L Cadilhac
email: leonard.cadilhac@u-psud.fr

B Collins
email: collins@math.kyoto-u.ac.jp

A METRIC CHARACTERISATION OF FREENESS

Introduction

The von Neumann conjecture formulated by Day in 1957 says that a group is not amenable if and only if it contains a non amenable free group. It was first disproved by A. Ol'shanskii in 1980 [START_REF] Ju | On the question of the existence of an invariant mean on a group[END_REF] and since then, the family of counterexamples has been expanded. A similar question can be asked at the level of von Neumann algebras: if a finite factor is not amenable, does it necessarily contain a free group factor? Little is known in that direction except for a breakthrough of Gaboriau and Lyons [START_REF] Gaboriau | A measurable-group-theoretic solution to von Neumann's problem[END_REF], who show that for certain wreath product groups G (which may not contain F 2 ), L(F 2 ) ⊂ L(G). Note that the authors of [START_REF] Gaboriau | A measurable-group-theoretic solution to von Neumann's problem[END_REF] are in fact mainly interested in a version of the von Neumann conjecture for measure preserving actions for which they provide a positive answer.

A difficulty in trying to tackle this problem is that there are no known abstract properties of M which would characterize the fact that L(F 2 ) embeds in M. This remark is what first motivated us to write this note on a metric characterisation of freeness. Although it is not clear that the results obtained (see Corollary 1.2) can be used in the study of the von Neumann conjecture, we believe that they are of independent interest. Indeed, they generalize at the operator level a well-known result of Kesten [START_REF] Kesten | Symmetric random walks on groups[END_REF] who showed that given g 1 , . . . , g N in a countable group G the freeness of the g i 's is characterized by the norm of the Markov operator associated to a random walk on G supported by the g i 's and their inverses. Let us denote by λ : G → L(G) the left-regular representation. In a von Neumann algebraic point of view, Kesten's result can be reformulated as follows:

(1.1) g 1 , . . . , g N are free in G ⇔ N i=1 λ(g i ) + λ(g i ) * L(G) = 2 √ 2N -1.
We extend this result by replacing the λ(g i )'s by any finite family of unitary operators in a finite von Neumann algebra M. The notion of freeness and Haar unitaries (unitaries with null moments) will be considered with respect to a fixed normal faithful tracial state τ . We obtain the following:

Theorem 1.1. Let N ∈ N, N > 1.
Let u 1 , . . . , u N be unitaries in M. Then, the following assertions are equivalent:

(1) the operators u 1 , . . . , u N are free Haar unitaries, [START_REF] Cohen | Cogrowth and amenability of discrete groups[END_REF] we have the equality:

N i=1 u i ⊗ (u op i ) * + u * i ⊗ u op i M⊗M op = 2 √ 2N -1.
Note that (1) ⇒ (2) is a consequence of (1.1). Let us also mention that the inequality

N i=1 u i ⊗ (u op i ) * + u * i ⊗ u op i M⊗M op ≥ 2 √ 2N -1,
is verified for any family of unitaries (see [START_REF] Pisier | Quadratic forms in unitary operators[END_REF]). This leads to the following corollary:

Corollary 1.2. Let M be a finite von Neumann algebra. Then the following are equivalent:

(1)

L(F 2 ) embeds in M, (2) inf u 1 ,u 2 ∈U (M) u 1 ⊗ ū1 * + u * 1 ⊗ ū1 + u 2 ⊗ ū2 * + u * 2 ⊗ ū2 M⊗M = √ 3
, and this infimum is achieved.

Let us make a few remarks in relation to this result. Firstly, amenability can also be characterized via the consideration of the same quantity. Indeed, in the spirit, of [START_REF] Connes | Classification of injective factors. Cases II 1 , II ∞ , III λ , λ = 1[END_REF]Theorem 5.1], we know that a factor N is hyperfinite if and only if for every finite family of unitaries u 1 , . . . , u n in N ,

n i=1 u i ⊗ (u * i ) op N ⊗ min N op = n.
In this sense, it is at the extreme opposite of freeness. Secondly, it is worth pointing out that the problem considered in this manuscript complements results of [START_REF] Lehner | A characterization of the Leinert property[END_REF], and also of [START_REF] Grigorchuk | Symmetrical random walks on discrete groups[END_REF][START_REF] Cohen | Cogrowth and amenability of discrete groups[END_REF], who consider generators of a group instead of general unitaries. Thirdly, in view of the above papers, it is natural to wonder what are the possible values of

N i=1 u i ⊗ (u op i ) * + u * i ⊗ u op i M⊗M op
when the unitaries range on all possible choices in any tracial von Neumann algebra. This is clearly a subset of the interval [2 √ 2N -1, 2N], and it can easily be seen that this is the whole interval. Although many other approaches seem to be possible, let us just outline one way to prove this assertion: take N free unitary Brownian motions {t → u i (t), i ∈ {1, . . . , N}} as defined in [START_REF] Voiculescu | The analogues of entropy and of Fisher's information measure in free probability theory. VI. Liberation and mutual free information[END_REF]. Using explicit descriptions of the free unitary Brownian motion (see [START_REF] Biane | Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems[END_REF]) one can show that it is norm continuous, converges to free Haar unitaries, and that this convergence holds in norm for t In section 2, we introduce our combinatorial approach to Theorem 1.1. Section 3 contains the core technicalities: we use free group combinatorics in order to obtain a suitable lower bound on positive characters of F N which allows us to conclude in Section 4.

→ N i=1 u i (t) ⊗ (u op i (t)) * + u * i (t) ⊗ u op i (t)
Acknowledgements: This work was initiated during the visit of LC to Kyoto University early 2020, after preliminary discussions at MF Oberwolfach in 2018. BC was supported by JSPS KAKENHI 17K18734 and 17H04823. The authors are grateful to Mikael de la Salle, Cyril Houdayer, Eric Ricard, Adam Skalski and Narutaka Ozawa for inspiring comments and discussions.

A combinatorial approach

2.1. Reformulation of Theorem 1.1. Let N ∈ N. Let ϕ be a positive definite function on the free group F N . We extend ϕ linearly to C[F N ] and keep the same notation, i.e. for any finitely supported family (a g ) g∈F N ∈ C,

ϕ g∈F N a g • g = g∈F N a g • ϕ(g).
Let s 1 , . . . , s N ∈ F N be free generators of F N and set

a := N i=1 s i + s -1 i ∈ C[F N ].
We aim to prove the following:

Theorem 2.1. Assume that: • ϕ is constant on the conjugacy classes of F N (it is a character), • ϕ(e) = 1, • ∃g ∈ F N , g = e, ϕ(g) = 0, • ∀g ∈ F N , ϕ(g) ≥ 0. Then, lim n→∞ ϕ(a 2n ) 1 2n > 2 √ 2N -1. Lemma 2.2. Theorem 2.1 implies Theorem 1.1. Proof. Consider the representation π of F N determined by π(s i ) = u i ⊗ (u op i ) * in U(M⊗M op ) for any i ∈ {1, . . . , N}. Define ϕ := (τ ⊗ τ op ) • π. Note that ϕ is a positive character on F N and that N i=1 u i ⊗ (u op i ) * + u * i ⊗ u op i M⊗M op = lim n→∞ N i=1 u i ⊗ (u op i ) * + u * i ⊗ u op i 2n = lim n→∞ ϕ(a 2n ) 1 2n .
Assume that the u i 's are not free Haar unitaries. This means that there exists g ∈ F N such that g = e and ϕ(g) = 0. So ϕ satisfies the conditions for Theorem 2.1 and hence

N i=1 u i ⊗ (u op i ) * + u * i ⊗ u op i M⊗M op > 2 √ 2N -1.
Remark 2.3. To prove Theorem 2.1, it suffices to show that ∃n, ϕ(a 2n )

1 2n > 2 √ 2N -1,
by Hölder's inequality.

2.2. Some notations. We always assume F N to be equipped with a set of distinguished generators S = {s 1 , . . . , s N }. Denote by W N the set of words on the alphabet S ∪ S -1 . For any w ∈ W N , denote by l(w) its length. We identify elements of F N with their writing as a reduced word. Define, for any k ∈ N and w = w 1 . . . w k ∈ W N of length k, the circular permutation of w by

σ 1 (w) = w k w 1 . . . w k-1 .
For any t ∈ Z, let σ t := σ t 1 . We say w ∈ W N is cyclically reduced if σ t (w) is reduced for any t (note that it is equivalent to w and σ 1 (w) being reduced). If w is a reduced word, w can uniquely be written as

w = uvu -1 ,
where v is cyclically reduced and u is any reduced word. The element v will be referred to as the root of w an denoted by r(w).

3.

Lower bound for positive characters of F N 3.1. Overview of the section. Let ϕ be as in Theorem 2.1 and g 0 = e such that ϕ(g 0 ) = α > 0. Using Lemma 3.2 (in the following subsection), without loss of generality, g 0 can be chosen to be of even length 2l 0 , l 0 ∈ N. This section is devoted to proving the following proposition:

Proposition 3.1. There exists a constant C > 0 such that for any k ≥ l 0 ,

l(g)=2k ϕ(g) ≥ Ck 2 (2N -1) k .
Note that the constant C that we obtain above may be explicitly computed from N, l 0 and α but its precise value is inconsequential. Our goal is to exhibit a large enough family of words in F N on which we can bound ϕ from below. To do so, we combine two different operations:

• The first operation consists in conjugating elements thanks to the tracial property. However, by a simple calculation using the estimate given in Lemma 4.1, it can be seen that considering only conjugates of g 0 is not enough to obtain a sufficiently good lower bound on ϕ(a 2n ). • The second operation is to multiply elements for which we already have a lower bound and apply Lemma 3.2. This is however a bit cumbersome because simplifications may occur and estimating the length of the products obtained this way (to be able to use Lemma 4.1) requires some caution. The proof is divided into 4 different steps:

• Step 0 -we show that we can obtain new elements of positive trace by multiplication (Lemma 3.2). • Step 1 -we fix an integer i and apply Lemma 3.2 to the conjugates of g 0 of length 2i to obtain a set C

(2) i of new elements of quantified positive trace which have length approximately 4i and are roots. The control on the length is essential to guarantee that our sets C

(2) i do not overlap too much when i varies. This will be key during step 3.

• Step 2 -we apply circular permutations to the elements of C

(2) i to construct a bigger set D i of roots with positive traces.

• Step 3 -we consider the sets D i (k) of elements of length 2k with roots in D i , show that they are essentially disjoint and big enough (in terms of trace) to conclude. To conclude the introduction to the section, fix an integer k 0 ≥ 6 such that (3.1) k 0 ≥ l 0 + ln(4)ln(α 2 ) ln(2N -1) .

The main reason for this k 0 is linked to step 1 above. When multiplying a family of elements of F N (as in Lemma 3.2) of length 2i, we would like to keep only the products obtained of length 4i. Unfortunately, by doing so, we would not obtain enough elements for the argument to work and hence we have to allow for a certain number of simplifications to occur. This number of simplifications will be given by k 0 which, crucially (see the proof of Lemma 3.4), does not depend on i.

3.2.

Step 0: locating more positivity by multiplication. The following lemma allows to obtain new elements of positive trace through multiplication.

Lemma 3.2. Let k ∈ N and a > 0. Let h 1 , . . . , h k be such that ϕ(h i ) = α for any j ∈ {1, . . . , k}. Then, 1≤i,j≤k

ϕ(h -1 i h j ) ≥ k 2 α 2 .
Proof. Set h 0 = e and consider the matrix

A = (ϕ(h -1 i h j )) 0≤i,j≤k . Since ϕ is positive definite, A is positive. Let ε > 0. Consider the vector v = (1, -ε, . . . , -ε) ∈ C k . Denote by b the mean of (ϕ(h -1 i h j )) 0<i,j≤k . Note that v ⊥ Av = 1 -2kαε + k 2 bε 2 . Set ε = 1
kα . By positivity of A, we have:

0 ≤ 1 -2 + b α 2 . Hence, b ≥ α 2 .

3.3.

Step 1: products of conjugates of g 0 . Let i ∈ N, i > l 0 + 6k 0 and consider the set C i of all conjugates of g 0 or g -1 0 of length 2i. Denote by R 0 the set of conjugates of g 0 or g -1 0 of length 2l 0 . Note that: (3.2)

|C i | = |R 0 | (2N -2)(2N -1) i-l 0 =: c 0 (2N -1) i .
In the course of the proof, we have to apply circular permutations σ j to the words we obtain to generate new words. In order to guarantee at that future step that we indeed obtain new words, we need to remove from the beginning some pathological elements of C i . For any j ∈ N, define:

(3.3) C i (j) := {g ∈ C i : ∀n, m ∈ {k 0 + 1, . . . , i -l 0 } (|n -m| = 2j) ⇒ g n = g m } ,
where g n denotes the n-th letter of g. When j << i, these are the words of length 2i which have been obtained by conjugating g 0 by a word which is 2j-periodic except maybe on a small part (at the beginning of the word). This means that

|C i (j)| ≤ c 0 (2N -1) 2j+k 0 . Define (3.4) C ′ i := C i \ j≤ i-l 0 k 0 C i (j).
Since k 0 ≥ 6, jk 0 ≤ il 0 and il 0 ≥ 6k 0 , we have 2j + k 0 ≤ (il 0 )/2. Hence,

k 0 ≤ i-l 0 k 0 |C i (j)| ≤ c 0 (i -l 0 )(2N -1) (i-l 0 )/2 ≤ c 0 2 (2N -1) i-l 0 .
Then, by (3.2), (3.5)

|C ′ i | ≥ c 0 2 (2N -1) i-l 0 .
Now that we have selected suitable conjugates of g 0 , let us multiply them to obtain new elements. Let C

(2) i

:= {r(g) : g ∈ C ′ i • C ′ i , l(g) > 4i -2k 0 } . Remark 3.3.
Let us keep in mind the form of the elements that we are dealing with. Let h, h ′ ∈ C ′ i . This means that there exists r 0 and r ′ 0 in R 0 and u, u ′ in F N such that h = ur 0 u -1 and h ′ = u ′ r ′ 0 (u ′ ) -1 . Write u = wv and u ′ = wv ′ such that no cancellation occur in the product v -1 v ′ . Then r(hh ′ ) belongs to C

(2) i if and only if l(w) < k 0 and r(hh

′ ) = vr 0 v -1 v ′ r ′ 0 v ′-1 . Lemma 3.4.
The following estimate holds:

g∈C (2) i ϕ(g) ≥ c 0 8 α 2 (2N -1) 2i-2l 0 -k 0 .
Proof. For any g ∈ C ′ i , there are at most c 0 (2N -1) i-k 0 elements g ′ of C ′ i for which l(gg ′ ) ≤ 4i -2k 0 . Indeed, the first k 0 letters of g ′ must coincide with the first k 0 letters of g in order for at least k 0 simplifications to occur. This means that:

g,g ′ ∈C ′ i ϕ(gg ′ ) ≤ c 0 (2N -1) i-k 0 |C ′ i | + g,g ′ ∈C ′ i l(gg ′ )>4i-2k 0 ϕ(gg ′ ).
Then, by Lemma 3.2 (note that considering gg ′ or g -1 g ′ does not modify the following sum since

C ′ i = (C ′ i ) -1 ): g,g ′ ∈C ′ i l(gg ′ )>4i-2k 0 ϕ(gg ′ ) ≥ |C ′ i | 2 α 2 -c 0 (2N -1) i-k 0 |C ′ i | ≥ c 2 0 α 2 4 (2N -1) 2i-2l 0 - c 2 0 2 (2N -1) 2i-l 0 -k 0 ≥ c 2 0 α 2 8 (2N -1) 2i-2l 0 ,
where we used the fact that by definition of k 0 ,

(2N -1) -k 0 ≤ α 2 4 (2N -1) -l 0 .
Finally, note that at most (2N -1)

k 0 factors h, h ′ ∈ C ′ i can give rise to the same element r(hh ′ ) ∈ C (2) i . This means that (2N -1) k 0 g∈C (2) i ϕ(g) ≥ g,g ′ ∈C ′ i l(gg ′ )>4i-2k 0 ϕ(gg ′ ),
which is the desired estimate.

3.4.

Step 2: circular permutations. Define:

D i := σ j (g) : 0 ≤ j ≤ i -l 0 k 0 , g ∈ C (2) i .
Lemma 3.5. The following estimate holds:

g∈D i ϕ(g) ≥ i -l 0 k 0 c 0 8 α 2 (2N -1) 2i-2l 0 -k 0 .
Proof. Given the estimate obtained in Lemma 3.4, it suffices to prove that for any 0 ≤ j, j ′ ≤ i/k 0 and g, g ′ ∈ C

(2)

i , σ j (g) = σ j ′ (g ′ ) ⇒ (g = g ′ and j = j ′ ).

Let us assume that σ j (g) = σ j ′ (g ′ ). If j = j ′ then immediately g = g ′ and we get the expected conclusion. So assume by contradiction that j = j ′ . We can also assume without loss of generality that j = 0.

According to Remark 3.3, g and g ′ can be written as follows:

g = ur 0 u -1 vs 0 v -1 g ′ = u ′ r ′ 0 (u ′ ) -1 v ′ s ′ 0 (v ′ ) -1
where r 0 , s 0 , r ′ 0 , s ′ 0 belong to R 0 and u, u ′ , v, v ′ are words of the same length j with il 0k 0 < j ≤ il 0 . Write u = u 1 . . . u j and u ′ = u ′ 1 . . . u ′ j . Since g = σ t (g ′ ), we obtain by looking at the occurrences of u and u ′ in g and g ′ ,

u ′ n = u n+t ∀n ∈ [1, j -t].
And, looking at the occurrences of u -1 and (u ′ ) -1 ,

u n = u ′ n+t ∀n ∈ [1, j -t].
This means that u is 2t-periodic. Since g is an element of C

i , g comes from a product hh ′ , h, h ′ ∈ C ′ i . Write h = wv 0 w -1 . We have w = w 1 . . . w i-l 0 -j u. By construction of C ′ i , w cannot be 2t-periodic starting from its k 0 -th letter, which is a contradiction (see (3.4), (3.3), and recall that by assumption t ≤ il 0 k 0 ).

3.5.

Step 3: conjugation. For any k ≥ 2i, define:

D i (k) := {g ∈ F N : |g| = 2k, r(g) ∈ D i } . Lemma 3.6. Let k ∈ N and i, i ′ ≥ l 0 + 6k 0 . Assume that 2i, 2i ′ ≤ k and |i -i ′ | ≥ k 0 .
Then,

• D i (k) and D i ′ (k) are disjoint,
• There exists C ′ > 0 independent of k or i such that:

g∈D i (k) ϕ(g) ≥ C ′ i(2N -1) k .
Proof. For the first point, simply remark that elements in D i have length between 4i and 4i -4k

0 + 4. So if |i -i ′ | ≥ k 0 then D i and D i ′ are disjoint so D i (k) and D i ′ (k) are disjoint.
Let g be a cyclically reduced element of length 4i in F N . Note for k > 2i, there are (2N -2)(2N -1) k-2i-1 elements of F N of length 2k and root g. This means that any element of D i is the root of at least (2N -2)(2N -1) k-i-1 elements in D i (k). Hence,

g∈D i (k) ϕ(g) ≥ (2N -2)(2N -1) k-2i-1 g∈D i ϕ(g).
By Lemma 3.5, we obtain the expected estimate.

Proof of Proposition 3.1. Let k ∈ N, k large enough for the sums below to be non-empty (for example k ≥ 20k 0 + 2l 0 ). By Lemma 3.6, we have:

|g|=2k ϕ(g) ≥ 6k 0 +l 0 ≤i≤k/2 k 0 |i g∈D i (k) ϕ(g) ≥ 6k 0 +l 0 ≤i≤k/2 k 0 |i C ′ i(2N -1) k ≥ Ck 2 (2N -1) 2k ,
where C is a small enough constant independent of k.

Proof of the main theorem

For any w ∈ W N , denote by g(w) the corresponding element of F N .

Lemma 4.1. Let n ∈ N and k < n. Let g ∈ F N such that l(g) = 2k. Then:

|{w ∈ W N : l(w) = 2n, g(w) = g}| ≥ 2n n -k - 2n n -k -1 (2N -1) n-k =: N n,k .
Proof. We interpret words as paths on the Cayley graph of F N . Note that the quantity we consider only depends on l(g). A way to generate paths going from e to any element of length 2k is to first choose at which times the path is going to go away from e and at which times the path is going to come back towards e. Since the path is going to an element of length 2k, it has to go n + k times away from e and nk back to e. The number of possible choices there, for a path of length 2n, is given by the Catalan triangle

C n,k = 2n n -k - 2n n -k -1 .
Moreover, when we chose to go away from e, there are at least 2N -1 possible directions (2N if the path is at e and 2N -1 otherwise), and 2N possible directions for the first time, thus obtaining C n,k 2N(2N -1) n+k-1 paths. Finally, since for now all we have fixed is the length of the target of the path and not a particular point, we have to divide this result by the number of elements of length 2k in F N i.e. 2N(2N -1) 2k-1 , to get the desired estimate.

We are now ready to prove our main theorem.

Proof of Theorem 2 . 1 .

 21 Let n ∈ N. First remark that by Lemma 4.1 and Proposition 3.1,C n,k k 2 (2N -1) n .Note that given the expression of C n,k ,l 0 ≤k≤n k 2 C n,k ≥ sup l 0 ≤k≤n k 2 2n nk . Now chose k = n 1/3

  . For n large, 2n (2N -1) n > 2 2n (2N -1) n .Which concludes the proof by Remark 2.3.Proof of Corollary 1.2. Let τ be a faithful trace on M and consider L 2 (M, τ ). It follows from Theorem 1.1 that all reduced words in u i and their inverses form an orthonormal family. Since the GNS representation of the von Neumann subalgebra generated by u 1 , u 2 is faithful, it allows to conclude that is is isomorphic to L(F 2 ).

	2n n -k	∼ ∼	2 2n √ πn 2 2n √ πn	n n + k e -k+o(1) e k+o(1) n+k	n n -k	n-k
		∼	2 2n √ πn	.
	Hence, for n large enough,
	ϕ(a 2n ) ≥	C √ π	n 1/6 2


Idriss Miganeh Waiss

André Kimbonguila

Mohamed Fatouma

Laurette Brigelia Abdoul-Latif

Joël Nkeletela

Jérémy Scher

Petit

Fatouma Mohamed

Abdoul-Latif

Laurette Brigelia Nkeletela

Joël Scher

Jérémy Petit
email: jeremy.petit@univ-lorraine.fr

powder: influence of milling and sieving processes 3

Keywords: particle size, Polyphenols, Antioxidant Activity, preservation, powder reconstitution, okra seed powder okra seed powder, particle size, polyphenols, antioxidant activity, preservation, 28 powder reconstitution. 29

come

Introduction 30

Okra fruit (Abelmoschus esculentus (L.) Moench) is widely consumed around the world and it ABTS +• radical was produced by a reaction between an aqueous solution of ABTS at 7 mM and 159 a potassium persulfate K 2 S 2 O 8 solution at 4.9 mM. This mixture was stirred for 16 h in the dark 160 at room temperature (22 °C), leading to a blue green-colored solution containing the ABTS +• 161 cation radical (Cai et al., 2015). Afterwards, the ABTS +• solution was then diluted with ethanol 162 until reaching between 0.65 and 0.70 absorbance at 734 nm. For analysis, 100 µL powder 163 sample hydromethanolic extract was mixed with 900 µL ABTS +• radical solution and the 164 mixture was vortexed. After 10 min incubation at room temperature (22 °C), the absorbance of 165 the mixture was measured at 734 nm against a blank solution (mixture of 100 µL phosphate 166 buffer at pH 7.4 and 900 µL ABTS +• radical solution). The results were expressed in terms of A 0 ] × 100

(3)

169
with A 0 (-) representing the absorbance of the blank solution and A 1 (-) the absorbance of the 170 solution obtained from okra seeds powder extract after the DPPH ABTS +• assay. Where masses of wet and dry sediments were expressed in grams.

181

The following protocol was used to determine the water solubility index of powder samples: 182 2.5 g (M 1 ) okra powder and 20 mL distilled water were mixed in a centrifuge tube, then the 183 mixture was stirred at 300 rpm (Variomag Poly) for 30 min at 20 °C. After centrifugation 184 (Thermo Scientific, Heraeus Megafuge 8R Centrifuge) at 1 500 × g for 20 min at 20 °C, the 185 supernatant was carefully removed and the pellet was dried in an oven at 103 ± 2 °C for 6 h, 186 cooled in a desiccator, and weighed (M 2 ). WSI was calculated according to equation 5: The analysis of the particle size distribution was performed in order to check if the successive 201 milling and sieving processes were efficient in micronizing okra seeds and separating unsieved 202 powders into sufficiently different granulometric classes. Ttable 1 shows that the median 203 particle size decreased when the milling frequency was increased: for instance, for unsieved 204 powders, median particle size was lowered value, from 723 µm at 6 000 rpm to 478 µm at 205 12 000 rpm and 229 µm at 18 000 rpm. The median particle size of granulometric classes was 206 found between the mesh sizes of their respective upper and lower sieves as expected when 207 sieving is completed, confirming the efficiency of the sieving process to separate powders in 208 granulometric classes differing in median particle size.

209

Span of unsieved powders obtained at 6 000, 12 000, and 18 000 rpm were equal to 2.11 ± 0.02, 210 2.03 ± 0.01, and 2.96 ± 0.16, respectively, showing that the width of the particle size distribution 211 was larger at higher milling frequency. This could be explained by the higher proportion of fine 212 particles when using a higher milling frequency, leading to increased heterogeneity of unsieved 213 powders.

214 influence of granulometric characteristics of these powders on their proximate composition 231 revealed that higher protein and fat contents were found in the smaller particle size fractions of 232 okra powder seeds (Waiss et al., 2020). Thus, the discriminating power of the sieving process 233 in matter of total phenolic content and proximate composition was enhanced at higher milling 234 frequency.

235 The results of ABTS et al., 2007;Becker et al., 2016;Deli, et al., 2019).

285

Indeed, the phenolic compounds such as flavonoids, polyphenols, and tannins are the main The representation of okra powder samples on the F1 × F2 biplot revealed a separation of 356 powder samples according to median particle size (Figure 4). Smaller granulometric classes 357 (≤ 180 µm at 6 000, 12 000, and 18 000 rpm milling frequency and 180 -315 µm at 18 000 rpm 358 milling frequency) were located at the bottom right of the F1 × F2 biplot and characterized by 359 their high total phenolic content and WSI, and obviously by their low median particle size.

360

Higher granulometric classes (≥ 500 µm at 6 000, 12 000, and 18 000 rpm milling frequency 361 and 315 -500 µm at 18 000 rpm milling frequency) were found at the left of the F1 × F2 biplot

Conclusion 369

The combination of milling and sieving processes was successful in generating granulometric 2.96 ± 0.16 i For each milling speed separately, means ± standard deviations followed by the same superscripted letters in the same column were not significantly different at p < 0.05 according to Tukey's HSD test.

Institute of Food Science and Technology

International Journal of Food Science & Technology F o r P e e r R e v i e w The objective of this work is to determine the impact of the particle size of okra seed powders on physicochemical, water interaction, and phytochemical properties. To our knowledge, this is the first time that a study focuses on characterization of okra seed powders with granulometric method.

The results of this study permitted to identify a granulometric class achieving a compromise between particle size of powders and bioactive compound content, powders reconstitution with interaction of water.

We confirmed that the manuscript or its contents in some other form have not been published

previously by any of the authors and/or is not under consideration for publication in another journal at the time of submission. The authors wish to thank the reviewers for their helpful comments. Detailed point-by-point response can be found hereafter, with reviewers' comments in black (line numbers associated with reviewers' comments correspond to submitted manuscript) and answers in blue (line numbers related to manuscript revision correspond to revised manuscript). In revised manuscript, additions were performed in blue and deletions in red crossed words.

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Reviewer #1

This is an interesting work that contributes with important results considering the influence of the particle size of milling in relation to the antioxidant activity and extractable phenolic levels. This is then related to the accessibility of these important compounds as well as of proteins and polysaccharides into the gut. This work complements a previous one of these authors where a complete determination of the chemical composition of these fractions with determined particle size ranges was performed.

This manuscript is acceptable for publication after correction according to the following revision.

Please, have the manuscript grammar and writing revised.

Amendments can be also found in the pdf file of the manuscript. This should be corrected along the manuscript.

This was corrected in the whole manuscript.

ABSTRACT.

The Abstract was in general well written.

Please, better write: "The influence of the particle size derived by milling of okra seeds at three different speed velocities (6,000; 12,000 and 18,000 rpm) on functionality (total phenolic content and antioxidant activity), reconstitutability (water absorption capacity, water solubility index), and preservation (water activity) of powders was studied. Okra seed powders were sieved with 180, 315, and 500 μm sieves. An increased ....."

The abstract was clear, complete and grammatically sound, thus the authors preferred to keep it unchanged. In particular, 6 000, 12 000, and 18 000 rpm are values of milling frequencies, not milling velocities. Also, reconstitutability and preservation ability are powder functionalities. Please, could you express the water activity value here and along the manuscript with three figures after the decimal point? Water activity meters are supposed to measure the water activity with three figures after the decimal point. It is important this complete value.

This was corrected in the whole manuscript.

"preservation" (not preservability)

"Preservability" was replaced by "preservation ability" in L21.

KEYWORDS.

Please, replace "okra seeds" by "okra seed powder".

This was corrected in L27.

INTRODUCTION.

It was well written indeed.

L38. Please, correct to: "of oil rich in unsaturated fatty acids including linolenic and linoleic acids, which are both essential for humans ....."

Concerned sentence was corrected (cf. L35).

L40. Please, correct to: "contents in polysaccharides and phenolic compounds including flavonoids (Arapitsas, 2008)."

Concerned sentence was corrected (cf. L37-38).

L41. "Phenolic compounds are known to ...."

Concerned sentence was corrected (cf. L38).

L42. Please, correct to: "Although the high phytochemical content of .... What is phytochemical "activity"?"

Concerned sentence was corrected (cf. L39-40).

All compounds in a plant are "phytochemicals", many of which are not precisely healthy (e.g. curare, valerian, alkaloids, etc., etc.).

L64. Please, correct to: "okra seed powder" ("seed" in singular because here it functions as an adjective).

Concerned sentence was corrected (cf. L61). L76. "as well as on preservation ...."

Concerned sentence was corrected (cf. L73).

L92. "The powder obtained was then collected...."

As concerned sentence appeared clear and grammatically sound, it was kept unchanged.

L104-106. These two sentences should be passed to the Results and Discussion section.

Sentence in L102-104 was used to introduce the interest of analysing particle size distribution of powders. So, authors judged preferable to keep it at the beginning of the method subsection related to granulometry.

L106. Please, define clearly here in the text the meaning of 'unsieved' powders. It is really not understandable.

This was added in L92-94 when defining how unsieved powders were prepared.

L113-115. Please, re-write to: "d10, d50, and d90, which signifies the point in the size distribution, up to and including which, 90% or 50% or 10% of the total volume of material in the sample is 'contained'. For example, if the d90 is 315 micrometres, this means that 90%

of the sample has a size of 315 micrometres or smaller. The span of a volume-based size distribution is defined as indicated by equation (1), which quantifies the...."

The authors preferred to redefine granulometric parameters from a statistical point of view for accuracy purposes. This was kept unchanged, as it was grammatically sound (cf. the numerous occurrences of "X mL distilled water" for instance : https://www.linguee.com/englishfrench/search?source=auto&query=ml+distilled+water).

L136. "2.67 mL of distilled" Idem.

L137. "500 μL of Folin-Ciocalteu" Idem.

L137. "and 300 μL of sodium carbonate (Na2CO3, 20 % w/v) were added."

Idem.

L141. "for calibration curve (R2 = 0.99)"

The initial sentence was grammatically sound; thus, it was kept unchanged. Also, the "2" should be in exponent to mean "power 2", it was also kept unchanged.

L145-148. "The 1,1-diphenyl-2-picrylhydrazyl (DPPH) free radical scavenging method was used according to Matsuda et al. (1999). Five hundred microlitres of 0.1 mM DPPH"

Concerned sentence was corrected (cf. L144-146).

L148. "with 500 μL of okra seed powder"

As previously indicated, this was kept unchanged, as it was grammatically sound.

L151. "of DPPH solution"

Idem.

L151. "and 150 μL of distilled water" Idem.

L155. "with ABS"

The authors did not understand this comment. "(-)" means dimensionless and unit should be given in brackets for parameters used in presented equations.

L171. "after the ABTS+• assay" (no "DPPH")

This was corrected in L170.

L173-174. "(WAC) was determined by addition of 10 mL of distilled water to 1 g of okra seed powder contained into a centrifuge tube. The mixture was stirred at 300 rpm" L201. Please, also clarified here what is the meaning of "unsieved"

Unsieved is now defined in L91-94.

L202-203. Please, clarify the meaning of d50, at least between parentheses to remember it.

"d50" was replaced by "median particle size" in L202-203.

L205-206. Please, indicate here the mesh size of the 'upper' and the mesh size of the 'lower' sieves used.

"their respective" was added in L206 to indicate that upper and lower mesh sizes depend on the considered granulometric class.

L206. It was written: "confirming the efficiency of the sieving process." Why? Please, these explanations and conclusions are not understandable. Please, explain first these facts, based on the literature too.

The efficiency of the sieving process concerns its ability to separate powders into granulometric classes differing in median particle size (this was added L207-208). The fact that median particle size is found between upper and lower mesh sizes indicates that sieving was completed, i.e. particles were separated in granulometric classes according to their size comprised between the upper and lower mesh sizes ("sieves as expected when sieving is completed" was added in L203-204 to improve clarity).

L208. "in Table 1."

Concerned sentence was corrected (cf. L202).

L208. Nothing has been here reported and explained about the span (equation 1) results obtained and shown in Table 1. The values of span obtained are high, low, (????).

A discussion about span results was added in L209 -213. L239. "All extracts were able to scavenge DPPH• radicals and this capacity increased with the speed of milling from 6,000 to 18,000 rpm, that is, with the decrease in particle sizes."

Concerned sentence was kept unchanged because it was grammatically sound, and the influence of particle size was discussed in L251,254.

L248-249. As observed in Figure 2, this is not true when comparing >=500 microns granulometry and <=180 microns for 6,000 and 12,000 rpm.

"at a given milling frequency" was added in L251 to remind that this observation concerns granulometric classes and unsieved powder obtained at the same milling frequency.

L251. "The general increase in DPPH• scavenging"

Concerned sentence was corrected (cf. L 254).

L260. "unsieved powders (Fig. 2) shows" "(Figure 2)" was added in concerned sentence (cf. L264).

L269. "in Figure 3." Indeed, moisture content is indirectly related to food stability during storage, as moisture content and food structure both impact food water activity, and the latter parameter is the one that is determinant for food stability during storage. Thus, it is not false to write that "moisture content is an important parameter that determines powdered food ingredient stability during storage" but it can be misleading. Concerned sentence was reformulated (cf. L290, 293), and additions were made in L300-302.

Please, begin this part (3.4) with the information of the water activity results obtained.

Afterwards, report the moisture contents. The authors preferred to keep the original order: moisture content results first, and water activity results after. Indeed, this permitted to remind that water activity is the good parameter to evaluate water availability for microorganisms and alteration reactions.

L291-292. It was written: "are known to limit microbial growth and extend food shelf-life" Again, this is the same mistake. The 'water activity' is which determines the growth of bacteria, yeasts and molds. Please, see the stability map of Labuza (1977). Be careful, this is a physicochemical concept. Labuza (1977) -Figure 10: https://doi.org/10.1111/j. 1745-4549.1977.tb00321.x Indeed, as indicated in a previous answer, water activity is the good parameter to evaluate water availability for microorganisms and alteration reactions.

However, it is true that "low levels of moisture content are known to limit microbial growth and extend shelf-life". The threshold of 12 % (w/w) is often reported in the literature as a global limit for all types of food. Thus, it was decided to keep concerned sentence unchanged, as additions were made in L290 and 299-302 to better define the link between water moisture, water activity, and food preservability (Tapia, et al., 2007;Bhandari et al., 2013).

L293-295. Please, re-analyzed all results with the water activity of each granulometry.

The influence of median particle size (directly linked to granulometry) on water activity was already discussed in L290-293, and the influence of particle size was similar for all milling frequencies. Thus, no addition was made in the previous discussion of the influence of median particle size on water activity.

The moisture content is a function of the water activity of equilibration and the sorption capacity of the solid material, where the latter depends on the chemical composition and porosity.

Indeed, this was briefly added with the concept of "food structure" in L308-311. L340-341. The total explained varianza obtained is certainly lower than that recommended by the PCA method (80% to 90%).

L293. "in

It was added that this was sufficient to outline the main links between parameters in L348-349.

CONCLUSIONS.

L371-372. It was written: "Powders with smaller median particle sizes had lower moisture content and water activity, ....." I'm thinking about: If powders with smaller median particle sizes had lower moisture content and also lower water activity, hence the conclusion is not that 'as the particle size was reduced bellow XXX microns, the aDsorption capacity of the solid matrix of the powder decreased.' I believe that the true conclusion is that the porous were in part collapsed by the milling process. Please, treat to indicate possible explanations for the results observed.

This was described in more details in L374-382.

L374. "food spoilage by microbial"

Concerned sentence was corrected (cf. L382).

L375. Why biochemically???

Biochemically was indicated here to refer to enzymatic reactions.

TABLES.

Footnotes: Please, delete "according to Tukey's HSD test" since this was already explained under 'Statistical analyses' section.

This information must be indicated for each table where samples were sorted by Tukey's HSD test, as it explains the meaning of superscripted letters. Referee # 2

Comments to the Author I think this is an interesting and well-written study.

A few questions:

-what is the definition of superfine milling in terms of particle size?

Superfine milling permits to reach particle sizes up to tens of micrometers. This term has been removed from the manuscript.

-why didn't you repeat with more than one okra sample? How general are the results?

Okra seeds were imported from the Republic of Congo by the first author of the article and only a limited amount of material was allowed to cross borders. It appeared important to work with a single lot of okra seeds in order to limit the impact of proximal composition variability among lots. Thus, the authors chose to work with a single lot and to produce a single replicate of each sample, but to replicate analyses three times to ensure result accuracy.

-How did you control and measure moisture content during milling and sieving? It must have a very big effect on results.

Equipments employed for milling and sieving were not equipped to control moisture content of product. This was achieved after milling and sieving steps by a thermogravimetric method (cf. L122-123). Milling and sieving steps were performed the same day for all powders in a close room where air thermohygrometry was controlled (20 °C, 40 -50 % relative humidity). This permitted to limit the influence of surrounding air relative humidity on moisture content of produced powders. The laboratory is not equipped with HPLC able to separate and quantify phenolic compounds, thus it was chosen to determine the total phenolic content only.

Please format every reference accurately and consistently --Scientific names in italics, and words like 'in vitro'. Genus name -upper case initial letter, species name, lower case initial letter.

This was corrected in the whole manuscript.

-Upper and lower case initial letters as required for article title, and for journal name.

This was corrected in the whole manuscript.

-Pay attention to accurate format for book or chapter citation This was corrected in the whole manuscript.

Referee #3

Comments to the Author

The scientific data is relevant and should be shared with others. Nonetheless, the manuscript has several problems.

1 . All the affiliation of authors should be in English.

Affiliations are standardised by the universities of authors and should be kept in their native language.

2. In "3.4. Moisture content and water activity", ≥ 500 µm granulometric classes at 6 000, 12 000, and 18 000 rpm milling speed have higher moisture content, but why did the powders have lower water activity in table 2? Indeed, > 500 µm granulometric classes have the highest moisture contents but intermediate values of water activity. This may be due to the fact that components of > 500 µm granulometric classes interact more with water and decrease its bioavailability. This is consistent with the hypothesis that these higher size granulometric classes may contain more fibres, which are known to strongly retain water. results of all granulometric classes and unsieved samples of okra seed powders are presented in table 2.". However, no WAC and WSI in table 2. The data was not found in the manuscript.

WAC and WSI were added in table 2 and discussed in L323-335.

%

  Inhibition of ABTS radical = [ A 0 -A 1

  Water absorption capacity (WAC) and water solubility index (WSI) were evaluated according 173 toDeli et al. (2019). Water absorption capacity (WAC) was determined as follows: first, 1 g 174 okra seeds powder and 10 mL distilled water were mixed in a centrifuge tube, then the mixture 175 was stirred at 300 rpm (Variomag Poly) for 30 min at 20 °C and centrifuged (Thermo Scientific, 176 Heraeus Megafuge 8R Centrifuge) at 1 500 × g for 30 min at 20 °C. The supernatant was 177 carefully removed. The pellet was weighed, dried in an oven at 103 ± 2 °C for 6 h, cooled in a 178 desiccator, and weighed again. WAC was evaluated as follows (equation 4):

  analysis of variance, the calculation of Pearson's correlation coefficients, and 190 principal components analysis (PCA) were performed by using Microsoft Excel 2016 and 191 significant differences were assessed with Tukey's HSD at a 5 % confidence level with the 192 DSAASTAT add-on (Andrea ONOFRI, Department of Agricultural, Food and Environmental 193 Sciences, University of Perugia, Italy). The determination of Pearson's correlation coefficients 194 and PCA were performed to examine correlations between characterized properties of okra seed 195 powders: median particle size, span, total phenolic content, results of DPPH and ABTS assays, 196 water activity, moisture content, WAC, and WSI. Results were expressed as means ± standard

Figure 2 .

 2 Figure1

362

  Figure 4

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: Inhibition of DPPH radical by hydromethanolic extracts of granulometric classes (≤

Figure 2 :

 2 Figure 2

Figure 3 :

 3 Figure 3: Inhibition of ABTS•+ by hydromethanolic extracts of granulometric classes (≤ 180 µm; 180 -315 µm; 315 -500 µm; ≥ 500 µm) and unsieved powders of okra seed. Bars topped by different letters were statistically different at p < 0.05 according to Tukey's.

Figure 4 :

 4 Figure 4: Representation of all granulometric classes (≤ 180; 180 -315; 315 -500; ≥ 500 µm) and unsieved samples of okra seed powders on the principal components F1 and F2 deduced from PCA.

•

  Figures and Tables should be uploaded as separate files. Tables should be in editable format and Figures. Scans (TIFF only) should have a resolution of at least 300 dpi (halftone) or 600 to 1200 dpi (line drawings) in relation to the reproduction size This was included in the submission file.

  When a Figure number and a Table number is mentioned along the text of the manuscript, they also should begin written in Capital letter: Examples: "as shown in Figure 2....." "as summarized in Table 2, ...."

  L73. Please, change to: "predict their powder behavior during storage and reconstitution for an efficient use as functional ...." Concerned sentence was corrected (cf. L70-71).

  figures after the decimal point. Please, report the water activity values in the manuscript with three decimal figures after the decimal point, including in the Abstract.Indeed, an additional significant digit was added to water activity values in the abstract and in the results and discussion sections.L125. With this method only the extractable phenolics are being obtained, but not the chemically bound phenolics. Please, indicate this fact under 'Results and Discussion' section. This was added in L125-127. L136. "In test tubes, 30 μL of okra powder"

  Concerned sentences are grammatically sound, thus authors preferred to keep it unchanged.L192. "at a 5 % of confidence level" "X % confidence level" can be employed: cf. https://www.linguee.fr/anglaisfrancais/traduction/confidence+level.html L193. "Please, report the version, trade mark and country of origin of DSAASTAT add-on software of Excel."It was added inL192-193.

  Figure was capitalised in L242.

F

  of figure was capitalised in L271. L290-291. It was written: "Moisture content is an important parameter that determines powdered food ingredient stability during storage." This is not true!!! It is a wrong physicochemical concept. The 'water activity' is which determines the stability and NOT the water content or moisture content. Please, correct it.

Figure 4

 4 Figure 4 was deleted. The comments were added in L345-349.

-

  It is very much preferable, if you report total phenolics, to also measure and report the chemical composition of the phenolics.

  Line 308-309, "The water absorption capacity (WAC) and water solubility index (WSI)

Table 1 215 3

 13 Figure1presents the total phenolic content of unsieved samples and granulometric classes of 217 okra seed powders obtained at the three different milling frequencies. Higher total phenolic 218 contents were observed for samples composed of smaller particles. Moreover, a greater milling 219 frequency induced an increase in total phenolic content: for example, for unsieved powders, the

	.2. Total phenolic content

226

negative correlation between total phenolic content and median particle size d50 evidenced by 227 a Pearson's correlation coefficient of r = -0.71 (p < 0.05). Indeed, the bioactive compounds are 228 often associated with proteic and fatty fractions in plant extracts (Becker et al., 2016; Deli, 229 Baudelaire, et al., 2019). As indicated in the introduction, a previous study focused on the 230

. Moisture content and water activity 290

	312	WSI significantly differed according to the milling frequency. When milling frequency was
	313	increased, the median particle size of powders was decreased, enhancing water solubility index.
	314	This was confirmed by the highly negative correlation between WSI and median particle size
	315	(Pearson's correlation coefficient of r = -0.89, p < 0.05), indicating that interaction with water
	316	was improved for smaller particles (cf. Ttable 2). This can be explained by the fact that milling
	317	induces particle size reduction, leading to the improvement of material functionalities such as
	286 318	bioactive molecules that contribute to antioxidant activity of plant extracts (Rosa et al., 2013; particle surface specific area and surface energy, enhancing the interactions of plant material
	287 319	Brewer et al., 2014; Becker et al., 2017). with water (Ye et al., 2016). In addition, this may result from the exposure of hydrophilic groups
	288 320	Figure 3 (coming from cellulose and hemicellulose) at the surface of particles obtained by plant
	321	micronization (Zhao et al., 2009; Ting et al., 2014). Zhao et al. (2009) also reported increased
	289 291 292 293 294 295 322 323 324 325 326 327 328	3.4Moisture content is an important parameter that is indirectly related to powdered food ingredient stability during storage. Low levels of moisture content (up to 12 % (w/w)) are known to limit microbial growth and extend food shelf-life (Aryee et al., 2006; Kaur et al., 2013). Results displayed in Ttable 2 show that the moisture content of granulometric classes and unsieved samples of okra seed powders ranged from 8.61 ± 0.08 to 12.07 ± 0.05 (w/w), presuming a good stability during storage. The moisture content was generally decreased at dispersibility and solubility for ginger powders produced with a more intense milling, owing to F improved interactions with water evidenced by a higher WSI (Zhao et al., 2009). Water o absorption capacity represents the ability of a product to absorb water. Table 2 shows that all r powders had a high WAC whatever the granulometric class and the milling frequency. This can P be explained by the high proportion of mucilage of okra seeds that highly contribute to water e absorption (Aboubakar et al., 2008). No clear influence of median particle size on WAC was e r denoted, except for a small trend of higher WAC for intermediate granulometric classes. Thus,
	296 297 298 299 300 301 329 330 331 332 333	smaller median particle size and/or at higher milling frequency (cf. Table 2), as evidenced by the moderate positive correlation between moisture content and median particle size (Pearson's correlation coefficient of r = 0.73, p < 0.05). Moisture content is not sufficient to characterize food stability during storage, as food structure also contributes to water availability for microorganisms and alteration reactions: the concept of water activity has been developed to account for these combined influences of moisture content and food structure (Tapia, et al., performed milling was sufficient to break cell walls and expose hydrophilic groups of all okra R seed components. The results of the current study are consistent with the work of (Deli, Petit, e et al., 2019), who reported that WAC and WSI were enhanced for smaller particles of Boscia v senegalensis seeds, Dichostachys glomerata fruits, and Hibiscus sabdariffa calyxes. Hence, it i w e can be hypothesized
	302	2007; Bhandari et al., 2013). Water activity ranged between 0.493 ± 0.010 and 0.583 ± 0.010
	303	for investigated okra seed powders (cf. Ttable 2), which was below the threshold of 0.650
	304	permitting the growth of microorganisms like xerophilic mold and bacteria (Tapia et al., 2007).
	305	Water activity was generally decreased at lower median particle size, similarly to water content.
	306	This was expected, as a more intense milling leads to higher stress, more friction, and thus
	307	higher heating of the plant material, leading to smaller particles with a lower moisture content
	308	(Becker et al., 2017)
	309	3.5. Water interactions
	310	The water absorption capacity (WAC) and water solubility index (WSI) results of all
	311	granulometric classes and unsieved samples of okra seed powders are presented in Table 2.

that okra seed powders obtained at higher milling frequency should interact 334 more easily with water and be more soluble, which can facilitate their intestinal absorption 335 during digestion. 336 Table 2 337 3.6. Principal component analysis of physicochemical, water interaction, and 338 phytochemical properties

  inhibition and water activity. It is worth noting that median particle size was negatively 353 correlated to total phenolic content, WSI, and ABTS •+ inhibition percentage, and positively to 354 moisture content, confirming the previous discussion of experimental results.

	355	F
		o r
		P
		e e r
		R e v
		i e w

339

PCA is used to highlight similarities or oppositions between variables and identify the variables 340 that are most correlated between them. All granulometric classes and unsieved samples of okra 341 seed powders obtained at the three milling frequencies were compared using PCA on the basis 342 of their physicochemical (moisture content, median particle size), water interaction (water 343 activity, water solubility index, and water absorption capacity), and phytochemical (total 345 result of this PCA is presented in Figure 4. Variables are presented in red color and principal 346 components are represented on F1 and F2 axes which respectively explained 43.4 % and 20.7 347 % variability among powder properties, allowing the PCA to explain a total variability of 64.1 348 % which was sufficient to outline the major links between physicochemical and phytochemical 349 properties of okra seed powders. F1 axis was positively correlated with ABTS •+ inhibition, WSI, 350 and total phenolic content, and negatively correlated with median particle size and moisture 351 content. F2 axis was positively correlated with span and WAC, and negatively correlated with 352 DPPH •

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	440 441 442 443 444 496 497 500 501 502 503 504 505 545 546	F o r F o r F o r of Industrial Cork Residue for Use as a Growing Medium', HortScience, 38(6), pp. 1242-1246. doi: 10.21273/hortsci.38.6.1242. Ćujić, N. et al. (2015) 'Optimization of polyphenols extraction from dried chokeberry using maceration as traditional technique', Food Chemistry, 194, pp. 135-142. doi: 10.1016/j.foodchem.2015.08.008. F o r F o r gumbo (Abelmoschus esculentus L.) Seed and Oil', Research Journal of Environmental and Earth Sciences, 2(1), pp. 49-54. Rahman, M. A. A. and Moon, S. S. (2007) 'Antioxidant polyphenol glycosides from the plant Draba nemorosa', Bulletin of the Korean Chemical Society, 28(5), pp. 827-831. doi: 10.5012/bkcs.2007.28.5.827. Räsänen, E., Antikainen, O. and Yliruusi, J. (2003) 'A new method to predict flowability 2 4 6 8 10 12 6 000 rpm 12 000 rpm 18 000 rpm b ab ab a a bc a ab a ab d ab ab c a Total phenolic content (mg GAE/g DM) using a microscale fluid bed', Acta Chimica SlovenicaPharmSciTech, 4(4), pp. 418-424. doi: 0 Unisieved ≥ 500 µm 315 -500 µm 180 -315 µm ≤ 180 µm 10.1208/pt040453. F o r F Granulometric class o r Figure1: Total phenolic content of granulometric classes (≤ 180 µm; 180 -315 µm; 315 -500
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		e e r e e r e e r e e r e e r e e r e e r
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370

classes with significantly different physicochemical, water interaction, and phytochemical 371 properties. The results evidenced a considerable influence of median particle size of okra seeds 372 powder on its total phenolic content, antioxidant activity deduced from ABTS and DPPH tests, 373 moisture content, water activity, and water solubility index. Powders with smaller median 374 particle sizes (i.e. lower size granulometric classes and/or powders obtained at higher milling 375 frequency) had lower moisture content, water activity, and water absorption capacity, and 376 higher total phenolic content, antioxidant activity, and water solubility index. This can result 377 from the higher level of disintegration of plant matrix structure induced by a more intense 378 milling (i.e. undergone by smaller particles and/or at higher milling frequency) which makes 379 its compounds more accessible to solvents, thus improving phenolic compounds extraction and 380 powder solubility, and decreases the number of pores of the solid matrix, thus lowering water 381 absorption capacity. For all samples, moisture content and water activity were sufficiently low 382 to avoid food spoilage. Hence, okra seed powders were biochemically and microbiologically 383 stable, ensuring a good shelf-life during storage. Water interaction parameters showed high 384 levels of water absorption and solubility, which seems to indicate that the reconstitutability of 385 okra powder could be improved by the successive milling and sieving process. Finally, all 386 obtained results suggest that okra seed powders can be used in food manufacturing as functional 387 ingredients enhancing antioxidant activity and water interactions. 406 AFNOR (1976) 'Détermination de la teneur en eau-méthode par étuvage', in INRA/EDP 407 Sciences, pp. 743-756. 408 Ahmed, B. T. and Kumar, S. A. (2016) 'Antioxidant and antidiabetic properties of 409 abelmoschus esculentus extract -An in vitro assay', Research Journal of Biotechnology, 436 Cai, L. et al. (2015) 'Purification and characterization of three antioxidant peptides from 437 protein hydrolysate of grass carp (Ctenopharyngodon idella) skin', Journal of Functional 438 Foods. 439 Carmona, E. et al. (2003) 'Granulometric Characterization and Alteration during Composting 445 Deli, M., Petit, J., et al. (2019) 'Effect of sieved fractionation on the physical, flow and 446 hydration properties of Boscia senegalensis Lam., Dichostachys glomerata Forssk. and 447 Hibiscus sabdariffa L. powders', Food Science and Biotechnology. Springer Singapore, 22, 448 pp. 1-15. doi: 10.1007/s10068-019-00597-6. 449 Deli, M., Baudelaire, E., et al. (2019) 'Successive grinding and sieving as a new tool to 450 fractionate polyphenols and antioxidants of plants powders: Application to Boscia 451 senegalensis seeds, Dichrostachys glomerata fruits, and Hibiscus sabdariffa calyx powders', 452 Food Science and Nutrition, 7(5), pp. 1795-1806. doi: 10.1002/fsn3.1022. 453 Floegel, A. et al. (2011) 'Comparison of ABTS/DPPH assays to measure antioxidant capacity 454 in popular antioxidant-rich US foods', Journal of Food Composition and Analysis. Elsevier 455 Inc., 24(7), pp. 1043-1048. doi: 10.1016/j.jfca.2011.01.008. 456 Forny, L., Marabi, A. and Palzer, S. (2011) 'Wetting, disintegration and dissolution of 457 agglomerated water soluble powders', Powder Technology, 206(1-2), pp. 72-78. doi: 458 10.1016/j.powtec.2010.07.022. 459 Hemery, Y. M. et al. (2010) 'Dry-fractionation of wheat bran increases the bioaccessibility of 460 phenolic acids in breads made from processed bran fractions', Food Research International, 464 doi: 10.1155/2014/341291. 465 Jarret, R. L., Wang, M. L. and Levy, I. J. (2011) 'Seed oil and fatty acid content in okra 498 Petit, J. et al. (2017) 'Aptitude à l'écoulement de poudres alimentaires: impact des propriétés 499 physicochimiques des particules', Industries alimentaires et agricoles(IAA), pp. 26-30. 506 Rosa, N. N. et al. (2013) 'Ultra-fine grinding increases the antioxidant capacity of wheat 507 bran', Journal of Cereal Science. Elsevier Ltd, 57(1), pp. 84-90. doi: 508 10.1016/j.jcs.2012.10.002. 509 Sabitha, V. et al. (2011) 'Antidiabetic and antihyperlipidemic potential of Abelmoschus 510 esculentus (L.) Moench. in streptozotocin-induced diabetic rats', Journal of Pharmacy and 511 Bioallied Sciences, 3(3), pp. 397-402. doi: 10.4103/0975-7406.84447. 512 Salameh, C. et al. (2016) 'Physico-chemical and rheological properties of Lebanese kishk 513 powder, a dried fermented milk-cereal mixture', Powder Technology. Elsevier B.V., 292, pp. 514 307-313. doi: 10.1016/j.powtec.2016.01.040. 515 Siliveru, K. et al. (2017) 'Granular bond number model to predict the flow of fine flour 516 powders using particle properties', Journal of Food Engineering. Elsevier Ltd, 208, pp. 11-517 18. doi: 10.1016/j.jfoodeng.2017.04.003. 518 Tapia MS, Alzamora SM, A. L.-M. (2007) 'Effects of water activity (aw) on microbial 519 stability: as a hurdle in food preservation', in Fundamentals and Applications. Woodhead 523 112-128. doi: 10.1016/j.jff.2013.12.010. 524 Torkpo, S. et al. (2009) 'Esterase, total protein and seed storage protein diversity in Okra 525 (Abelmoschus esculentus L. Moench)', West African Journal of Applied Ecology, 9(1), pp. 1-547 µm; ≥ 500 µm) and unsieved samples of okra seed powders. DM: dry matter, GAE: gallic acid 548 equivalents. Bars topped by different letters were statistically different at p < 0.05 according to 549 Tukey's HSD test.

Table 1 :

 1 Granulometric parameters of granulometric classes (≤ 180 µm, 180 -315 µm, 315 -500 µm, ≥ 500 µm) and unsieved samples of okra 609 seed powders obtained at 6 000, 12 000, and 18 000 rpm milling frequencies(Waiss et al., 2020).

			International Journal of Food Science & Technology		Page 24 of 49
	Milling speed	Granulometric classes	d10 (µm)	d50 (µm)	d90 (µm)	Span (-)
	(rpm)
		≤ 180 µm	23.1 ± 0.2 ab	85.3 ± 0.6 a	181.3 ± 0.5 a	1.87 ± 0.01 f
		180 -315 µm	152.0 ± 1.0 de	257.0 ± 1.0 c	406.0 ± 1.0 b	0.97 ± 0.01 d
		315 -500 µm	326.6 ± 6.0 g	498.3 ± 2.5 e	745.3 ± 2.5 c	0.87 ± 0.03 bcd
	6 000	≥ 500 µm	589.3 ± 18.5 i	1100.0 ± 43.5 h	2173.3 ± 73.7 g	1.44 ± 0.02 e
		Unsieved powder	141.0 ± 7.8 d	723.3 ± 29.9 f	1670.0 ± 65.5 f	2.11 ± 0.02 h
		≤ 180 µm	21.9 ± 0.1 ab	75.6 ± 2.8 a	167.3 ± 6.3 a	1.95 ± 0.06 fg
		180 -315 µm	165.6 ± 1.5 e	264.6 ± 1.5 c	409.3 ± 9.8 b	0.92 ± 0.03 cd
		315 -500 µm	312.0 ± 1.0 g	463.3 ± 1.1 de	671.6 ± 4.9 c	0.78 ± 0.01 bc
	12 000	≥ 500 µm	542.6 ± 5.6 h	801.3 ± 25.0 g	1243.3 ± 75.0 e	0.90 ± 0.05 bcd
		Unsieved powder	52.9 ± 0.5 c	457.6 ± 5.1 de	983.0 ± 7.8 d	2.03 ± 0.01 gh
		≤ 180 µm	17.8 ± 0.2 a	52.1 ± 0.6 a	149.3 ± 1.5 a	0.77 ± 0.01 b
		180 -315 µm	37.4 ± 0.1 bc	181.0 ± 0.0 b	332.6 ± 2.0 b	0.48 ± 0.01 a
	18 000	315 -500 µm	290.0 ± 4.3 f	442.6 ± 12.8 d	644.6 ± 25.9 c	0.80 ± 0.02 bc
		≥ 500 µm	527.6 ± 4.7 h	781.0 ± 14.1 g	1186.6 ± 37.8 e	0.84 ± 0.02 bcd
		Unsieved powder	25.7 ± 0.3 ab	228.6 ± 7.5 bc	702.67± 55.7 c	2.96 ± 0.16 i
	612
	613
	614
	615
	616
	617
			Institute of Food Science and Technology

610

Means ± standard deviations followed by the same superscripted letters in the same column were not significantly different at p < 0.05 according 611 to Tukey's HSD test.

Table 2 :

 2 Moisture content, water activity, water absorption capacity, and water solubility index of granulometric classes (≤ 180 µm, 180 -315 619 µm, 315 -500 µm, ≥ 500 µm) and unsieved samples of okra seed powders obtained at 6 000, 12 000, and 18 000 rpm milling frequencies.

		Milling frequency	Powder samples	Moisture	content	Water activity (-)	Water	absorption	Water	solubility
		(rpm)		(g/g) (Waiss et al.,		capacity, WAC (g/g)	index, WSI (g/100
				2020)					g)
			≤ 180 µm	9.59 ± 0.12 cd		0.555 ± 0.001 ef	4.16 ± 0.46 abc	32.71 ± 0.53 f
			180 -315 µm	9.01 ± 0.67 ab		0.556 ± 0.002 ef	6.46 ± 0.20 e		24.06 ± 0.10 cd
		6 000	315 -500 µm	9.21 ± 0.11 bc		0.540 ± 0.001 cd	5.58 ± 0.25 cde	22.74 ± 0.74 bc
			≥ 500 µm	12.07 ± 0.05 i		0.524 ± 0.003 b	4.76 ± 0.24 abcd	15.98 ± 0.23 a
			Unsieved powder	11.10 ± 0.07 fg		0.583 ± 0.007 h	5.07 ± 0.75 cde	22.75 ± 0.37 bc
			≤ 180 µm	9.15 ± 0.10 abc		0.565± 0.008 efg	4.09 ± 0.28 ab	37.75 ± 0.46 g
			180 -315 µm	8.61 ± 0.08 a		0.524 ± 0.002 b	5.77 ± 0.41 de	32.53 ± 1.06 f
		12 000	315 -500 µm	9.22 ± 0.05 bc		0.503 ± 0.004 a	5.12 ± 0.30 bcde	25.05 ± 0.57 d
			≥ 500 µm	11.79 ± 0.06 hi		0.531 ± 0.004 bc	4.98 ± 0.23 bcde	17.64 ± 0.16 a
			Unsieved powder	11.27 ± 0.08 fgh	0.550 ± 0.001 de	5.19 ± 0.52 bcd	22.00 ± 0.78 b
			≥ 500 µm	9.56 ± 0.08 bc		0.572 ± 0.002 gh	3.52 ± 0.12 a		41.12 ± 1.16 h
			315 -500 µm	10.15 ± 0.02 de	0.564 ± 0.005 fgh	4.06 ± 0.87 abc	36.77 ± 0.66 g
		18 000	180 -315 µm	11.32 ± 0.06 gh	0.560 ± 0.002 efg	5.60 ± 0.50 cde	23.50 ± 0.07 bc
			≤ 180 µm	13.20 ± 0.17 j		0.550 ± 0.010 ef	4.67 ± 0.12 abc	17.00 ± 0.14 a
			Unsieved powder	10.69 ± 0.09 ef		0.493 ± 0.009 a	5.55 ± 0.55 bcde	30.58 ± 0.80 e
	620	Means ± standard deviations followed by the same superscripted letters in the same column were not significantly different at p < 0.05 according to Tukey's
	621	HSD test.

Figure1: Total phenolic content of granulometric classes (≤ 180 µm; 180 -315 µm; 315 -500 µm; ≥ 500 µm) and unsieved okra seed powders. DM: dry matter, GAE: gallic acid equivalents. Bars topped by different letters were statistically different at p < 0.05 according to Tukey's HSD test.

Table 1 :

 1 Granulometric parameters of granulometric classes (≤ 180 µm, 180 -315 µm, 315 -500 µm, ≥ 500 µm) and unsieved samples of okra seed powders obtained at 6 000, 12 000, and 18 000 rpm milling speed(Waiss et al., 2020).

	Milling speed	Granulometric classes	d10 (µm)	d50 (µm)	d90 (µm)	Span (-)
	(rpm)
		≤ 180 µm	23.1 ± 0.2 ab	85.3 ± 0.6 a	181.3 ± 0.5 a	1.87 ± 0.01 f
		180 -315 µm	152.0 ± 1.0 de	257.0 ± 1.0 c	406.0 ± 1.0 b	0.97 ± 0.01 d
		315 -500 µm	326.6 ± 6.0 g	498.3 ± 2.5 e	745.3 ± 2.5 c	0.87 ± 0.03 bcd
	6 000	≥ 500 µm	589.3 ± 18.5 i	1100.0 ± 43.5 h	2173.3 ± 73.7 g	1.44 ± 0.02 e
		Unsieved powder	141.0 ± 7.8 d	723.3 ± 29.9 f	1670.0 ± 65.5 f	2.11 ± 0.02 h
		≤ 180 µm	21.9 ± 0.1 ab	75.6 ± 2.8 a	167.3 ± 6.3 a	1.95 ± 0.06 fg
		180 -315 µm	165.6 ± 1.5 e	264.6 ± 1.5 c	409.3 ± 9.8 b	0.92 ± 0.03 cd
		315 -500 µm	312.0 ± 1.0 g	463.3 ± 1.1 de	671.6 ± 4.9 c	0.78 ± 0.01 bc
	12 000	≥ 500 µm	542.6 ± 5.6 h	801.3 ± 25.0 g	1243.3 ± 75.0 e	0.90 ± 0.05 bcd
		Unsieved powder	52.9 ± 0.5 c	457.6 ± 5.1 de	983.0 ± 7.8 d	2.03 ± 0.01 gh
		≤ 180 µm	17.8 ± 0.2 a	52.1 ± 0.6 a	149.3 ± 1.5 a	0.77 ± 0.01 b
		180 -315 µm	37.4 ± 0.1 bc	181.0 ± 0.0 b	332.6 ± 2.0 b	0.48 ± 0.01 a
	18 000	315 -500 µm	290.0 ± 4.3 f	442.6 ± 12.8 d	644.6 ± 25.9 c	0.80 ± 0.02 bc
		≥ 500 µm	527.6 ± 4.7 h	781.0 ± 14.1 g	1186.6 ± 37.8 e	0.84 ± 0.02 bcd
		Unsieved powder	25.7 ± 0.3 ab	228.6 ± 7.5 bc	702.67± 55.7 c

Table 2 :

 2 Moisture content, aw (-): water activity, WAC: Water Absorption Capacity(g/) and WSI: Water Solubility Index (g/100g) of ≥500 µm; 500-315µm, 315-180 µm,180 µm ≥ and for unsieved seeds. Means ± standard deviations followed by the same superscripted letters in the same row were not significantly different at P < 0.05 according to Tukey's HSD test.

	Milling speed (r.p.m.) granulometric classes	Moisture content (g/g)	aw (-)	WAC (g/g)	WSI (g/100 g)
			(Waiss et al., 2020)
		≥ 500 µm	12.07 ± 0.05 c	0,524 ± 0,00 a	4,76 ± 0,24 ab	15,98 ± 0,23 b
		315 -500 µm	9.21 ± 0.11 a	0,540 ± 0,00 b	5,58 ± 0,25 b	22,74 ± 0,74 b
	6 000	180 -315 µm	9.01 ± 0.67 a	0,556 ± 0,00 c	6,46 ± 0,20 c	24,06 ± 0,10 b
		≤ 180 µm	9.59 ± 0.12 a	0,555 ± 0,00 c	4,16 ± 0,46 a	32,71 ± 0,53 c
		Unsieved powder	11.10 ± 0.07 b	0,583 ± 0,01 d	5,07 ± 0,75 b	22,75 ± 0,37 b
		≥ 500 µm	11.79 ± 0.06 d	0,531 ± 0,00 b	4,98 ± 0,23 b	17,64 ± 0,16 a
		315 -500 µm	9.22 ± 0.05 b	0,503 ± 0,00 a	5,12 ± 0,30 b	25,05 ± 0,57 b
	12 000	180 -315 µm	8.61 ± 0.08 a	0,524 ± 0,01 b	5,77 ± 0,41 b	32,53 ± 1,06 c
		≤ 180 µm	9.15 ± 0.10 b	0,565 ± 0,00 d	4,09 ± 0,28 a	37,75 ± 0,46 d
		Unsieved powder	11.27 ± 0.08 c	0,550 ± 0,00 c	5,19 ± 0,52 b	22.00 ± 0,78 ab
		≥ 500 µm	13.20 ± 0.17 e	0,550 ± 0,01 b	4,67 ± 0,12 ab	17,00 ± 0,14a
		315 -500 µm	11.32 ± 0.06 d	0,560 ± 0,00 c	5,60 ± 0,50 b	23,50 ± 0,07 b
	18 000	180 -315 µm	10.15 ± 0.02 b	0,564± 0,00 cd	4,06 ± 0,87 ab	36,77 ± 0,66 d
		≤ 180 µm	9.56 ± 0.08 a	0,572 ± 0,00 d	3,52 ± 0,12 a	41,12 ± 1,16 e
		Unsieved powder	10.69 ± 0.09 c	0,493 ± 0,01 a	5,55 ± 0,55 b	30,58 ± 0,80 c

LATIF, Laurette Brigelia NKELETELA, Joël SCHER, Jérémy PETIT which we wish to submit for publication as an original article in 'International Journal of Food science and Technology'.

  'How do grinding and sieving impact on physicochemical properties, polyphenol content, and antioxidant activity of Hieracium pilosella L. powders?', Journal of Functional Foods, 35, pp. 666-672. doi: 10.1016/j.jff.2017.06.043.

	Becker, L. et al. (2017) This reference was cited in the introduction as matters the impact of grinding and sieving on
	physicochemical and phytochemical properties of resulting powders. It was useful to discuss
	results of chemical composition and antioxidant activity.
	Deli, et al. (2019) 'Effect of sieved fractionation on the physical, flow and hydration properties
	of Boscia senegalensis Lam., Dichostachys glomerata Forssk. and Hibiscus sabdariffa L.
	powders', Food Science and Biotechnology. Springer Singapore, 22, pp. 1-15. doi:
	F 10.1007/s10068-019-00597-6. F
	o r o r
	P P
	e e r e e r
	R e v Arapitsas, P. (2008) 'Identification and quantification of polyphenolic compounds from okra seeds and skins', Food Chemistry, 110(4), pp. 1041-1045. doi: 10.1016/j.foodchem.2008.03.014. R e v
	i e w This reference, cited in introduction, reports information regarding contents of phenolic compounds in Ours seed. i e w
	Tapia MS, et al. (2007) 'Effects of water activity (aw) on microbial stability: as a hurdle in
	food preservation', in Fundamentals and Applications. Woodhead Publishing Limited, pp. 599-
	630. doi: 10.1533/9781845690243.3.599.
	Kind regards,
	This article was useful in the discussion section. It was cited for interpreting okra seed powders
	Jérémy PETIT stability during storage from water activity values.

This reference was cited because it reports information on the influence of sieving and grinding on the physical properties (particle size, shape, and specific surface area), aptitude for preservation (water activity), and reconstitutability (solubility, water absorption capacity) of plant powders. Similar grinding and sieving conditions and methods for the determination of proximal composition were employed.

Table 2

 2

	...."
	Concerned sentence was corrected (cf. L293).
	L309. "in Table 2."
	Concerned sentence was corrected (cf. L296).
	L310-312. WAC and WSI results are not reported in Table 2. Please, complete the Table. It
	is then not possible to revise this part.

The result of WAC and WSI were added in table 2. The comments were added in

L323-335. L312. "high" (not 'highly', an adverb)

Table 2 .

 2 It was written in the legend: "and water solubility index of granulometric classes" Indeed, the water solubility index was not reported in the Table. Please, add a column with these results. Please, also comment and discuss them in the text.WAC and WSI were added in table 2 and discussed inL323-330 and 333-335.

	F
	o r
	P
	e e r
	R e v
	i e w

Data availability statement

Research data are not shared.

Institute of Food Science and Technology

International Journal of Food Science & Technology

Conflict of interest

The authors declare that they have no conflict of interest.