Cristina G Fernandes

José C De Pina

Jorge Luis

Ramírez Alfonsín
email: jorge.ramirez-alfonsin@umontpellier.fr

Sinai Robins
email: srobins@ime.usp.br

Cubic graphs, their Ehrhart quasi-polynomials, and a scissors congruence phenomenon *

The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes. The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation.

Introduction

A cubic graph is a graph whose vertices have degree three and a {1, 3}-graph is a graph whose vertices have degree either one or three. The graphs are allowed to have loops and parallel edges. Motivated by a result of Mochizuki [START_REF] Mochizuki | A Theory of Ordinary p-Adic Curves[END_REF], Liu and Osserman [START_REF] Liu | Mochizuki's indigenous bundles and Ehrhart polynomials[END_REF] associated a polytope P G to each {1, 3}-graph G and studied its Ehrhart quasi-polynomial.

For each degree three vertex v of a {1, 3}-graph G = (V, E), let a, b, and c be the three edges incident to v. Denote by S(v) the linear system consisting of a perimeter inequality and three metric inequalities defined on the variables w a , w b , and w c as follows:

w a + w b + w c ≤ 1 w a ≤ w b + w c w b ≤ w a + w c w c ≤ w a + w b .
From the three metric inequalities, one can immediately conclude that w a , w b , and w c are nonnegative. Consider the union of all the linear systems S(v), taken over all degree three vertices v of G. The polytope P G is defined by the set of all real solutions for this linear system.

Let E = {1, . . . , m} and w : E → R be a weight function defined on the edges of G. We use the vector notation w = (w 1 , . . . , w m ) ∈ R m . In particular, when w is a solution for the linear system defining P G , we write w ∈ P G .

Given a rational polytope P, Eugène Ehrhart defined the function L P (t) := |tP ∩ Z m |, which is the number of lattice points in the closed dilated polytope tP, for a nonnegative integer parameter t. Ehrhart showed that this function is a polynomial in t when P is an integral polytope. More generally, if P is a rational polytope, the function L P (t) is a quasi-polynomial whose period is closely related to the denominators appearing in the coordinates of the vertices of P [START_REF] Ehrhart | Polynômes arithmétiques et méthode des polyèdres en combinatoire[END_REF][START_REF] Beck | Computing the Continuous Discretely[END_REF].

Liu and Osserman conjectured ( [START_REF] Liu | Mochizuki's indigenous bundles and Ehrhart polynomials[END_REF]Conj. 4.2]) that polytopes associated to connected {1, 3}-graphs with the same number of vertices and edges have the same Ehrhart quasi-polynomial. They partially proved their conjecture, by showing that these quasipolynomials coincide for all nonnegative odd values of the dilation parameter t. In 2013, Wakabayashi [15, Thm. A(ii)] proved their conjecture.

An ingredient in Wakabayashi's proof for Liu and Osserman's conjecture is a local transformation performed in {1, 3}-graphs. This transformation is called an A-move by Wakabayashi and it is also known as a nearest neighbor interchange (NNI). The NNI has been studied mainly for binary trees [START_REF] Culik | A note on some tree similarity measures[END_REF][START_REF] Robinson | Comparison of labeled trees with valency three[END_REF], cubic graphs [START_REF] Tsukui | Transformations of cubic graphs[END_REF], and {1, 3}-graphs [START_REF] Wakabayashi | Spin networks , Ehrhart quasi-polynomials, and combinatorics of dormant indigenous bundles[END_REF]. We present the following general result for connected graphs with the same degree sequence, which might of interest on its own. We refer to a vertex of degree one in a graph simply as a leaf. An edge is external if it is incident to a leaf, otherwise it is internal. Theorem 1. Let G and G ′ be connected graphs with the same degree sequence and the same set of external edges. Then (a) G can be transformed into G ′ through a series of NNI moves.

(b) One can choose a spanning tree in G and a spanning tree in G ′ and require that all the pivots of the NNI moves are internal edges of both of these spanning trees.

In particular, for the special case of {1, 3}-graphs, the proof provided for Theorem 1(a) constitutes an alternative graph theoretic proof of a proposition by Wakabayashi [START_REF] Wakabayashi | Spin networks , Ehrhart quasi-polynomials, and combinatorics of dormant indigenous bundles[END_REF]Prop. 6.2].

One of the concerns of this paper is on the scissors congruence conjecture for the unimodular group, which is an analogue of Hilbert's third problem (equidecomposability). Concretely, this was stated as the following question by Haase and McAllister [START_REF] Haase | Quasi-period collapse and GL n (Z)-scissors congruence in rational polytopes[END_REF]. An integral matrix U is unimodular if it has determinant ±1. An affine unimodular transformation is defined by x → Ux + b, where U is a unimodular matrix and b is a real vector.

Question 2. [7, Question 4.1] Suppose that P and P ′ are polytopes with the same Ehrhart quasi-polynomial. Is it true that there is a decomposition of P into relatively open simplices P 1 , . . . , P k and affine unimodular transformations U 1 , . . . , U k such that P ′ is the disjoint union of U 1 (P 1 ), . . . , U r (P r )?

We show that for polytopes associated to {1, 3}-graphs such a scissors congruence decomposition holds. Namely, we have the following. Theorem 3. Let G and G ′ be two connected {1, 3}-graphs with the same number of vertices and edges. Then there is a dissection of P G into smaller polytopes P 1 G , . . . , P k G and affine unimodular transformations U 1 , . . . , U k such that P G ′ is the union of

U 1 (Q 1 G ), . . . , U k (Q k G ).
The proof of Theorem 3 relies on a piecewise unimodular transformation associated to a weighted version of the NNI move.

The rational polytope P G , associated to a {1, 3}-graph G, enjoys some fascinating symmetry. Linke [START_REF] Linke | Rational Ehrhart quasi-polynomials[END_REF] considered the extension of L P (t) for all nonnegative real numbers t. Royer [START_REF] Royer | Semi-reflexive polytopes[END_REF] defined a polytope to be semi-reflexive if L P (s) = L P (⌊s⌋) for every nonnegative real number s. One can verify that P G is semi-reflexive. A polytope is reflexive if it is integral, the origin is in its interior, and it is semi-reflexive [START_REF] Beck | Computing the Continuous Discretely[END_REF]. We prove the following. Theorem 4. For each {1, 3}-graph G, the polytope 4P G -½ is reflexive.

The paper is organized as follows. Section 2 contains the results involving the NNI move in graphs with the same degree sequence, including Theorem 1, while Section 3 discusses the extension of the NNI move to weighted graphs. Section 4 describes the unimodular decomposition of P G and presents the proof of Theorem 3. In Section 5, we prove Theorem 4. Finally, Section 6 contains some concluding remarks.

Nearest neighbor interchange

With an eye towards Section 3, where we consider weighted graphs, we think of a graph as defined by Bondy and Murty [START_REF] Bondy | Graph Theory[END_REF]. A graph G is an ordered pair (V, E) consisting of a set V of vertices and a set E, disjoint from V , of edges, together with an incidence function ψ G that associates with each edge of G an unordered pair of (not necessarily distinct) vertices of G. The degree sequence of G is the monotonic nonincreasing sequence consisting of the degrees of its vertices.

A nearest neighbor interchange (NNI) is a local move performed in G on a trail W of length three. This move interchanges one end of the two extreme edges of W on the central edge (Figure 1). We refer to the central edge of W as the pivot of the NNI move. The result of the move is another graph G ′ on the same number of connected components, with the same degree sequence. We consider the graph G ′ as having the same set of vertices and edges, that is, G ′ = (V, E), and only the incidence function ψ G ′ is adjusted accordingly. Intuitively, one can think of the edges as sticks that are being moved from G to G ′ . We think of an NNI move as a function γ W that associates the graph G to the graph

G ′ . In symbols, γ W (G) = G ′ . Observe that W is also a trail in G ′ and γ W (G ′ ) = G.
The well-known rotation, used in data structures to balance binary trees, is a particular NNI move, performed on a {1, 3}-tree (Figure 2). An NNI move does not affect the incidence to leaves, thus it preserves the partition of the edge set into internal and external edges.

Culik and Wood [4, Thm. 2.4] proved that any two {1, 3}-trees with ℓ (labelled) leaves can be transformed into one another through a finite series of NNI moves. They additionally gave an upper bound of 4ℓ -12 + 4ℓ log 2 (ℓ/3) on the number of NNI moves needed for this transformation. In this section, first we extend Culik and Wood's theorem to trees with the same degree sequence (Lemma 8), which we then use to extend their result further to connected graphs with the same degree sequence (Theorem 1).

A caterpillar is a tree for which the removal of all leaves results in a path, called its central path, or in the empty graph. For the later, we define that the central path is empty.

Lemma 5. Any tree can be transformed into a caterpillar with the same degree sequence through a series of NNI moves.

Proof. Let T be a tree. If T is a caterpillar, there is nothing to prove. So, we may assume T is not a caterpillar. Let P be a longest path in T and uv an internal edge of T not in P such that u is a vertex in P . The vertex u has two neighbors in P , otherwise P would not be a longest path, as v is not in P . Let w and z be the two neighbors of u in P . Let W be a trail with edges wu, uv, and vv ′ , where v ′ is a neighbor of v other than u. Perform an NNI on the trail W as in Figure 3, to insert v in P , obtaining another tree T ′ with the same degree sequence and a path longer than P . By repeating this process, we obtain a desired caterpillar after a finite number of NNI moves. The spine of a caterpillar is the sequence of the degrees of the vertices in the central path. We say the caterpillar is ordered if its spine is a monotonic nonincreasing sequence (Figure 4). Lemma 6. Any caterpillar can be transformed into an ordered caterpillar with the same degree sequence through a series of NNI moves.

Proof. An NNI can be used to swap any two adjacent vertices in the central path of a caterpillar, as in Figure 5(a). So we use an NNI to decrease, one by one, the number of inversions in the spine of a caterpillar until we obtain an ordered caterpillar.

Lemma 7. The external edges of a caterpillar can be sorted arbitrarily through a series of NNI moves.

Proof. An NNI can be used to swap any two external edges incident to two adjacent vertices in the central path of a caterpillar, as in Figure 5(b).

The next lemma is an extension of previous works in the literature [START_REF] Culik | A note on some tree similarity measures[END_REF][START_REF] Robinson | Comparison of labeled trees with valency three[END_REF][START_REF] Tsukui | Transformations of cubic graphs[END_REF][START_REF] Wakabayashi | Spin networks , Ehrhart quasi-polynomials, and combinatorics of dormant indigenous bundles[END_REF] that might be of independent interest. Proof. Let T and T ′ be two trees with the same degree sequence and the same set of external edges. Using Lemmas 5 and 6, we obtain a series of NNI moves that transforms T into an ordered caterpillar with the same degree sequence and the same set of external edges of T . Similarly, we obtain another series for T ′ . Using Lemma 7, we extend the series of NNI moves for T to sort the external edges of the caterpillar coming from T into the order they appear in the caterpillar coming from T ′ . The composition of these two series of NNI moves, with the series for T ′ inverted, gives a series of NNI moves that transforms T into T ′ .

Let G be a connected graph that is not a tree, and let e be an edge that is in a cycle. The graph obtained from G by cutting e is the graph G ′ resulting from the splitting of e into two edges, each connecting one of the ends of e to one of two new leaves (Figure 6). Choose an edge in a cycle of G and an edge in a cycle of G ′ . Let us denote these two edges by e. Let H and H ′ be the graphs obtained from G and G ′ , respectively, by cutting their edge e. The number of vertices in H and H ′ is n + 2 and the number of edges is m + 1. Call e ′ and e ′′ the two new edges in both H and H ′ . Since e is in a cycle in G and in a cycle in G ′ , the graphs H and H ′ are connected and the dimension of their cycle space is r -1. Also, H and H ′ have the same degree sequence and the same set of external edges. By induction, H can be transformed into H ′ through a series of NNI moves. The same series of NNI moves transforms G into G ′ . Indeed, the set of external edges in all graphs obtained during the application of this series of NNI moves, transforming H into H ′ , contain e ′ and e ′′ . Glue e ′ and e ′′ into an edge in each of these graphs, obtaining a sequence of connected graphs that is the result of a series of NNI moves starting at G and ending at G ′ . This ends the proof of (4).

G G ⇓ cut ⇓ cut H = H 0 NNI ←→ • • • NNI ←→ H k = H ′ ⇓ glue ⇓ glue ⇓ glue G = G 0 NNI ←→ • • • NNI ←→ G k = G ′
Similarly, by induction in the cycle space of G and G ′ , one can prove (b). Indeed, it is sufficient to choose in each step of the induction an edge to be cut which is not in either of the spanning trees.

For the case of cubic graphs, the proof of Theorem 1 provides an alternative proof for a theorem by Tsukui [14, Thm. II], which refers to NNI moves as S-transformations, where the 'S' stands for slide. For the slightly more general case of {1, 3}-graphs, the proof of Theorem 1 provides an alternative proof for a proposition by Wakabayashi [15, Prop. 6.2], which refers to NNI moves as A-moves. Wakabayashi's proof uses a topological pants decomposition for compact, oriented surfaces of finite genus.

Figure 7 shows a series of NNI moves from the complete graph K 4 to a tree with a loop added to each of its leaves.

Weighted NNIs for {1, 3}-graphs

To deal with weights on the edges of a {1, 3}-graph, we now enhance the NNI move. This was achieved by a bijection defined by Wakabayashi [START_REF] Wakabayashi | Spin networks , Ehrhart quasi-polynomials, and combinatorics of dormant indigenous bundles[END_REF]Prop. 6.3].

Let G = (V, E) be a {1, 3}-graph and w be a weight function defined on the edges of G. A weighted NNI is a local move performed in (G, w) on a trail W of length three induced by a NNI move in G on W . The result of the move is the graph G ′ obtained from G by applying an NNI move on W , and the weight function w ′ defined on the edges of G ′ as follows.

+2 +1 -3 -2 +3 -1 +2 +1 -3 -2 +3 -1 +2 +1 -3 -2 +3 -1 +2 +1 -3 -2 +3 -1 +2 +1 -3 -2 +3 -1 +2 +1 -3 -2 +3 -1 NNI NNI NNI NNI NNI K 4 T 4
Let e be the central edge of W , that is, the pivot of the NNI move. Let a and b be the other edges in W , and c and d be the remaining edges adjacent to e, as depicted in Note that, if w is integer valued, then so is w ′ . Moreover, since pivots are always internal edges and an NNI move does not affect the partition of the edges into internal and external, a weighted NNI move may only change the weights of internal edges. We think of a weighted NNI move as a function φ W (G, w) = (G ′ , w ′ ), which extends the previously defined NNI move, γ

W (G) = G ′ . Note that φ W (G ′ , w ′ ) = (G, w), because γ W (G ′ ) = G and w ′ e + max{w ′ b + w ′ c , w ′ a + w ′ d } -max{w ′ a + w ′ c , w ′ b + w ′ d } = w ′ e + max{w b + w c , w a + w d } -max{w a + w c , w b + w d } = w e + max{w a + w c , w b + w d } -max{w b + w c , w a + w d } + max{w b + w c , w a + w d } -max{w a + w c , w b + w d } = w e .
Wakabayashi [START_REF] Wakabayashi | Spin networks , Ehrhart quasi-polynomials, and combinatorics of dormant indigenous bundles[END_REF]Prop. 6.3] proved that w ∈ tP G if and only if w ′ ∈ tP G ′ for every integer t ≥ 0, which implies Liu and Osserman's conjecture. We observe that this holds also for every real t ≥ 0, and state it below as we will use it in the next section.

Lemma 9.

Let G be a {1, 3}-graph and w be a weight function defined on the edges of G. Let W be a trail in G of length three and suppose that φ W (G, w) = (G ′ , w ′ ). Then w ∈ tP G if and only if w ′ ∈ tP G ′ for every t ≥ 0.

Therefore there are distinct rational polytopes whose Ehrhart quasi-polynomials coincide for all real t.

As a weighted NNI changes only the weight of the pivot, which is always an internal edge, the series of NNI moves from G to G ′ changes only the weights of internal edges.

In fact, by Theorem 1(b), one can choose a spanning tree T in G and a spanning tree T ′ in G ′ and require that the series of NNI moves uses as pivots only internal edges of T and T ′ . As a consequence, the bijection from P G to P G ′ keeps fixed the majority of the coordinates of the points. Namely, it changes only coordinates that correspond to internal edges of the chosen spanning trees. Formally, the latter discussion provides the following as a corollary of Theorem 1(b) and Lemma 9. Let E = {1, . . . , m} and w : E → R be a weight function defined on the edges of G. If X is a subset of E, then the restriction of w to X is the function w| X : X → R such that w| X (x) = w(x) for all x ∈ X.

Corollary 10. Let G and G ′ be connected {1, 3}-graphs with the same number of vertices and on the same set E of edges. Then there exists a bijection φ between P G and P G ′ such that, for w ′ = φ(w), we have that

w ′ | X = w| X
where E \ X is the set of internal edges of arbitrary spanning trees in G and G ′ .

Scissors congruence

Haase and McAllister [START_REF] Haase | Quasi-period collapse and GL n (Z)-scissors congruence in rational polytopes[END_REF] have raised Question 2 that can be thought of as an analogue of Hilbert's third problem (equidecomposability) for the unimodular group. We show that for polytopes associated to {1, 3}-graphs a statement similar to Question 2 holds.

Let G be a {1, 3}-graph with edge set E = {1, . . . , m}. Let W be a trail in G of length three and suppose that φ W (G, w) = (G ′ , w ′ ). As argued ahead, the function φ W (G, •) : R m → R m is associated to one or two hyperplanes in R m and two or four unimodular transformations. For short, let φ(w) = φ W (G, w) for every w ∈ R m . Let a, b, c, d, and e be as in Figure 8, with a, e, and b being the edges of W . Clearly φ is piecewise linear, namely, for w ∈ R m , w ′ f = w f for every f = e and

w ′ e = w e + w b -w d if w a + w b ≥ w c + w d and w a + w d ≥ w b + w c , (1a) w ′ e = w e + w a -w c if w a + w b ≥ w c + w d and w a + w d < w b + w c , (1b) w ′ e = w e + w c -w a if w a + w b < w c + w d and w a + w d ≥ w b + w c , (1c) w ′ e = w e + w d -w b if w a + w b < w c + w d and w a + w d < w b + w c . ( 1d
)
The hyperplanes associated to φ are w a + w b -w c -w d = 0 and w a -w b -w c + w d = 0, which are either the same hyperplane (if a = b or c = d) or two orthogonal hyperplanes. Moreover, the matrix that gives the linear transformation in each case is unimodular. Indeed, the matrix for case (1a) is obtained from the identity matrix by substituting the row corresponding to the edge e by the row χ e + χ b -χ d , the matrix for case (1b) is obtained from the identity matrix by substituting the row corresponding to the edge e by the row χ e + χ a -χ c , the matrix for case (1c) is obtained from the identity matrix by substituting the row corresponding to the edge e by the row χ e + χ c -χ a , and the matrix for case (1d) is obtained from the identity matrix by substituting the row corresponding to the edge e by the row χ e + χ d -χ b . Thus the determinant of each such matrix is always 1. Therefore each of them is unimodular. Figure 9 shows an example with the two 3-regular graphs on two vertices and their polytopes. The two graphs differ by one NNI move. The function = w ′ between the two polytopes in Figure 9 is defined by w ′ 1 = w 1 , w ′ 2 = w 2 , and

(0, 1 2 , 1 2 ) ( 1 2 , 1 2 , 0) ( 1 2 , 0, 1 2 ) ( 1 4 , 1 4 , 1 2 ) ( 1 2 , 1 2 , 0) 1 2 3 w 1 w 2 w 3 w 1 w 3 w 2 w 1 + w 2 + w 3 ≤ 1 2w 1 + w 3 ≤ 1 w 1 ≤ w 2 + w 3 w 3 ≤ 2w 1 w 2 ≤ w 1 + w 3 w 3 ≥ 0 w 3 ≤ w 1 + w 2 2w 2 + w 3 ≤ 1 w 3 ≤ 2w 2
w ′ 3 = w 3 + max{w 1 + w 2 , w 1 + w 2 } -max{2w 1 , 2w 2 } = w 3 + w 1 + w 2 -2 max{w 1 , w 2 } = w 3 -w 1 + w 2 if w 1 ≥ w 2 w 3 + w 1 -w 2 if w 1 ≤ w 2 ,
which is piecewise unimodular. In this case, only one hyperplane (the one containing the shaded triangle inside the polytopes, w 1 = w 2 ) splits the polytopes into two, each part being unimodularly equivalent to one of the parts of the other polytope. The corresponding unimodular transformations are given by

U w 1 ≤w 2 =    -1 0 -0 0 1 0 1 -1 1    U w 1 ≥w 2 =    1 -0 -0 0 1 0 -1 1 1    .
Note that the polytope on the right side of Figure 9 has one more vertex than the other one, so that the combinatorial types of these piecewise unimodularly equivalent

1 0 0 0 1 0 -1 -1 -1    U w 1 ≤w 2 U w 1 ≥w 2    1 0 0 0 1 0 -1 -1 -1    U w 1 ≥w 3    1 0 0 -1 -1 -1 0 0 1       1 0 0 -1 -1 -1 0 0 1    U w 1 ≤w 3
P G 2 = T 1 (Q 1 G 1 ) ∪ T 2 (Q 2 G 1 ) = T 1 (Q 1 G 1 ∩ U 1 (Q 1 G )) ∪ T 1 (Q 1 G 1 ∩ U 2 (Q 2 G )) ∪ T 2 (Q 2 G 1 ∩ U 1 (Q 1 G )) ∪ T 2 (Q 2 G 1 ∩ U 2 (Q 2 G )) = T 1 U 1 (P 11 ) ∪ T 1 U 2 (P 12 ) ∪ T 2 U 1 (P 21 ) ∪ T 2 U 2 (P 22 ) .
This completes the proof for the two first weighted NNI moves assuming that both are associated to only one hyperplane and consequently two unimodular transformations.

For the remaining cases, whenever a weighted NNI move is associated to two hyperplanes (and consequently four unimodular transformations), the polytopes would be dissected into up to four smaller polytopes, but the process would be essentially the same.

Reflexivity

An equivalent way to define a reflexive polytope is to require that it is integral and has the hyperplane description {x ∈ R d | Ax ≤ ½} for some integral matrix A. Another equivalent definition of reflexivity is to say that a polytope P is reflexive if and only if the origin is in P • and (t + 1)P • ∩ Z d = tP ∩ Z d for all t ∈ Z ≥0 , where P • is the interior of P.

Proof of Theorem 4. The polytope 4P G -½ consists of the vectors w ∈ R m satisfying

w a + w b + w c ≤ 1 w a -w b -w c ≤ 1 -w a + w b -w c ≤ 1 -w a -w b + w c ≤ 1,
for each degree three vertex v and edges a, b, and c incident to v, for a total of 4n 3 inequalities, where n 3 is the number of degree three vertices in G. From Liu and Osserman [10, Prop. 3.5], the polytope 4P G is integral, and thus so is 4P G -½.

The reflexivity of the integral polytope 4P G -½ has some interesting consequences, as follows. Let m denote the number of edges of G. Since 4P G is an m-dimensional integral polytope, we have that for t ≡ 0 mod 4:

L P G (t) = t + m m + h * 1 t + m -1 m + • • • + h * d-1 t + 1 m + h * d t d ,
where h * 1 , . . . , h * d-1 , h * d are the coefficients of the numerator of the rational function given by the Ehrhart series of 4P G [START_REF] Beck | Computing the Continuous Discretely[END_REF]Lemma 3.14].

The polytopes 4P G and 4P G -½ have the same Ehrhart polynomial. Hence it follows from the reflexivity of 4P G -½ and from Hibi's palindromic theorem [START_REF] Beck | Computing the Continuous Discretely[END_REF]Thm. 4.6] that h * k = h * m-k for all 0 ≤ k ≤ m/2. Therefore it follows that to describe L 4P G (t), we only need to compute half of its coefficients, thus lowering the computational complexity required to compute L 4P G (t).

Concluding remarks and open problems

In the process of proving the main theorem, a great deal of structure of the polytopes P G , and their corresponding cones has been revealed. The polytopes P G and their corresponding cones may be related to the well-known metric polytopes and metric cones [START_REF] Deza | Geometry of Cuts and Metrics[END_REF]. In particular, in the case that G is a planar graph, the polytope P G can be thought of as a restricted type of "metric polytope" [START_REF] Laurent | Graphic vertices of the metric polytope[END_REF], associated to the dual graph G * . It would be interesting to pursue connections to metric polytopes.

Wakabayashi [START_REF] Wakabayashi | Spin networks , Ehrhart quasi-polynomials, and combinatorics of dormant indigenous bundles[END_REF] gave a formula for two of the four constituent polynomials of the Ehrhart quasi-polynomial for P G and for the volume of P G when G is a connected cubic graph on n vertices. It is rather surprising that the Verlinde formula makes an appearance here, namely, Wakabayashi showed that, for odd t, where B n is the n-th Bernoulli number. Zagier [START_REF] Zagier | Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula[END_REF] showed that the above Verlinde formula is the polynomial

L P G (t) = (t + 2) n/2 2 n+1 n/2 k=0 (-1) k-1 2 2k B 2k (2k)! c k (t + 2) 2k ,
where c k is the coefficient of x -2k in the Laurent expansion of sin -n x at x = 0. It would be interesting to determine if a similar formula holds for all connected {1, 3}-graphs. Even when G is a {1, 3}-tree, it is completely open, and quite interesting, to determine the Ehrhart quasi-polynomial of P G . We determined the following quasi-polynomials for some small trees with LattE [START_REF] Baldoni | Software and user's guide for LattE integrale[END_REF].

Figure 1 :

 1 Figure 1: An NNI move on the trail marked in bold. One of the incidences of the edges a and b were interchanged.

Figure 2 :

 2 Figure 2: Right and left rotations applied to a binary tree.

Figure 3 :

 3 Figure 3: An NNI to insert v in the central path.

Figure 4 :

 4 Figure 4: (a) A caterpillar with spine (4, 5, 2, 3) and central path highlighted in bold. (b) An ordered caterpillar.

Figure 5 :Lemma 8 .

 58 Figure 5: (a) An NNI to swap two adjacent vertices in the central path of a caterpillar. (b) NNIs to swap labels on leaves hanging from adjacent vertices in the central path.

Figure 6 :

 6 Figure 6: Graph G ′ obtained from G by cutting an edge e.

Figure 7 :

 7 Figure 7: A series of NNI moves from K 4 to the graph T 4 .

Figure 8 .

 8 Possibly a, b, c, and d are not pairwise distinct. The weight function w ′ is such that w ′ f = w f for every f = e and w ′ e = w e + max{w a + w c , w b + w d } -max{w b + w c , w a + w d }.

Figure 8 :

 8 Figure 8: Edges and weights in a weighted nearest neighbor interchange.

Figure 9 :

 9 Figure 9: The polytopes of the two cubic graphs on two vertices. The shaded triangles are the intersection of the polytopes with the hyperplane w 1 = w 2 .

Figure 10 :

 10 Figure 10: Example for Theorem 3.

L

Acknowledgements

We thank Tiago Royer and Fabrício Caluza Machado for valuable comments.

CNPq 456792/2014-7 and 452507/2016-2, CAPES PROEX.

polytopes may differ. Observe how the extra vertex ( 14 , 1 4 , 1 2 ) is "formed" when going from the polytope on the left to the one on the right, and how it "disappears" when going in the other direction. Also, the edge between the origin and the vertex ( 12 , 1 2 , 0) in the polytope on the left is not an edge in the polytope on the right.

Theorem 3 extends this for graphs that differ by a series of NNI moves. It relates to Question 2.

Proof of Theorem 3. The proof is constructive. By Theorem 1, there exists a finite sequence of NNI moves that transforms

We shall explain the procedure for the first two NNI moves assuming that both underlying weighted NNI moves are associated to only one hyperplane and consequently two unimodular transformations. The other cases and the rest of the NNI moves follow in an analogous but more cumbersome fashion.

Let W 0 be the trail used in the first NNI move, that goes from G = G 0 to G 1 , and let φ 0 (w) = φ W 0 (G 0 , w), for w ∈ R E , be the corresponding weighted NNI move. Let H 0 be the hyperplane associated to φ 0 , and let U 1 , U 2 be the two unimodular transformations, one for each side of the hyperplane H 0 . The hyperplane H 0 dissects the polytope

In words, we presented a dissection of P G into two smaller polytopes and two affine unimodular transformations U 1 and U 2 that, if applied to the two smaller polytopes, result in P G 1 . Now we will proceed one more step, and present a dissection of P G into four smaller polytopes, and four affine unimodular transformations that, if applied to the four smaller polytopes, will result in P G 2 (Figure 10).

Let W 1 be the trail used in the second NNI move, that goes from G 1 to G 2 , and let φ 1 (w) = φ W 1 (G 1 , w), for w ∈ R E , be the corresponding weighted NNI move. Let H 1 be the hyperplane associated to φ 1 , and let T 1 , T 2 be the two unimodular transformations, one for each side of the hyperplane H 1 . The hyperplane H 1 dissects the polytope

) each into two smaller polytopes, obtaining the dissection

The latter naturally induces the following dissection:

So, if we let

( 1 2 , 0, 0)

NNI on W = (1, 2, 3)

)

( 1 2 , 0, 0) (0, Further generalizations to more general graphs and connections with other topics as manifold invariants would also be of interest.


I T Lucas
email: ivlucas@ccr.jussieu.fr

S Durand-Vidal

E Dubois

J Chevalet

P Turq

Surface charge density of maghemite nanoparticles: Role of electrostatics in the proton exchange

Keywords: nanocolloid, colloidal dispersions, amphoteric oxides, polyacid, surface potential, zeta potential

In the present work, we focus on the surface charge properties of well-defined maghemite (γ-Fe 2 O 3 ) nanoparticles dispersed in water. Due to the acidic properties of surface groups, the nanoparticles surface can be positively or negatively charged depending on the pH.

Consequently there are electrostatic repulsion forces between particles that allow stabilizing the suspensions in water. Acid-base titrations of the surface groups starting from acidic, alkaline and neutral pH are performed to access the dependence of the surface charge density with the pH. Titrations are followed by potentiometric and conductimetric measurements.

Simultaneous zetametry measurements by acoustophoresis are implemented. The

experimental curves are analyzed using several hypotheses, in particular we assume that the sites on the surface of the particles behave as weak diacids. The interpretation of the experimental determinations of the charge implies to take into account an electrostatic term due to the potential developed at the surface of the particles. Thanks to the complementarity of the potentio-conductimetric titrations and of the zetametry measurements, it is possible to better understand the phenomena of protonation and deprotonation of the colloidal particles as a function of the medium of dispersion (acidic, alkaline, neutral pH and nature of the counterions) and of the nature of the reagent.

I-Introduction

In aqueous medium, colloidal particles are usually stabilized by electrostatic interactions, which strongly depend on the surface charge of the particles. The stability of the suspension is often described in terms of a zeta potential that depends both on the magnitude of the surface charge of the particles and on the ionic strength (the latter determines the range of the electrostatic interactions). The knowledge of the charge properties is therefore important to prepare stabilized dispersions of particles. In this article, magnetic nanocolloidal suspensions are studied in an intermediate concentration range (1.5 volume percent) close to the conditions used for preparing mixed magnetic systems 1 . These nanocolloids have a typical radius of 6nm and are electrostatically stabilized in aqueous solutions. Their surface charge arises from the inherent properties of the oxides on their surface: charges are positive at acidic pH and negative at alkaline pH (at intermediate pH, the charge of the particles decreases, then aggregation process appears). Acid-base titrations can give an access to the surface charge density σ 0 of the particles and thus information on the acidic properties (pKa) of surface oxide groups. We are interested here in the influence on the magnitude of σ 0 of several parameters, as the pH, the nature of the counter ions of the particles and the reagent used for the titration.

For that, acid-base titrations are followed by simultaneous potentiometric, conductimetric and zetametry measurements.

The determination of the surface charge density is not straightforward and depends on experimental and theoretical approaches that will be discussed in details. A classical model describing the colloidal suspension as a melt of strong acid (the medium of dispersion) and weak diacids (the surface oxides) characterized by two pKa values is used in a first step to analyze the titration curves. However this leads to some differences between the values extracted from the different curves. We explain this in a second step considering an electrostatic contribution in the proton exchange that depends on the surface charge density and therefore on the pH. Complementary zetametry measurements allow us to decide whether the description of the charge of the particles is accurate. This article is divided in two major parts. The first one gives details on the sample preparation (synthesis and dispersion of the particles), on the characterization techniques implemented and on the theoretical tools used. The second one is dedicated to results and discussions on their interpretation.

II-Materials and methods

Colloidal suspensions

Synthesis and characterisation

The colloidal suspensions used in this study are composed of nanoparticles of maghemite (γ-Fe 2 O 3 ), a magnetic iron oxide. These nanoparticles are chemically synthesized according to the process described by Massart and al 2 (1987). Briefly, an aqueous mixture of iron(II) chloride and iron(III) chloride is alkalinized with concentrated ammonium hydroxide. The precipitate obtained, composed by magnetite (Fe 3 O 4 ) particles, is separated from the solvent and acidified with nitric acid in order to neutralize the excess of base. Eventually, the particles are oxidized in maghemite (γ-Fe 2 O 3 ) in a solution of ferric nitrate at 100 ºC. At this step, the size distribution of the particles dispersed in acidic solution lies between diameters d of 5nm and 20nm. To reduce the polydispersity of the colloidal suspension, the precipitation of the largest particles is achieved by addition of HNO 3 , which screens the electrostatic repulsion.

The supernatant composed of particles of smaller sizes is removed by suction. The fraction with the largest particles is carefully washed with water and acetone (see §2.2) in order to decrease the ionic strength by removing the excess of ions (H + , NO 3 -). The resulting colloids have an average radius of 6nm. The size distribution of such particles, which can be approximated by a lognormal law, is usually determined from the magnetic properties of the solutions by fitting magnetization curves [START_REF]MAGNETIC Fluids and applications handbook[END_REF][START_REF] Chantrell | [END_REF] and here also determined from the AFM measurements. The size distribution of particles obtained from the AFM pictures (≈ 2000 particles) exhibits a sharp peak between d = 11.5 and 12.5 nm (Figure1). A log normal fit yields an average particle diameter d 0 = 12 nm (with lnd 0 = <lnd>, d 0 being also the median diameter) and a polydispersity σ 0 = 0.280. The volume fraction Φ vol of the particles is evaluated from a chemical titration of iron 5 (eq. 1) and from flame emission spectroscopy after dissolution of the particles in HCl (12 mol L -1 ). Φ vol is given by:

[ ] [ ] ) . ( . 577 . 1 2 ). 3 2 ( (%) 1 3 2 - = + = Φ L mol Fe Fe M M O Fe O Fe vol ρ (1)
with M the molar weigh and ρ Fe2O3 the density of maghemite (4.9 g.cm -3 ). Due to the acidic properties of surface groups, the surface of the nanoparticles can be positively or negatively charged depending on the pH (table 1). Consequently, there are electrostatic repulsion forces between particles that allow stabilizing the suspensions in water at extreme pH. However, the suspensions are not stable for 3.5 < pH < 10.5 because the surface charge density is too low.

Influence of the nature of the counter-ions

For the experiment presented here, we use the following types of suspensions: the particles associated with their counter ions dispersed in both acidic and alkaline medium. The concentration of the ions in the suspensions (ionic strength) and their nature are parameters that also control the stability of the colloidal suspensions (see figure 2). For instance divalent cations like sulphate (SO 4

2-

) screen the electrostatic interactions (by adsorption) leading to the aggregation of the nanoparticles whereas univalent and larger ions like nitrate (NO ) are suitable for electrostatically stabilized suspensions. Moreover, chloride ion (Cl -) (very small) is a complexing agent of iron and leads to a fast dissolution of the particles in acidic medium. Therefore we dispersed the nanoparticles in perchloric acid (HClO 4 ) and nitric acid (HNO 3 ) between pH 1 and 3.5. (Below pH = 1.5, iron oxides are not thermodynamically stable and start to dissolve but the kinetics of dissolution is very slow). In order to obtain stabilized colloidal suspension of negatively charged particles, tetramethylammonium hydroxide (TMA + OH -: Sigma Aldrich) is used within the 10-13 pH range. In fact, NaOH cannot be used to prepare stabilized suspensions because sodium (Na + ) counter-ions (very small) are flocculating agents of our system in alkaline medium. Thus the number of counter ions that can be used is limited because of the constraints related to the colloidal stability.

Particles pH range of stability

Stabilizing Counterions

Non-Stabilizing Counterions

Reagent

Determination of the surface charge density of the nanoparticles

When dispersed in aqueous media, iron oxide nanoparticles acquire a surface charge as a result of the Brönsted acid-base behaviour of the surface oxides (surface sites). The simplest hypothesis to describe the sites on the surface is to assume identical sites which behave as weak diacids. Their pH-dependent protonation/deprotonation process can then be described by the following equilibria (2) and (3) where M is the metal of the spinel type nanoparticles

M-OH 2 + = M-OH + H + (2)
M-OH = M-O -+ H + (3)
In order to determine the structural charge and the zeta potential of our nanoparticles we have performed simultaneously potentiometric-conductimetric acid-base titrations as proposed in reference 6 and zetametry measurements on each colloidal sample (80 mL) at a volume fraction Φ vol (%) = 1.5 under CO 2 -free atmosphere. The simultaneous measurements (zetaprobe, conductivity and pH electrode, temperature probe, burettes) impose the volume of 80ml. The volume fraction Φ vol (%) = 1.5 is a compromise : it is high enough in order to distinguish the titrations of strong and weak acids in the conductivity curves; it is low enough to keep an efficient homogenization whatever the pH. Indeed, at higher volume fraction and at intermediate pH, the particles precipitate and form a thixotropic phase which hinders homogenization.

Titration procedure

In this study we perform several titrations through distinct experiments (not successive) from acidic medium (pH = 2) to alkaline medium (forward titration), from alkaline (pH = 12)

to acidic medium (backward titration) and from the neutral pH (assimilated to the point of zero charge (PZC) of the particles) to acidic or alkaline media. The initial pH is adjusted to pH = 2 in the forward titration, to pH = 12 in the backward titration and to pH ∼ 7 in the case of the titration starting from the PZC of the particles. In the forward titrations we use tetramethylammonium hydroxide ([TMAOH] = 1 ± 0.05 mol L -1 ) as strong alkaline reagent (pKa > 13), whereas in the backward ones, we use nitric and perchloric acid at the same concentration. All reagents used in these investigations are of analytical grade (VWR International France). After each experiment, the concentrations of the reagents (TMAOH, HClO 4 and HNO 3 ) are checked by acid base titration with HCl and NaOH (1.0 mol L -1 , Normadose VWR). Before titrations, alkaline reagents are stirred and degassed by purified argon for 10 min to prevent the carbonation phenomena.

Potentiometric-conductimetric method

Both the potentio-conductimetric techniques and reagent additions in the suspension are carried out by using a DT 1200 analyzer (Dispersion technology, USA) which includes electrochemical probes, thermometer and electronic burettes. The immersion-type conductimetric cell is calibrated by using KCl solutions of known conductivities while the pH electrode is used after calibration by three buffer solutions (pH 4, 7 and 10). As particles coagulate in the intermediate range of pH, the diffusion of the potential determining ions in the medium, and therefore the electrochemical readings, can be affected. Therefore, we apply an efficient magnetic stirring for all experiments in order to minimize these sources of inaccuracy.

Zetametry

Maghemite based "ferrofluids" absorb in the visible spectrum (solutions are black reddish)

and appear opaque at high volume fractions. Thus we cannot apply standard techniques based on Doppler effect to measure the zeta potential of our particles like in laser zetametry. We use the acoustophoresis, a technique based on the signal produced between two electrodes when the charged suspension is exposed to an ultrasonic wave. The instrument, DT 1200, measures the induced current (amplitude and phase) named CVI (Colloidal Vibration Intensity) between the electrodes. The software deduces the electrophoretic mobility and therefore the zeta potential (classical theory: Helmholtz-Smoluchowski equation 7 ) from CVI. It has to be notified that the zeta potential values used in this study are the ones given directly by the apparatus. The determination of the electrophoretic mobility of the particles from the CVI requires a calibration of the apparatus by a colloid of well known electrokinetic properties.

The calibration is performed using a colloidal suspension of ludox ® (TM-50 Sigma Aldrich) at Φ weight (%)=10 (silica particles: 30nm, KCl (0.01 mol L -1 ), pH = 9, ζ(mV) = -38). Several parameters, measured or taken from tables, are necessary: the volume fraction of the particles, the dielectric constant, the sound speed, the density of both the particles and the solvent, as well as the compressibility and the viscosity of the solvent. Concerning our systems, the relationship between CVI and zeta potential is independent of the size as far as the size is lower than 100nm [START_REF] Dukhin | Ultrasound for Characterizing[END_REF] .

Eventually, the signal of the supported salt is an additional factor which could influence the value of the calculated zeta potential. A procedure exists to correct the zeta potential by measuring the signal which comes from the salt. In our system, such measurements show that the effect of salt (here the ionic strength is less than 0.08 mol L -1 ) on the zeta potential value is negligible.

Preparation of the samples

Throughout all experiments carried out in this study, the same sample of ferrofluid is used and no additional salt is dissolved in the suspension (the ionic strength depends only on the concentration of the acidic or alkaline species in the suspension, the charged particles plus the counter-ions in their vicinity). Maghemite particles once synthesized are stabilized in nitric acid within the pH range 1.5-1.8. Consequently, in order to disperse the particles in the appropriate medium (acidic: HClO 4 , HNO 3 , alkaline: TMAOH or neutral) and finally to adjust the pH and therefore the ionic strength values by eliminating undesirable ions, the suspensions undergo chemical and physical treatments. The protocols used are described under.

-Sample at pH=12: transfer from acidic medium HNO 3 toward alkaline medium

TMAOH

TMAOH is added to a suspension of particles in a nitric acid solution until a concentration of 0.5 mol L -1 is reached in order to reverse the surface charge of the particles. The suspension is vigorously stirred and heated at 100°C during 15 minutes. The particles are then dispersed in alkaline medium, their counterions are TMA + , and TMA + NO 3 -pairs remain in the solution.

To remove these ions, acetone (Normapure grade) is added in order to destabilise the suspension. In fact the change in the dielectric constant of the medium weakens the repulsion forces between particles and leads to the destabilisation of the suspensions. Under the action of the gravitational and magnetic forces, the aggregated particles sediment at the bottom of the container. The supernatant containing water, ions and acetone is removed by suction. The precipitate is carefully washed by successive additions of acetone followed by suction in order to eliminate ions TMA + , NO 3 and OH -in excess. Diethyl ether is poured on the precipitate to remove acetone by precipitation/suction. Then water is added: the precipitate disperses immediately in water, and the diethyl ether, non miscible with water, separates. This organic phase is removed by suction and the remaining solvent dissolved in water is eliminated by slight heating. Finally, a solution of TMAOH (10 -2 mol L -1 ) is poured in the concentrated suspension in order to adjust the pH and the volume fraction of the sample.

-Sample at pH=2: transfer from alkaline medium TMAOH toward acidic medium HClO 4

We do not perform the transfer of the particles from the nitric acid medium to the perchloric acid medium directly. In a first step, the suspension in HNO 3 is transferred into alkaline medium as described above. In a second step, a perchloric acid solution is added to the suspension until a concentration of 0.5 mol L -1 is reached. The particles are then dispersed in acidic medium, their positive charges are compensated by ClO 4 -counter ions, and TMA + ClO 4 -pairs remain in the solution. Contrary to suspensions of negatively charged particles in TMAOH (pH = 13) and of positively charged particles in HNO 3 at pH = 1, which can be easily destabilized by addition of acetone, it is very difficult to precipitate particles dispersed in HClO 4 at pH = 1. Therefore, suspensions in HClO 4 are destabilized by addition of concentrated perchloric acid (12 mol L -1 ) until precipitation. Indeed, the ionic strength becomes so large, that the electrostatic interactions between particles are highly screened, leading to the precipitation of particles. However it is difficult to reduce this high ionic strength by successive washing of the precipitate with acetone. In this system, it is thus easier, although longer, to control the pH and the ionic strength I of the suspension with the technique of dialysis (note that this technique can also be used for the other systems). The suspension is packed into a dialysis tube (Spectra/Por-12000-14000) plunged in a solution adjusted to the selected pH. The wall of the dialysis tube is a membrane which is permeable to the small molecules and the ions, whereas the particles and the macromolecules cannot pass through this membrane. The osmotic pressure outside the tube has to be adjusted to prevent the outside solution to pass inside the tube and make it inflate and burst. A polymer Dextran: 500000 g mol -1 (Amersham Biosciences) is dissolved (Φ weight = 5%) in the outer solution.

After several days, the inner pH in the dialysis tube stabilizes at a value which depends on the pH of the outside solution.

-Sample at the neutral pH

Starting from alkaline or acidic medium, the colloidal suspensions are neutralized respectively with HNO 3 or NaOH till pH ∼ 7 is reached. At pH ∼7 (near the PZC) the magnitude of the structural charge of the particles tends toward zero, thus particles flocculate. The precipitate is carefully washed with water till the conductivity of the supernatant (removed by suction) is close to that of de-ionized water.

Determination of the surface charge density of the particles

The total concentration of protons bound to the oxides at the surface of the particles can be experimentally determined from potentiometric-conductimetric titrations. The colloidal suspensions are a mixture of weak acidities (the surface sites of the nanoparticles) and of strong acidities in the medium of dispersion (or strong base in alkaline medium). In a common acid-base titration of a mixture of a strong acid with weak polyacids, measurements of conductivity make it possible to determine with precision the equivalence points corresponding to the neutralization of all the acidic and basic functions if the pKa values are distinct (∆pKa = pKa 1 -pKa 2 > 2). A similar interpretation can be applied in colloidal suspensions if we assume that: (I) all surface sites are amphoteric oxides, so that they exchange protons according to equations ( 2) and (3) (II) all the surface sites are identical (III)

the sites are weak acids (IV) the two acidities of each site are weak enough (pKa 1 >> 2, pKa 2 << 12) and they are distinct (∆pKa > 2) (V) the sites are independent, which means that there is no influence of one site on the other sites (VI) at pH = 2 and pH = 12, the charge is at saturation (all sites are charged) thus the number of charges per particle is the same at pH = 2 and pH = 12. We shall discuss the validity of these hypotheses later on.

Titration of surface acidities

Considering these assumptions, there are therefore at least two equivalence points (EP) in the forward and the backward titration. First an EP that corresponds to the neutralization of the strong acid or base and then a second EP that corresponds to the neutralization of all weak diacid groups. These EP are usually determined from first derivative of the pH curve, or from the abrupt changes of slope of the conductivity curves. The total concentration of adsorbed protons [H + ] ads is determined from the amount of reagent (C reagent = 1 mol L -1 ) which is added to the suspension between the first and the second equivalence point.

In the case of titrations starting from acidic, alkaline and neutral pH, it is also possible to evaluate the number of protons bound to the surface of the particle by comparing the pH measured after the addition of the reagent to the solution and the pH expected if no particles are present. For an alkaline reagent (TMAOH), this difference ∆nOH -is calculated according to the equation ( 4):

nOH dropped - + nOH initial - = (nOH free -) + (nH initial + -nH free + ) + ∆nOH - (4)
nOH - initial and nH + initial are respectively the amount of hydroxide ions and of protons (mol) that are initially present in solution, nOH - free and nH + free are the hydroxide ions and the protons that remain free after the addition of reagent (nOH - dropped ). Symmetrical expressions can be obtained for an acidic reagent.

For the titrations performed between acidic and alkaline pH (pH = 2 → 12), the amount of protons removed at the surface of the particles (nH + ads ) corresponds to ∆nOH -. The total concentration C T of the charges at the surface of the particle is therefore the half of the total number of bound protons determined from the titration (C T = [H + ] ads / 2) if we assume the hypotheses formulated above.

For the titrations performed from the neutral pH, C T is directly accessible (C T = [H + ] ads )

without considering the assumptions (III), (V) and (VI) described above. We thus consider that the neutral pH corresponds to a point of zero charge at which no charge subsists (PZC) on the surface of the particles and that the protons or hydroxides can reach the whole surface of the particles.

Surface charge density calculation

In the framework of the previous hypotheses, the surface charge density σ 0 is a function of pH and depends on the relative amount of sites positively (MOH 2

+

) and negatively charged (MO -) at the surface of the particles according to the equation ( 5) with F the faraday constant, V solution the volume of the solution, S particles the total surface of the particles.

[

] [ ] ( ) - + - × × = MO MOH S V F particles solution 2 0 σ (5)
The maximum surface charge density of particles (σ 0 sat : C m -2 ) obtained at pH = 2 or 12 is determined from the C T (mol m -3 ) value:

( ) ( )(
)

(%) 3 100 100 3 / 4 (%) ² 4 3 0 vol T solution vol solution T particules total sat a C F a V a F V C S Q Φ × × = × × Φ × × × = = π π σ (6)
with Φ vol (%) the volume fraction of particles and a(m) the radius of particles. As the sizes of the particles are polydisperse, a is a mean size, therefore σ 0 sat is a mean charge density. This is a first approach which will be improved later on, however other studies are necessary before to be able to really take into account the size polydispersity in the determination of the surface charge density because we do not know yet the evolution of the surface charge density with the size. Nevertheless, to avoid an impact of this polydispersity on the comparison of the different experiments performed, the same particles have been used here for all the experiments, and it has been checked regularly that no evolution of their size distribution occurred.

III-Determination of the surface charge density: Results and Discussion

Mixture of strong and weak acids: classic analysis

In a first approach, we use the assumptions (I) to (VI) to interpret the titration experiments and deduce the surface charge density.

Potentio-conductimetric titrations from pH=2

Titrations in figure 2 (ii) The shape of the conductivity curves between the two equivalence points EP1 and EP2 changes depending on whether TMAOH or NaOH is used as the reactant. Firstly, the increase of conductivity between EP1 and PZC is higher with NaOH than with TMAOH, because the individual molar conductivity of Na + is greater than that of TMA + (λ°Na + = 5.01 mS mol -1 m² , λ°TMA + = 4.49 mS mol -1 m² at 25°C). Secondly, increasing the pH above pH > 7, the particles remain aggregated whatever the pH if one uses NaOH, whereas the particles can be individually redispersed for high pH if one uses TMAOH. This may be due to the higher condensation of the small Na + ions compared with the big TMA + ions. As the condensed Na + no longer contribute to the conductivity χ, the slope of χ between PZC and EP2 is smaller with Na + than with TMA + . Despite this difference of aggregation between the two systems (NaOH and TMAOH), the same surface charge density is obtained from both titrations, which means that aggregation does not impede charge determination.

The change in the slope of the conductivity around the neutral pH is assumed to be related to the end of the titration of the first surface acidity of the particles as described in the experimental part. By considering independent acidic groups on the surface of the particles, the pKa value for the first and the second acidity correspond to the pH at the half-equivalence.

For particles dispersed in HNO 3 at pH = 2.0, the pKa determined from the potentiometric curve are pKa 1 = 4.4 and pKa 2 = 9.3. For free acidic or alkaline species, such a difference in the pKa values should lead to high variations of pH with the volume of reactant dropped into the solution. It is not the case with the acidic groups located on the surface of the particles.

Influence of the nature of the counter ions

We probe here the influence of the nature of the counter ions on the magnitude of the structural charge of the particles. For that we compare the different systems consisting in particles dispersed in HClO In the present study, the results seem similar: from the conductivity curves (see figure 5) performed in colloidal suspensions (dispersed in acidic media before the addition of TMAOH 1 mol L -1 ) without any salt added, different values of [H + ] ads are evaluated in HNO 3 and HClO 4 , respectively 47 mmol L -1 and 32 mmol L -1 . Note that the titrations were performed twice for each acid and gave reproducible results. This value 32mmol L -1 corresponds to a charge density at saturation σ 0 sat = 0.20 C m -2 in HClO 4 at pH =1.9 (σ 0 sat = 0.30 C m -2 in HNO 3 ). However, the amount of TMAOH added to neutralize the free protons in the suspension dispersed in HClO 4 is higher than in HNO 3 (see Figure 3). Using V TMAOH at EP1, we expect ∆pH = 0.2 and ∆χ = 0.24 Sm -1 between the two colloidal suspensions before titrations, values higher than the experimental ones (∆pH = 0.1 and ∆χ = 0.13 Sm -1 ). Moreover the amount of reagent dropped from EP 2 (V = 4.35mL, pH = 9.8) to the end of the titration at pH = 12.4 does not correspond to the variation of pH measured in the suspension. For both systems, the incoherences between the variations of pH and conductivity with the amount of reactant dropped question the fact that all the sites on the surface of the particles are titrated between EP1 and EP2.

Determination of σ 0 (C m -2 ) versus pH. Comparison with zeta potential

Knowing the surface charge density at saturation σ 0 sat in HNO 3 and HClO 4 , it is possible to deduce the variation of σ 0 with the pH of the medium according to (7) using the assumptions (I) to (VI) described in §2.4 :

σ o = V solution ⋅ F S particles 10 -2 pH -Ka 1 Ka 2 10 -2 pH + Ka ⋅10 -pH + Ka 1 Ka 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⋅ C T ( 7
)
with V solution the volume of the solution (m 3 ), S particles the surface of the particles (m²), Ka 1 and Ka 2 the acidity constants related to the equilibria ( 2) and ( 3

Validity of the previous analysis

Potentio-conductimetric titrations from pH=12

To elucidate the existence of an influence of the nature of the counterion on the charge of the particles, we performed titrations of a colloidal suspension (Φ Vol (%) = 1.5) in TMAOH pH = 12.0 by HClO 4 and HNO 3 (1 mol L -1 ). The influence of the ions ClO 4

-or NO 3 -on the potentio-conductimetric curves should appear when the particles are positive (pH < 7). For both titrations with HClO 4 and HNO 3 , although the first equivalence point is not accurately determined given that the initial conductivity is low at pH = 12.0 (data not shown), the amount of protons, determined by the titration, that adsorb on negatively charged particles leads to the same surface charge density. Moreover, the magnitude of the surface charge density determined is 0.3 C.m -2 , as previously found for the particles dispersed in HNO 3 at pH = 2.0. These results confirm that there is no influence of the nature of these counter-ions on the magnitude of the surface charge density.

Equivalent system with independent acidities and two distinct pKa.

As a comparison, we performed a titration of a solution of CH On the contrary, in the case of particles titrated with NaOH, the potentiometric curve does not exhibit any marked variation and the changes in the slope of the conductivity curve are not well marked either. This is the reason why it becomes difficult to determine accurately the position of the EP even from conductivity. Moreover, for the particles, the volume of reagent dropped at EP1 corresponds to an amount of alkaline reagent much greater than the initial amount of free protons determined by the pH measurement before the titration. This means that, before EP1 one probably titrates not only the strong acid in solution but also some sites on the particles.

Interpretation of the potentiometric curves

As the equivalence points determined from conductimetry are not accurate, we analyze the potentiometric curves with Eq (4) (see § 2.4.) in order to determine ∆nOH -, which gives the number of protons linked to the particles.

Influence of the electrostatics in the proton exchange process.

In order to explain the discrepancy between the titration curves performed from extreme pH values and the interpretations of the conductivity curves, and also to describe more accurately the variation of σ 0 with pH, another titration protocol is used and an additional term taking into account the electric field developed by the particles is introduced.

Experimental determination of σ 0 (C m -2 ) versus pH: titrations from the PZC

The titration between pH = 2 and pH = 12 does not allow us to access the experimental variation of σ ο with the pH, even from the potentiometric curves. Only a calculated σ 0 can be obtained assuming several hypotheses among which the symmetrical charge between pH = 2 and 12. To bypass this assumption, we perform titrations from an initial pH close to the neutral pH to acidic and alkaline pH, as detailed in the experimental part. The surface charge density at each pH is calculated assuming that no charge subsists at the initial pH, therefore no counter ions subsist in the solution either. This assumption is reasonable since the conductivity of the precipitated suspension is close to the conductivity of water at the initial pH. On figure 7, surface charge densities determined from titrations from the PZC using HNO 3 , HClO 4 and TMAOH as reagent are presented and compared to σ 0 (Cm -2 ) calculated using Eq. ( 7) and a symmetrical charge density σ 0 = 0.3 C m -2 at pH = 2 and 12. The profile of σ 0 (C m -2 ) between the PZC and the acidic pH (protonation curve) appears to be the same whatever the reagent used (HNO 3 or HClO 4 ) as already observed with the forward and the backward titrations analyzed as in Figure 6 (backward titrations not shown). However the evolution of the experimental σ 0 with the pH is not well predicted by the profile of σ 0 calculated with Eq (7).

In the alkaline part, the evolution of the experimental σ 0 versus the pH is different, and the surface charge density at extreme pH close to pH=12 seems to be larger than that near pH=2, meaning that the charge is not symmetrical with respect to the PZC. Moreover, the charge densities σ 0 = 0.22 C m -2 and σ 0 = 0.35 C m -2 obtained respectively at pH = 2.0 and pH = 12.0 are not exactly similar to the surface charge densities determined from the potentiometric curves in the forward and the backward titrations (σ 0 = 0.30 C m -2 ). Note that the error on the pH measurements performed near the extreme values and in colloidal suspensions may lead to an important error on the determination of the amount of protons that are bound to or removed from the particles. Therefore, we limit our analysis to the range 2 < pH < 12. Despite these possible errors, the total amount of protons linked to the particles deduced from Figure 7 corresponds to an equivalent concentration [H + ] ads = 50 mmol L -1 , is in good agreement with the value of 47 mmol L -1 obtained in § 3.1. HNO 3 1 mol L -1 , HClO 4 1 mol L -1 and TMAOH 1 mol L -1 were used as reagent.

-calculated (dashed line) with Eq (7) with: pKa 1 = 5.4, pKa 2 = 9.3, a surface charge density at pH = 2.0 and 12.0, σ 0 = 0.3 C m -2 .

Theoretical determination of σ 0 (C m -2 ) versus pH

Let us consider a model based on a site dissociation model taking into account the potential developed at the surface of the particles. It implies that the sites on the particles are equivalent, however no longer independent. Several authors interested in the dissociation properties of acid groups at the surface of colloidal silica 12 , latex particles 13 and acrylamideacrylic acid copolymers 14 , emphasize the existence of such an additional contribution of electrostatic nature ∆G el in the dissociation process of poly(-base, -acid). We note Ka 1 ° and Ka 2 ° the thermodynamic constants (8) associated to the acid base equilibria ( 2) and ( 3), constants which correspond to the acidity of the surface when the global charge tends toward zero, thus here near the PZC. Note that we consider here concentrations and not activities. α 1 and α 2 are the dissociation coefficients.

K 1 0 = -MOH [ ]H + [ ] -MOH 2 + [ ] = α 1 H + [ ] 1-α 1 and K 2 0 = -MOH [ ] H + [ ] -MOH 2 + [ ] = α 2 H + [ ] 1-α 2 (8)
The changes in the free energy of the dissociation process ∆G el in the case of poly-acids induce a change in the apparent dissociation constants Ka 1 and Ka 2 according to the equation ( 9) with R the molar gas constant:

pKa = pK a 0 + 0.4343 ∆G el R ⋅ T (9)
∆G el can be expressed in terms of the electrostatic potential 15,16 , ψ d , at the surface of the particle, with e the elementary charge and N A the Avogadro number:

d A el eN G ψ - = ∆ (10)
Although ψ d is called here the surface potential, it is actually localized on the outer Helmholtz plane (oHp) in a description of the ionic atmosphere of the particles with surface amphoteric oxide groups 17 according to a double layer model (Gouy-Chapman-Stern model). The plane oHp is the plane defined by the counter ions condensed on the surface of the particles.

Considering a spherical particle of radius a in a symmetrical 1-1 type electrolyte solution, Ohshima et al 18 proposed a method to determine the electric potential ψ(r) at a distance r (r≥a) from the centre of the particle.

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ ⋅ ⋅ = + T k e c
( ) ( ) ( ) ( ) ( ) 2 1 2 2 2 0 0 2 cosh 4 cosh ln 8 4 cosh 2 1 2 sinh 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ + ⋅ ⋅ + ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ d d d d r A Y a Y Y a Y T R F κ κ κ ε ε σ (12) with Y d the reduced potential T R F Y d A d ⋅ ⋅ = ψ
, F A the Faraday constant, R the molar gas constant and κ the Debye parameter for a 1-1 electrolyte given by (13):

2 1 0 2 3 10 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = - T R F c r A ε ε κ (13)
Thus it is possible to calculate σ 0 (C m -2 ) as a function of ψ d for each acid base equilibrium, and then to deduce the dissociation coefficients α 1 and α 2 according to (14):

S N e ⋅ ⋅ - = ) 1 ( 1 0 α σ and S N e ⋅ ⋅ - = 2 0 α σ (14)
N s (number.m -2 ) corresponds to the total amount of sites binding protons at the surface of the particle. α 1 =0 signifies that all sites are (-M-OH 2 + ) and σ 0 is positive, conversely α 2 =1 signifies that all sites are (-M-O -) and σ 0 is negative.

Knowing the dependence between σ 0 and (α 1 ,α 2 ), it is possible to establish the dependence of σ 0 with the pH according to the equation (15), with the adjustable parameters pKa pKa 1 ° and pKa 2 ° relative to the equilibrium (2) and (3) can be extrapolated from pKa=f(α) (α is obtained from σ exp with (14), pKa from (15), suffix 1 is for the acidic part, suffix 2 for the alkaline part). This gives pK 1 ° = 4.5 and pK 2 ° = 9 for particles initially dispersed in HNO 3 .

However, the extrapolation is not accurate because of the strong variations of pKa close to pK a °. Therefore we prefer to determine pK a ° from the fit of σ 0 =f(pH). This gives pKa 1 ° = 5.4 and pKa 2 ° = 9, values not so far from those determined in the potentio-conductimetric titrations considering independent acidic groups. The good agreement of the simulation and the experimental results on figure 8 emphasizes the necessity to take into account the electrostatic contribution in order to accurately determine the surface charge density of the particles.

Conclusion and perspectives

In this study, we determined the surface charge density of well defined nanocolloids by implementing simultaneous potentio-conductimetric titration technique and zetametry. The physico-chemical conditions of synthesis and dispersion of the colloidal particles are very well defined and allow preparing suspensions in acidic, alkaline or neutral pH under controlled conditions of ionic strength, volume fraction and of the nature of the counterions.

Titrations starting from acidic, alkaline and neutral pH and simultaneous zetametry measurements, allowed us to accurately determine the surface charge density of the particles.

For the titrations starting from acidic and alkaline media, potentiometric measurements as well as zetametry measurements showed that the surface charge density of the nanoparticles is neither influenced by the nature of the counter ions (ClO 4 -, NO 3 -), nor by the nature of the reagent (TMA + , Na + ). Additional titrations starting from the neutral pH give us a direct access to the variation of the surface charge density with the pH. It confirms that the colloidal particles are not fully charged at pH = 2 and pH = 12 and that the surface charge density at pH=12 is more important than at pH = 2 as raised by zetametry measurements.

The properties of the surface oxide groups are studied as a function of the pH of the medium of dispersion (acidic, alkaline, neutral), of the nature of the counterions, and of the nature of the reagent. Different thermodynamical models are applied to describe the proton exchanges between the surface of the particles and the solution. In a first step, we show that a model with independent acidities does not adequately describe the experimental titrations curves and the experimentally determined charge of the particles. In a second step, we take into account the electrostatic field generated by the particle itself during the protonation. This model well explains the gradual variation of the charge with the pH experimentally determined: it is indeed more difficult to add charges on an already charged particle than on an uncharged or weakly charged particle.

The complementarity of the techniques used (simultaneous potentio-conductimetric titrations and zetametry) allowed us to understand the protonation and the deprotonation of the surface of the colloidal particles and therefore to define their stability range. The next step will be to understand the link between the structural charge, the effective charge, which is the overall charge after condensation of the counter ions, and the zeta potential.

Aknowledgements

We thank D. Talbot for help with chemical titrations of iron.

Figure 1 :

 1 Figure 1: Left: Histogram representing the size distribution of maghemite nanoparticles

Figure 2 :

 2 Figure 2: Potentio-conductimetric titration of colloidal suspensions Φ vol (%) = 1.5 dispersed in HNO 3 at pH = 2.0, with TMAOH (1mol L -1 ) (open symbols) and in HNO 3 pH = 1.9 with

3 -> ClO 4 -.

 34 4 and HNO 3 titrated by TMAOH or NaOH, and particles dispersed in TMAOH and titrated by HClO 4 or HNO 3. In the literature, a difference in the magnitude of the structural charge of the particles while changing NO 3 -for ClO 4 -has already been observed on Goethite (α-FeOOH)10 , and on rutile nanoparticles (TiO 2 , rod-like: length 200nm, width 45nm, 05 C m -2 )11 . In these studies, the influence of several anions on the surface charge density is investigated on systems positively charged below the point of zero charge (PZC). Titrations performed on these systems emphasize that proton adsorption decreases in the order Cl -> NO This order corresponds to the Hofmeister series describing the difference in the charge density of colloidal particles according to the nature of the counter ion.

3 Figure 3 :

 33 Figure 3: Potentio-conductimetric titration of colloidal suspensions (Φ vol (%) = 1.5) dispersed

6 for HClO 4 .Figure 4 :

 644 Figure 4: Surface charge density of the particles σ 0 (C m -2 ) dispersed in HNO 3 (full line) and

  3 COONH 4 and HNO 3 at pH = 2.0 by NaOH. The salt CH 3 COONH 4 is chosen because the pKa values of the pairs CH 3 COOH/CH 3 COO -(pKa = 4.75) and NH 4 + /NH 3 (pKa = 9) are very close to those of the particles. At pH = 2, the weak acidic groups CH 3 COOH and NH 4 + are not dissociated. The concentration of CH 3 COONH 4 (23 mmol L -1 ) dissolved in HNO 3 is equivalent to the concentration of the protons (46 mmol L -1 ) bound to the particles (Φ vol (%) = 1.5) dispersed in HNO 3 at pH = 2. In figure 5, three high variations of pH appear on the potentiometric curve and three marked changes in the slope of the conductivity curve. The EP associated to these variations is in good agreement with the amount of alkaline reagent required to neutralize first 10 -2 mol L -1 of strong acid (HNO 3 ), then 23 mmol L -1 of CH 3 COOH, and finally 23 mmol L -1 of NH 4 + .

Figure 5 : 4 +/

 54 Figure 5: Potentio-conductimetric titrations with NaOH (1 mol L -1 ): Comparison between a colloidal suspension (open symbols) Φ vol (%) = 1.5 dispersed in HNO 3 at pH=2.0 and a solution of CH 3 COO/NH 4 (23 mmol L -1 ) and HNO 3 at pH= 2.0 (full symbols): pKa

Figure 6 plotsFigure 6 :

 66 Figure 6: Amount of alkaline reagent that has reacted with the particles in the suspension or

Figure 7 :

 7 Figure 7: Surface charge density of particles as a function of the pH:-determined from potentiometric titration of colloidal suspension Φ vol (%) = 1.5, from the

  The equation(11) satisfies the spherical Poisson Boltzmann (P.B.) equation, with c electrolyte concentration, ε r : relative permittivity of the solution (78.5: relative permittivity of water), ε 0 : the permittivity of a vacuum and k B : Boltzmann constant. A simple approximate analytical solution to the spherical P.B. equation allows linking the surface charge density of the particle σ 0 to the surface potential ψ d :

Figure 8 : 2 .

 82 Figure 8: Dependence of the structural charge σ 0 (C m -2 ) with the pH: open symbols: σ 0

Table 1 :

 1 Characteristics

of the colloidal dispersion in acidic medium (positively charged particles) and in alkaline medium (negatively charged particles): pH range of stability, stabilizing counterions, non-stabilizaing counterions. Fourth column: reagents used for the titrations.