence on A 0 and A 1 , as shown on Figure 5.c.

The problem is then to determine an interval B* in between Bo and B1 (which restricts the possible values for the variable Y, on the output universe V), such that A* ➔ B* is a valid rule w.r.t. A 0 ➔ Bo and A 1 ➔ B 1 according to the principle proposed in the introduction.

For the sake of simplicity, the case of one input and one output rules is first detailed. Moreover, in all this section, on ly the increasing case is considered, i.e., the case where A 1 is "greater" than A 0 , and B 1 is "greater" than Bo (see for in stance Figure 1). Ali the results obtained in this case can be easily carried over to the decreasing case.

A. One input, one output rules

In this first case, the premises of the two rules in K, namely Thus, given that A ,\ ➔ BÀ are the rules obtained from K by linear interpolation, the idea is to compute a union or an intersection of A. À s that corresponds to ( or best fils) . ..\ESmax ÀESma.x

Clearly, only the last two cases need to be detailed. Proposition 1: These three cases are mutually exclusive. Proof: As previously said, each interval A is denoted by its lower and upper bounds [ !!,a]. The values l!. and X given by:

• g,* = l!.•!! 1 +(1-l!.)•g, 0 ,and then : l!.= ( g, *-Q:o ) /(g,1 -Q:o),
• a* = X-a1 + (1-°X) •ao, and then: X= (a* -ao)/(a1 -ao), are such that the left side of A ,1 matches the left side of A*, and the right side of A 5: matches the right side of A*. Then, from the values l!. and X, three cases can occur (see Figure 2):

• l!. = X corresponds to the first case, as A ,1 = A x = A*, • l!. < X means that A ,1 CA* and A x CA* (case number 2), • !l > X means that A ,1 ::) A* and A x ::) A* (case number 3), and these three cases are obviously exclusive. As a consequence, when .C is a potentially incoherent set of rules, intersections of A>.s (i.e., case 3) shouldn't be used. A more cautious approach could even consider that the interpola tive inference method proposed here may not be used when .C is not coherent. Indeed, il will be shown in the following that this method is equivalent to the CRI applied to C only when .C is coherent. Thus, for incoherent cases, a slightly different method will be proposed.

B. Case of A*s bigger than A>.s

First consider the case of a large A*, i.e., an interval such that A* :> A>. for some À E (0, l]. Proposition 3: Given A* such that A* :> A >. , for one À E [0, 1], the set Sm ax = {.,\, A>. :::> A*} is the maximal (for inclusion) subset of (0, lJ which verifies (1). This set Smax is the interval (1, À], where 1 = (g_* -[4))/(g_ 1 -fJ-0) and À = ( a * -ao)/ ( a1ao)-Proof: Clearly, ,,:\ (resp. À) is the smallest (resp. the greatest) A such that A>. :) A*. Then, il can be easily shown that VA E

■

As in section II-B, the set Smin is infinite, and is not very useful for the computation of B*. But in case of A*s smaller than A>.s, only two values in Sare necessary to obtain A* .

Indeed, A* = A� n A x , as shown on Figure 2.

Proposition 6:

The set S m i n = {X,::\} is the minimal (for in clusion) set S which verifies [START_REF] K6czy | Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases[END_REF]. Proof: It is clear that S m in verifies [START_REF] K6czy | Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases[END_REF]. The set S min is min imal since the suppression of one of its elements leads to a set which clearly do not verify [START_REF] K6czy | Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases[END_REF]. And it is the minimal set since the replacement of one element in Smi n also leads to a set which do not verify (2), as clear from Figure 2.

■

Here, the set B* can be computed as An equivalent way to compute B* is given by the counterpart of Proposition 4 for A*s smaller than A.\s-Proposition 7: Given the two sparse rules in /C, and an obser vation A* in between A 0 and A 1 , such that A* CA>., for one À E [0, l], the corresponding output B* is given, when it ex ists, by [!t, V], with l2.* = Q� and V = bx, or equivalently:

B * _ [ Q1 -l2.o a * �hQo -[4JQ1 ÏÏ1 -ÎÏo a" a1bo -aob1 ] - -+ -=--"----::......, - - + - - . !!1 -fJ-0 Q.1 -fJ-0 a1 -ao a1 -ao Proof: It cornes straightforwardly from the computation of the intersection B x n B ô. == [h, bx), since ,,:\ > X ■
Clearly, the expression of B* IS the same in Propositions 4 and 7. However, in this case, the interval B* does not always exist.

D. Detection of an incoherent situation

The two rules A� ➔ B ô. and A x ➔ B-5:. are said to be incoherent when A* = A� n A x # 0 and B* == BA n B r = 0.

These cases of incoherence depend on properties on the an tecedent (Ao and A1) and conclusion parts (Bo and B1) of the two rules in K,. Thus, it is possible to deterrnine a necessary and sufficient coherence condition for the set I:,.

■

Note that when the two lines [ and r intersect, [, is poten tially incoherent on a subset of [ao, g,_ 1 ] only. As a corollary of Proposition 8, one can compute the interval of potential inco herence in this case, which corresponds to the subinterval of [ ao , g,_ 1 ] for which [ is above r.

Beside the calculations, it is important to understand the meaning of potential incoherence for sparse rules. Indeed, Potential incoherence Coherence Fig. 3. Coherence and potential incoherence cases considering the CRI (as a generalization of the Modus Ponens) for the reasoning mechanism, it has been shown that sparse rules cannot lead to incoherence (see [START_REF] Dubois | Checking the coherence and redundancy of fuzzy knowledge bases[END_REF]). Strictly speaking, K, is never incoherent. Only C may be incoherent. However, when C is incoherent, K, can be said incoherent w.r.t. the in terpolative inference mechanism proposed here, according to a generalization of the notion of coherence for rule-based sys tems: "for any allowed input, the system must give a nonempty output".

From the equations, it can be seen that potential incoherence may occur either when the Es are sparser than the As, or when the Es are more precise than the As (or a combination of these two effects since they can compensate). The Es being sparser than the As means that they are divergent, as the conclusion parts are Jess similar than the condition parts of the rules. It seems a natural extension of the notion of incoherence from a similarity point of view ( see Figure 4). Thus, K, can be said po tentially incoherent if two rules in C are potentially incoherent in the sense of [START_REF] Dubois | Checking the coherence and redundancy of fuzzy knowledge bases[END_REF], as shown on with a well-chosen interval A* could lead to a singleton for E*. In this particular case, obtaining a singleton, i.e., a pre cise value for B*, from anîmprecise observation A* and two (even more) imprecise rules seems hardly acceptable. In such a case, if A* would be made still more precise, B* becomes empty. By contrast, it can be shown that B*s obtained in the coherent case are always more imprecise than Ùle correspond ing A*. This is why this method, and especially the intersec tion of ru les shouldn' t be used in the incoherent case.

E. Comparisons and an inference algorithm based on the CRI

It is interesting to compare this interpolative reasoning method with previously proposed ones, and especially the two ones in [START_REF] Dubois | Graduai rules and the approx imation of control laws[END_REF], shown on Figure 5.a-b. These two methods are similar to the one proposed in this paper (for coherence cases) since they are all equivalent to the definition of a relation on the Cartesian product of the input output universes U x V in between the rules, and the application of the CRI as the rea soning mechanism. From a semantic point of view, the first method considers the pair (A, B) (from the rule A ➔ B), as a fuzzy point and applies the extension principle (see [START_REF] Dubois | Graduai rules and the approx imation of control laws[END_REF], [START_REF] Dubois | On fuzzy interpolative reasoning[END_REF]). It means that the relation is the set of all segments of the form [m 0 , mi), where m 0 is a point in Ao x Bo, and m 1 a point in A 1 x B 1 . The obtained relation is the convex hull of ( A 0 x B 0 ) U ( A 1 x B 1 ) .

The second method considers the pair (A, B) as a part of a fuzzy fonction from U to V. Thus, for a point In the incoherent case, the CRI cannot be applied, since the relation represented by the set of rules ,C on U x V is empty.

(u, v) E U x V, if u E A,
However, if the entire set ,C is incoherent in this case, consid ering only one rule from ,C still makes sense, as well as consid ering one rule, obtained by union of rules in .C. Indeed, a rule A >. ➔ B >. , for one>. E [O, 1) such that A>. overlap neither A 0 , nor A 1 is always coherent w.r.t. K.. And it is clear that the case 1 of Section Il-A (for which 3>. E [O, 1) such that A* = A>.) corresponds to the CRI with A* and such a rule. Similarly, a rule U >.E S A>. ➔ u >. E S B>. such that U>.E S A>. overlap nei ther A 0 , nor A 1 is always coherent w.r.t. JC. And it can be eas ily shown that the case 2 of Section II-A (for which :lÀ E [0, 1) such that A* 2 A>.) corresponds to the CRI with A* and such a rule. Thus it is sound to use the proposed method for these two cases, while it is not for the third one.

To summarize, the proposed method is equivalent to the CRI applied to ,C in the coherent case (as previously said). In the incoherent case, the method is sound (and still corresponds to the CRI, but only with one well-chosen rule instead of the set C) only with input sets A* greater or equal to one A>.. Smaller A*s should then be approximated by a well-chosen A>.. A good choice can be the A>. centered on A*.

F. Rules with several inputs

This section briefly deals with the case of rules with sever al inputs, i.e., the case of a multi-dimensional input universe U = U 1 x ... x U p . Two main approaches can be considered, depending on the number of mies available for the interpola tion process.

First consider that the set of mies JC still contains two mles. The generalization of the method proposed in the previous sec tions consists in representing A* by means of Â>.S, and then computing B* using the >.s. It cornes down to the computa tion, for each dimension i, of the conclusion Bt obtained from the two mles with a mono-dirnensional input Ao , i ➔ Bo and A 1 i ➔ B 1 . And then to the computation of the convex union of the Bt s. ln fact, it is acceptable when the (imprecise) point A* is on the (imprecise) line defined by the two (imprecise) points A 0 and A 1 in the universe U, but in the general case it gives very irnprecise results. This is why a better method consists in considering more rules (if available) to define an imprecise hyper-plane of U x V (as well as the imprecise line defined by [.[(u}, Ï'(u)] depicted on Figure 5 for one input). If U has p dimensions (and V one) it requires p + 1 mies whose condition parts A; are "around" A.*. This method resembles the one described at the end of [START_REF] Dubois | Fuzzy logic, control engineering and artifi cial intelligence[END_REF] and in [START_REF] Dubois | On fuzzy interpolative reasoning[END_REF]. These two methods are not developed here for the sake of brevity. Note however that they involve some non trivial problerns of classical interpolation in multidimensional spaces.

III. Fuzzy CASE

In this section, the sets involved in the considered rules may be fuzzy instead of just intervals. The considered fuzzy sets are trapczoïdal-shaped, not only because they arc simple to deal with, but rather because they are the most commonly used in practical applications. The extension to more general fuzzy sets is left for further research.

In order to generalize the proposed interpolation method to fuzzy sets, let us first study some properties of the union and intersection of the corresponding A >. s and the B >,. s. In this case, A>. is obtained from A 0 and A 1 , applying the Jin ear interpolation mechanism of the previous section on each of its a-cuts (which are supposed to be intervals): (A >. ) <> = À• (A 1 ) <> + (1-À) • (Ao) a , where (A) a is the a-eut of A. The following properties are obvious.

• If the fuzzy sets A 0 and A 1 are trapezoïdal-shaped, then the fuzzy sets A>., for À E [O, 1] are also trapezoïdal-shaped.

Then, it can be shown th�t this property is also true for the union or intersections of A>.s considered by the interpolation mechanism.

• If the fuzzy sets A 0 and A 1 are trapezoïdal-shaped, then the union U ÀEI A>., where I Ç [O, 1] is an interval, is also trape zoïdal-shaped.

The requirement for I to be an interval is not always nec essary. The union of the cores of the A>., for >. E I being an interval is sufficient for the union to be trapezoïdal-shaped.

However, as in 11-B, the set I = S max which is used in the following is an interval.

• If the fuzzy sets Ao and A 1 are trapezoïdal-shaped, then the intersection n >.EI A>., for any I Ç [O, l], is also trape zoïdal-shaped, which may be degenerated into a triangle, a sub-normalized triangle, or the empty set.

Although these results give interesting properties on the shapes of unions or intersections of Â>.S, it is rather clear that, in the general case, a trapezoïdal-shaped input A* cannot be recovered exactly by such union or intersections. It would re quire very particuiar properties on the si opes of A*.

To generalize the interpolation method to the fuzzy case, the next section proposes an approximation of A* by a union or an intersection of A >,. s. Then, Section III-B shows that applying the method proposed for intervals to an o:-cuts decomposition of the fuzzy sets in order to obtain a-cuts of B* correspond to the method proposed by K6czy and Hirota [START_REF] K6czy | Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases[END_REF] and thus raises the same problems.

A. Approximation of A*

Since A* cannot be exactly retrieved in general, a first solu tion consists in computing an approximation of A*. However, this approximation should be meaningful. Considering the in terpretation of fuzzy mies based on possibility theory (see for instance [START_REF] Ughetto | lmplicative and conjunctive fuzzy rules -A tool for reasoning from knowledge and examplcs[END_REF]), the conclusion of a mie modeled with a genuine implication is a restriction of the set of possible values for the output variable. Thus, for this restriction imposed by B* not being arbitrary, ils approximation bas to be a superset of B*, which means that the approximation of A* bas also to be a su perset of A*. Moreover in order not to Jose information, this superset has to be the smallest possible one. For a given A* in between A 0 and A 1 , such an approximation obtained either by union, or by intersection of A"s is unique.

For trapezoïdal-shaped fuzzy sets, the inclusion A* Ç ÂA, defined by A* ÇA>. if\>'u EU, µA,(u):::; µA,(u), is equiv alent to the classical inclusion of the core and of the support (C(A*) Ç C(AA) and S(A*) Ç S(A>.)).

As well as in Section Il, there are 3 different cases: Then, considering the interval Smax = [�, X], the computa tion of the trapezoïdal-shaped fuzzy set B* can be done as in Section Il, computing the left and right sides of the trapezoïd. There are also incoherent cases, in particular (but not only) when the obtained B* is sub-normalized.

B. Computation on the a-eut decomposition of A*

This approach consists in i) decomposing the fuzzy sets into their a-cuts, ii) using the method proposed in Section II on the a-cuts, iii) and then computing B*, also from its a-cuts. This method seems natural since it provides an exact representation of A*. Moreover, computation with level cuts are often used in fuzzy logic, and especially for fuzzy interpolation.

However, it can be shown that it presents the same draw backs as the method proposed by K6czy and Hirota [START_REF] K6czy | Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases[END_REF]. In particular, the computed a-cuts of B* may not fulfill the nest edness requirement. Figure 6 shows an example. Although it corresponds to a coherent case, it shows the relations induced by this method for the support (in light grey) and core (in a darker grey). For the proposed A*, it is immediate that the core and support of the B* computed this way are disjoint ! Support ■ Core of the interpolated relation Fig. 6. Interpolation by a-eut decomposition Now, considering that ail the fuzzy rules in K, or[, are grad uai rules, modeled with Rescher-Gaines implication, the fami ly of rules ( A A )a-+ (BA) ,. ,a E [0,1],obtainedfrom one rule by a-cutting A >. and B >. , gives an infinity of classical rules whose input and output parts are intervals. Computation with this (double) infinity of rules (or only somc of them) and the CRI could lead to an acceptable result, while also processing by a-eut decomposition. However, the probability for incon sistcncy between rulcs is increased. This is a tapie for further research.

IV. CONCLUDING REMARKS

This paper has outlined a method to interpolate between sparse fuzzy data, which has been also compared to approach es based on the Compositional Rule of Inference. Although this method corresponds to the CRI applied to one rule, de fined by union or intersection of linearly interpolated rules, it is different from the CRI applied to the whole set of interpo lated rules, which may be incoherent. This lead to a notion of incoherence for sparse rules which ought to be detailed in fur ther developments. Besides, when three rules are considered, Ai -+ B;, i = 1, ... , 3, such that A 1 is in between A 0 and A2, taking A* = A 1 is a possible way to check whether or not this set of 3 rules is compatible with the linear interpolation principle in the sense that B* 2 B 1 . If B* Ç B 1 , removing the rule A 1 -+ B 1 could be thought of, since the interpolation provides a more precise result. Otherwise, and in particular if B* n B 1 = 0, the interpolation should only be done between Ao and A 1 , or A 1 and A2, and not between Ao and A2.

, 4 0

 4 and A 1 , as well as their conclusions Bo and B 1 , are com pact subsets of the real line IR. Assuming a linear interpolation mode! between the two rules in K leads to the family of rules A.\ ➔ B.\, for ail À E [O, 1], where A" = À • A 1 + (1 -À) • Ao and B.\ = À• B1 + (1 -À) • Bo. An example of such an A.\ and a B>., with >. = 0.5, is shown on Figure 1.

Fig. 1 .

 1 Fig. 1. Linear interpolation between two non fuzzy rules

  4. *. Then, the corresponding union or intersection of B À s gives the output interval B*. This idea cornes from the fact that the two rules (.4. n A.') ➔ (B n B') and (.4. UA') ➔ (BU B') are logical consequences of a set of two rules {A ➔ B, A' ➔ B'}, and the generalization of this property to a set of more than 2 rules. Then, given the set of two rules K, and an observation A* in between the premises of the rules, three different situations can occur: 1. A* = A>. for some À E [O, l]. Then, the rule A>. ➔ B>.. obtained by linear interpolation directly gives the conclusion B* = B>.. 2. A* ::) A>. for some À E [O, l]. Then, with S max = {À, A* ::) A.>.} Ç [O, l], the set A* is obtained by: LJ A>. = A*, and then B* = LJ B>.. (1) >..ESm•x >.eSm•• 3. A* C A>. for some >. E [O, 1). Then, with Smax = {À, A* CA.>.} Ç [O, 1], the set A* is obtained by: n A>.. = A*, and then B* = n B>.. (2)

Fig. 2 .Proposition 2 :

 22 Fig. 2. The 3 cases depending on l!. and X Now, since the method can be viewed as the replacement of the initial set of two rules K = {Ao ➔ Bo, A 1 ➔ Bi}, with the (infinite) set of rules .C = { A>. ➔ B.\, À E [O, 11} (obtained from the two rules in K), the following proposition gives a fondamental result. Proposition 2: The set of rules .C = {A.\ ➔ B>., >. E [O, 1]}, obtained by linear interpolation from /C = { A 0 ➔ Bo, A1 ➔ B 1 }, may be potentially incoherent, in the sense of coherence given in [14]. It means that two rules in .C can be simultaneous ly triggered (i.e., A>. nA,\, =/= 0), while they give incompatible conclusions (i.e., BÀ n BÀ, = 0). Proof: See an example of coherence and incoherence cases on Figure 3 and a characterization of the incoherence cases given by Proposition 8. ■ It is clear from the definition that the incoherence can be espe cially detected in the third case, when n ÀES A>, = A* ( # 0) and n ÀES B,\ = 0 in (2).

4 :

 4 [::\,X], A>. :::> A*. Thus, Smax = [::\, "X], and this set is maximal for inclusion.■This proposition gives one possible set S which verifies (1), and thus such that B* = U>.ES B>,. However, this set does not really enable the computation of B*, since it is an infinite subset of(0, l].A firs' t way to solve this problem could consist in comput ing ajinite subset S of S m ax such that A* == U >.ES A>,. How ever, as one can get a set S such that U >-ES A>. = A* and U >.ES B>. f; B* (wilh A* and B* defined by (1)) because of Jack of convexity of U>-ES B>., this approach is not developed here.Another approach is based on the fact that B* is an interval on the continuous universe V, namely[!t, b]. Thus, in order to define B*, it is sufficient to compute its two boundaries !t andb. Proposition Given the two sparse rules in K,, and an obser vation A* in between A 0 and A1, such that A* :::> A>., for one A E (0, 1], the corresponding output B* is given by [!!*, b], with 1/ = Q� and ,;* = bx, or equivalently: * [ l2,1 -l2.o * g_ 1 f2.o -Q0Q1 ÎÏ1 -ÎÏo = * a1ÎÏ0 -aob1 ] B = ---g,_ +��-�,=----=-a + _ _ !!1 -Q.o . !!1 -Q.o a1 -ao a1 -ao Proof: The set B* is given by the interval U>-ESmax B>., and S max is the interval [::\, "X]. The lower bound of B* is then the smallest value of the lower bounds of the B;.s, for À E S max• And it is clear that this value is the lower bound of B;.. Thus !/ = !!. � -The proof is the same for the upper bound. -■ C. Case of A* s smaller than A_,s Now consider the third case, wherc A* is included in one ,4,. The next proposition is the counterpart of Proposition 3 for small A*s (the same notations are used). Proposition 5: Given A* such that A* C A.\ 0 , for one >.o E [0, 1], the set S rnax == {,\, ,4;. C A*} is the maximal (for in clusion) subset of [0, 1] which verifics (2). This set Smax is the interval ['X,::\], where ::\ = (g_ 1 -g_*)/(g,_ 1 -[4)) and X= (a1 -a *) /(a1 -ao)-Proof: It is similar to the proof of Proposition 3.

Proposition 8 :

 8 Let I: be the line defined by the two points (fJ-0,Qo) and Üh,!!. 1 ), and Ï' the line defined by the two points (ao , b o ) and (ïï1, ÏÏ1), as shown on Figure 3. The set .C is po tentially incoherent if the line [ is above the Iine f for at least one point in [ao,g,_ i ], i.e., if3u E (ëio d ! i l, r.(u) > I'(u). Proof: Consider a value u E U such that [(u) > I'(u), and A*= {u}. The computation of B 1,_ and E x (as defined in Sec tion II-C) shows that B 1,_ = [I:(u), bA] and that B r == [f!.x, f(u)]. And thus B* == B� n B r = 0 (since I:(u) > Ï'(u)), which means incoherence.

Figure 4 .Fig. 4 .

 44 Fig. 4. Generalized notion of incoherence

VFig. 5 .

 5 Fig. 5. Three different interpolative mechanisms

  then v E B. The interpolation between these two rules only considers the right side of A 0 and the Ieft side of A 1 , and takes the corresponding convex hull. It leads to the relation depicted on Figure 5.b. Although this relation is more restrictive than the previous one, it is more in accordance with the notion of rule. Indeed, if u E A, then v cannot be outside B, while it is the case in Figure 5.a (see the grey region above the rectangle representing Ao x Bo on U x V). The relations defined by these interpolation processes can be represented by two bounding lines. It is also the case with the third figure (5.c), which corresponds to the method pro posed here in a coherent case. It can be represented by the two lines I: and f defined above, and corresponds to the relation n >. e( o , i ] A>. ➔ B>., defined by the set ofrules C. This relation is more restrictive than the two others. In these three cases, the reasoning mechanism can be the CRI applied lo the relation defined on U x V by two bounding lines, as shown on Figure 5. Thus, it cornes down Lo the eval uation of a fonction in two points. For the proposed method, it is B* = [l'(Q*), Ï'(a*)] for the case of an increasing interpola tive fonction (as on Figure 5.c). This result is in accordancc with B* = A* o n ,\ E[O , l ] A" ➔ B>.., i.e., with the CRI.

  1. A*= A>., for one>. E [O, l], 2. A* C A >. , for one>. E [O, 1]. In this case A* is approxi mated by n>.ESm•x A>. with Smax = { À E [O, 1], A* C A>.}. 3. A* i A >. ,\>'>. E [O, 1] (i.e., ail the other cases). Then, A* is approximated by UÀESm•x Â>., where Smax = [�, X] is given by � = sup A'Ç ( A ,. ,A,) >., (denoting [A>., A 1 ] the convex hull of Â>. and A1) and X= infA•Ç(A 0 ,A,.] >..


Gabin Guélou

Christophe Couder

Charles Manière

Christophe Candolfi

Bertrand Lenoir

Lucile Lallemant

Christophe Coureau

Emmanuel Guilmeau
email: emmanuel.guilmeau@ensicaen.fr

Issues and opportunities from Peltier effect in functionally-graded colusites: from SPS temperature modeling to enhanced thermoelectric performances

Keywords: thermoelectric, sulfide, colusite, scale up, ball milling

The quaternary sulphide V-Sn colusite, Cu 26 V 2 Sn 6 S 32 , is one of the most promising costefficient thermoelectric materials to date because of the low toxicity, wide availability and low cost of the composing elements. Recent studies have demonstrated the potential of this environmentally-friendly material and its transport properties are now well understood. In the present work, we take the next step of producing large quantities of optimised V-Sn colusite using industrial-grade precursors and investigating the effect of Spark Plasma Sintering (SPS) in the production of large cylindrical pucks of up to 30 mm in diameter and 10 mm in thickness. In the process, we identified and solved several key issues including the generation of temperature gradients during SPS, porosity and defect formation. The generation of radial and axial temperature gradients within the sample during SPS has been modeled using modified Fourier and Ohm laws and confirmed experimentally thanks to the T SPS -dependent transport properties of V-Sn colusite and EDS analysis. We demonstrate that large pucks of colusite with enhanced thermoelectric properties can be produced using a combination of SPS and High-temperature Isostatic Press (HIP). Overall, our work experimentally and theoretically demonstrates that the production of both homogeneous and functionally-graded bulk materials can be easily up-scaled through a careful control of the SPS conditions.

Introduction

Thermoelectricity (TE) relies on the ability of optimised materials to generate a potential gradient from a temperature difference high enough so that a TE device can effectively convert a sizeable portion of thermal energy into a useful electrical current. The conversion efficiency of a TE material directly relies on a particular combination of electrical and thermal properties embedded in the dimensionless thermoelectric figure of merit, 𝑍𝑇 = 𝑆²𝑇 𝜌𝜅 ⁄ , where S is the Seebeck coefficient (or thermopower), ρ is the electrical resistivity and κ is the total thermal conductivity. So far, decades of research, involving a broad range of materials, have seen TE devices restricted to niche applications such as space exploration for which the cost and toxicity combined to the relatively low efficiency of such devices are outweighed by their reliability and lifespan. [START_REF] He | Advances in thermoelectric materials research: Looking back and moving forward[END_REF][START_REF] Snyder | Complex thermoelectric materials[END_REF][START_REF] Rowe | Thermoelectrics, an environmentally-friendly source of electrical power[END_REF][START_REF] Sootsman | New and old concepts in thermoelectric materials[END_REF][START_REF] Min | Chapter 38. Peltier Devices as Generators[END_REF][START_REF] Champier | Thermoelectric generators: A review of applications[END_REF] In order to popularise the use of TE devices and significantly contribute to reducing our greenhouse gas emission, the production of large amount of TE materials from low-cost, widely available, non-toxic precursors using costeffective synthesis and manufacturing techniques is the next milestone.

Amongst the promising families of materials, ternary and quaternary n-and p-type copper sulphides stand out by their high performance-to-cost ratio with many of the best performing compositions being free of heavy and toxic elements. This is in particular the case of bornite Cu 5 FeS 4 , [START_REF] Qiu | Sulfide bornite thermoelectric material: A natural mineral with ultralow thermal conductivity[END_REF][START_REF] Guélou | Ball milling as an effective route for the preparation of doped bornite: Synthesis, stability and thermoelectric properties[END_REF][START_REF] Kumar | The crucial role of selenium for sulphur substitution in the structural transitions and thermoelectric properties of Cu5FeS4 bornite[END_REF], stannoidite Cu 8 Fe 3 Sn 2 S 12 , [START_REF] Kumar | Copper Hyper-Stoichiometry: The Key for the Optimization of Thermoelectric Properties in Stannoidite Cu8+xFe3-xSn2S12[END_REF] germanite derivative Cu 22 Fe 8 Ge 4 S 32 , [START_REF] Kumar | Designing a Thermoelectric Copper-Rich Sulfide from a Natural Mineral: Synthetic Germanite Cu22Fe8Ge4S32[END_REF][START_REF] Kumar | Crossover from germanite to renierite-type structures in Cu22-xZnxFe8Ge4S32 thermoelectric sulfides[END_REF] mohite Cu 2 SnS 3 , [START_REF] Shen | Ecofriendly p-type Cu2SnS3 thermoelectric material: crystal structure and transport properties[END_REF] kesterite Cu 2 ZnSnS 4 , [START_REF] Liu | A wide-band-gap p -type thermoelectric material based on quaternary chalcogenides of Cu2ZnSnQ4 (Q=S,Se)[END_REF][START_REF] Yang | Nontoxic and abundant copper zinc tin sulfide nanocrystals for potential high-temperature thermoelectric energy harvesting[END_REF] Cu 4 Sn 7 S 16 , [START_REF] Bourgès | Low thermal conductivity in ternary Cu4Sn7S16 compound[END_REF] chalcopyrite CuFeS 2 [START_REF] Chen | Thermoelectric Enhancement of Ternary Copper Chalcogenide Nanocrystals by Magnetic Nickel Doping[END_REF],

tetrahedrites Cu 12-x T x Sb 4 S 13 (T = Mn, Fe, Ni, Zn). [START_REF] Lu | High performance thermoelectricity in earth-abundant compounds based on natural mineral tetrahedrites[END_REF][START_REF] Suekuni | Thermoelectric properties of mineral tetrahedrites Cu 10Tr 2Sb 4S 13 with low thermal conductivity[END_REF][START_REF] Suekuni | High-performance thermoelectric mineral Cu12-xNixSb4S13 tetrahedrite[END_REF][START_REF] Bouyrie | Crystal structure, electronic band structure and high-temperature thermoelectric properties of Te-substituted tetrahedrites Cu12Sb4-xTexS13 (0.5 ≤ x ≤ 2.0)[END_REF][START_REF] Chetty | Tetrahedrites as thermoelectric materials: An overview[END_REF][START_REF] Barbier | Structural stability of the synthetic thermoelectric ternary and nickelsubstituted tetrahedrite phases[END_REF][START_REF] Barbier | Thermoelectric Materials: A New Rapid Synthesis Process for Nontoxic and High-Performance Tetrahedrite Compounds[END_REF] and colusites Cu 26 T 2 M 6 S 32 (T = V, Nb, Ta, Cr, Mo, W; M = Sn, Ge). [START_REF] Suekuni | High-performance thermoelectric minerals: Colusites Cu 26 V 2 M 6 S 32 (M = Ge, Sn)[END_REF][START_REF] Suekuni | Tunable electronic properties and low thermal conductivity in synthetic colusites Cu26-xZnxV2M6S 32 (x ≤ 4, M = Ge, Sn)[END_REF][START_REF] Bourgès | Structural analysis and thermoelectric properties of mechanically alloyed colusites[END_REF][START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Kumar | High Power Factors of Thermoelectric Colusites Cu26T2Ge6S32 (T = Cr, Mo, W): Toward Functionalization of the Conductive "Cu-S" Network[END_REF][START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF][START_REF] Kumar | Copper-rich thermoelectric sulfides: size mismatch effect and chemical disorder in the [TS4]Cu6 complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) colusites[END_REF] In our previous works, we reported a figure of merit, ZT, of 0.9 at 673 K, for the environmentally-friendly V-Sn colusite, Cu 26 V 2 Sn 6 S 32 , a highly promising mineral-derivative that rivals the state-of-the-art PbTe-based compounds for mid-temperature range applications in power generation. [START_REF] Shi | Advanced Thermoelectric Design: From Materials and Structures to Devices[END_REF] Such performance was achieved through careful control over the formation of short-to-medium range structural defects and sulphur loss promoted by the sintering temperature. [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF] This enabled the simultaneous optimisation of electrical and thermal properties via charge carrier concentration tuning and thermal conductivity reduction. Following this work, we demonstrated the feasibility of an up-scaled production using industrial binary precursors and large-batch mechanical alloying, while retaining excellent TE properties. [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] Here, we now turn towards the production of large monoliths of V-Sn colusite using Spark Plasma Sintering (SPS) and address the key issues related to the inhomogeneity of the temperature within the puck during consolidation. A temperature gradient spontaneously forms during current-activated, pressure-assisted densification (CAPAD) because of the Peltier effect. [START_REF] Maizza | Peltier effect during spark plasma sintering (SPS) of thermoelectric materials[END_REF][START_REF] Becker | The effect of Peltier heat during current activated densification[END_REF] This is particularly important in the present case where the sintering temperature is of paramount importance to obtain the desired TE properties. [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF] We combined computational Joule heating/thermoelectric modeling and experimental data in order to determine the extent of this issue in V-Sn colusite and to develop a reproducible procedure to cancel the temperature gradient. Of note, promoting a carefully-controlled temperature gradient during sintering may also offer a way of producing functionally-graded materials. [START_REF] Guillon | Field-Assisted Sintering Technology/Spark Plasma Sintering: Mechanisms, Materials, and Technology Developments[END_REF] Using this optimised procedure, we demonstrate the production of large monoliths of V-Sn colusites with enhanced TE properties.

Results and discussions

Modeling of SPS heating affected by Peltier effect

In a typical SPS configuration using graphite mould and punches, the powdered sample is placed directly between carbon tabs. This allows the electrical current to flow through or around the sample depending on the difference of electrical resistivity between the sample and the mould, as well as on the contact resistance at the junction between the different components of the set-up. In all performing thermoelectric materials like colusite, the combination of high electrical conductivity and high Seebeck coefficient cannot be ignored when attempting to control the temperature during SPS. This is particularly important as we demonstrated the dramatic impact of temperature over the TE properties of colusite. [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] In order to obtain a qualitative map of the temperature distribution in a large consolidated colusite sample, we investigated the multiphysics modeling of SPS Joule heating explicitly taking into account thermoelectric effects. The construction of the Joule heating model and all electric/thermal contact resistance at each SPS interfaces is detailed elsewhere. [START_REF] Manière | Contact resistances in spark plasma sintering: From in-situ and ex-situ determinations to an extended model for the scale up of the process[END_REF] The thermoelectric effect is added through the Seebeck coefficient, which affects the Joule The modeling of heat flux within various configurations of the mould and punches, using the aforementioned simulation, are shown in Figure 1 and S1. A "typical" SPS configuration (Figure 1a) considering the thermoelectric performance of the fully ordered stoichiometric colusite, [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF] reveals the generation of two distinct (axial and radial) temperature gradients.

The axial component, ΔT P , is a direct consequence of the Peltier effect caused by the relatively high Seebeck coefficient in p-type colusite. Because the electrical current flows from bottom to top in the SPS press used in this work, the Peltier effect implies a heat flow from the lower section to the top. Considering the impact of the thermal contact resistance at the upper/lower powder/punch interfaces, these partial thermal barriers tend to reduce the lower punch heat flow (by Peltier effect) and conversely confine the heat at the upper powder interface. Consequently, the horizontal thermal contact resistance exaggerates the Peltier thermal gradient in the powder. Another inhomogeneity in the temperature arises from heat loss through radiation from the mould resulting in a radial temperature difference, ΔT R and an erroneous estimation of the powder temperature from the temperature measured on the die (ΔT R ~ 75 K, Figure 1a). In Figure 1a, the temperature difference between the powder/punch column and the mould is clear and worsened by the contact resistance between them. This radial temperature gradient adds up to the axial component and results in an underestimation of the average temperature in the sample area. In order to dispose of these temperature gradients and to monitor the sintering temperature more accurately, we apply two simple corrections that can be easily scaled-up. First, in order to equilibrate the temperature in the punches and the mould, we thermally insulated the latter using a graphite felt. To simulate its impact on heat flux, we assumed a thermal insulation on the lateral die surface (Figure 1b).

From this simulation, it appears that, despite the contact resistance, between the punches and mould, the temperature is rather homogeneous across the set-up, with only the Peltier thermal gradient left through the sample remaining. From here, we can either exaggerate this ΔT P , by displacing the mould downward (Figure 1c), partially compensate it by displacing the mould upward (Figure 1d) or almost cancel it by electrically insulating the upper/lower powder interfaces via a boron nitride spray (Figure 1e). Exaggerating ΔT P by moving the die downward (Figure 1c) enables to produce samples with functionally-graded properties so long as these depend on the sintering temperature, as it is the case for the V-Sn colusite. The compensated configuration (Figure 1d) is poorly efficient as the thermal contact resistance still implies the development of ΔT P ~ 20K despite the higher lower punch temperature.

Conversely, the electrically insulated configuration (Figure 1e) forces the current flux out of the sample and through the mould, subsequently cancelling ΔT P in a so-called "hot press" configuration. Note that the temperature fed back to the PID is calculated to be much closer to the temperature experienced by the sample than in the previous configurations (Figure 1a).

In the following sections, the thermoelectric properties of large pucks (30 mm diameter, 10 mm thick) of V-Sn colusite prepared from up-scaled mechanical alloying [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] are discussed.

First, we compare the thermoelectric properties of the top, middle and bottom segments between two pucks prepared in either a "typical" SPS (Figure 1a) or "hot press" (Figure 1e) configuration. In a second time, we discuss the density issues related to induced sulphur loss in large samples and demonstrate that an additional High temperature Isostatic Pressing (HIP) is successful in obtaining excellent density and controlled thermoelectric performances.

Experimental confirmation of thermal gradients

The mechanochemical synthesis of the powders used for the SPS sintering of large samples is described elsewhere. [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] The pre-reacted powders are then allowed to crystallise in-situ during the SPS sintering, forming a range of structural defects that depend on the temperature and sulphur volatilisation. The unit cell parameter, a, of the sintered material varies significantly with the level of disorder in the crystal structure. With binary sulphide precursors at low sintering temperature, the unit cell remains at around 10.76 Å with the tendency of forming vanadium-rich core shell microstructure from incomplete crystallisation. [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] The unit cell can be as high as ca. 10.83 Å for hot-pressed samples prepared from high purity elements, and containing a large proportion of structural defects. [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF] Regardless of the choice of precursors, it appears that high temperature sintering (1023 K -1083 K) yield the best results with a higher ZT value. It is therefore of prime importance to reproduce high-temperature sintering on a larger scale. In order to monitor the temperature experienced by the sample during sintering, and to confirm our theoretical model, we carried out Rietveld refinements of the XRPD data acquired at the top, middle and bottom cross sections of 30 mm pucks sintered at 873 K from typical SPS and "hot press" configurations (Table 1, Figure 2 and Figure S2 and S3 of the supplementary information). sintering temperature. This is further confirmed by the measured thermoelectric properties of the sample (Figure 3 and4). In the typical SPS configuration, the electrical resistivity, Seebeck coefficient and resulting power factor of up-scaled colusite depend on the position of the measured cross section. The top section, predicted to be experiencing higher temperatures during SPS, exhibits the largest unit cell parameter and the most semiconducting electrical transport properties, with electrical performances close to those of highly-disordered colusite. [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF][START_REF] Candolfi | Disorder-driven glasslike thermal conductivity in colusite Cu26V2Sn6S32 investigated by Mössbauer spectroscopy and inelastic neutron scattering[END_REF] As the cross section approaches the bottom part, the transport properties and unit cell parameters are consistent with a progressively more ordered colusite, evidencing a progressively lower effective sintering temperature. The higher sulphur volatilisation occurring as we move closer to the upper punch of the SPS, i.e. closer to the top section, can be clearly seen from point-by-point EDX analysis, carried out across the sample every 500 μm (Figure S4). In the top section of the puck, the electrical resistivity displays values ranging between 3.0 mΩ cm and 4.2 mΩ cm at 300 K and 700 K, respectively, with the distinctive minimum at around 350 K. [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] Similarly, the Seebeck coefficient is high with values between 110 μV K -1 and 185 μV K -1 over the investigated temperature range. The electrical transport properties then become increasingly metallic at the middle and bottom sections of the puck with room-temperature electrical resistivity values of 1.5 mΩ cm and 1.0 mΩ cm, respectively, up to 2.2 mΩ cm and 1.7 mΩ cm at 700 K. Meanwhile, the Seebeck coefficient also decreases from 77 μV K -1 and 55 μV K -1 at 300 K to 142 μV K -1 and 118 μV K -1 at 700 K for the middle and bottom sections, respectively. In "hot press" configuration, the variations between the different sections are completely cancelled out and the electrical performances are those of the bottom, i.e. the coldest section of the sample in a typical SPS configuration. A similar conclusion can be drawn from the thermal conductivity with very different values measured for the different cross sections in the typical SPS configuration (Figure 4). The top section exhibits thermal conductivity values ranging from 1.54 W m -1 K -1 to 1.12 m -1 K -1 at 300 K and 673 K, respectively, while the top of the puck shows a thermal conductivity about 35 % lower, consistent with a higher sintering temperature, larger unit cell parameter and higher level of disorder. This is consistent with a more homogeneous and more stoichiometric sample when using the "hot press" configuration. Overall, the peak ZT value is functionally graded in a typical SPS configuration while the addition of the insulating barriers leads to a large homogeneous puck with a constant ZT across the sample (Figure 4). Right column: in "hot press" configuration (i.e. with insulating spacers and jacket).

Finally, the unit cell parameters and transport properties of V-Sn colusite, sintered at 873 K in "hot press" configuration, are slightly different than those of lab-scale samples, should they be synthesised from high-purity elements [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF] or industrial grade binary precursors. [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] This evidences a discrepancy between the measured temperature in the mould and the actual sintering temperature, confirmed by the electrical transport properties measurements. By comparison, the unit cell parameter and thermoelectric performance of the "hot press"

SPS873 sample seem to be consistent with an actual sintering temperature between 1023 K and 1073 K. From our calculations, this seems to suggest that a better thermal insulation of the mould would be needed for better control over the temperature. However, it must be noted that these discrepancies might also arise from differences in sulphur loss and/or defect formation caused by the much lower surface-to-volume ratio in the large pucks. This possibility seems more likely, as evidenced by the expected trend with respect to sintering temperature and the density issues addressed in the following section.

Consolidation and TE optimisation

As demonstrated previously, [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF][START_REF] Candolfi | Disorder-driven glasslike thermal conductivity in colusite Cu26V2Sn6S32 investigated by Mössbauer spectroscopy and inelastic neutron scattering[END_REF] increasing the sintering temperature is crucial to obtaining high thermoelectric performance. We prepared 3 additional pucks of 30 mm in diameter and 10 mm in thickness sintered in the "hot press" configuration, established in the precedent section, at 923 K, 973 K and 1023 K (noted hereafter SPS923, SPS973 and SPS1023, respectively). The differences in surface-to-volume ratio in such large pellets, compared to the 10 mm typically used in lab-scale experiments, led to issues regarding sulphur volatilisation. Indeed, as evidenced by the much lower densities obtained for SPS973 and SPS1023, with closed porosity, it appears that sublimated sulphur remained trapped in the sintered sample and led to high levels of porosity on cooling (Table 2). This is particularly detrimental for higher sintering temperature as attempts to sinter samples at 1073 K led to the formation of millimetre-scale pores within the samples, making them unusable.

For samples sintered below 1023 K however, owing to the closed porosity, it was possible to correct for the loss of density by using Hot Isostatic Pressing (HIP). Thus, we produced two additional series including an additional HIP step at 1500 bar and 873 K or 1023 K for 1h.

The crossed impact of SPS sintering temperature and HIP temperature on the microstructure and thermoelectric performances has been investigated. Regardless of the HIP temperature, this step was successful in correcting the density, up to 100 % for all samples (Table 2). The measured volume mass even reached values slightly over 100 % of the crystallographic value, the latter being slightly underestimated because of sulphur loss. This is clear from the microstructural analysis (Figure 5 and S5) where pores can be observed in the SPS873 samples while SPS1023 displays more but smaller pores. In both cases, the additional HIP treatment at 873 K is sufficient to obtain an extremely dense material. Aside from the effect on density, the HIP treatments also clearly affect the microstructure of the materials and in particular the concentration and composition of the microstructural defects. In a previous work, we already established the nature of these defects and the effect of sintering temperature on these defects in smaller SPS samples. [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] We demonstrated that low sintering temperatures led to an incomplete crystallisation of the colusite phase, leaving V-rich coreshell microstructures (CSM) as artefacts of the formation mechanism of colusite. These It is thus not surprising to observe a large quantity of V-rich CSM in the SPS873 sample (Figure 5 and6) while very few could be seen in the SPS1023 sample. However, the latter still exhibits V-rich regions with a composition close to that of colusite with an equal amount of V and Sn. Interestingly, the observation of the defects in SPS873 and SPS1023 samples with and without HIP treatments evidences a systematic grain growth in V-rich regions, inside the area marked by a beige line. In the SPS873 sample (Figure 6a), the grain size remains small (sub-micron) and homogeneous across the sample, even around the CSM marked by the red and orange lines. After HIP at 873 K, it is clear that the outer layers of the CSM have a significantly larger grain size than the colusite matrix, of around 1 μm (Figure 6b). At the same time, these CSM are scarcer and many have diffused into the colusite matrix leaving V-rich spot-regions without a vanadium metal core. After HIP treatment at 1023 K, the homogeneous grain size within the colusite matrix increases up to 2-3 μm (Figure 6c).

However, around the V-rich spot-regions and the few remaining CSM, the grain size can be as much as 5 times larger. A similar observation can be made for the SPS1023 sample where such grain growth, localised around the V-rich regions, can be seen before the HIP treatment (Figure 6d). After HIP at 873 K or 1023 K (Figure 6e and 6f, respectively), nearly all V-rich regions have diffused into the matrix, replaced by spot-regions of large grain sizes dispersed within the colusite matrix. These observations are consistent with the formation mechanism of colusite, through the diffusion of vanadium into a Cu-Sn-S matrix, and highlight the importance of the crystallisation process in shaping the microstructure of colusite materials.

At low sintering temperature and without HIP, the materials can be described as a composite of V-poor colusite matrix and V-rich CSM. Previously, in lab-scale experiments, we demonstrated that a complete diffusion process of vanadium and single-phase colusite can be obtained by increasing the sintering temperature. [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] In large consolidated samples however, the associated sulphur volatilisation leads to significant porosity and a HIP treatment is necessary to ensure a high density. In the process, we can clearly observe that the diffusion of vanadium into the colusite matrix is accompanied by a faster grain growth in the regions surrounding a V-rich CSM. The thermoelectric performances of the three series of samples are shown in Figure S6 and S7. Compared with previous transport property data, the electrical resistivity and Seebeck coefficient for all samples are generally high. [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF][START_REF] Candolfi | Disorder-driven glasslike thermal conductivity in colusite Cu26V2Sn6S32 investigated by Mössbauer spectroscopy and inelastic neutron scattering[END_REF] SPS873 is the most metallic sample with an electrical resistivity varying from 1.46 mΩ cm to 2.31 mΩ cm, and a Seebeck coefficient increasing from 65 μV K -1 to 129 μV K -1 at 298 K and 667 K, respectively. This corresponds to a threefold increase compared to the values obtained on smaller sintered pellets. This is likely caused by the differences in the SPS set up and, in particular, by the use of much larger moulds that increase the error in the measurement of the actual temperature experienced by the sample. By comparison with our previous work, [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] and considering that the powders have been prepared in the exact same scaled-up conditions, we can speculate that the temperature in the sample during SPS has been underestimated. However, the presence of a large amount of defects makes it very difficult to estimate the value of the temperature experienced by the sample and a qualitative comparison is more pertinent. Indeed, the observed increase in electrical resistivity and Seebeck coefficient with increasing sintering temperature is consistent with a decrease in the charge carrier concentration through sulphur volatilisation and the complete crystallisation of the colusite phase. This effect becomes nearly negligible for the samples with a HIP treatment at 1023 K. The samples prepared by SPS at temperatures higher than 973 K, without HIP, exhibit an electrical resistivity higher than expected with respect to the Seebeck coefficient, consistent with the low relative density observed for these samples. This issue has been fully eliminated by the additional HIP step, regardless of its temperature. Because the decrease in charge carrier concentration leads to an increased electrical resistivity and Seebeck coefficient, the power factor remains remarkably similar for all samples with high densities. Overall, the best performing sample is SPS923 followed by a HIP step at 1023 K (Figure 7 and8), although all samples with this HIP temperature have reasonably similar thermoelectric performances (Figure S6 and S7). The SPS873 and SPS923 samples already exhibited high relative density; therefore, the HIP step at 873 K had virtually no impact on the transport properties. However, for samples sintered by SPS at higher temperature, i.e. exhibiting a lower relative density after SPS, this HIP at 873 K was successful in increasing the relative density, thus decreasing the electrical resistivity, retaining a high Seebeck coefficient and increasing the thermal conductivity. As a result, these differences arising from the various sintering temperatures are retained and the electrical transport behaviour is increasingly semiconducting with increasing SPS temperature. The impact of the HIP step at 1023 K, however, is expected to be much more pronounced for all samples and to nearly fully suppress the differences made by the SPS temperature. This is consistent with previous studies where it was determined that high temperatures lead to sulphur volatilisation and the formation of atomic-scale defects. [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF][START_REF] Candolfi | Disorder-driven glasslike thermal conductivity in colusite Cu26V2Sn6S32 investigated by Mössbauer spectroscopy and inelastic neutron scattering[END_REF][START_REF] Suekuni | Atomic-Scale Phonon Scatterers in Thermoelectric Colusites with a Tetrahedral Framework Structure[END_REF] Regardless of the SPS temperature, a HIP step at 1023 K ensured high density, complete crystallisation of the colusite phase and the removal of most microscale defects. In the same time, as extensively investigated previously, [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF][START_REF] Candolfi | Disorder-driven glasslike thermal conductivity in colusite Cu26V2Sn6S32 investigated by Mössbauer spectroscopy and inelastic neutron scattering[END_REF][START_REF] Suekuni | Atomic-Scale Phonon Scatterers in Thermoelectric Colusites with a Tetrahedral Framework Structure[END_REF] the high temperature leads to the formation of atomic-scale defects that ultimately reduce the thermal conductivity. This is clear from the significant decrease measured for SPS923 + HIP at 1023 K (Figure 8) from 1.04 W m -1 K -1 to 0.81 W m -1 K -1 at 300 K and from 0.86 W m -1 K -1 to 0.65 W m -1 K -1 at 673 K. This reduction occurs while the power factor (Figure 7) remains high and unchanged compared to before and after HIP at 873 K. A maximum value of 0.83 mW m -1 K -2 is obtained for all three samples as a result of the simultaneous increase in electrical resistivity and Seebeck coefficient. This evidences a decrease in charge carrier concentration caused by the highest level of sulphur loss induced by the HIP step at 1023 K. Data for the remaining samples are given in the supplementary information (Figure S6 andS7).

As a result, while all samples subjected to a HIP step at 1023 K reached a maximum ZT value of around 0.75 at 673 K, the best performance was achieved for the SPS923 sample with ZT = 0.85 at 673 K. These results both confirm the need for high temperature in order to improve the performances of the V-Sn colusite and the nonessential nature of a high temperature sintering step, so long as it is followed by an annealing process such as HIP. The fact that SPS923, after HIP at 1023 K, exhibits the highest ZT most likely depends on the type of disorder already present in the sample before annealing. Indeed, this sample still showed significant levels of microstructural defects, illustrated by detectable amounts of sulvanite in the XRPD pattern. For samples sintered at 973 K and 1023 K, the issue related to density and the high level of sulphur volatilisation might play a role in the performance after the HIP step. In these two samples, the charge carrier concentration has been exceedingly reduced, leading to a lower power factor and ultimately lower ZT.

Conclusion

The production of large cylindrical pucks of industrial-grade colusite, Cu 26 V 2 Sn 6 S 32 , using 100g scale mechanical alloying and spark plasma sintering has been successfully demonstrated. In the process, several issues related to the formation of temperature gradients in the sample during SPS have been successfully modeled, experimentally confirmed and easily prevented by adding insulating barriers forcing the current out of the sample area. It was previously established that optimising the thermoelectric properties of V-Sn colusite required temperature-induced disorder, in particular atomic-scale defects promoted by high temperatures. In this scaled-up process, producing pucks of 30 mm in diameter and 10 mm in thickness, we evidenced new issues related to in-situ sulphur volatilisation, leading to a low relative density and a lower TE performance. We demonstrated that an additional HIP step at 873 K was sufficient in correcting this density issue; however, a higher temperature of 1023 K yields better results directly from samples sintered at lower SPS temperatures. Overall, the best performing sample, sintered at 923 K and subjected to HIP at 1023 K, exhibits a peak ZT of 0.85 at 673 K. This value is on par with the best lab-scale high-purity V-Sn colusite reported to date and can be obtained in very large amounts. Finally, this approach can be of interest for a wide range of applications involving Spark Plasma Sintering. These include the optimisation of TE materials and applications that will benefit from obtaining functionallygraded materials.

Experimental Section

Synthesis and consolidation

Polycrystalline samples of Cu 26 V 2 Sn 6 S 32 were synthesized by mechanical alloying using industrial-grade binary precursors SnS and CuS (Tribotecc) and V (99.9%, Alfa Aesar).

Large batches of products (100g) were obtained from the ball-milling of the precursors in a 250 ml tungsten carbide jar containing 15 balls of 20 mm under argon atmosphere. This was carried out in a Fritsch Pulverisette 5 Premium line planetary ball-mill operating at room temperature. The stoichiometric mixture of precursors was milled for a total of 12 h at 400 rpm in sequences of 30 min with 5 min intervals. The obtained powders were then loaded in graphite dies of 30 mm in diameter and densified by spark plasma sintering in a SPS-FCT HPD 25. The different sintering temperatures have been selected by considering previously reported TGA/DSC analysis and adapted following the difficulties encountered during the scale up procedure. [START_REF] Guélou | A scalable synthesis route for multiscale defect engineering in the sustainable thermoelectric quaternary sulfide Cu26V2Sn6S32[END_REF] Regardless of the sintering temperature, heating and cooling rates of respectively 50 K min -1 and 20 K min -1 were used with a 30 min isotherm at the sintering temperature under a pressure of 64 MPa. All milling were carried out in inert atmosphere (argon) but storage and manipulation of the precursors and the products of mechanical alloying were done in air. All samples appear to be stable in air.

Heating simulation

The calculations of SPS Joule heating were performed by finite-element analyses using the software COMSOL Multiphysics. The governing equations of Joule heating with thermoelectric effect, the electro-thermal boundary conditions, and the graphite and electrothermal contact resistance are described elsewhere. [START_REF] Maizza | Peltier effect during spark plasma sintering (SPS) of thermoelectric materials[END_REF][START_REF] Manière | Contact resistances in spark plasma sintering: From in-situ and ex-situ determinations to an extended model for the scale up of the process[END_REF][START_REF] Manière | Proportional integral derivative, modeling and ways of stabilization for the spark plasma sintering process[END_REF] 4.3. Electrical and thermal properties measurements.

The electrical resistivity (ρ) and Seebeck coefficient (S) were measured simultaneously on bar-shaped samples, cut from the obtained SPS samples (as described in the main text), from 300 K up to 700 K on heating and cooling using an ULVAC-ZEM3 instrument under partial helium pressure. A NETZSCH LFA-457 instrument was used for the measurement of the thermal diffusivity under argon flow. The thermal conductivity (κ) was determined as the product of the geometrical density (confirmed by Archimedes' method), thermal diffusivity, and theoretical heat capacity using the Dulong-Petit approximation. The lattice contribution to the thermal conductivity (κ L ) was determined by subtracting the estimated electronic component (κ e ) from the measured total thermal conductivity (κ). The estimated measurement uncertainties are 6% for the Seebeck coefficient, 8% for the electrical resistivity, 11% for the thermal conductivity, and 16% for the final figure of merit, ZT. [START_REF] Alleno | Invited Article: A round robin test of the uncertainty on the measurement of the thermoelectric dimensionless figure of merit of Co0.97Ni0.03Sb3[END_REF]

X-ray powder diffraction

Rapid acquisition of X-ray powder diffraction (XRPD) were carried out on reactive powders (before SPS) using two X'PERT Pro MPD PANalytical (Phillips) equipped with a Cu (Kα 1 = 1.5406 Å and Kα 2 = 1.5444 Å) anticathode. The source was fitted with a PIXcel3D detector and a nickel filter to eliminate K β copper radiation. High-resolution X-ray powder diffraction (XRPD) data were collected at room temperature using a Bruker D8 Advance Vario 1 twocircle diffractometer (θ-2θ Bragg-Brentano mode) using Cu Kα radiation (λ = 1.5406 Å) with a Ge (111) monochromator (Johansson type) and a Lynx Eye detector. The XRPD data analyses were performed by Rietveld refinement using the GSAS and ExpGUI software packages. [START_REF] Larson | General Structure Analysis System (GSAS)[END_REF][START_REF] Toby | EXPGUI, a graphical user interface for GSAS[END_REF] Background contribution to the XRPD patterns were estimated manually.

Zero-point shift, peak shape parameters, asymmetry parameters, lattice parameters, fractional atomic coordinates and isotropic displacement parameters (i.e. Debye-Waller factors B iso )

were refined.

Figure 1 .

 1 Figure 1. Numerical investigation of SPS heating in (a) "typical" SPS configuration, (b,c,d,e)

Figure 2 .

 2 Figure 2. XRPD room temperature data of scaled-up sintered colusite Cu 26 V 2 Sn 6 S 32

Figure 3 .

 3 Figure 3. Temperature dependence the electrical resistivity, Seebeck coefficient and power

Figure 4 .

 4 Figure 4. Temperature dependence of the thermal conductivity and figure of merit for

Figure 5 .

 5 Figure 5. Back-scattered scanning electron micrographs (× 500) of polished surfaces of

Figure 6 .

 6 Figure 6. Scanning electron micrographs (× 5000) of polished surfaces of SPS873 and

Figure 7 .

 7 Figure 7. Temperature dependence the electrical resistivity, Seebeck coefficient and power

Figure 8 .

 8 Figure 8. Temperature dependence of the thermal conductivity and figure of merit for

Table 1 .

 1 Results from Rietveld refinements from XRPD room temperature data of scaled up sintered colusite Cu 26 V 2 Sn 6 S 32 .

			Classic SPS Φ30 -SPS873	"hot press" Φ30 -SPS873
			Bottom	Middle	Top	Bottom	Middle	Top
		a (Å)	10.7734(2)	10.8083(1)	10.8098(1)	10.7868(1)	10.7851(1)	10.7890(1)
		Rwp (%)	8.06	9.54	7.73	9.39	9.05	9.65
		χ²	2.063	2.025	1.891	2.039	1.825	2.144
	V Cu(1)	ρ (g cm -3 ) Biso (Å 3 ) Biso (Å 3 )	4.637 4.9(4) 1.84(4)	4.593 1.5(2) 2.17(3)	4.591 0.3(1) 2.22(2)	4.62 2.1(2) 1.96(3)	4.622 1.9(2) 1.55(3)	4.617 1.9(2) 1.65(3)
	Cu(2)	x Biso (Å 3 )	0.2514(4) 1.84(4)	0.2527(3) 2.17(3)	0.2539(2) 2.22(2)	0.2540(3) 1.96(3)	0.2526(3) 1.55(3)	0.2529(3) 1.65(3)
	Cu(3) Sn	x Biso (Å 3 ) Biso (Å 3 )	0.2529(6) 1.84(4) 6.4(1)	0.2535(5) 2.17(3) 3.01(5)	0.2548(3) 2.22(2) 2.91(4)	0.2517(8) 1.96(3) 3.93(6)	0.2529(5) 1.55(3) 3.86(7)	0.2530(5) 1.65(3) 4.40(7)
	S(1)	Biso (Å 3 )	1.23(5)	1.70(5)	1.81(3)	1.63(4)	1.14(4)	1.36(4)
		x	0.1355(6)	0.1318(6)	0.1270(6)	0.1299(7)	0.1324(5)	0.1321(6)
	S(2)	Biso (Å 3 )	1.23(5)	1.70(5)	1.81(3)	1.63(4)	1.14(4)	1.36(4)
		x	0.3867(4)	0.3798(3)	0.3798(2)	0.3821(3)	0.3835(3)	0.3836(3)
		y	0.3655(4)	0.3654(3)	0.3672(3)	0.3659(3)	0.3656(3)	0.3649(3)
		z	0.1196(5)	0.1221(6)	0.1243(5)	0.1225(6)	0.1204(5)	0.1216(5)
	Impurities	-	2% Cu3VS4	-	-	-	-

Table 2 .

 2 Measured volume mass, relative density and detected impurities for colusite samples sintered at 873 K, 923 K, 973 K and 1023 K, before and after HIP at 873 K and 1023 K.

	Sample	THIP (K

) ρ (g cm -3 ) d (%) Impurities (wt%)

	SPS873	-	4.57	99.8	Sulvanite (2.5 %) Vanadium (0.5 %)
	SPS923	-	4.52	98.8	Sulvanite (2.2 %)
	SPS973	-	4.38	93.0	-
	SPS1023	-	4.20	91.3	-
	SPS873	873	4.71	102	-
	SPS923	873	4.70	102	-
	SPS973	873	4.69	102	-
	SPS1023	873	4.68	102	-
	SPS873	1023	4.64	101	-
	SPS923	1023	4.64	101	-
	SPS973	1023	4.64	101	-
	SPS1023	1023	4.63	101	-

Acknowledgements

The authors would like to thank Christelle Bilot and Jerôme Lecourt for technical support. This work was supported by the French Agence Nationale de la Recherche (ANR) through the programs Energy Challenge for Secure, Clean and Efficient Energy (Challenge 2, 2015, ANR-15-CE05-0027), FEDER and Normandy Region. The authors would like to extend their thanks to Tribotecc for supplying large amounts of industrial grade precursors.

Supplementary Information

The Supplementary Information is available at DOI: Figures S1-S7;

Author Information

The authors declare no conflict of interest.