The simplest Cremona transformation of projective 3-space is the involution σ : (x 0 :

x 1 : x 2 : x 3 ) → 1 x 0 : 1 x 1 : 1 x 2 : 1 x 3 ,
which is a diffeomorphism outside the tetrahedron (x 0 x 1 x 2 x 3 = 0). More generally, if L i := j a ji x j are linear forms defining the faces of a tetrahedron, we get the Cremona transformation σ L : (x 0 :

x 1 : x 2 : x 3 ) → 1 L 0 : 1 L 1 : 1 L 2 : 1 L 3 • a ij -1 ,
which is a diffeomorphism outside the tetrahedron (L 0 L 1 L 2 L 3 = 0). The vertices of the tetrahedron are called the base points. If Q is a quadric surface in P 3 , its image under a Cremona transformation is, in general, a sextic surface. However, if Q passes through the 4 base points, then its image σ L (Q) is again a quadric surface in P 3 passing through the 4 base points. In many cases, we can view σ L as a map of Q to itself. The aim of this paper is to show that these Cremona transformations generate both the group of automorphisms and the group of diffeomorphisms of the sphere, the torus and of all non-orientable surfaces.

Let us start with the sphere S 2 := (x 2 + y 2 + z 2 = 1) ⊂ R 3 and view this as the set of real points of the quadric Q := (x 2 + y 2 + z 2 = t 2 ) ⊂ P 3 in projective 3-space. Pick 2 conjugate point pairs p, p, q, q on the complex quadric Q(C) and let σ p,q denote the Cremona transformation with base points p, p, q, q. As noted above, σ p,q (Q) is another quadric surface. The faces of the tetrahedron determined by these 4 points are disjoint from S 2 , hence σ p,q is a diffeomorphism from S 2 to the real part of σ p,q (Q). Thus Q and σ p,q (Q) are projectively equivalent and the corresponding Cremona transformation σ p,q can be viewed as a diffeomorphism of S 2 to itself, well defined up to left and right multiplication by O(3, 1). It is also convenient to allow the points p, q to coincide; see (9) for a precise definition. Let us call these the Cremona transformations with imaginary base points. Our first result is that, algebraically, these generate the automorphism group.

Theorem 1. The Cremona transformations with imaginary base points σ p,q and O(3, 1) generate Aut(S 2 ).

Most diffeomorphisms of S 2 are not algebraic, so the best one can hope for is that these Cremona transformations generate Diff(S 2 ) as topological group. Equivalently, that Aut(S 2 ) is a dense subgroup of Diff(S 2 ). The results of [START_REF] Lukackiī | The structure of Lie algebras of spherical vector fields and the diffeomorphism groups of S n and RP n[END_REF][START_REF] Lukackiī | The system of generators of the group of diffeomorphisms of an n-dimensional torus[END_REF], pointed out to us by M. Zaidenberg, imply that the group of algebraic automorphisms is dense in the group of diffeomorphisms for the sphere and the torus. His methods, reviewed in (15), use the SO(3, R) action on the sphere and the torus action on itself.

In order to go futher, first we need to deal with diffeomorphisms with fixed points. Building on , it is proved in that Aut(S 2 ) is n-transitive for any n ≥ 1. Using this, it is easy to see (19) that the density property also holds with assigned fixed points.

Corollary 2. Aut(S 2 , p 1 , . . . , p n ) is dense in Diff(S 2 , p 1 , . . . , p n ) for any finite set of distinct points p 1 , . . . , p n ∈ S 2 , where Aut( ) denotes the group of algebraic automorphisms of S 2 fixing p 1 , . . . , p n and Diff( ) the group of diffeomorphisms fixing p 1 , . . . , p n .

Note that, for a real algebraic variety X, the semigroup of algebraic diffeomorphisms is usually much bigger than the group of algebraic automorphisms Aut(X). For instance, x → x + 1

x 2 +1 is an algebraic diffeomorphism of R (and also of RP 1 ∼ S 1 ), but its inverse involves square and cube roots. The difference is best seen in the case of the circle S 1 = (x 2 + y 2 = 1).

Essentially by the Weierstrass approximation theorem, any differentiable map φ : S 1 → S 1 can be approximated by polynomial maps Φ : S 1 → S 1 . By contrast, the group of algebraic automorphisms of S 1 is the real orthogonal group O(2, 1) ∼ = P GL(2, R), which has real dimension 3. Thus Aut(S 1 ) is a very small closed subgroup in the infinite dimensional group Diff(S 1 ).

The Cremona transformations with real base points do not give diffeomorphisms of S 2 ; they are not even defined at the real base points. Instead, they give generators of the mapping class groups of non-orientable surfaces.

Let R g be a non-orientable, compact surface of genus g without boundary. Coming from algebraic geometry, we prefer to think of it as S 2 blown up at g points p 1 , . . . , p g ∈ S 2 . Topologically, R g is obtained from S 2 by replacing g discs centered at the p i by g Möbius bands. Up to isotopy, a blow-up form of R g is equivalent to giving g disjoint embedded Möbius bands M 1 , . . . , M g ⊂ R g .

There are two ways to think of a Cremona transformation with real base points as giving elements of the mapping class group of R g .

Let us start with the case when there are four real base points p 1 , . . . , p 4 . We can factor the Cremona transformation σ p1,...,p4 as σ p1,...,p4 :

Q π1 ← B p1,...,p4 Q π2 → Q
where on the left π 1 : B p1,...,p4 Q → Q is the blow up of Q at the 4 points p 1 , . . . , p 4 and on the right π 2 : B p1,...,p4 Q → Q contracts the birational transforms of the circles Q ∩ L i where the {L i } are the faces of the tetrahedron with vertices {p i }. In Figure 1, the • represent the 4 base points. On the left hand side, the 4 exceptional curves E i lie over the four points marked •. On the right hand side, the images of the E i are 4 circles, each passing through 3 of the 4 base points. Since σ p1,...,p4 is an involution, dually, the four points marked • on the right hand side map to the 4 circles on the left hand side.

A Cremona transformation σ p1,p2,q,q with 2 real and a conjugate complex pair of base points act similarly. Here only two Möbius bands are altered.

In general, we can think of the above real Cremona transformation σ p1,...,p4 as a topological operation that replaces the set of g Möbius bands M 1 , . . . , M g by a new set M ′ 1 , . . . , M ′ 4 , M 5 , . . . , M g . In this version, σ p1,...,p4 is the identity on the surfaces but acts nontrivially on the set of isotopy classes of g disjoint Möbius bands. One version of our result says that the transformations σ p1,...,p4 and σ p1,p2,q,q act transitively on the set of isotopy classes of g disjoint Möbius bands. The other way to view σ p1,...,p4 is as follows. First, we obtain an isomorphism σ ′ p1,...,p4 : B p1,...,pg S 2 ∼ = B q1,...,qg S 2 for some q 1 , . . . , q g ∈ S 2 . Under this isomorphism, the exceptional curve E(p i ) ⊂ B p1,...,pg S 2 is mapped to the exceptional curve E(q i ) ⊂ B q1,...,qg S 2 for i ≥ 5 and to the circle passing through the points {q j : 1 ≤ j ≤ 4, j = i} for i ≤ 4. As we noted above, there is an automorphism Φ ∈ Aut(S 2 ) such that Φ(q

i ) = p i for 1 ≤ i ≤ n. Thus Φ • σ ′ p1,...,p4 : B p1,...,pg S 2 ∼ = -→ B p1,...,pg S 2
is an automorphism of B p1,...,pg S 2 which maps E(p i ) to E(q i ) for i ≥ 5 and to a simple closed curve passing through the points {p j : 1 ≤ j ≤ 4, j = i} for i ≤ 4.

Theorem 3. For any g, the Cremona transformations with 4, 2 or 0 real base points generate the (non-orientable) mapping class group M g .

Finally, we can put these results together to obtain a general approximation theorem for diffeomorphisms of such real algebraic surfaces.

Theorem 4. Let R be a compact, smooth, real algebraic surface birational to P 2 and q 1 , . . . , q n ∈ R distinct marked points. Then the group of algebraic automorphisms Aut(R, q 1 , . . . , q n ) is dense in Diff(R, q 1 , . . . , q n ).

As a topological manifold, here R can be the sphere, the torus or any nonorientable surface

RP 2 # • • • #RP 2 .
5 (Other algebraic varieties). Similar assertions definitely fail for most other algebraic varieties. Real algebraic varieties of general type have only finitely many birational automorphisms. (See [START_REF] Ueno | Classification theory of algebraic varieties and compact complex spaces[END_REF] for an introduction to these questions.) For varieties whose Kodaira dimension is between 0 and the dimension, every birational automorphism preserves the Iitaka fibration. If the Kodaira dimension is 0 (e.g., Calabi-Yau varieties, Abelian varieties), then every birational automorphism preserves the canonical class, that is, a volume form, up to sign. The automorphism group is finite dimensional but may have infinitely many connected components. In particular, using [START_REF] Comessatti | Sulla connessione delle superfizie razionali reali[END_REF], for surfaces we obtain the following. Proposition 6. Let S be a smooth real algebraic surface. If S(R) is an orientable surface of genus ≥ 2 then Aut(S) is not dense in Diff S(R) .

If X has Kodaira dimension -∞, then every birational automorphism preserves the MRC fibration [START_REF] Kollár | Rational curves on algebraic varieties[END_REF]Sec.IV.5]. Thus the main case when density could hold is when the variety is rationally connected [START_REF] Kollár | Rational curves on algebraic varieties[END_REF]Sec.IV.3]. It is clear that the analog of (1) fails even for most geometrically rational real algebraic surfaces. Consider, for instance, the case when R → P 1 is a minimal conic bundle with at least 8 singular fibers. Then Aut(R) is infinite dimensional, but every automorphism of R preserves the conic bundle structure [START_REF] Iskovskikh | Factorization of birational mappings of rational surfaces from the point of view of Mori theory[END_REF]Thm. 1.6(iii)]. Conic bundles with 4 singular fibers are probably the only other case where the analog of (4) holds.

The results of [START_REF] Lukackiī | The structure of Lie algebras of spherical vector fields and the diffeomorphism groups of S n and RP n[END_REF] imply that Aut(S n ) is dense in Diff(S n ) for every n ≥ 2 and, similarly, Aut(T n ) is dense in Diff(T n ) for every n ≥ 2, where T n is the n-dimensional torus. It is not clear to us what happens with other varieties birational to P n .

(History of related questions)

. There are many results in real algebraic geometry that endow certain topological spaces with a real algebraic structure or approximate smooth maps by real algebraic morphisms. In particular real rational models of surfaces were studied in , [START_REF] Mangolte | Real algebraic morphisms on 2-dimensional conic bundles[END_REF] and approximations of smooth maps to spheres by real algebraic morphisms were investigated in , , [START_REF] Kucharz | Algebraic morphisms into rational real algebraic surfaces[END_REF], [Joglar-Kollár03], [Joglar-Mangolte04], [START_REF] Mangolte | Real algebraic morphisms on 2-dimensional conic bundles[END_REF].

An indication that Aut(S 2 ) is surprisingly large comes from [Biswas-Huisman07], with a more precise version developed in .

(Plan of the proofs).

In Section 1 we prove that the Cremona transformations with imaginary base points generate Aut(S 2 ). Next, in Section 2, we prove (4) for the identity components. If φ : R → R is homotopic to the identity, then φ can be written as the composite of diffeomorphisms φ i : R → R such that each φ i is the identity outside a small open set W i ⊂ R. Moreover, we can choose the W i in such a way that for every i there is a morphism π i : R → S 2 that is an isomorphism on W i . The map φ i then pushes down to a diffeomorphism of S 2 . We take an approximation there and lift it to R.

The case R = S 1 × S 1 follows from [START_REF] Lukackiī | The system of generators of the group of diffeomorphisms of an n-dimensional torus[END_REF]. Generators of the mapping class group of non-orientable surfaces have been written down by [START_REF] Chillingworth | A finite set of generators for the homeotopy group of a non-orientable surface[END_REF] and [START_REF] Korkmaz | Mapping Class Groups of Nonorientable Surfaces[END_REF]. In Section 3 we describe a somewhat different set of generators. We thank M. Korkmaz for his help in proving these results.

Theorem 3 is proved in Section 4. We show by explicit constructions that our generators are given by Cremona transformations.

1. Generators of Aut(S 2 ) Max Noether proved that the involution (x, y, z) → 1

x , 1 y , 1 z and P GL 3 generate the group of birational self-maps Bir(P 2 ) over C. Using similar ideas, proved that Aut(P 2 (R)) is generated by linear automorphisms and certain real algebraic automorphisms of degree 5. In this section, we prove that Aut(S 2 ) is generated by linear automorphisms and by the σ p.q . The latter are real algebraic automorphisms of degree 3.

Example 9 (Cubic involutions of P 3 ). On P 3 take coordinates (x, y, z, t). We need two types of cubic involutions of P 3 . Let us start with the Cremona transformation

(x, y, z, t) → 1 x , 1 y , 1 z , 1 t = 1
xyzt (yzt, ztx, txy, xyz) whose base points are the 4 "coordinate vertices". We will need to put the base points at complex conjugate point pairs, say (1, ±i, 0, 0), (0, 0, 1, ±i). Then the above involution becomes

τ : (x, y, z, t) → (x 2 + y 2 )z, (x 2 + y 2 )t, (z 2 + t 2 )x, (z 2 + t 2 )y . Check that τ 2 (x, y, z, t) = (x 2 + y 2 ) 2 (z 2 + t 2 ) 2 • (x, y, z, t),
thus τ is indeed a rational involution on P 3 . Consider a general quadric passing through the points (1, ±i, 0, 0), (0, 0, 1, ±i).

It is of the form

Q = Q abcdef (x, y, z, t) := a(x 2 + y 2 ) + b(z 2 + t 2 ) + cxz + dyt + ext + f yz. By direct computation, Q abcdef τ (x, y, z, t) = (x 2 + y 2 )(z 2 + t 2 ) • Q abcdf e (x, y, z, t).
(Note that ef changes to f e. Thus, if e = f , then τ restricts to an involution of the quadric (Q = 0) but not in general.)

Assume now that we are over R. We claim that τ is regular on the real points if a, b = 0. The only possible problem is with points where (x 2 + y 2 )(z 2 + t 2 ) = 0. Assume that (x 2 + y 2 ) = 0. Then x = y = 0 and so Q(x, y, z, t) = 0 gives that b(z 2 + t 2 ) = 0 hence z = t = 0, a contradiction.

Whenever Q has signature (3, 1), we can view (Q = 0) as a sphere and then τ gives a real algebraic automorphism of the sphere S 2 , which is well defined up to left and right multiplication by O(3, 1). A priori the automorphisms depend on a, b, c, d, e, f , so let us denote them by τ abcdef .

Given S 2 , the above τ abcdef depends on the choice of the base points, that is, 2 conjugate pairs of points on the complex quadric S 2 (C). The group O(3, 1) has real dimension 6. Picking 2 complex points has real dimension 8. So the τ abcdef should give a real 2-dimensional family of automorphisms modulo O(3, 1).

We also need a degenerate version of the Cremona transformation when the 4 base points come together to a pair of points. With base points (1, 0, 0, 0) and (0, 1, 0, 0), we get (x, y, z, t) → (xz 2 , yt 2 , zt 2 , z 2 t).

If we put the base points at (1, ±i, 0, 0) then we get the transformation

σ ′ : (x, y, z, t) → y(z 2 -t 2 ) + 2xzt, x(t 2 -z 2 ) + 2yzt, t(z 2 + t 2 ), z(z 2 + t 2 ) .
Take any quadric of the form

Q = Q ′ abcdef (x, y, z, t) := a(x 2 + y 2 ) + bz 2 + czt + dt 2 + e(xt + yz) + f (xz -yt). By direct computation, Q ′ abcdef σ ′ (x, y, z, t) = (z 2 + t 2 ) 2 • Q ′ adcbef (x, y, z, t).
As before, if Q ′ has signature (3, 1), we can view (Q ′ = 0) as a sphere and then σ ′ gives a real algebraic automorphism of the sphere S 2 , which is well defined up to left and right multiplication by O(3, 1). Let us denote them by σ abcdef . Despite the dimension count, the group O(3, 1) does not act with a dense orbit on the set of complex conjugate point pairs and complex conjugate directions. Indeed, after complexification, the quadric becomes P 1 × P 1 and we can chose the two points to be p 1 := (0, 0) and p 2 := (∞, ∞). The subgroup fixing these two points is C * × C * and the diagonal acts trivially on the tangent directions at both of the points p i . Thus the σ abcdef form a 1-dimensional family.

Theorem 10. The group of algebraic automorphisms of S 2 is generated by O(3, 1), the τ abcdef and σ abcdef .

Remark 11. It is possible that the τ abcdef alone generate Aut(S 2 ). In any case, as the 4 base points come together to form 2 pairs, the τ abcdef converge to the corresponding σ abcdef . Thus the τ abcdef generate a dense subgroup of Aut(S 2 ) (in the C ∞ -topology.)

One reason to use the σ abcdef is that, as the proof shows, the τ abcdef and σ abcdef together generate Aut(S 2 ) in an "effective manner." By this we mean the following.

Any rational map Φ : S 2 S 2 can be given by 4 polynomials

Φ(x, y, z, t) = Φ 1 , Φ 2 , Φ 3 , Φ 4 .
Note that Φ does not determine the Φ i uniquely, but there is a "minimal" choice. We can add any multiple of x 2 + y 2 + z 2 -t 2 to the Φ i and we can cancel common factors. We choose max i {deg Φ i } to be minimal and call it the degree of Φ. It is denoted by deg Φ. (It is easy to see that these minimal Φ i are unique up to a multiplicative constant.) Note that deg Φ = 1 iff Φ ∈ O(3, 1).

By "effective" generation we mean that given any Φ ∈ Aut(S 2 ) with deg Φ > 1, there is a ρ which is either of the form τ abcdef or σ abcdef such that deg Φ • ρ) < deg Φ.

12 (Proof of (10)). The proof is an application of the Noether-Fano method. See for details.

Let k be a field and

Q ⊂ P 3 a quadric defined over k. Assume that Pic(Q) = Z[H]
where H is the hyperplane class. Let Q ′ be any other quadric and Φ :

Q Q ′ a birational map. Then Γ := Φ * |H Q ′ | is a 3-dimensional linear system on Q and Γ ⊂ |dH Q | for some d.
Let p i be the (possibly infinitely near) base points of Γ (over k) and m i their multiplicities. As in 2.8], we have the equalities

Γ 2 - m 2 i = deg Q ′ and Γ • K Q + m i = deg K Q ′ .
In our case, these become

m 2 i = 2d 2 -2 and m i = 4d -4.
Next we see how the transformations τ abcdef and σ abcdef change the degree of a linear system Γ.

Example 13 (Cremona transformation on a quadric). For the τ abcdef series, pick 4 distinct points p 1 , . . . , p 4 ∈ Q such that no two are on a line in Q, not all 4 on a conic and assume that s := m 1 + • • • + m 4 > 2d. Blow up the 4 points and contract the 4 conics that pass through any 3 of them. The p i are replaced by 4 base points of multiplicities 2d-s+m i . Their sum is 8d-4s+s = 8d-3s. Thus 4d-4 = m i is replaced by

m i -s + (8d -s), hence d becomes d -(s -2d) < d.
For σ abcdef , pick 2 distinct points p 1 , p 2 ∈ Q and 2 infinitely near points p 3 → p 1 and p 4 → p 2 such that no two are on a line in Q, not all 4 on a conic and assume that s

:= m 1 + • • • + m 4 > 2d.
Blow up the points p 1 , p 2 and then the points p 3 , p 4 . After this, we can contract the two conics that pass through p 1 + p 2 + p 3 (resp. p 1 + p 2 + p 4 ) and we can also contract the birational transforms of the exceptional curves over p 1 and p 2 . The rest of the computation is the same. The p i are replaced by 4 base points of multiplicities 2d-s+m i . Their sum is 8d-4s+s = 8d-3s. Thus 4d -4 = m i is replaced by

m i -s + (8d -s) hence d becomes d -(s -2d) < d.
Thus, as long as we can find p 1 , . . . , p 4 ∈ Q (or infinitely near) such that m 1 + • • • + m 4 > 2d, we can lower deg Φ by a suitable degree 3 Cremona transformation.

In order to find such p i , assume first to the contrary that m i ≤ d/2 for every i.

Then 2d 2 -2 = m 2 i ≤ d 2 m i = d 2 (4d -4) = 2d 2 -2d
. This is a contradiction, unless d = 1 and Φ is a linear isomorphism.

If we work over R and we assume that there are no real base points, then we have at least one complex conjugate pair of base points with multiplicity m i > d/2. We are done if we have found 2 such pairs.

In any case, up to renumbering the points, we have m 1 = m 2 = d 2 + a for some d/2 ≥ a > 0. Assume next that all the other

m j ≤ d 2 -a. Then 2d 2 -2 = m 2 i ≤ 2 d 2 -a 2 + d 2 -a m i -d + 2a = 2 d 2 -a 2 + d 2 -a 4d -4 -d + 2a
. By expanding, this becomes (a + 2)(d -4) ≤ -6.

Thus d ∈ {1, 2, 3}. If d = 3 then a + 2 ≥ 6 so d/2 ≥ a ≥ 4 gives a contradiction. If d = 2 then we get a = 1. Thus Γ consists of quadric sections with singular points at p 1 , p 2 . These are necessarily reducible (they have p a = 1 with 2 singular points), again impossible.

We also need to show that no two of the points lie on a line and not all 4 are on a conic. For any line

L ⊂ Q(C), (L • Γ) = d gives that i:pi∈L m i ≤ d.
In particular, m i ≤ d for every i and if p i , p j are on a line then m i + m j ≤ d. Thus out of p 1 , . . . , p 4 only p 3 , p 4 could be on a line. But p 3 , p 4 are conjugates, thus they would be on a real line. There is, however, no real line on S 2 .

Similarly, for any conic C ⊂ Q(C), (C • Γ) = 2d gives that i:pi∈C m i ≤ 2d. Thus not all 4 points are on a conic.

Remark 14. Note that we started the proof over an arbitrary field, but at the end we had to assume that that we worked over R. For a quadric surface Q with Picard number one, the above method should give generators for the group Bir * (Q) of those birational self-maps that are regular along Q(k). However, for other fields k, other generators also appear if there are more than 2 conjugate base points.

The identity component

The purpose of this section is to prove (4) for the identity components. For the sphere and the torus these were done by Lukackiȋ. Next we prove (4) for the identity components in the case R is the non-orientable surface R g .

15 (The results of Lukackiȋ). The paper [Lukackiī77, Thm. 2] proves that SO(n + 1, 1) is a maximal closed subgroup of Diff 0 (S n ). In particular, O(n + 1, 1) and anything else generate a dense subgroup of Diff(S n ).

Since this result seems not to have been well known, let us give a quick review of the steps of the proof.

We start with the Lie algebra of polynomial vector fields H 0 (S n , T S n ). Its structure as an so(n + 1) representation was described by [START_REF] Kirillov | The representations of the group of rotations of an n-dimensional Euclidean space by spherical vector fields[END_REF], including the highest weight vectors.

As we go from so(n + 1) to so(n + 1, 1), we get extra unipotent elements and their action on the highest weight vectors can be computed explicitly. One obtains that so(n + 1, 1) is a maximal Lie subalgebra of H 0 (S n , T S n ). This implies that SO(n + 1, 1) is a maximal connected closed subgroup of Diff 0 (S n ). It is easy to check that SO(n + 1, 1) is its own normalizer, which rules out all disconnected subgroups as well.

The paper [START_REF] Lukackiī | The system of generators of the group of diffeomorphisms of an n-dimensional torus[END_REF] gives generators of the Lie algebra H 0 (T n , T T n ) where T n denotes the n-dimensional torus

T n := x 2 1 + y 2 1 -1 = • • • = x 2 n + y 2 n -1 = 0 ⊂ R 2n
. This is again through explicit Lie theory. Up to coordinate changes by GL(n, Z), the generators are the shears

g(x 1 , . . . , x n-1 ) • ∂ ∂x n - ∂ ∂y n and y n • ∂ ∂x n - ∂ ∂y n .
(Using polar angles φ i , the latter is the vector field sin φ n • ∂/∂φ n .) Up to a factor of 2, this is exactly the tangent vector field corresponding to the unipotent group

1 t 0 1 ⊂ P GL(2, R) ∼ = O(2, 1) acting on S 1 .
Definition 16. Let X and Y be real algebraic manifolds and let I be any subset of

X. A map f from I into Y is algebraic if there is a Zariski open subset U of X containing I such that f is the restriction of an algebraic map from U into Y .
Consider the standard sphere S 2 ⊂ R 3 and let L be a line through the origin. Choose coordinates such that L is the x-axis and S 2 := (

x 2 + y 2 + z 2 = 1) ⊂ R 3 . Let M : [-1, 1] → O(2) be a real algebraic map. Then Φ M : S 2 → S 2 given by (x, y, z) → x, (y, z) • M (x)
is an automorphism of S 2 , called the twisting map with axis L and associated to M . A conjugate of a twisting map by an element of O(3, 1) is also called a twisting map.

The following results are proved in .

Theorem 17. Notation as above.

(1) Any C ∞ map M 0 : [-1, 1] → O(2) can be approximated by real algebraic maps M s : [-1, 1] → O(2). Moreover, given finitely many points t i ∈ [-1, 1], we can choose the M s such that M s (t i ) = M 0 (t i ) for every i.

(2) Given distinct points p 1 , . . . , p m and q 1 , . . . , q m there are two twisting maps (with different axes) Φ 1 and Φ 2 such that Φ 1 • Φ 2 (q i ) = p i for every i.

Moreover, (a) if p j = q j for some values of j then we can assume that Φ 1 (q j ) = Φ 2 (q j ) = q j for these values of j, and (b) if p i is near q i for every i then we can assume that the Φ 1 , Φ 2 are near the identity.

(3) Let R be any real algebraic surface that is obtained from S 2 by repeatedly blowing up m real (possibly infinitely near) points and let r 1 , . . . , r n be points in R. Then there are (nonunique) distinct points p 1 , . . . , p m and q 1 , . . . , q n and an isomorphism φ : R → B p1,...,pm S 2 such that φ(r i ) = q i .

By adding more points in (17.3) and compactness, we obtain the following stronger version:

Corollary 18. Let R be any real algebraic surface that is obtained from S 2 by repeatedly blowing up m real (possibly infinitely near) points and let r 1 , . . . , r n be points in R. There is a finite open cover R = ∪ j W j such that for every j there are distinct points p 1j , . . . , p mj , q 1j , . . . , q nj ∈ S 2 and an isomorphism φ j : R → B p1j ,...,pmj S 2 such that φ j (r i ) = q ij and φ j (W j ) ⊂ S 2 \ {p 1j , . . . , p mj }.

19 (Proof of (2)). Let p 1 , . . . , p n , q ∈ S 2 be any finite set of distinct points, and let φ ∈ Diff(S 2 , p 1 , . . . , p n ). By (15) there are automorphisms ψ s ∈ Aut(S 2 ) such that ψ s converges to φ.

For any s and i, set q s i := ψ s (p i ). As ψ s converges to φ, the q s i converge to p i for every i. By (17.2.b) there are automorphisms Φ s such that Φ s (q s i ) = p i and Φ s converges to the identity. Thus the composites Φ s • ψ s are in Aut(S 2 , p 1 , . . . , p n ) and they converge to φ.

Proposition 20. Let R be any real algebraic surface that is obtained from S 2 by repeatedly blowing up g real (possibly infinitely near) points and let r 1 , . . . , r n be points in R. Then the group Aut 0 (R, r 1 , . . . , r n ) of algebraic automorphisms homotopic to identity is dense in Diff 0 (R, r 1 , . . . , r n ).

Proof. Let φ : R → R be a diffeomorphism fixing r 1 , . . . , r n , and homotopic to the identity. Choose R = ∪ j W j as in (18). By a partition of unity argument, φ can be written as the composite of diffeomorphisms φ ℓ : R → R fixing r 1 , . . . , r n such that each φ ℓ is the identity outside some W j ⊂ R.

In particular, each φ ℓ descends to a diffeomorphism φ ′ ℓ of S 2 which fixes the points p 1j , . . . , p gj and q 1j , . . . , q nj . By (2), we can approximate φ ′ ℓ by algebraic automorphisms Φ ′ ℓ,s fixing all the points p 1j , . . . , p gj and q 1j , . . . , q nj . Since the Φ ′ ℓ,s fix p 1j , . . . , p gj , they lift to algebraic automorphisms Φ ℓ,s of R ∼ = B p1j ,...,pgj S 2 fixing the points r 1 , . . . , r n . The composite of the Φ ℓ,s then converges to φ.

Generators of the mapping class group

Definition 21. Let R be a compact, closed surface and p 1 , . . . , p n distinct points on R. The mapping class group is the group of connected components of those diffeomorphisms φ : R → R such that φ(p i ) = p i for i = 1, . . . , n.

M(R, p 1 , . . . , p n ) := π 0 Diff(R, p 1 , . . . , p n ) .
Up to isomorphism, this group depends only on the orientability and the genus of R. The orientable case has been intensely studied. Recent important results about the non-orientable case are in [START_REF] Korkmaz | Mapping Class Groups of Nonorientable Surfaces[END_REF][START_REF] Wahl | Homological stability for the mapping class groups of non-orientable surfaces[END_REF].

(In the literature, M g,n is used to denote both the mapping class group of an orientable genus g (hence with Euler characteristic 2 -2g) surface with n marked points and the mapping class group of a non-orientable genus g (hence with Euler characteristic 2 -g) surface with n marked points.)

In preparation for the next section, we establish a somewhat new explicit set of generators in the non-orientable case.

Write R as B p1,...,pg S 2 , the blow up of S 2 at g points. We start by describing some elements of the mapping class group. For more details see [START_REF] Lickorish | On the homeomorphisms of a non-orientable surface[END_REF][START_REF] Chillingworth | A finite set of generators for the homeotopy group of a non-orientable surface[END_REF][START_REF] Korkmaz | Mapping Class Groups of Nonorientable Surfaces[END_REF].

Definition 22 (Dehn twist). Let R be any surface and C ⊂ R a simple closed smooth curve such that R is orientable along C. Cut R along C, rotate one side around once completely and glue the pieces back together. This defines a diffeomorphism t C of R, see Figure 2.

The inverse t -1

C corresponds to rotating one side the other way. Up to isotopy, the pair {t C , t -1 C } does not depend on the choice of C or the rotation. Either of t C and t -1

C is called a Dehn twist using C. On an oriented surface, with C oriented, one can make a sensible distinction between t C and t -1 C . This is less useful in the non-orientable case.

Definition 23 (Crosscap slide). Let D be a closed disc and p, q ∈ D two points. Take a simple closed curve C in D passing through p, q and let C ′ denote the corresponding curve in B q D. Let M p be a small disc around p. Let {φ t : t ∈ [0, 1]} be a continuous family of diffeomorphisms of B q D such that φ 0 is the identity, each φ t is the identity near the boundary and as t increases, the φ t slide M p once around C. At t = 1, M p returns to itself with its orientation reversed, as in Figure 3. In particular, φ 1 (p) = p. Thus φ 1 can be lifted to a diffeomorphism of B p,q D which is not isotopic to the identity but is the identity near the boundary.

Let R be any surface, U ⊂ R a closed subset with C ∞ boundary and τ : U → B p,q D a diffeomorphism. Then τ -1 φ 1 τ : U → U is the identity near the boundary of U , hence it can be extended by the identity on R \ U to a diffeomorphism of R. Up to isotopy, this diffeomorphism does not depend on the choice of C, φ t and τ . It is called a cross-cap slide or a Y -homeomorphism using U . Note that for a cross-cap slide to exist, R must be non-orientable and of genus at least 2.

24 (Generators of the mapping class group). Let R g be a non-orientable surface of genus g ≥ 1. We write R g := B p1,...,pg S 2 with exceptional curves E i ⊂ R g and let π : R g → S 2 be the blow down map.

The map π gives a one-to-one correspondence between • simple closed smooth curves C R ⊂ R g whose intersection with any exceptional curve E i is transversal, and • immersed curves C = π(C R ) ⊂ S 2 whose only self-intersections are at the p i and no two branches are tangent. Generators of the mapping class group were first established by [START_REF] Lickorish | On the homeomorphisms of a non-orientable surface[END_REF] and simplified by [START_REF] Chillingworth | A finite set of generators for the homeotopy group of a non-orientable surface[END_REF]. The case with marked points was settled by [START_REF] Korkmaz | Mapping Class Groups of Nonorientable Surfaces[END_REF].

The generators are the following (1) Dehn twists along C R for certain smooth curves C ⊂ S 2 that pass through an even number of the p 1 , . . . , p g . (No self-intersections at the p i .) (2) Cross-cap slides using a disc D ⊂ S 2 that contains exactly 2 of the p 1 , . . . , p g . The results of [START_REF] Chillingworth | A finite set of generators for the homeotopy group of a non-orientable surface[END_REF] and of [START_REF] Korkmaz | Mapping Class Groups of Nonorientable Surfaces[END_REF] are more precise in that only very few of these generators are used. In the unmarked case, the above formulation is established in the course of the proof and stated on [START_REF] Chillingworth | A finite set of generators for the homeotopy group of a non-orientable surface[END_REF]p.427].

We will need somewhat different generators. We thank M. Korkmaz for answering many questions and especially for pointing out that one should use the lantern relation ( 26) to establish the following.

Proposition 25. The following elements generate the mapping class group of the marked surface B p1,...,pg S 2 , q 1 , . . . , q n ).

(1) Dehn twists along C R for certain smooth curves C ⊂ S 2 that pass through 0, 2 or 4 of the points p 1 , . . . , p g . (No self-intersections at the p i .) (2) Cross-cap slides using a disc D ⊂ S 2 that contains exactly 2 of the points p 1 , . . . , p g .

Proof. We have included all the cross-cap slides from ( 24). Thus we need to deal with Dehn twists along C R where C ⊂ S 2 is a simple closed curve passing through m of the points p 1 , . . . , p g with m > 4.

Using induction, it is enough to show that the Dehn twist along C R can be written as the product of Dehn twists along curves C ′ R where each C ′ ⊂ S 2 is a simple closed curve passing through fewer than m of the points p 1 , . . . , p g .

Assume for simplicity that C passes through p 1 , . . . , p m with m > 4 (and even). For I ⊂ {1, . . . , m} let t I be a Dehn twist using a simple closed curve C I passing through the {p i : i ∈ I} but none of the others. The precise choice of the curve will be made later. We show that, with a suitable choice of the curves, t 12345...m is a product of the

(Lantern relation of Dehn

). [START_REF] Dehn | Die Gruppe der Abbildungsklassen[END_REF][START_REF] Johnson | Homeomorphisms of a surface which act trivially on homology[END_REF] Fix 4 points q 0 , . . . , q 3 ∈ S 2 . Let t i be the Dehn twist using a small circle around q i and for i, j ∈ {1, 2, 3}, let t ij be the Dehn twist using a simple closed curve that separates q i , q j from the other 2 points. Then, with suitable orientations, t 0 t 1 t 2 t 3 = t 12 t 13 t 23 , (26.1)

where the equality is understood to hold in M(S 2 , q 0 , . . . , q 3 ).

Automorphisms and the mapping class group

The main result of this section is the following.

Theorem 27. Let R be a real algebraic surface that is obtained from S 2 by blowing up points and p 1 , . . . , p n ∈ R distinct marked points. Then the natural map

Aut(R, p 1 , . . . , p n ) ։ M(R, p 1 , . . . , p n ) is surjective.
Proof. We prove that all the generators of the mapping class group listed in (25) can be realized algebraically. There are 4 cases to consider:

(1) Dehn twists along C R ⊂ R for smooth curves C ⊂ S 2 that pass through either (a) none of the points p i , (b) exactly 2 of the points p i , or (c) exactly 4 of the points p i . (2) Cross-cap slides using a disc D ⊂ S 2 that contains exactly 2 of the points p i . We start with the easiest case (27.1.a).

28 (Algebraic realization of Dehn twists). Let C ⊂ S 2 be a smooth curve passing through none of the points p i . After applying a suitable automorphism of S 2 , we may assume that C is the big circle (x = 0).

Consider the map g : 2) be an algebraic approximation of g such that the corresponding twisting (16) Φ M is the identity at the points p i . Then Φ M is an algebraic realization of the Dehn twist around C.

[-1, 1] → O(2) where g(t) = 1 for t ∈ [-1, -ǫ] ∪ [ǫ, 1] and g(t) is the rotation by angle π(1 + t/ǫ) for t ∈ [-ǫ, ǫ]. Let M : [-1, 1] → O(
On the torus, the same argument works for either of the S 1 -factors. Up to isotopy and the natural GL(2, Z)-action, this takes care of all simple closed curves.

Next we deal with the hardest case (27.1.c).

29 (4 pt case). After applying a suitable automorphism of S 2 , we may assume that C is close to a circle in S 2 but the 4 points do not lie on a circle. that φ(E i ) = E i . That is, φ is lifted from a diffeomorphism of the g-pointed sphere (S 2 , p 1 , . . . , p n ). By (2), any such diffeomorphism is isotopic to an algebraic automorphism. Hence the Dehn twist along C R is also algebraic. 30 (2 pt case). The proof is the same as in the 4 point case but the description is easier.

A neighborhood of C gives an annulus with 2 blown-up points. After the Dehn twist we get two curves E ′ 1 , E ′ 2 as in Figure 10. We can assume that the two curves E ′ 1 , E ′ 2 are close to being circles, that is, close to the intersections S 2 ∩ H i for some planes for i = 1, 2. Let q, q be the 2 (complex conjugate) points where these 2 planes H i intersect the complexified sphere Q. Then the Cremona transformation with base points p 1 , p 2 , q, q is the inverse of the Dehn twist, again up to a diffeomorphism of S 2 . 31 (Crosscap slides). Here the topological picture is given by Figure 11. Note that E 1 is mapped to itself and E 2 is mapped to the (almost) circle E ′ 2 . Up to isotopy, we can replace E 1 with a small circle E ′ 1 passing through p 1 . As in (30), we obtain q, q such that the Cremona transformation with base points p 1 , p 2 , q, q is the inverse of the Dehn twist, up to a diffeomorphism of S 2 . 32 (Proof of ( 4)). Let φ : (R, q 1 , . . . , q n ) → (R, q 1 , . . . , q n ) be any diffeomorphism. By (20), there is an automorphism Φ 1 ∈ Aut(R, q 1 , . . . , q n ) such that Φ -1 1 • φ is homotopic to the identity.

By ( 27), we can approximate Φ -1 1 • φ by a sequence of automorphisms Ψ s ∈ Aut(R, q 1 , . . . , q n ). Thus Φ 1 • Ψ s ∈ Aut(R, q 1 , . . . , q n ) converges to φ.

  ÁR AND FR ÉD ÉRIC MANGOLTE

Figure 1 .

 1 Figure 1. Cremona transformation with four real base points.

Figure 2 .

 2 Figure 2. The effect of the Dehn twist around C on a curve.

Figure 3 .

 3 Figure 3. Cross-cap slide.

  . The shaded region is a sphere with four holes, and corresponds to a neighborhood of the lift to R 8 of C 12345678 . On each side of the picture are drawn the curves corresponding to the Dehn twists of the same side in (26.1): a) C 12 , C 34 , C 5678 , C 12345678 , b) C 1234 , C 125678 , C 345678 .

Figure 4 .

 4 Figure 4. Lantern relation for m = 8.

  Let us take an annular neighborhood of C and blow up the 4 points p 1 , . . . , p 4 . The resulting open surface is denoted by W ⊂ B p1,...,p4 S 2 . It contains the curve C R and the 4 exceptional curves E 1 , . . . , E 4 . If we cut the blown-up annulus W along the 5 curves A 1 , . . . , A 4 , D as indicated of the left hand side of Figure 5, we get the contractible surface U indicated on the right hand side of Figure 5. The left and right hand sides of U are identified to form a cylinder, giving a neighborhood of the curve C R ⊂ B p1,...,p4 S 2 . The big rectangle with lighter shading in U on the right corresponds to the lighter shaded are in W on the left. The 4 top and 4 bottom line segments of U are identified to form 4 Möbius bands.

Figure 5 .

 5 Figure 5. Two models of the annulus blown up in 4 points.

  Figure 7 shows the images of the curves E i after the Dehn twist around C R .

Figure 6 .

 6 Figure 6. The 4 exceptional curves.

Figure 7 .

 7 Figure 7. Effect of the Dehn twist around C R .These images can be deformed to obtain a configuration as in Figure8. Note that now E i intersects E ′ j iff i = j.

Figure 8 .

 8 Figure 8. Deformation of Figure 7.Next we convert this back to the annulus model W on the left hand side of Figure5.We obtain Figure9.The images of the exceptional curves E 1 , . . . , E 4 under the standard Cremona transformation with base points p 1 , . . . , p 4 are shown in Figure1.We see by direct inspection that the two quartets of curves in Figures 1 and 9 are isotopic. Thus, if we first apply the Dehn twist and then the (inverse) Cremona transformation and a suitable isotopy, we get a diffeomorphism φ : R n → R n such

Figure 10 .

 10 Figure 10. Cremona transformation with 2 real base points.

Figure 11 .

 11 Figure 11. Cross-cap slides.

  Dehn twists t 125...m , t 345...m , t 1234 , t 5...m , t 12 , t 34 . This is best shown by a picture for m = 8. In Figure 4, t 12345678 is a product of the Dehn twists t 125678 , t 345678 , t 1234 , t 5678 , t 12 , t 34

Acknowledgments . We thank W. Browder, N. A'Campo, S. Cantat, D. Gabai, J. Huisman, I. Itenberg, V. Kharlamov and A. Okounkov for many useful conversations. We are especially grateful to M. Korkmaz for his help with understanding the non-orientable mapping class group. M. Zaidenberg called our attention to the papers of Lukackiȋ. These enabled us to shorten the proofs and to improve the results.

Partial financial support for JK was provided by the NSF under grant number DMS-0500198. The research of FM was partially supported by the ANR grant "JCLAMA" of the french "Agence Nationale de la Recherche." He also benefited from the hospitality of Princeton University.


A Fourmont

A S Rogachev

S Le Gallet

O Politano

D Yu Kovalev

N A Kochetov

N F Shkodich

S G Vadchenko

Florence Baras

D Yu

Kovalev

Thermal Stability of Medium-and High-Entropy Alloys of 3d-Transition Metals

published or not. The documents may come

Introduction

Since their discovery by Yeh and Cantor [START_REF] Yeh | Nanostructured High-Entropy Alloys with Multiple Principal Elements: Novel Alloy Design Concepts and Outcomes[END_REF][START_REF] Cantor | Microstructural Development in Equiatomic Multicomponent Alloys[END_REF] , HEAs have become a major research topic in material science. HEAs are characterized by outstanding properties including corrosion resistance, limited diffusion, increased plasticity at low temperatures, and other special mechanical and magnetic properties that depend on structure and composition. Nonetheless, their usage for technological purposes remains limited [START_REF] Lu | Directly Cast Bulk Eutectic and Near-Eutectic High Entropy Alloys with Balanced Strength and Ductility in a Wide Temperature Range[END_REF] . Prior to their future exploitation, a fundamental study of their stability is a required step. The stability of materials is of primary importance per se; indeed, reliable stability opens up new perspectives for applications including coatings, interconnects (fuel cell) [START_REF] Yeh | Development of an Advanced Bond Coat for Solid Oxide Fuel Cell Interconnector Applications[END_REF] , and heat engine components [START_REF] Chen | A Review on Fundamental of High Entropy Alloys with Promising High-Temperature Properties[END_REF] .

Typically, HEAs are characterized by a single-phase disordered solid solution. We may wonder if the absence of short-and long-range chemical order is maintained whatever the conditions. Is limited diffusion in a distorted crystal lattice the guarantee of this stability? What happens in the presence of defects or nanostructures that are inevitably formed during the elaboration process?

One way to study the stability of a material is to submit it to annealing at high temperatures and look for possible structural transformations.

Thermal stability in cast and mechanically synthesized HEAs is debatable because experimental data are incomplete. Among the main high-entropy families, the family composed of 3d-transition metals is the most thoroughly studied. The CoCrFeNi single-phase alloy with an FCC crystal structure was characterized as stable at 1173 K for 600 h [START_REF] Praveen | Exceptional Resistance to Grain Growth in Nanocrystalline CoCrFeNi High Entropy Alloy at High Homologous Temperatures[END_REF] , but the phase decomposition occurred at 1073 K for 800 h 7 . As-cast AlxCoCrFeNi alloys commonly consist of FCC dendrites and a BCC interdendritic phase, depending on the Al content: only the FCC phase was obtained for x = 0-0.3, a FCC+BCC mixture for x = 0.5-0.8, and BCC alone for x = 0.9-3.0 [START_REF] Wang | Microstructure and Mechanical Properties of AlxCoCrFeNi High-Entropy Alloys at Elevated Temperatures[END_REF][START_REF] Butler | Oxidation Behavior of Arc Melted AlCoCrFeNi Multi-Component High-Entropy Alloys[END_REF] . At temperatures above 873 K, the as-cast equiatomic AlCoCrFeNi BCC phase transforms into FCC, with simultaneous precipitation of B2 and s phase inclusions [START_REF] Meshi | Retardation of the σ Phase Formation in the AlCoCrFeNi Multi-Component Alloy[END_REF] . A mechanically synthesized AlCoCrFeNi BCC phase transformed into a partially disordered B2 phase after 1 h annealing at 1073 K [START_REF] Shivam | Alloying Behaviour, Thermal Stability and Phase Evolution in Quinary AlCoCrFeNi High Entropy Alloy[END_REF][START_REF] Shivam | Evolution of Phases, Hardness and Magnetic Properties of AlCoCrFeNi High Entropy Alloy Processed by Mechanical Alloying[END_REF] .

Experimental data concerning phase transformations in MnCoCrFeNi, CuCoCrFeNi and other HEAs have been reported in several reviews [START_REF] Miracle | A Critical Review of High Entropy Alloys and Related Concepts[END_REF][START_REF] Yeh | Physical Metallurgy of High-Entropy Alloys[END_REF][START_REF] Rogachev | Structure, Stability, and Properties of High-Entropy Alloys[END_REF] . For instance, it has been shown that two similar FCC structures co-existed in an annealed milled MnCoCrFeNi [START_REF] Rogachev | Structure and Properties of Equiatomic CoCrFeNiMn Alloy Fabricated by High-Energy Ball Milling and Spark Plasma Sintering[END_REF] ; short-term annealing led to the release of Cr and Mn from the matrix, which shrunk and formed a second metastable FCC solid solution.

Equimolar TiCoCrFeNi is less often studied than the other compositions of the considered family.

One would expect a very complex microstructure with several intermetallic phases due to Ti. However, TiCoCrFeNi after laser cladding was reported to exhibit one main FCC phase and two secondary Simple Cubic (SC) and Laves phases [START_REF] Zhang | Investigation on Solid Particles Erosion Resistance of Laser Cladded CoCrFeNiTi High Entropy Alloy Coating[END_REF] .

In addition, the confrontation of experimental results with powerful modeling tools such as the Calphad method has proven useful. In their work, Zhang and Gao demonstrated that configurational entropy alone does not grant the stability of high-entropy alloys; thus, quantities such as phase enthalpy (whose calculation is made possible by Calphad) are of great interest in evaluating the stability of alloys in regard to chemical and temperature variations [START_REF] Zhang | CALPHAD Modeling of High-Entropy Alloys[END_REF][START_REF] Zhang | Understanding Phase Stability of Al-Co-Cr-Fe-Ni High Entropy Alloys[END_REF] . The chemical stability of the FCC solid solution in the MnCoCrFeNi system was thoroughly assessed and demonstrated to be very well described by a database such as TCHEA1, although an accurate secondary phase prediction was perfectible [START_REF] Bracq | The Fcc Solid Solution Stability in the Co-Cr-Fe-Mn-Ni Multi-Component System[END_REF] . Thanks to this combined experimental and thermodynamic approach, conclusive results have been obtained by Rao et al. [START_REF] Rao | Secondary Phases in AlxCoCrFeNi High-Entropy Alloys: An in-Situ TEM Heating Study and Thermodynamic Appraisal[END_REF] . They were able to efficiently identify nano-scale precipitates predicted by thermodynamic calculations and to confirm the role of Al in the phase formation in AlxCoCrFeNi alloys.

In this work, we considered the family of HEAs composed of 3d-transition metals: CoFeNiXZ where X=Cr and Z=Al,Ti. This family has been extensively studied since the discovery of the Cantor MnCoCrFeNi HEA. Starting from the system CoFeNi that forms the core of the alloy family, we considered the quaternary CoCrFeNi and finally quinary systems. We focused on HEAs produced by mechanical alloying. This production method has rapidly developed over the last decade [START_REF] Torralba | High-Entropy Alloys Fabricated via Powder Metallurgy. A Critical Review[END_REF][START_REF] Vaidya | High-Entropy Alloys by Mechanical Alloying: A Review[END_REF] . Here, the alloying is carried out by means of high-energy planetary ball milling (HEBM) of powder mixtures composed of elemental metallic powders. The alloyed powders were characterized with High Temperature X-ray Diffraction (HT-XRD) and room temperature XRD after annealing in order to estimate their stability.

We also carried out the thermodynamic description of ternary, quaternary and quinary systems using the Calphad approach. In this way, we were able to interpret the experimental results as compared to expected equilibrium phases and to evaluate the specificities of the alloying process followed by annealing.

Materials and methods

Elemental powders of Al, Ti, Cu, Co, Cr, Fe and Ni of purity greater than 98 wt% and of various particle sizes (all < 160 µm) were mixed in equimolar proportions and mechanically alloyed in high-energy planetary ball mills. Table 1 summarizes the structure and the lattice parameter of the metallic element used in this work. A water-cooling mill "Activator-2S" (Novosibirsk, Russia) with stainless-steel cylindrical jars and balls (7 mm in diameter) was used for most of the systems studied in this work. In these cases, the jars were evacuated and then filled with Ar gas at 4 bar.

The HEBM was run at a rotating speed of the sun wheel and the grinding drums at 694 and 1388 rpm, respectively. TiCoCrFeNi was obtained using the "AGO-2" (Novosibirsk, Russia) watercooling planetary mill with steel balls (9 mm in diameter) in air atmosphere, at a rotating speed of 912 rpm (for the mill) that corresponded to 2220 rpm for the jars. The second milling device was used in order to prove that presence of oxygen in a milling jar does not prevent formation of singlephase multicomponent alloys. The ball/powder weight ratio was 20:1 in all experiments. The milling time was systematically 90 minutes, or 120 minutes for the CuCoCrFeNi system. The X-ray diffraction analysis of powders was performed with a DRON-3M diffractometer (Russia). Both Cu-Ka radiation (wavelength λ = 0.15419 nm) and Fe-Ka radiation (λ = 0.19374 nm) were used in order to obtain complementary data. Indeed, the Fe-Ka ensures a better angle resolution. High-temperature XRD measurements were made in a vacuum at 10 -3 Pa, within the temperature range 298-1273 K using an ARL'XTRA diffractometer with the high-temperature accessory НТК2000 "Anton Paar" (Switzerland). The selected annealing temperatures were 873 K, 1073 K and 1273 K. For the accurate measurement of lattice parameters at room temperature (before and after annealing), crystalline silicon powder (NIST SRM 640b) admixed to the analyzing powder samples was used as inner marker (standard). The lattice parameters were determined with accuracy better than 0.002%. The SEM and EDS analyses were carried out using a Zeiss Ultra Plus field-emission scanning electron microscope (Carl Zeiss, Germany), equipped with an INCA Energy 350 XT energy dispersive spectrometer (Oxford Instruments, UK). Thermodynamic calculations were performed using Thermo-Calc® software based on the Calphad methodology with the TCHEA3 database. Phase fractions and phase composition as a function of temperature were calculated for the different systems. The lattice parameter of the cubic solid solution was estimated in the cases of ternary and quaternary systems.

Results

CoFeNi and CoCrFeNi

Three metals of the iron group, Co, Fe, and Ni constitute a ternary core of the 3d-transition metal family of HEAs. These metals have similar atomic radii, but different crystal structures.

Figures 1 and2 display the dependence of the equilibrium phase composition (expressed in volume ratio) on temperature in the case of the ternary CoFeNi and quaternary CoCrFeNi. In order to get a quick estimation of the composition of each phase, we also provide the distribution of elements in the different phases. The calculation was here performed in the case of equimolar systems. For the ternary CoFeNi, the main characteristics are as follows:

-Below 925 K, the system is composed of one ordered BCC phase and one disordered FCC phase. Nickel is the principal element in the FCC phase. The BCC ordered phase contains mainly Co and Fe elements.

-Above 925 K, a single FCC disordered phase is observed.

-The melting temperature is 1734 K.

In conclusion, the ternary CoFeNi may be considered as an HEA with only 3 elements. This typical behavior can be examined in light of the behavior of pure elements and binary phase diagrams. Nickel possesses an FCC structure from room temperature to its melting point. Although Co(Fe) has an HCP(BCC) structure at room temperature, Co(Fe) undergoes a phase transformation to FCC above 695 K (1184 K). The binary phase diagrams show that a single FCC solid solution Co+Ni or Fe+Ni can be obtained in a wide range of temperatures for any ratio between the elements. On the other hand, Co and Fe form a BCC solid solution below 1173 K if the mole fraction in Fe is greater than 0.2 or a FCC solid solution at a lower content in Fe. It is therefore not surprising that a single disordered FCC solid solution is observed at high temperatures.

Adding the element Cr to obtain the quaternary system CoCrFeNi leads to the following observations:

-Above 928 K, the system is composed of a single FCC disordered phase corresponding to the HEA.

-At lower temperatures [800 K-928 K], three phases coexist: a BCC-ordered phase with Co and Fe, a σ phase with a majority of Cr and the disordered FCC solid solution.

-The melting temperature is 1720 K.

Ternary and quaternary systems are characterized by the same melting temperatures. In equilibrium conditions, a single FCC solid solution is formed in a temperature range close to the melting temperature. For lower temperatures, the equilibrium corresponds to a mixture of different phases.

Fig. 3 displays the XRD patterns of CoFeNi and CoCrFeNi after 90 min of high-energy ball milling (HEBM). At the end of the process, Co, Fe and Cr peaks have disappeared. There remain only two peaks corresponding to an FCC structure. The broadening of these two peaks is easily noticeable; this broadening signifies that the powder most likely contains a high concentration of lattice defects and nanosized crystallites. Indeed, a crystalline size (coherent-scattering regions) was roughly estimated as 15 nm for CoFeNi and 21 nm for CoCrFeNi after HEBM. In addition, there is a strong shift on the FCC peaks; for instance, for CoFeNi, the (200) FCC peak jumps from 66.7° 2q (corresponding to the pure Ni lattice parameter a0=3.520Å) to 65.6° 2q. The lattice parameters of the FCC phase after 90 min of HEBM are estimated to be 3.57Å for CoFeNi and 3.58Å for CoCrFeNi. Hence, there is a strong lattice parameter increase associated with the formation of an FCC solid solution. The calculated lattice parameters are summarized in Table 2. Two minor peaks at 41.5° 2q and 50.25° 2q probably correspond to iron-chromium oxide (Fe,Cr)2O3 (card PDF 000-34-0412). Although the samples were heated in a vacuum during the high-temperature XRD analysis, some residual oxygen results in surface oxidation of the powder particles. There is no sign of any other phase formation during annealing, although we could have expected the formation of a σ phase in the case of the CoCrFeNi alloy. The direct comparison of thermodynamic calculation to experimental results thus leads to the conclusion that the HEBM process selects the HT phase: the FCC disordered solid solution. However, there is a slight decrease in lattice parameter with temperature in the case of CoFeNi whereas there is a slight increase in the case of CoCrFeNi (see Table 2). Note that this does not correspond to thermal expansion since these XRD measurements are performed at room temperature. Rogachev et al. suggested that the decrease could be due to the relaxation of the crystal structure [START_REF] Rogachev | Structure and Properties of Equiatomic CoCrFeNiMn Alloy Fabricated by High-Energy Ball Milling and Spark Plasma Sintering[END_REF] . The increase in lattice parameter after annealing of the milled powder seems to be related to Cr. It may suggest that some Cr-rich regions remained after milling, and that annealing led to a second chemical homogenization associated with a lattice parameter increase. Indeed, Cr has the highest melting temperature among the 4 considered elements and is known to be the last element to dissolve during milling. It could also be due to the formation of the Fe,Cr oxides during annealing which would lower the actual concentrations of these elements in the alloy. A third hypothesis is linked to the fact that the (111) FCC peak shifts to the left whereas the (200) peak seems to remain stationary. This could be the sign of a weak, thermally induced martensitic transformation; indeed Rogachev et al. suggested that this kind of transformation could occur in the AlCoCrFeNi alloy under similar constraints [START_REF] Rogachev | Evolution of Crystal Structure in High-Entropy AlCoCrFeNi Alloy: An in Situ High-Temperature X-Ray Diffraction Study[END_REF] .

In addition, the width of the peaks narrows. This narrowing means that lattice defects are relaxing and that grain size is increasing. After annealing at 1273 K, the sizes of coherent-scattering regions increase to 1340 nm and 680 nm for CoFeNi and CoCrFeNi, respectively.

The lattice parameter was estimated by XRD at annealing temperatures during the heat treatment for the ternary and quaternary (see Fig. 6). These experimental results have been compared to those obtained via two different theoretical approaches. The first estimation was directly calculated using Thermo-Calc. The second estimation relies on the well-known Vegard's law.

Vegard's law is the empirical rule that the lattice parameter of a solid solution of two constituents is approximately equal to a rule of mixtures of the two constituent lattice parameters at the same temperature. If generalized to a ternary equimolar system:

a0(CoFeNi) = ! " (a0(Co) + a0(Fe) + a0(Ni)) (1)
provided that all elements have the same crystallographic structure. Here, we used Thermo-Calc to obtain a0(FCC-Ni,T), a0(FCC-Co,T), and a0(FCC-Fe,T) for pure elements. In the case of the ternary, we noted that the Vegard's value best reflects the experimental observations, while the thermodynamic approach overestimates the lattice parameter at high temperatures. In the case of the quaternary, both estimations give greater values than the observed lattice parameter. Overall, both calculation methods give results that are relatively close to experimental measurements;

they have proven to be useful tools in evaluating lattice parameters at elevated temperatures.

Quinary systems

The volume ratios for X-CoCrFeNi HEAs were calculated using Thermo-Calc with the database TCHEA3 over a range of temperatures [773 K-1773 K]. Despite the fact that Mn-CoCrFeNi is not considered in the present study, the phase fraction of the Cantor HEA is depicted in Fig. 7 for the sake of comparison. At low temperatures, there is coexistence between a σ phase and a disordered FCC solid solution. At 773 K, the Mn element is equally distributed in the σ phase and FCC_dis. As the temperature increases, the volume fraction of the σ phase decreases until complete disappearance at 1100 K as compared to 900 K in the quaternary CoCrFeNi. The melting temperature, 1573 K, is lower than the melting temperature of CoCrFeNi.

As shown in Fig. 8a Note that the composition of the BCC_dis is similar to that of the σ phase. In the equimolar case, only a very small fraction of FCC_dis is formed (0.1 % in volume). A slight variation in the composition in Al leads to a very different result with the formation of an FCC_dis phase.

High Temperature XRD (HT-XRD) results on milled Al-CoCrFeNi powder are presented in Fig. 8b. Heating was carried out in a step-like manner with isothermal stages at 873, 1073, and 1273 K.

Starting from the bottom, the first pattern corresponds to room temperature XRD prior to annealing. The next three patterns correspond to XRD measurement at 873 K after 0h, 1h and 2h.

The next 6 patterns follow the same logic; the final XRD pattern corresponds to room temperature measurement after annealing.

After HEBM and prior to annealing, the Al-CoCrFeNi powder alloy seems to exhibit a single BCC phase, similarly to what was reported by Ji et al. with a far less powerful milling [START_REF] Ji | Mechanical Alloying Synthesis and Spark Plasma Sintering Consolidation of CoCrFeNiAl High-Entropy Alloy[END_REF] . However, a mixture of FCC+BCC has also been reported in several studies on the elaboration process of Al-CoCrFeNi by powder metallurgy [START_REF] Rogachev | High-Energy Ball Milling and Spark Plasma Sintering of the CoCrFeNiAl High-Entropy Alloy[END_REF][START_REF] Fourmont | Effects of Planetary Ball Milling on AlCoCrFeNi High Entropy Alloys Prepared by Spark Plasma Sintering: Experiments and Molecular Dynamics Study[END_REF] . In addition to the influence of the mechanical alloying on the phase constitution of the alloys, initial elemental particle sizes could play a major role in determining the final crystalline structure after HEBM. It is worth noting that the only peak remaining after HEBM is very broad. This peak may actually correspond to two peaks: the (111)

FCC and (110) BCC, both situated around 44.5° 2q; however, the (200) FCC peak is completely absent from the XRD pattern.

The HT-XRD patterns show that the single BCC phase is stable up to 1273 K. From this temperature, an FCC and an intermetallic σ phase begin to form. Then, close analysis of the 2h annealing pattern at 1273 K may suggest that there is a partial dissolution of σ in the FCC phase.

After cooling, three phases remain: FCC, BCC and σ. The agreement between experimental and calculation results is not excellent but leads to several remarks. Indeed, the calculated domain of stability of σ seems to be slightly overestimated since there is no sign of σ at 873 K and since σ begins to disappear at 1273 K although its calculated domain was [817 K-1276 K]. Obviously, one could argue that 2h is not enough to reach thermodynamic equilibrium and that the σ phase prediction is known to be perfectible [START_REF] Bracq | The Fcc Solid Solution Stability in the Co-Cr-Fe-Mn-Ni Multi-Component System[END_REF] . However, the formation of an FCC phase is barely expected according to the database and this constitutes a major discrepancy. The formation of an FCC phase could be partially explained by a composition modification due to oxide formation [START_REF] Fourmont | Dual-Phase AlCoCrFeNi HEA: Reactive Spark Plasma Sintering[END_REF] .

Indeed, oxide formation is a well-known drawback of the powder metallurgy route and could lead to an Al depletion in the main phase.

As shown in Fig. 9a, the volume ratios for Cu-CoCrFeNi HEA were calculated using Thermo- The HT-XRD patterns reveal that a small amount of a BCC phase appears at 873 K but disappears at 1073 K and above. This BCC phase is predicted by Calphad but at a lower temperature and could be the sign that its stability domain is shrunk by the consequences of the elaboration process (such as contamination or high concentration of lattice defects). Indeed, a high concentration of lattice defects tends to promote new phase formation. A second FCC phase, noted FCC-1, appears at 1073 K after 2h of annealing, remains at 1273 K after 1h but disappears for longer annealing times and is not visible after cooling. However, the two FCC phases could have very similar lattice parameters and appear as one XRD pattern. According to our thermodynamic calculations, a minor Cu-rich FCC is supposed to exist below 1393 K. The lattice parameter of the Cu-rich FCC phase is estimated at 3.70 Å at 1273 K by Thermo-Calc, whereas the lattice parameter of the main FCC phase is estimated at 3.64 Å. Hence, the hypothesis of two lattices of the same parameter seems unlikely.

Calc
In the case of the Ti-CoCrFeNi, Calphad showed the coexistence of a disordered FCC_dis solid solution and a BCC_ord phase close to the melting temperature (see Fig. 10a). The σ phase persists upon melting. At 1273 K, the composition of the three coexisting phases reads:

-BCC_ord: Ti28 Co21 Cr14 Fe16 Ni21.5;

-FCC_dis: Ti5 Co20 Cr28 Fe28 Ni19;

σ: Co15 Cr53 Fe24. The HT-XRD patterns show that the peaks corresponding to the BCC and FCC phases at 44.5° 2q appear clearly as two separate peaks from 873 K and above. At 873 K, the BCC phase is predominant, whereas the FCC phase becomes the main phase at 1073 and 1273 K. Note that the W peak corresponds to the measurement apparatus. In this case, there is also a noticeable peak refinement with temperature, corresponding to the increase in grain size and relaxation of the lattice defects.

Most of the

The superlattice peak associated with an ordered BCC phase B2 should be close to 32° 2θ.

However, the main BCC peak, (110), is already so small that the superlattice reflection would not be visible either way. The inversion between BCC and FCC phases as the main phase of the Ti-CoCrFeNi alloy as compared to calculated results is interesting; this discrepancy could be explained in a similar way to that noted in the Al-CoCrFeNi case. Indeed, Ti is known to be even more reactive with oxygen than Al and could form a significant amount of nano-oxide not discernible by XRD. However, our experimental results are coherent with laser cladded Ti-CoCrFeNi which exhibited a main FCC phase [START_REF] Zhang | Investigation on Solid Particles Erosion Resistance of Laser Cladded CoCrFeNiTi High Entropy Alloy Coating[END_REF] . These results thus seem to point out a weakness in the TCHEA3 database.

Discussion

The stability of HEAs is typically associated to the relation

Δ𝐺 mix = Δ𝐻 mix -𝑇Δ𝑆 mix (2)
It is commonly accepted that the entropic term ensures the stability of the alloy at high temperatures because of the increase in configurational entropy for a large enough number of constitutive elements. This simple and elegant point of view is currently called into question, justifying the name change of HEAs to complex concentrated alloys (CCAs) [START_REF] Zhang | CALPHAD Modeling of High-Entropy Alloys[END_REF][START_REF] Gorsse | From High-Entropy Alloys to Complex Concentrated Alloys[END_REF] . In order to appreciate the contribution of enthalpy and entropy in HEA stability (eq. ( 2)), the mixing behavior in Gibbs free energy, enthalpy, and entropy is presented in Fig. 11. We compared the mixing properties at high temperature (1273 K) of the quaternary CoCrFeNi and the quinaries Al-CoCrFeNi, and Ti-CoCrFeNi for a given phase. In this analysis, the content in Cr has been varied. The diagrams depicted in Fig. 11 are pseudo-binary: there is an equal repartition of the other elements whatever the content in Cr. In our previous analysis, we have remarked that the distribution of Cr in the different phases is an important factor in the occurrence of disordered phases. In the case of the CoCrxFeNi system, we noted that the stability Δ𝐺 mix < 0 is ensured by eq. ( 2) because Δ𝐻 mix > 0 in almost the whole composition range.

But the Gibbs free energy of mixing Δ𝐺 mix decreases monotically over the whole composition range with the increase in the number of components (4 to 5). As shown in Fig. 11b(11e), the addition of Al(Ti) radically changes the behavior of Δ𝐻 mix . Achieving a more negative Δ𝐻 mix implies the formation of stronger bonds among elements. This effect is very important at a low content in Cr (including the equimolar 𝑥(Cr) = 0.2). In the case of the equimolar Al-CoCrFeNi system, Δ𝐻 mix is about -30 kJ/mol for the BCC phase and -15 kJ/mol for the FCC phase. More surprisingly, we noted a decrease in Δ𝑆 mix as compared to the quaternary system and the entropic term 𝑇Δ𝑆 mix = 4.67 kJ/mol. The stability of the quinary HEA thus cannot be solely attributed to the entropic effect. The same conclusions can be drawn in the case of the Ti-CoCrFeNi In other words, the elaboration process selects the FCC_dis phase that remains stable upon annealing. Over the duration of annealing at 873 K, no transformation of FCC_dis to BCC_ord was observed, while this ordered phase, mainly composed of Fe and Co, is predicted in equilibrium conditions. Our hypothesis is that such discrepancies could be explained by the out of equilibrium process associated with powder metallurgy. The situation is completely different in the case of CoFeNi films prepared by reverse pulse electrodeposition with ultra-high magnetic saturation [START_REF] Tabakovic | Preparation of Metastable CoFeNi Alloys with Ultra-High Magnetic Saturation[END_REF] . In this case, the Co20Fe66Ni14 alloy is unstable for 2h annealing at 673 K with a significant reduction in magnetic saturation.

The quaternary CoCrFeNi system also fulfils relation (1). Fig. 11 demonstrates the High Entropy effect on alloy stability. The thermal stability of CoCrFeNi has recently attracted serious scrutiny in the literature. Indeed, this system can be considered as the base alloy for the class of X-CoCrFeNi HEAs where X is an element of different atomic size (Al or Ti in the present study). In a temperature range close to its melting temperature, at equilibrium, the system is characterized by a single FCC disordered phase (see Fig. 5). The thermal stability of the FCC_dis phases with a slight difference in lattice parameter (0.001 Å) has also been observed in non-equimolar as-cast CoCrFeNi with no change after 3h-annealing at 1373 K 33 .

In the case of HEAs synthesized by mechanical alloying followed by spark plasma sintering (SPS), the situation is quite different. S. Praveen et al. reported that elemental powders of Co, Cr, Fe, and Ni milled for 15h transformed into a major FCC_dis phase with a minor BCC_ord phase [START_REF] Praveen | Exceptional Resistance to Grain Growth in Nanocrystalline CoCrFeNi High Entropy Alloy at High Homologous Temperatures[END_REF] . After SPS, the major FCC phase was retained and the BCC_ord phase disappeared with the formation of tiny second phase particles (carbide Cr7C3 and oxide Cr2O3). Even for a long annealing duration of 600h at 1173 K, these authors observed sluggish grain growth that did not affect the stability of the nanocrystalline HEA alloy. The same behavior has been observed for long annealing at the lower temperature of 973 K [START_REF] Sathiyamoorthi | Thermal Stability and Grain Boundary Strengthening in Ultrafine-Grained CoCrFeNi High Entropy Alloy Composite[END_REF] . The authors attributed the strong resistance to grain growth to the presence of carbides and oxides. In the present study, no such secondary phases were observed. Grain growth, estimated by the increase in crystallite size, is thus not inhibited by such oxides and carbides.

Nanocrystalline CoCrFeNi thin films deposited on Si/SiO2 and c-sapphire substrates by magnetron co-sputtering have recently been investigated [START_REF] Kini | Nanocrystalline Equiatomic CoCrFeNi Alloy Thin Films: Are They Single Phase Fcc? Surf[END_REF] . In addition to the FCC phase, the films contained an additional σ phase even at room temperature and in as-deposited films due to a large number of nucleation sites and accelerated kinetics as compared to bulk systems.

Annealing at temperatures between 573 K and 1073 K leads to the formation of Cr oxides at the expense of the σ phase.

The powder of CoCrFeNi-HEA produced by HEBM (see Fig. 1) is characterized by a single FCC_dis solid solution of uniform chemical composition (see Fig. 2). The annealing at 873 K, 1073 K and 1273 K over 5h did not affect the stability of the disordered FCC phase, even at 873 K where thermodynamics predicts the formation of a small amount of σ phase. The only noticeable change was an increase in the lattice parameter that may be related to the complete dissolution of Cr in the solid solution upon heat treatment or to the other hypotheses raised in section 3. In conclusion, two main factors may induce discrepancies between experimental and numerical results: the metastable state attained after powder metallurgy route and contamination by minor elements. Both HEBM preparation and annealing conditions are of primary importance in the observed phase stability.

The equimolar Al-CoCrFeNi system is among the most studied HEAs after the Cantor alloy.

Nevertheless, discrepancies between experimental results and thermodynamic calculations have continuously been reported. In our study, a suspicious amount of FCC phase was found as soon Equimolar Ti-CoCrFeNi is among the least studied HEAs. Indeed, several groups evaluated the influence of Ti on the CoCrFeNi base [START_REF] Cui | Effect of Ti on Microstructures and Mechanical Properties of High Entropy Alloys Based on CoFeMnNi System[END_REF][START_REF] Shun | Microstructures and Mechanical Properties of Multiprincipal Component CoCrFeNiTix Alloys[END_REF][START_REF] Shun | Age Heat Treatment of the CoCrFeNiTi0.3 High-Entropy Alloy[END_REF] and they concluded that Ti often leads to complex phases such as laves and rhombohedral phases. Surprisingly enough, in our study we found only two simple phases: FCC and BCC. The main difference is related to grain size and lattice defect concentration. Indeed, milled powders are known to exhibit a loose crystalline structure and complex phases could be kinetically inhibited. Hence, long term annealing could be mandatory to reveal the actual microstructure of this alloy.

There is no consensus in the literature on the relevant duration of annealing. Indeed, annealing studies are performed over various durations. Long term annealing has been reported by several groups for a minimal annealing time of 50h or even 100h to ensure that the thermodynamic equilibrium is reached in HEAs concluded that their experimental results matched well with the Calphad predictions of the phase evolution for the B2 phase. Hence, short annealing times are an efficient way to ensure that the database and experimental results match well enough to proceed to a thorough analysis characterized by hundreds of hours annealing time.

The Calphad method, based on free energy minimization, gives the dependence of the equilibrium phase composition on temperature. For the compositions investigated in the present study, the disordered phase characteristic of high entropy alloys is formed in a temperature range close to the melting point, while ordered phases correspond to lower temperatures. We observed that some of these secondary phases transitorily appeared during heat treatment while others were inhibited. In order to report these observations, the kinetics of phase transformations should be taken into account with a detailed description of precipitation and diffusion mechanisms together with thermodynamic parameters (driving forces) [START_REF] Gorsse | Current and Emerging Practices of CALPHAD toward the Development of High Entropy Alloys and Complex Concentrated Alloys[END_REF] .

Conclusion

The investigation of thermal stability in High Entropy Alloys is a vast and complex domain. On one hand, the name of these alloys has been shown to be somewhat misleading since the high entropy effect has been demystified by several research groups. On the other hand, the field of High Entropy Alloys includes a wide range of compositions and elaboration processes. In our study, we chose to start from ternary medium entropy alloys to progress to the well-known Al-CoCrFeNi and the less studied Ti-CoCrFeNi. The three studied quinary compositions are relatively simple since they are equimolar with only one element varying. The thermal stability of mechanically alloyed powders is of great interest since they embody the perfect disordered solid solution; indeed, they do not exhibit contamination such as carbon from the sintering medium or segregation occurring during solidification.

In the present study, we found that medium entropy alloys CoFeNi and CoCrFeNi were the most stable of the alloys studied. In addition, our results revealed that adding a fifth element irremediably led to the formation of secondary phases after and/or during annealing. In a further finding of relevance, even the most stable HEA, the Mn-CoCrFeNi Cantor alloy, has recently been reported to exhibit two FCC solid solutions after short time annealing [START_REF] Rogachev | Structure and Properties of Equiatomic CoCrFeNiMn Alloy Fabricated by High-Energy Ball Milling and Spark Plasma Sintering[END_REF] .

The limited diffusion in a distorted crystal lattice does not seem to guarantee the thermal stability of High Entropy Alloys. Indeed, the single-phase microstructure is lost after even a short annealing time. In addition, those secondary phases were not often predicted by the Calphad method based on TCHEA3 (Thermo-Calc). It seems likely that the presence of defects related to the elaboration process led to some sort of kinetic shifting of their stability domain. However, in the case of medium entropy alloys, by confronting the thermodynamic calculations and the HT-XRD results, we observed that some phases were inhibited at relatively low temperatures. And, interestingly enough, in the case of quinary high entropy alloys, we found that complex phases such as Laves and other ordered phases could also be inhibited.

Experimental results demonstrated that phase transformations and/or precipitations of secondary phases took place in all considered equimolar HEAs. Thermodynamic calculations also showed the co-existence of two or more phases at room and moderate temperatures, which is in agreement with eq. ( 2) since the entropy impact decreases with decreasing temperature. At the same time, we never observed complete decay of the initial HEA phase into simple ordered compounds (intermetallics, ternary ordered alloys, etc.). High entropy FCC and BCC phases remain after annealing in all studied materials, despite the transformations. Our findings allow us to conclude that the search for stable high-temperature alloys can be continued.

  SEM images of milled powder cross sections are shown in Fig. 4. The lack of contrast demonstrates that only one chemically uniform phase remains after milling. EDS microanalysis reveals that the global composition is still equimolar in CoFeNi and CoCrFeNi. There is no preferential adherence to the milling apparatus by any one element in particular. From XRD measurements and SEM analyses, we can conclude that the mixtures of the individual powders Co+Fe+Ni and Co+Cr+Fe+Ni have completely transformed into the single-phase alloys CoFeNi and CoCrFeNi. The CoFeNi and CoCrFeNi milled powders were annealed at 873 K, 1073 K and 1273 K over 5.1 h. The XRD patterns at room temperature after annealing are presented in Fig. 5.

  , Al-CoCrFeNi is characterized by the presence of a BCC_ord phase over the whole temperature range [773 K-1773 K]. At 1073 K, the composition of the two coexisting phases reads: -BCC_ord: Al28 Co22 Fe17 Ni27 and -σ: Co15 Cr55 Fe26. Above 1273 K, the σ phase disappears in favor of a BCC_dis phase. The chromium element is redistributed between the two BCC phases. At 1273 K, we have -BCC_ord: Al25 Co21 Cr11 Fe18 Ni25 -BCC_dis: Co17 Cr45 Fe26.

  with the database TCHEA3 over a range of temperatures [773 K-1773 K]. This HEA is characterized by the coexistence of two disordered FCC solid solutions at high temperatures (T > 973 K):-FCC_dis#1 with a majority of Cu (Cu89 Co1 Fe2 Ni8) -FCC_dis#2 without Cu (Cu2 Co25 Cr25 Fe25 Ni23).The FCC_dis#1 phase melts prior to the appearance of the? FCC_dis#2 phase. The amount of FCC_dis#1 is directly related to the fraction of Cu. The ordered BCC_ord phase and σ phase are observed at relatively low temperatures.HT-XRD results on milled Cu-CoCrFeNi powder are presented in Fig.9b. Measurements were carried out slightly differently than in the Al-CoCrFeNi case, the sample were heated at the selected annealing temperature over 5h and then cooled down for each temperature of interest. For the sake of comparison, the first pattern corresponds to XRD results after 1h of annealing and the second after 2h. Mechanical treatment of the five-component Cu+Co+Cr+Fe+Ni mixtures results in the formation of a single-phase FCC structure as shown in the bottom pattern.

  Ti is in the BCC_ord phase. At low temperatures (T < 1000 K), Laves phases C14 and C15, intermetallic Ni3Ti (D024) and the σ phase are stable. Laves phases and intermetallic Ni3Ti rearrange at 1000 K into FCC_dis and BCC_ord. The melting of FCC_dis takes place prior to the melting of the BCC_ord. Over 40 K, the ordered BCC_ord phase coexists with the liquid phase.HT-XRD results on milled Ti-CoCrFeNi powder are presented in Fig.10b. Heating was carried out similarly to the previous Al-CoCrFeNi alloy, except that the XRD patterns after 0h of annealing at the selected temperature are not displayed. Mechanical treatment of the fivecomponent Ti+Co+Cr+Fe+Ni mixtures results in the formation of a single-phase BCC structure as shown in the bottom pattern. In this case, the last remaining peak is also very broad. It could correspond to two peaks, one FCC and one BCC.

  system, often named medium-entropy alloy, in fact satisfies the relation(2) defining the notion of high-entropy alloy despite the limited number of constituents: Δ𝐻 mix = 32.5 𝑘𝐽/𝑚𝑜𝑙 > 0 ; -𝑇Δ𝑆 mix = -109 𝑘𝐽/𝑚𝑜𝑙 at 𝑇 = 1273 𝐾.In the present work, the ternary elaborated by HEBM is composed of a single FCC disordered phase whose lattice parameter 𝑎 & =3.57 Å is greater than the lattice parameter of Ni (𝑎 & =3.52 Å). Although the three constituents have similar atomic diameters, the host FCC-Ni matrix is affected by the substitution of Ni by Co and Fe. This solid solution obtained by mechanical treatment is chemically uniform. The annealing of CoFeNi at 873 K, 1073 K, and 1273 K has no effect on the phase stability except a slight relaxation of the lattice parameter measured at room temperature (𝑎 & = 3.56 Å after the annealing at 1273 K for instance).

FCC_dis phase was investigated

  in systems produced by means of different techniques. In the case of HEAs prepared by arc-melting, Vaidya et al. did not observe any modification in structure or composition for a heat treatment in the temperature interval [1073 K-1373 K] for 96 h 31 . On the other hand, Feng He et al. reported that the CoCrFeNi HEA is thermally metastable at 1023 K for a long annealing of 800 h 32 . The phase decomposition is similar to the clustering of GP zones with the formation of an FCC phase with a different lattice parameter (𝑎 & =3.55 Å as compared to the matrix value 𝑎 & =3.66 Å). The thermal stability is thus revealed to be very sensitive to the duration and temperature of annealing. The existence of two very similar

  as the temperature reached 873 K. The hypothesis of Al oxide depleting the alloys of Al may partially explain this observation, but it is not sufficient. Indeed, we can turn to Munitz et al. who focused their work on the effect of heat treatment on the arc-melted Al-CoCrFeNi alloy[START_REF] Munitz | Heat Treatment Impacts the Micro-Structure and Mechanical Properties of AlCoCrFeNi High Entropy Alloy[END_REF] . The as-cast alloy exhibited a BCC phase (A2+B2) with an Al-Ni rich dendrite core and Co-Cr-Fe rich interdentritic regions. A σ phase and an FCC phase formed in the interdendritic regions with heat treatments between 923 K and 1248 K over 3h. The formation of Al oxide is easier to control in the case of the liquid-state process; the influence of oxygen is thus most likely negligible in this study. Yet the amount of FCC after annealing is significant; the database seems unable to predict the correct equilibrium in this case. It is interesting to note that TCNI9, a database very similar to HEA3, predicts a greater amount of FCC.Studies of Cu-CoCrFeNi HEAs are mainly limited to thin films and coatings. An et al. have found that thin films were composed of FCC grains with preferential (111) orientation as compared to as-cast Cu-CoCrFeNi HEAs where two FCC phases were detected (3.58 Å and 3.61 Å): a Cu-rich phase in interdendritic regions and a Cu-poor phase in dendritic morphology[START_REF] An | Solid-Solution CrCoCuFeNi High-Entropy Alloy Thin Films Synthesized by Sputter Deposition[END_REF] . The in-situ annealing in TEM of Cu-CoCrFeNi thin films prepared by magnetron sputtering has been evaluated in[START_REF] Arfaoui | Transformations in CrFeCoNiCu High Entropy Alloy Thin Films during In-Situ Annealing in TEM[END_REF] in the temperature range [673 K-973 K] for 5 min. Up to 673 K, a single FCC phase was observed. At 823 K, the film had a two-phase structure: the original FCC phase and a BCC phase with an epitaxial relationship. This type of transformation can be attributed to diffusionless processes. From 823 K, fast morphological and structural changes occur, controlled by volume diffusion processes with the separation of Cr-rich areas and Cu-rich ones. In the present study, the annealing of Cu-CoCrFeNi HEA produced by HEBM also revealed transient phases: a BCC phase and a second FCC phase as a function of temperature and annealing time.

Figure 1 .Figure 2 .

 12 Figure 1. Phase fractions (volume fraction) in the CoFeNi system and element distributions in the

Figure 4 .

 4 Figure 4. SEM images of the CoFeNi (a) and CoCrFeNi (b) alloys after 90 min of HEBM. Image

Figure 5 .

 5 Figure 5. Room temperature XRD patterns of the CoFeNi (a) and CoCrFeNi (b) alloys after HEBM

Figure 6 .

 6 Figure 6. Lattice parameters for ternary and quaternary HEAs as a function of temperature. Only

Figure 7 .

 7 Figure 7. Volume ratios of the different phases in Mn-CoCrFeNi HEA calculated by Thermo-Calc.

Figure 8 .

 8 Figure 8. Volume ratios for the Al-CoCrFeNi alloy calculated using Thermo-Calc (a). High

Figure 9 .

 9 Figure 9. Volume ratios of phases for HEA Cu-CoCrFeNi calculated using Thermo-Calc (a). HT-

Figure 10 .

 10 Figure 10. Volume ratios of phases for the Ti-CoCrFeNi alloy calculated using Thermo-Calc (a).

Figure 11 .

 11 Figure 11. Calculated mixing properties of the FCC(BCC) phase at 1273 K: Gibbs free energy of

  [START_REF] Zhang | Understanding Phase Stability of Al-Co-Cr-Fe-Ni High Entropy Alloys[END_REF][START_REF] Pickering | Precipitation in the Equiatomic High-Entropy Alloy CrMnFeCoNi[END_REF] . Reynolds et al. performed high-energy synchrotron radiation X-ray diffraction measurements during in situ annealing (SR-XRD) and noted that 3h annealing was sometimes not long enough[START_REF] Reynolds | Comparing CALPHAD Predictions with High Energy Synchrotron Radiation X-Ray Diffraction Measurements during in Situ Annealing of Al0.3CoCrFeNi High Entropy Alloy[END_REF] . As compared to the above-mentioned studies, the stable until a temperature just below the bulk HEA degradation temperature using HT-XRD with short time annealing[START_REF] Dolique | Thermal Stability of AlCoCrCuFeNi High Entropy Alloy Thin Films Studied by In-Situ XRD Analysis[END_REF] . Even in their SR-XRD study,Reynolds et al.

annealing times of a few hours as used in the present work may seem a little short. However, conclusive remarks have been drawn in the literature reporting studies with short annealing times. Indeed, Dolique et al. demonstrated that HEA thin films deposited by DC magnetron sputtering were

Table 1 .

 1 Structure and lattice parameter of the metallic element used in this work.

	Element Structure Lattice parameter
			(Å)
	Al	FCC	4.05
	Co	HCP	2.51 / 4.07
	Cr	BCC	2.88
	Cu	FCC	3.61
	Fe	BCC	2.87
	Ni	FCC	3.52
	Ti	HCP	2.95 / 4.68

Table 2 .

 2 Calculated lattice parameter (Å) of the CoFeNi and CoCrFeNi milled powders before and after annealing at different temperatures. The number in parentheses is the estimated standard deviation. For instance, 3.571(2) means 3.571 ± 0.002.

	System	Initial, at room	Lattice parameter after annealing and cooling down to
	(alloy)	temperature		room temperature
			873K	1073K	1273K
	CoFeNi	3.571(2)	3.5711(2)	3.5621(3)	3.5620(3)
	CoCrFeNi	3.575(4)	3.5706(2)	3.5748(1)	3.5752(3)

Acknowledgements

The experimental portion of this work was undertaken with the financial support of the Russian Science Foundation (project no. 20-13-00277). Thermo-Calc calculations were supported by the Pilot-Hy project funded by the Regional Council of Bourgogne Franche-Comté, the FEDER and the EIPHI Graduate School (Contract ANR-17-EURE-0002). Pr. G. Caboche is also gratefully acknowledged for helpful discussions.