Ilya Bogaevsky
email: ibogaevsk@gmail.com

Michel Rouleux
email: rouleux@univ-tln.fr

Semi-classical Lagrangian intersections

We investigate normal forms of germs of glancing Lagrangian intersections, and their semiclassical counterparts. This is motivated by singularities of Bessel beams, or diffraction problems by an obstacle with a non convex boundary.

0/ Introduction

Let M = R n , we restrict essentially to the case n = 2.

Parametrices of linear PDE's on M make use of symbolic calculus adapted to Lagrangian intersection, see [MelUhl] for high frequency asymptotics, and [AnDoNaRo] for semi-classics in the framework of Maslov canonical operator.

Let H(x, hD x ; h) be a h-PDO whose symbol has the asymptotic expansion H(x, p; h) ∼ H 0 (x, p)+ hH 1 (x, p) + • • •, and assume that H 0 (x, p) = 0 is a smooth, non critical hypersurface, fibrated by Hamiltonian curves. To start with, we shall denote H 0 simply by H.

On the other hand, let Λ → T * M be a smooth embedded Lagrangian manifold, and f h be a Lagrangian semi-classical distribution microlocally supported on Λ.

The main concern is to construct the asymptotic outgoing solution of (0.2)

H(x, hD x ; h)u h (x) = f h (x), u h (x) = E + f h (x; h) = ∞ 0 e -itH/h f h (x) dt
and in particular to solve Cauchy problem (0.3)

hD t v h + H(x, hD x )v h = 0, v h | t=0 = f h
The local theory amounts to study germs of Lagrangian manifolds and hypersurfaces.

At the classical level the question is about a pair consisting of a Lagrangian submanifold Λ (we call the "initial manifold") and a hypersurface G = {H = 0}. We are interested in the set Λ 1 of characteristics of H intersecting Λ. Global aspects of Lagrangian intersection theory were greatly motivated by Arnold conjectures [Ar] and have been extensively studied (see [ElGr]). Here we simply focus on the germs of the generating functions near an intersection point.

The generic case is when v H is transverse to Λ at t = 0, then Λ 1 is precisely the union of the outgoing/incoming flows Λ ± of Λ by the Hamilton vector field v H in G, see Theorem 1.1 below. This is the situation considered in [MelUhl] and [AnDoNaRo] : Λ and Λ + intersect cleanly along Λ ∩ Λ + = ∂Λ + , and (Λ, Λ + ) is called an intersecting pair of Lagrangian manifolds.

Here we want to consider a point z ∈ Λ where the intersection is not clean, i.e. v G ∈ T z Λ. We call such a point a glancing point, and denote their set by G(Λ, G). We answer first the question :

• What is the classification of such pairs (Λ, G) up to symplectic diffeomorphisms ?

Normal forms which deals with isolated points of non-transversality, called a kiss ( [START_REF] Eliashberg | Lagrangian intersections theory. A finite dimensionnal approach[END_REF]p.19].

This is "the generic case", which has been investigated in [ZaMy], using Morse families. We recall the most simple type of kiss in Theorem 1.2 below.

In case G is tangent to Λ along a non-singular hypersurface ℓ ⊂ Λ, we call (Λ, ℓ, H) a triad.

Of course, such a pair uniquely defines the Lagrangian submanifold Λ 1 .

Next we consider the semi-classical level, and try to represent the solution of (0.3) and (0.2) near a glancing point. The first step is to construct the solution of Hamilton-Jacobi equation parametrizing Λ 1 , which is generically singular.

A similar problem arises in diffraction theory for the wave equation A theorem of Melrose [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators I-IV[END_REF]Theorem 21.4.8] says that such a pair, near the glancing point z, is

H = ∂ 2 t -∆ x in M = (R n x \ O) × R t outside
symplectically equivalent to F = {x 1 = 0} and G = {g = ξ 2 1 -x 1 -ξ 2 = 0}.
The second main question we adress here is the following:

• Given a pair of glancing hypersurfaces (F, G) in T * M at some point z, find the germs at z of some Lagrangian manifolds Λ transverse to F at z and such that v g ∈ T z Λ ?

There are indeed such one-parameter families of Λ, for which we compute

A z (Λ, G), B z (Λ, G), see Propositions 3.1-3.2 below.
Next we construct solutions of Hamilton-Jacobi equation for hD t +(hD x 1 ) 2 -x 1 -hD x 2 with initial condition on Λ. This, conveniently adapted to dimensions n ≥ 3, could be applied to the diffractive Cauchy problem above when the obstacle is not necessarily convex at a point of the boundary, see [Le], [Sm].

1/ Lagrangian intersections

First recall the situation where v H is transverse to Λ. Let ι 0 : Λ 0 → T * M be a smooth embedded Lagrangian manifold, and ι 1 : Λ 1 → T * M be a smooth embedded Lagrangian manifold with smooth boundary ∂Λ 1 (isotropic manifold). Following [MelUhl] we say that (Λ 0 , Λ 1 ) is an intersecting pair of Lagrangian manifolds iff Λ 0 ∩ Λ 1 = ∂Λ 1 and the intersection is clean, i.e.

(1.1)

∀z ∈ ∂Λ 1 T z Λ 0 ∩ T z Λ 1 = T z ∂Λ 1
(in particular Λ 0 and Λ 1 cannot be transverse). On the set of intersecting pairs of Lagrangian manifolds we define an equivalence relation by saying that (Λ 0 , Λ 1 )

∼ (Λ ′ 0 , Λ ′ 1 ) iff near any z ∈ ∂Λ 1 , z ′ ∈ ∂Λ ′ 1 , there is a symplectic map κ such that κ(z) = z ′ , and a neighbhd V ⊂ T * M of z such that κ(Λ 0 ∩ V ) ⊂ Λ ′ 0 , κ(Λ 1 ∩ V ) ⊂ Λ ′ 1 .
We will call the equivalence class a Lagrangian pair. The following result readily extends [MelUhl,Prop.1.3] in the special case of homogeneous Lagrangian manifolds : Theorem 1.1 [MelUhl]: All intersecting pairs of manifolds in T * M are locally equivalent. More precisely near each z ∈ T * M , there exists a canonical map κ : T * M → T * R n such that κ(z) = (0, 0), κ(Λ 0 ∩ V ) ⊂ T * 0 R n (vertical fiber at 0), and κ(Λ 1 ∩ V ) ⊂ Λ 0 + , Λ 0 + being the flow-out of T * 0 R n by the Hamilton vector field v ξ n = ((0, • • • , 0, 1), 0) of ξ n , passing through some (0; ξ) = (0; (ξ ′ , 0)) i.e.

(1.5)

Λ 0 + = {(x, ξ) ∈ T * R n : x = (0, x n ), ξ = (ξ ′ , 0), x n > 0}
Lemma 1.1 has its semi-classical counterpart : near a point z of intersecting pair the normal form for Λ is given by {x = 0} and this of H is (say) hD x n . This solves locally problem (0.2). Namely, let f h be a Lagrangian distribution defined by its amplitude a(x ′ , x n -t, ξ; h), and supported microlocally on the "vertical plane" Λ = {x = 0}. Let θ T ∈ C ∞ (R + ) vanish near +∞ and equal to 1 for t ≤ T , then

(1.6) u(x, h) = i h ∞ 0 θ T (t) dt e i(x ′ ξ ′ +(x n -t)ξ n )/h a(x ′ , x n -t, ξ; h) dξ satisfies (0.2) mod O(h ∞ ) whenever x n ≤ T /2.
Assume instead that Hamilton foliation of H is simply tangent to Λ at some z 0 ∈ Λ ∩ {H = 0}.

Then Λ 1 is a singular Lagrangian manifold near z 0 , and the intersection Λ ∩ Λ 1 is not clean. Such a point z 0 of non-transversality is called glancing.

Near a glancing point Theorem 1.1 does not apply. So we need to extend (locally) Theorem 1.1 to this situation, by finding the normal form of non Lagrangian (or "glancing") intersection for (Λ, G)

near z 0 .
We have the following :

Theorem 1.2 [ZaMy]: Assume the germs (Λ, G) are non-transversal at an isolated point z 0 ∈ T * M , then (Λ, G) has generically the following normal form, up to a symplectic transformation κ ±

(1.7)

Λ = {ξ = ∂φ ± ∂x }, G = {g = 0}
where g(x, ξ) = ξ 1 and the generating functions φ ± (x) = ±

x 3 1 3 + x 1 x 2 2 .
The proof is based on the theory of Morse families, see e.g. [BaWe].

In the case of φ -(x) we get the two isotropic lines

x 1 = ±x 2 , ξ 2 = ±2x 2 2 , ξ 1 = 0 in the intersection of Λ with the Hamiltonian level ξ 1 = 0. Their extension along the characteristics ξ 1 = ξ 2 = x 2 = 0 (v H = (1, 0; 0, 0) at z 0 ) is the union of two Lagrangian submanifolds {ξ 1 = 0, ξ 2 = ±2x 2 2 }.
In the case of φ + (x), the intersection of Λ with ξ 1 = 0 reduces to x = 0.

We are now to define some invariants of (Λ, G) at z = z 0 .

Definition 1.3: Define Λ locally near z by f 1 = f 2 = 0, with {f i , f j } = 0. Consider the symmetric matrix

A z = A z (Λ, G) = {f 2 , {f 2 , g}} -{f 1 , {f 2 , g}} -{f 2 , {f 1 , g}} {f 1 , {f 1 , g}} (z)
and the vector

B z = B z (Λ, G) = {g, {g, f 1 }} {g, {g, f 2 }} (z)
Let z be a glancing point for the pair (Λ, G) i.e.

g(z) = f 1 (z) = f 2 (z) = 0, {g, f 1 }(z) = {g, f 2 }(z) = 0
We distinguish the following 10 possibilities for the 2-jets of (Λ, G) z :

det A z > 0, B z = 0 (1) det A z > 0, B z = 0 (2) det A z < 0, t B z A z B z = 0 (3) det A z < 0, t B z A z B z = 0, B z = 0 (4) det A z < 0, B z = 0 (5) det A z = 0, t B z A z B z = 0 (6) det A z = 0, A z = 0, t B z A z B z = 0, B z = 0 (7) det A z = 0, A z = 0, B z = 0 (8) A z = 0, B z = 0 (9) A z = 0, B z = 0 (10)
Remark 1.1: Of course, A z and B z are invariant under symplectic transformations, but they are not uniquely defined. However each of the cases above is preserved by a non-degenerate linear transfor-

mation of the type f ′ 1 f ′ 2 = P f 1 f 2 , where P ∈ GL(2) has variable coefficients. Namely if A ′ z is defined as A z with f ′ 1 , f ′ 2 instead of f 1 , f 2 , then A ′ z = (det P z ) 2 t P -1 z A z P -1 z , B ′ z = P z B z
However A z and B z cannot account for higher order terms occurring in the classification of [ZyMa].

Under the hypothesis of Theorem 1.2, an elementary computation shows that

f 1 = ξ 1 ∓ x 2 1 - x 2 2 , f 2 = ξ 2 -2x 1 x 2 , g = ξ 1 : A ± 0 = 2 0 0 ±2 , B ± (0) = ∓2 0
so we are in case (1) for +, and in case (3) for -.

We turn now to the semi-classical level, and try to solve (0.2) or (0.3). Recall Λ t = exp tv H (Λ)

verifies Λ t+s = exp tv H (Λ s ) for all t, s ≥ 0. Let Γ t be the set of points of Λ t such that v H is tangent to Λ t , we have also Γ t+s = exp tv H (Γ s ). So the glancing property is invariant under the Hamiltonian flow.

We can trivially solve in the (ξ 1 , x 2 ) coordinates Hamilton-Jacobi equation ∂Ψ ∂t

+ ∂Ψ ∂x 1 = 0, Ψ| t=0 = φ(x 1 , x 2 ). This gives Ψ(t, x 1 , x 2 ) = φ(-t + x 1 , x 2 ), so the condition Ψ| t=0 = φ ± gives (1.10) Ψ ± (t, x 1 , x 2 ) = ± 1 3 (-t + x 1 ) 3 + (-t + x 1 )x 2
By definition

Λ 1 = {ξ 1 = ∂Ψ ∂x 1 = 0, ξ 2 = ∂Ψ ∂x 2 } = {ξ 1 = ±(-t + x 1 ) 2 + x 2 2 = 0, ξ 2 = 2(-t + x 1 )x 2 }.
For Ψ + we get:

Λ 1 = {ξ 1 = 0, t = x 1 , x 2 = 0, ξ 2 = 0}
This is a one-dimensional isotropic submanifold because the intersection Λ and G is a point. For Ψ - we get:

Λ 1 = {ξ 1 = 0, t = x 1 ± x 2 , ξ 2 = ∓2x 2 2 }.
This is a Lagrangian submanifold with a singular line. (1) For Tricomi Hamiltonian, H(x, p) = x 2 p 2 1 + p 2 2 , the residual points are those for p 1 = 0, the special points those for p 1 = 0 but p 2 = 0, and the ordinary points those for p 1 p 2 = 0.

(2) For Métivier Hamiltonian, H(x, p) = p 2 1 + (x 2 1 + x 2 2 )p 2 2 , the residual points are given by p 2 = 0 or x = (x 1 , x 2 ) = 0, the special points by p 2 = 0 and x = 0, but p, x = 0, and the ordinary points by p 2 x, p = 0.

(3) Let H be the "conformal metric" given by (2.1)

H(x, p) = |p| m 1 ρ(x)
where ρ is a smooth positive function on M , m ≥ 1. The residual points are the critical points of ρ ; at a special point, ∇ρ, p = 0, i.e. v H is tangent to the level curves of ρ. This example of practical interest was extensively studied in [AnDoNaRo2,[START_REF]Applications to the propagation of Bessel beams in nonhomogeneous media[END_REF].

Together with H positively homogeneous of degree m, consider an initial manifold Λ. In the following we try to link Lagrangian intersections with the classification above.

We take first Λ

= T * x 0 M ≈ {x = x 0 } (conormal bundle to {x 0 } in T * M ). Then Λ intersects cleanly Λ + along ∂Λ + .
Consider next the "initial" Lagrangian manifold

(2.3) Λ = {x = X(ϕ, ψ) = ϕω(ψ), p = P (ϕ, ψ) = ω(ψ), ϕ ∈ R}
which is called the "cylinder" ; here ω ∈ S n-1 is the unit vector parametrized by ψ. This is the wave-front set of semi-classical distributions related with Bessel functions.

Let us check first the Lagrangian intersection in this case. The tangent space T z Λ has the parametric equations

δX = ω(ψ)δϕ + ϕω ⊥ (ψ)δψ, δP = ω ⊥ (ψ)δψ, δϕ, δψ ∈ R so v H ∈ T z Λ iff there exist (δψ, δϕ) such that ∂ p H = ω(ψ)δϕ + ϕω ⊥ (ψ)δψ, -∂ x H = ω ⊥ (ψ)δψ
Taking scalar products with ω(ψ), ω ⊥ (ψ), and using Euler identity, we get δϕ

= ∂ p H, P (ψ) = mH = mE, δψ = -∂ x H, ω ⊥ (ψ) . Then, for z ∈ Σ E , (2.5) ∂ p H + ϕ∂ x H, ω ⊥ (ψ) = 0, -∂ x H, ω(ψ) = 0, H(z) = 0
Second relation (2.5) means that z = ϕω(ψ), ω(ψ) is a special point (Definition 1.2). Since

(ω(ψ), ω ⊥ (ψ) form a basis of R 2 , these relations are necessary and sufficient for v H ∈ T z Λ, i.e.

z to be glancing.

Examples 2.2:

(4) When H = p 2 , all points are glancing. When H = |p| m ρ(x) , z(0) is a glancing point iff (2.6) either : ϕ = 0 and ∇ρ = 0, or : ϕ = 0 and ∇ρ(0), ω(ψ) = 0

Second condition means that if z(0) = (0, ω(ψ)) is a special point.

(

) Assume again M = R 2 . For a, b ∈ R, let ρ(x) = 1 + ax 2 1 + bx 2 2 . Consider Hamiltonian on T * R 2 of the form H(x, p) = µ, p ρ(x) 5
with µ = (1, 0). A computation shows that, with

x 1 = ϕ cos ψ, x 2 = ϕ sin ψ -∂ x H, ω(ψ) = 2ϕ cos ψ ρ 2 (x) (a cos 2 ψ + b sin 2 ψ) -ρ 2 (x) -∂ x H, ω(ψ) = ρ(x) + 2ϕ 2 cos 2 ψ(a -b) sin ψ so choosing a > b > 0 and E > 1, we see that G(Λ, G) = ∅.
Otherwise, there may be a unique glancing point near x = 0.

(6) Consider Hamiltonian with constant coefficients H(x, p) = µ, p , then when ω(ψ) is parallel to µ, the corresponding ray is contained entirely in Λ, so the intersection is not transverse. This corresponds to a triad.

(7) Here is another example of a triad. Consider the scattering problem by an obstacle, e.g. H = -∆ on L 2 (R 2 \ O) with Dirichlet boundary conditions on ∂O, and E = 1 so that the characteristics are parametrized by arc-length.

Let ℓ be a Lagrangian submanifold in {T * ∂O : p 2 = 1} and Λ be its preimage under the natural projection T * ∂O R 2 → T * ∂O. Then (Λ, ℓ, {p 2 = 1}) is a triad and the union of all characteristics in p 2 = 1 intersecting ℓ is a Lagrangian submanifold Λ 1 with singularities.

For example, Λ 1 has cusps along the ray being tangent to an inflection point of ∂O. If n = 3 then Λ 1 can have more interesting singularities: open swallow-tails and open Whitney umbrellas (see [Arn], [Gi], [Sch]).

The submanifold Λ 1 gives a short-wave approximation of solutions for the obstacle problem.

When Λ 1 is singular this is of particular interest since we cannot resort to standard Maslov canonical operator.

Remark 2.2: When (Λ, G) are given by ( 2.3) and ( 2.1) and m = 1, G = {H = E} there is a general Morse family parametrizing Λ t , of the form Φ(x, t, ψ, ϕ, λ) = mEt + ϕ + λ P (t, ϕ, ψ), x -X(t, ϕ, ψ)

where mEt+ϕ is an eikonal coordinate, and (X, P ) the solution of Hamilton equations Ẋ = ∂ x H, Ṗ = -∂ x H with prescribed value on Λ, see [AnDoNaRo3]. But for a glancing Lagrangian intersection, this formula is of little help.

3/ Glancing hypersurfaces and Lagrangian intersections.

Let F, G be two smooth hypersurfaces of T * M intersecting transversally at z. Recall from [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators I-IV[END_REF]Definition 21.4.6] By the theorem of equivalence of glancing hypersurfaces of Melrose [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators I-IV[END_REF]Theorem 21.4.8] there are local symplectic coordinates (x, ξ) vanishing at z such that F, G are defined resp. by x 1 = 0 and ξ 2 1x 1ξ 2 = 0. Then g = ξ 2 1x 1ξ 2 = 0 will be the "normal form" of H in these coordinates. (We use the notation g for the normal form of H, or H -E).

We apply this theorem to G being the energy surface H = E (i.e. g = 0) and F an auxiliary hypersurface intersecting G transversally at a glancing point z. We want to find germs of Lagrangian manifolds Λ such that Λ is transverse to F at z but (Λ, G) has glancing intersection at z. Again, this means that T z Λ ∩ (T z F ) σ = {0} and Rv H (z) = (T z G) σ ⊂ T z Λ, where superscript σ denotes symplectic orthogonal.

All computations below will be carried out in the local symplectic coordinates (x, ξ) vanishing at z.

Constructing some phase functions parametrizing Λ.

The Hamiltonian vector field for g = ξ 2

1x 1ξ 2 takes the form

v g = 2ξ 1 ∂ x 1 -∂ x 2 + ∂ ξ 1 , or using coordinates (δx, δξ) in T T * M , as (3.2) v g = (δx 1 , δx 2 ; δξ 1 , δξ 2 ) = (2ξ 1 , -1; 1, 0)
We know that a general Lagrangian manifold can be parametrized in the mixed representation, so when n = 2 by one of the following cases

(3.3) (I) Λ = {ξ = ∂φ ∂x }, (II) Λ = {x = - ∂φ ∂ξ } (III) Λ = {x 1 = - ∂φ ∂ξ 1 , ξ 2 = ∂φ ∂x 2 , (IV) Λ = {x 2 = - ∂φ ∂ξ 2 , ξ 1 = ∂φ ∂x 1 }
To determine Λ, we write that the glancing intersection of (Λ, G) at z should take place at z = (x 1 , x 2 , ξ 1 , ξ 2 ) = 0, i.e. for z small enough, v g (z) ∈ T z Λ implies z = 0. Then we check that Λ is transverse to F at z = 0. There will be of course many possibilities for Λ, and we content ourselves with a few significant examples.

The linear situation is the case where φ is quadratic. Cases (I),(II),(III) give one-parameter families of Lagrangian manifolds. We show that Case (IV) does not occur.

Proposition 3.1: Consider first a quadratic phase φ = φ 0 .

• In Cases (I),(II),(III) φ 0 are one-parameter families taking values

(3.4) (I) φ 0 (x) = 1 2 (ax 2 1 -2x 1 x 2 ), (II) φ 0 (ξ) = 1 2 (2ξ 1 ξ 2 + cξ 2 2 ), (III) φ 0 (x 2 , ξ 1 ) = 1 2 b(ξ 1 + x 2 ) 2
respectively, where a, b, c = 0. The manifold Λ is transverse to F at z = 0, and the corresponding matrices A z (Λ, G), B z (Λ, G) in Definition 1.3 are then given by

(3.5) (I) A z = 2 1 a a a 2 , B z = 2 -a 1 , (II) A z = 2 0 0 0 1 , B z = 2 1 0 (III) A z = 2 0 0 0 1 , B z = 2 1 0
So with the notations of Definition 1.3, case (I) is of type ( 6) or ( 7) according to a 3 + 2a 2 + 1 = 0 or not, while cases (II) and and (III) are of type (6).

• Case (IV) does not occur.

Proof:

• Case (I). The quadratic phase takes the form

φ 0 = 1 2 (ax 2 1 + 2bx 1 x 2 + cx 2 2 )
. That v g be tangent to Λ at z express as

(3.6) (δx 1 , δx 2 ; aδx 1 + bδx 2 , bδx 1 + cδx 2 ) = (2ξ 1 , -1; 1, 0)
By substitution we find

ξ 1 = b+1 2a = c 2b , so condition ξ 1 = 0 gives b + 1 = c = 0. Substituting in ξ 1 = ax 1 + bx 2 gives ax 1 -x 2 = 0. Then the condition z ∈ G, namely g = 0 gives -x 1 -ξ 2 = 0, so altogether x 1 = -ξ 2 = x 2
a , ξ 1 = 0, and z = (x 1 , ax 1 , 0, -x 1 ). Finally the condition z ∈ F gives x 1 = 0, hence z ∈ F ∩ G gives z = 0. So the phase φ 0 defining Λ is given by φ 0 = 1 2 (ax 2 1 -2x 1 x 2 ). It is also clear that Λ is transverse to F at z = 0. Let us compute the matrix elements of A z , B z . With

f j = ξ j -∂φ ∂x j , we have {f 1 , {f 1 , g}} = 2a 2 , {f 2 , {f 2 , g}} = 2, {f 1 , {f 2 , g}} = -2a, {g, {g, f 1 }} = -2a, {g, {g, f 2 }} = 2
which gives (3.5)(I).

• Case (II). The phase is of the form φ 0 = 1 2 (aξ 2 1 + 2bξ 1 ξ 2 + cξ 2 2 ), Hamilton vector field v g = (-aδξ 1bδξ 2 , -bδξ 1cδξ 2 ; δξ 1 , δξ 2 ) = (2ξ 1 , -1; 1, 0), so necessary condition ξ 1 = 0 gives a = 0, b = 1. The condition z ∈ G gives z = (-ξ 2 , -cξ 2 ; 0, ξ 2 ), and the condition z ∈ Λ gives -x 2 = 0, so we get z = 0.

The phase defining Λ is given by φ 0 = 1 2 (2ξ 1 ξ 2 + cξ 2 2 ). Again Λ is transverse to F at z = 0. With f j = x j + ∂φ ∂ξ j , we have

{f 1 , {f 1 , g}} = 2, {f 2 , {f 2 , g}} = 0, {f 1 , {f 2 , g}} = 0, {g, {g, f 1 }} = 2, {g, {g, f 2 }} = 0
which gives (3.5)(II).

• Case (III). The phase is of the form

φ 0 = 1 2 (aξ 2 1 + 2bx 2 ξ 1 + cx 2
2 ), and Hamilton vector field

v g = (-aδξ 1 -bδx 2 , δx 2 ; δξ 1 , bδξ 1 + cδx 2 ) = (2ξ 1 , -1; 1, 0), so necessary condition ξ 1 = 0 gives a = b = c.
The condition z ∈ G gives z = (x 1 , -x 1 a ; 0, -x 1 ), and the condition z ∈ Λ gives x 1 = 0, so we get z = 0 as expected. The phase defining Λ is thus given by φ

0 = a 2 (ξ 1 + x 2 ) 2 . Again Λ is transverse to F at z = 0. With f 1 = x 1 + ∂φ ∂ξ 1 , f 2 = ξ 2 -∂φ ∂x 2 we have {f 1 , {f 1 , g}} = 2, {f 2 , {f 2 , g}} = 0, {f 1 , {f 2 , g}} = 0, {g, {g, f 1 }} = 2, {g, {g, f 2 }} = 0
which gives (3.5)(III).

• Case (IV). The quadratic phase takes the form φ = 1 2 (ax 2 1 + 2bx 1 ξ 2 + cξ 2 2 ), but Hamilton vector field v g = (δx 1 , -bδx 1cδξ 2 ; aδx 1 + bδξ 2 , δξ 2 ) = (2ξ 1 , -1; 1, 0) cannot be tangent to Λ near z = 0. ♣ Next we consider φ = φ 0 + φ 1 , where φ 1 is a cubic homogeneous polynomial. The role of the cubic term is suggested by the normal form in Theorem 1.2.

Proposition 3.2:

The phase with a cubic term takes the form φ = φ 0 + φ 1 . The manifold Λ is again transverse to F at z = 0.

• In Case (I) φ 0 fixes entirely the one-parameter family

(3.8) (I) φ(x) = 1 2 (ax 2 1 -2x 1 x 2 ) - 1 3 (ax 1 -x 2 ) 3 , A z = B z = 0 which has type (10) in Definition 1.3. • In Case (III), φ 0 (x 2 , ξ 1 ) = 1 2 b(ξ 1 + x 2 ) 2 , and φ 1 is of the form φ 1 (x 2 , ξ 1 ) = 1 3 (a ′ ξ 3 1 + 3b ′ ξ 2 1 x 2 + 3c ′ ξ 1 x 2 2 + d ′ x 3 2
). Let ∆ be as in (3.17). If ∆ = 0 we have

A z = 2 d ′ -b ′ + c ′ -b ′ + c ′ 1 + b ′ -a ′ , B z = 2 1 -c ′ + 2b ′ b ′ + d ′ -c which has type (1) in Definition 1.3. If ∆ = 0, z = 0 is glancing iff either (1-c ′ )b ′ = 0 or (1-b ′ )c ′ = 0. • In Case (II) (3.10) (II) φ(ξ) = 1 2 (2ξ 1 ξ 2 + cξ 2 2 ) + 1 3 (-ξ 3 1 + 3c ′ ξ 1 ξ 2 2 + d ′ ξ 3 2 )
• Case (IV) does not occur.

Proof:

• Case (I). The phase defining Λ is φ = φ 0 + O(|x| 3 ) with φ 0 = 1 2 (ax 2 1 -2x 1 x 2 ), and

φ 1 = 1 3 (a ′ x 3 1 + 3b ′ x 2 1 x 2 + 3c ′ x 1 x 2 2 + d ′ x 3 2 )
To this order, the condition v g (z) ∈ T z Λ takes the form

δx 1 , δx 2 ; (a + 2a ′ x 1 + 2b ′ x 2 )δx 1 + (-1 + 2b ′ x 1 + 2c ′ x 2 )δx 2 , (-1 + 2b ′ x 1 + 2c ′ x 2 )δx 1 + (2c ′ x 1 + 2d ′ x 2 )δx 2 = (2ξ 1 , -1; 1, 0)
which yields the system (3.12)

-aξ 1 + b ′ x 1 + c ′ x 2 = 2ξ 1 (a ′ x 1 + b ′ x 2 ), ξ 1 + c ′ x 1 + d ′ x 2 = 2ξ 1 (b ′ x 1 + c ′ x 2 )
Eliminating ξ 1 on the LHS gives

(3.13) (b ′ + ac ′ )x 1 + (c ′ + ad ′ )x 2 = 2ξ 1 (a ′ + ab ′ )x 1 + (b ′ + ac ′ )x 2
The condition z ∈ F gives x 1 = 0. Substituting into (3.13) we get

x 2 = 0 or c ′ + ad ′ = 2ξ 1 (b ′ + ac ′ ). Assume x 2 = 0, then the condition z ∈ G gives ξ 2 1 = x 1 + ξ 2 = b ′ x 2 1 + 2c ′ x 1 x 2 + d ′ x 2 2 = 0, so z = 0. Assume instead c ′ + ad ′ = 2ξ 1 (b ′ + ac ′ ).
Then as before c ′ + ad ′ = 0, so either ξ 1 = 0 and (3.12) gives again x 2 = 0, so z = 0. Finally we are left with the condition b ′ + ac ′ = 0. Altogether we

find φ = φ 0 + φ 1 = 1 2 (ax 2 1 -2x 1 x 2 ) -1 3 (ax 1 -x 2 ) 3 .
Again Λ is transverse to F at z = 0. With f j = ξ j + ∂φ ∂x j , a little computation shows that all second brackets vanish at z = 0, which gives (3.8). • Case (III). We recall ψ 0 from (3.4)(III), and try

φ 1 = 1 3 (a ′ ξ 3 1 + 3b ′ ξ 2 1 x 2 + 3c ′ ξ 1 x 2 2 + d ′ x 3 2 ). With φ = φ 0 + φ 1 , we have x 1 = -∂φ ∂ξ 1 , ξ 2 = ∂φ ∂x 2 , i.e.
(3.15)

x 1 = -b(ξ 1 + x 2 ) -a ′ ξ 2 1 -2b ′ ξ 1 x 2 -c ′ x 2 2 ξ 2 = b(ξ 1 + x 2 ) + b ′ ξ 2 1 + 2c ′ ξ 1 x 2 + d ′ x 2 2
with b = 0. The condition for v g (z) = (2ξ 1 , -1; 1, 0) = (δx 1 , δx 2 ; δξ 1 , δξ 2 ) ∈ T z Λ gives δξ 1 = 1 and δx 2 = -1, together with the linear system in (x 2 , ξ 1 ) shows that ξ 1 = 0 (which means z ∈ G). So z = 0 as is claimed.

(3.16) (1 + a ′ -b ′ )ξ 1 + b ′ (1 -c ′ )x 2 = 0 (b ′ -c ′ )ξ 1 + (c ′ -d ′ )x 2 = 0 with determinant (3.17) ∆ = (1 + a ′ )(c ′ -d ′ ) + b ′ (d ′ -b ′ ) + b ′ c ′ (b ′ -c ′ ) If ∆ = 0,
functions φ = φ 0 + φ 1 of inequivalent types (I)-(III) (case (IV) cannot occur), φ 0 , φ 1 are quadratic and cubic polynomials respectively. Moreover all cases (1)-( 10) in Definition 1.3 can occur.

3.2 Solving Hamilton-Jacobi (HJ) equation in (ξ 1 , x 2 ) coordinates.

As before our purpose is to parametrize Λ 1 by a phase function Ψ(x 1 , x 2 ) with prescribed value on Λ. This is the solution of HJ equation for (hD

x 1 ) 2 -x 1 -hD x 2 , namely (3.25) ∂Ψ ∂x 1 2 -x 1 - ∂Ψ ∂x 2 = 0
This equation cannot be solved directly since the Hamilton vector field v g is not transversal to Λ1.

We search Ψ as the critical value of another phase function Φ(x 1 , x 2 , t). This will lead to multivalued solutions of (3.25).

Instead of solving directly HJ for hD t + g(x, hD x ) = hD t + (hD x 1 ) 2x 1 -hD x 2 , we make a partial Fourier transform quantizing the metaplectic transformation (x 1 , x

2 ; ξ 1 , ξ 2 ) → (ξ 1 , x 2 , -x 1 , ξ 2 ),
and consider instead HJ equation for hD

t + hD ξ 1 -hD x 2 + ξ 2 1 . A first change variables (t, x 2 , ξ 1 ) → (s ′ = 1 2 (t -x 2 ), r ′ = 1 2 (t + x 2 ), ξ 1 ), is followed by (s ′ , r ′ , ξ 1 ) → (s = 1 2 (s ′ -ξ 1 ), r = 1 2 (s ′ + ξ 1 ), r ′ ). The inverse map is given by (3.30) x 2 = r ′ -r -s, ξ 1 = r -s, t = r ′ + r + s and operator hD t + hD ξ 1 -hD x 2 + ξ 2 1 is changed to hD s ′ + hD ξ 1 + ξ 2 1 , then to hD r + (r -s) 2 .
In the new variables, HJ equation takes the form (3.31) ∂Φ ∂r + (rs) 2 = 0 and its general solution is given by

(3.32) Φ(r, r ′ , s) = - 1 3 (r -s) 3 + Φ 0 (r ′ , s)
where Φ 0 (r ′ , s) is determined by the initial value at t = 0, corresponding to s = -(r + r ′ ). This yields the condition

(3.33) Φ| t=0 = Φ r, r ′ , -(r + r ′ ) = - 1 3 (2r + r ′ ) 3 + Φ 0 (r ′ , -r -r ′ )
so we need to express the initial value Φ| t=0 on Λ in terms of variables r, r ′ , s.

We discuss again according to Cases (I)-(III) listed in Proposition 3.1.

• Case (I). We start with the linear case φ(x) = φ 0 (x) = 1 2 (ax 2 1 -2x 1 x 2 ). Consider partial Legendre transform x 1 → φ 0 (x)x 1 ξ 1 , the critical point is given by x 1 = 1 a (x 2 + ξ 1 ), and the critical value φ crit = -1 2a (ξ 1 + x 2 ) 2 . Substituting (3.30) at t = 0 (i.e. s = -rr ′ ) gives (3.34)

Φ| t=0 = φ = - 1 2a (3r ′ + 2r) 2
which has a real solution iff q = x 2 + ξ 1 = 0. Then t = ξ 1 = -x 2 is a degenerate critical point, with Summing up, we have proved so far:

∂ t Φ = ∂ 2 t Φ = 0, ∂ 3 t Φ = -3/2,
Proposition 3.4: Let Λ be paramerized by a quadratic phase φ 0 or with a cubic term φ 0 + φ 1 . The critical point of t → Φ(t, ξ 1 , x 2 ) is given by t = ξ 1 .

In Case (I) (quadratic phase φ 0 ) the critical value of t → Φ(t, ξ 1 , x 2 ) given in (3.36) is

(3.51) Φ crit (ξ 1 , x 2 ) = - 1 3 ξ 3 1 - 1 2a (ξ 1 + x 2 ) 2
For φ = φ 0 + φ 1 , Φ(t, x 1 , x 2 ) is independent of t, and equal to (3.51) mod O(ξ 4 1 ). In Case (II) (quadratic phase φ 0 ) the critical value of t → Φ(t, ξ 1 , x 2 ) is again (3.51), with another constant a.

In Case (III) (quadratic phase φ 0 ) the critical value of t → Φ(t, ξ 1 , x 2 ) is again (3.51), with another constant a. For φ = φ 0 + φ 1 , t → Φ(t, ξ 1 , x 2 ) has a real critical point iff q = x 1 + ξ 2 = 0, and the critical value is given by (3.40).

Case (IV) does not occur.

Remarks:

1) Let J = F ∩ G = {x 1 = 0, ξ 2 = ξ 2 1 }, Q = {q = 0}. We note that q = x 2 + ξ 1 Poisson commutes with g = ξ 2

1x 1 + ξ 2 , so G = {g = 0} is the flow out of J through v q . 2) In relevant Cases (I)-(III), Φ crit = -1 3 ξ 3 1 on Q.

3.3

The phase function Φ in (x 1 , x 2 ) coordinates.

As in [Sj] we pass from (ξ 1 , x 2 ) to (x 1 , x 2 ) coordinates by Legendre transform, so Ψ is the critical value of ξ 1 → x 1 ξ 1 + Φ crit (ξ 1 , x 2 ), so within hypotheses of Proposition 3.4, and in general case (3.51) we have (3.52)

ξ 1 = ax 1 -x 2 + O (ax 1 -x 2 ) 2 )
which gives Ψ by substituting in Φ crit (ξ 1 , x 2 ) . . .

In case (3.40) we have the system ξ 2 1x 1 = 0, x 2 + ξ 1 = 0. or since the critical points belong to energy surface G = {g = 0} (3.53) x 1 = x 2 2 , ξ 2 = 0

The caustic is then given by the parabola x 1 = x 2 2 .

  an obstacle O : the question here is about the classification of glancing hypersurfaces (F, G) in T * M at some point z. Here G = {τ 2ξ 2 = 0} and F stands for the lift of the boundary of O in T * M . It turns out that F and G play symmetrical roles.

2 /

 2 Examples of non-Lagrangian intersections. Consider a positively homogeneous Hamiltonian H(x, p) of degree m with respect to p. The energy surface should then be specified as H(x, p) = E. We first classify points z = (x, p) ∈ G = {H = E} according to v H (z). Definition 2.1 [AnDoNaRo3]: Let H be positively homogeneous of degree m with respect to p. We call a point z = (x, p) such that -∂ x H(z) = 0 an ordinary point if -∂ x H(z), p = 0, and a special point otherwise. If -∂ x H(z) = 0 we call z a residual point. Examples 2.1:

F

  and G are said to be glancing at z iff the Hamilton foliation of F = {f = 0} and G = {g = 0} (locally near z) are simply tangent at z. Stated otherwise, we have f (z) = g(z) = {f, g}(z) = 0 (Poisson bracket), but the second Poisson brackets {f, {f, g}}(z), {g, {g, f }}(z) are non zero.

  which holds when (a ′ , b ′ , c ′ , d ′ ) vary in an open set of R 4 , then (3.15) has the trivial solution x 2 = ξ 1 = 0. First Eq. (3.15) then gives x 1 = 0 (which means z ∈ F ) and second Eq.(3.15)

Assume instead ∆ = 0. Then (3.16) reduces to the first Eq. which has the general solution (3.18) x 2 = λ(1

with λ ∈ R. Substituting into ξ 2 1x 1x 2 = 0 (z ∈ G) and x 1 = 0 (z ∈ F ) leads to the system

Summing up Eq. (3.19), we get rid of the linear term, and substituting (3.17) gives (provided 1 +

We discuss according to the cases (1

In the first case, (3.17) gives

x 1 = 0. Substituting into Eq. (3.19) eventually gives x 2 = 0, so z = 0 as claimed.

Assume and Eq. (3.19) to

Summing up again Eq. (3.20) gives the quadratic equation

, and the condition z ∈ F ∩ G is equivalent to z = 0, which proves our claim.

We are left with the case c ′ = 0, but b ′ = 1.

• Case (II).

• Case (IV).

♣

Let's summarize our construction so far:

Given a point of glancing intersection for the pair of hypersurfaces (F, G), where G takes the form {g = 0} with g(x, ξ) = ξ 2 1 -x 1 -ξ 2 , and F = {x 1 = 0} we can choose Λ transverse to F and such that (Λ, G) has a glancing intersection at z = 0. It is parametrized by phase with

We use (3.32)- (3.33) to express Φ in the original variables as

The map t → Φ(t, ξ 1 , x 2 ) has a degenerate critical point at t = ξ 1 , with

So the situation is similar with the diffraction by an obstacle for the wave equation [Sj], [Ta], [Ro] but Φ does not arise from a Morse family (with an auxiliary θ-parameter). We add now the cubic term

The corresponding critical value of Legendre transform is simply

which turns out to be independent of t, and equals, mod O(ξ 4 1 ), the critical value of (2.54) when the cubic term φ 1 is absent.

• Case (II). We consider only the linear case. We make Legendre transform with respect to ξ 2 variable.

The critical point is ξ 2 = -1 c (x 2 + ξ 1 ), and it turns out that the critical value of Legendre transform is -1 2c (x 2 + ξ 1 ) 2 which gives again the same result as in Case (I) (linear case).

which gives the same result as in Case (I) (linear case).

We add now the cubic term. A computation similar to Case (I) (with cubic term) gives Φ(x 2 , ξ 1 , t) = -

The critical point of t → Φ(t, ξ 1 , x 2 ) is given by (tξ 1 ) 2 + 1 2 (x 2 + ξ 1 ) 2 = 0


Gabriela Zanghelini

Violaine Athès

Martine Esteban-Decloux

Pierre Giampaoli

Stéphane Vitu

Isobaric vapor-liquid equilibrium of α-terpineol highly diluted in hydroalcoholic mixtures at 101.3 kPa: experimental measurements and thermodynamic modeling

Keywords: vapor-liquid equilibrium, infinite dilution activity coefficient, alpha-terpineol, hydroalcoholic mixture, NRTL, UNIQUAC

Terpenes are important varietal compounds responsible for the characteristic aromas of alcoholic beverages. Reliable vapor-liquid equilibrium (VLE) data for this class of aroma compounds in hydroalcoholic media is essential to understand their behavior during distillation and thus achieve a desired quality in the distilled products. In this work, experimental measurements for the VLE of α-terpineol highly diluted in ethanol-water mixtures were carried out in a recirculation ebulliometer operating at 101.3 kPa for boiling temperatures from (354.99 to 369.93) K. Equilibrium compositions were determined by gas chromatography for α-terpineol and by density measurements for ethanol. Results show that α-terpineol is expressively more volatile in the dilute region (x Et < 0.01), where it can be up to 30 times more abundant in the vapor phase and twice as volatile as ethanol. Inversely, when x Et > 0.15 α-terpineol can be 10 times richer in the liquid phase. The experimental data were correlated by semi-empirical models NRTL and UNIQUAC and compared with predictions by UNIFAC. The data regressed with the NRTL model showed the best agreement with the experimental data. The binary interaction parameters fitted by the model are suitable to be used in the design and simulation of distillation processes for the production of alcoholic beverages.

INTRODUCTION

Alcoholic beverages are essentially highly non-ideal hydroalcoholic mixtures containing hundreds of highly diluted aroma compounds, also known as congeners. Ethanol and water are the major compounds, accounting for approximately 96% of the total mass [START_REF] Valderrama | Advances on modeling and simulation of alcoholic distillation. Part 1: Thermodynamic modeling[END_REF], whereas congeners are present at trace amounts (with mass fractions between 10 -10 and 10 -3 ) and cover several chemical families including alcohols, carboxylic acids, aldehydes, norisoprenoids, esters and terpenes. Despite their low concentration, congeners are passible of volatilizing into the headspace where they can impart odor notes that vary in quality and intensity and build up the characteristic aroma of alcoholic beverages. The olfactory impact of an aroma compound depends on its distribution between the vapor and liquid phases, or absolute volatility, as well as physical, chemical and structural properties of each component in a mixture and the interactions between them [START_REF] Ammari | Flavor Retention and Release from Beverages: A Kinetic and Thermodynamic Perspective[END_REF][START_REF] Paravisini | Interactions between aroma compounds and food matrix[END_REF]. Understanding the intricate interactions between the different species in a system and their behavior during processing is essential to achieve a desired quality in the product. Amongst odorant compounds in wines and distilled beverages, terpenes are a large class of volatile compounds of various structural types, including monoterpenes (C 10 ) and sesquiterpenes (C 15 ). Often associated with floral notes, terpenes are important varietal compounds responsible for the characteristic aromas of wine, where they are present as volatile aglycones or non-volatile glycosides [START_REF] Marais | Terpenes in the Aroma of Grapes and Wines: A Review, South African[END_REF][START_REF] Gunata | The aroma of grapes I. Extraction and determination of free and glycosidically bound fractions of some grape aroma components[END_REF]. One of such aglycones is α-terpineol, a natural monoterpene present in grapes and wine that is considered a discriminant compound between white wines typically used for distillation [START_REF] Cantagrel | From vine to Cognac[END_REF]. It has been detected in cachaça [START_REF] Amorim | Sugar cane spirit (cachaça): Effects of mixed inoculum of yeasts on the sensory and chemical characteristics[END_REF], pisco [START_REF] Cacho | Characterization of the aromatic profile of the Italia variety of Peruvian pisco by gas chromatographyolfactometry and gas chromatography coupled with flame ionization and mass spectrometry detection systems[END_REF] and in French spirits cognac [START_REF] Awad | Evolution of Volatile Compounds during the Distillation of Cognac Spirit[END_REF], calvados and armagnac [START_REF] Ledauphin | Differences in the volatile compositions of French labeled brandies (Armagnac, Calvados, Cognac, and Mirabelle) using GC-MS and PLS-DA[END_REF], and is deemed to play a major role in the aroma of gin [START_REF] Vichi | Characterization of volatiles in different dry gins[END_REF], tequila [START_REF] González-Robles | The impact of maturation on concentrations of key odour active compounds which determine the aroma of tequila[END_REF] and grappa [START_REF] Giannetti | Flavour fingerprint for the differentiation of Grappa from other Italian distillates by GC-MS and chemometrics[END_REF].

In spirit distillation, congeners are extracted from the wine or fermented base and concentrated along with ethanol in the distillate fractions. Not all aroma compounds are desirable in the distillate, as some of them are associated with deleterious odors or a level of toxicity that might hamper the quality of the final product. Thereby, the challenge of distillation lies in achieving a balance between the concentrations of pleasant and undesirable odorant compounds in the distillate, which in turn requires a thorough knowledge of the behavior of the system and its components. Compound separation in 3 distillation processes is driven by heat-induced deviations of a given state from a state of vapor-liquid equilibrium (VLE). As such, the design and simulation of distillation processes for the production of alcoholic beverages entails accurate knowledge of VLE data for aroma compounds highly diluted in ethanol-water mixtures at the entire composition range.

Thermodynamic equilibrium implies an equality of temperature and pressure in the different phases and an equality of fugacities for each species in each phase. Thus, the equilibrium problem unfolds into the measurement and calculation of four variables: temperature, pressure and compositions of the vapor and liquid phases.

Because distillation is most often carried out at atmospheric pressure (~101.3 kPa), equilibrium is generally modeled following the gamma-phi approach, in which the vapor phase is described by equations of state and the non-ideality associated with the liquid phase is expressed by activity coefficient models, also known as molar excess Gibbs energy models [START_REF] Faúndez | Predictive models to describe VLE in ternary mixtures water + ethanol + congener for wine distillation[END_REF][15][START_REF] Abbott | Low-pressure phase equilibria: Measurement of VLE[END_REF][START_REF] Jaubert | Thermodynamic Models for Chemical Engineering: Design, Develop, Analyse and Optimize[END_REF]. Given the low concentrations of congeners in the wine and distillate, their influence in the thermodynamic behavior of the ethanol-water mixture is negligible and their interactions with the hydroalcoholic matrix are characterized by infinite-dilution activity coefficients ( ), also referred to as limiting activity coefficients [START_REF] Valderrama | Advances on modeling and simulation of alcoholic distillation. Part 1: Thermodynamic modeling[END_REF]. Reliable experimental values of allow to accurately describe VLE in the dilute regions of a mixture and can be correlated by thermodynamic models to predict VLE over the entire composition range [START_REF] Sandler | Infinite dilution activity coefficients in chemical, environmental and biochemical engineering[END_REF][START_REF] Eckert | Measurement and application of limiting activity coefficients[END_REF] and estimate binary interaction parameters that can be applied for process simulation.

Thermodynamic models that are suitable for low-pressure VLE of ethanol + water + congener mixtures include predictive models such as UNIFAC [START_REF] Fredenslund | Group-contribution estimation of activity coefficients in nonideal liquid mixtures[END_REF] and PSRK [START_REF] Holderbaum | PSRK: A Group Contribution Equation of State Based on UNIFAC[END_REF] and semiempirical models such as Wilson [START_REF] Wilson | Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing[END_REF], NRTL [START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF][START_REF] Renon | An empirical equation or an inspiring model for fluids mixtures properties?[END_REF] and UNIQUAC [START_REF] Abrams | Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems[END_REF]. While the first are entirely predictive models, the latter are correlating models, meaning that they require experimental VLE data to calculate certain empirical parameters and binary interaction parameters. However, the availability of such data in the literature is rather scarce for some systems, as is the case for multicomponent mixtures in the dilute region, especially for aroma compounds in aqueous media. Although predictive methods can be useful tools to represent VLE when empirical data is not available, they have been shown to provide limited accuracy for binary and multicomponent mixtures [START_REF] Raal | Vapour-Liquid Equilibrium at Low Pressure[END_REF]. As a result, experimental 4 measurements of infinite dilution activity coefficients still prove needed to accurately characterize the behavior of volatile compounds throughout distillation.

Despite the prominent role of terpenes in the typicality of wine distillates, the number of studies involving VLE data for terpenes in hydroalcoholic media is rather scarce in the literature. Low-pressure VLE data have been reported for some terpenes highly diluted in ethanol-water mixtures, including linalool (CAS no. 78-70-6) [START_REF] Athès | Vapour-liquid equilibria of aroma compounds in hydroalcoholic solutions: Measurements with a recirculation method and modelling with the NRTL and COSMO-SAC approaches[END_REF][START_REF] Deterre | Vapor-liquid equilibria measurements of bitter orange aroma compounds highly diluted in boiling hydro-alcoholic solutions at 101.3 kPa[END_REF] and α-pinene (CAS no. The experimental apparatus is a Gillespie-type recirculation ebulliometer operating at adiabatic and isobaric conditions [START_REF] Gillespie | Vapor-Liquid Equilibrium Still for Miscible Liquids[END_REF] and compositions are determined using a density meter and a gas chromatograph coupled to a flame-ionization detector (GC-FID). The experimental ternary data measured are then used to calculate infinite-dilution activity coefficients and binary interaction parameters for correlating models NRTL and UNIQUAC. The predictive capability of the UNIFAC model for the system is also evaluated.

EXPERIMENTAL MEASUREMENTS

Materials

The chemical compounds used in this work with their respective mass fraction purities, CAS numbers and some of their pure component properties are listed in Table 1. Deionized water (resistivity 18.2 MΩ-cm at 25 °C) was obtained from a Milli-Q purification system (Simplicity 185, Millipore Waters, France). For the VLE measurements, the different hydroalcoholic mixtures were prepared by precisely weighing ethanol, water and α-terpineol with a resolution of ±0.001 g (EG420-3NH, Kern). Ethanol mole fractions ranged from x Et = 5

(1.6•10 -2 to 3.0•10 -1 ) (or from w Et = (4.0•10 -2 to 5.2•10 -1 ) in mass fractions) and the mole concentration of α-terpineol in the initial solutions was fixed at 1.0•10 -4 , which corresponds to a mass composition ranging from wαT = (6.5•10 -4 to 8.4•10 -4 ).

Experimental VLE measurements

Vapor-liquid equilibrium measurements were carried out in a Gillespie-type recirculation still (Labodest VLE 602, iFisher GmbH, Germany) operating at isobaric and adiabatic conditions. The equipment has been used by our research team in the past years for VLE measurements of other aroma compounds [START_REF] Athès | Vapour-liquid equilibria of aroma compounds in hydroalcoholic solutions: Measurements with a recirculation method and modelling with the NRTL and COSMO-SAC approaches[END_REF][START_REF] Deterre | Vapor-liquid equilibria measurements of bitter orange aroma compounds highly diluted in boiling hydro-alcoholic solutions at 101.3 kPa[END_REF][START_REF] Puentes | Vapor-Liquid Equilibrium of Ethyl Lactate Highly Diluted in Ethanol-Water Mixtures at 101.3 kPa. Experimental Measurements and Thermodynamic Modeling Using Semiempirical Models[END_REF] and its principle has been thoroughly described in the literature [START_REF] Athès | Vapour-liquid equilibria of aroma compounds in hydroalcoholic solutions: Measurements with a recirculation method and modelling with the NRTL and COSMO-SAC approaches[END_REF][START_REF] Dias | Vapor-Liquid Equilibrium of Mixtures Containing the Following Higher Alcohols: 2-Propanol, 2-Methyl-1-propanol, and 3-Methyl-1-butanol[END_REF][START_REF] Voutsas | Measurements and thermodynamic modeling of the ethanol-water system with emphasis to the azeotropic region[END_REF][START_REF] Malanowski | Experimental methods for vapour-liquid equilibria[END_REF]. System pressure is assessed by a digital manometer with an accuracy of ±0.5 kPa (P-10 WIKA, iFisher GmbH, Germany) and the temperatures of liquid and vapor phases in the separation chamber are measured by a Pt-100 platinum 6 probe with an accuracy of ±0.05 K, periodically calibrated against a reference probe (Pt-100 RTD 712, Fluke, France). A schematic diagram of the apparatus is provided in Figure 1.

"Cottrell pump" or equilibrium chamber, (4) separation chamber, (5) Pt-100 temperature probe, sampling tubes for the [START_REF] Cantagrel | From vine to Cognac[END_REF] vapor and (7) liquid phases.

Eighteen independent equilibrium assays were conducted, during which pressure was maintained at 101.3 kPa under inert nitrogen atmosphere. For each run, 85 mL of mixture are inserted in the mixing chamber [START_REF] Valderrama | Advances on modeling and simulation of alcoholic distillation. Part 1: Thermodynamic modeling[END_REF] and brought to a boil by a glass quartz immersion heater [START_REF] Ammari | Flavor Retention and Release from Beverages: A Kinetic and Thermodynamic Perspective[END_REF]. Rising vapors transporting some liquid droplets flow through the Cottrell pump

(3), in which the vapor and liquid phases remain in intimate contact and maximum mass transfer is assured, then enter the separation chamber (4). The liquid and condensed vapor are separately recycled into the mixing chamber, where the mixture is under constant stirring. Equilibrium is considered to be reached when vapor temperature remains stable (±0.1 K) for 30 minutes. A small amount of sample (2 mL) is then collected through the sampling tubes (6,7) for each phase for analysis.

Determination of equilibrium compositions 7

The compositions of the liquid and condensed vapor phases were determined by density measurements, followed by gas chromatography. Density measurements were carried out in an Anton Paar oscillating tube density meter (DMA 4500 M, Anton Paar) set to 25 °C with a resolution of (±0.00001) g•cm -3 . A two-point calibration of the device was performed regularly using pure degassed water and dry air. The ethanol mass fractions were determined from the density measurements using the mathematical model developed by Wagenbreth

and Blanke [START_REF] Wagenbreth | Analytische Darstellung der Dichte von Äthanol-Wasser-Mischungen zur Berechnung der Interlationalen Alkoholtafeln der OIML[END_REF] and reported by the International Organization of Legal Metrology (OIML) [START_REF]International alcoholometric tables, OIML R22[END_REF]. ). Terpinen-4-ol (CAS 562-74-3; Sigma-Aldrich) was added as an internal standard at a fixed concentration of 4.2•10 -3 g•g -1 to minimize system variability between injections. All injections were performed in triplicate and results were multiplied by a dilution factor accounting for the initial adjustment. The detection and quantitation limits estimated for α-terpineol from the calibration curves were LOD = 3.4•10 -5 g•g -1 and LOQ = 1.0•10 -4 g•g -1 (or 6.3•10 -6 and 1.9•10 -5 in mole fraction, respectively).

Computation of uncertainties

The standard uncertainties related to experimental measurements were calculated according to the law of propagation of uncertainty. Sources of uncertainty considered include mass fraction purity, the mass measurements, the density determination, the temperature and pressure measurements and the repeatability of chromatographic analyses (based on three injections). The calculated values are presented with the experimental data in the results section and in the appendix.

THERMODYNAMIC MODELING

This section explores the basic equations of vapor-liquid equilibrium and introduces the thermodynamic models employed in this work. The choice of a suitable thermodynamic model is essential to accurately represent phase equilibria of multicomponent mixtures in the simulation of separation processes.

Vapor-liquid equilibrium theory

At low pressures (<1000 kPa), VLE can be computed by a modified Raoult's law, in which the vapor phase is considered an ideal gas mixture and all properties of the liquid phase can be presumed pressure independent [15].

The vapor pressure of a pure compound i at a temperature T/K can be calculated from the Riedel equation [START_REF] Vetere | The Riedel Equation[END_REF][START_REF] Riedel | Eine neue universelle Dampfdruckformel Untersuchungen über eine Erweiterung des Theorems der übereinstimmenden Zustände. Teil I[END_REF]: 9 in which A i , B i , C i , D i , E i are empirical coefficients specific to each compound. The Riedel coefficients for the studied system are listed in Table 2. a all coefficients were obtained from the DIPPR Database, available in the Simulis Thermodynamics package (ProSim) [START_REF] Prosim | Simulis Thermodynamics[END_REF].

T min /K and T max /K define the temperature interval in which the equation coefficients are valid.

The equilibrium behavior of an aroma compound in hydroalcoholic mixtures can be further described by its partition coefficient and its relative volatility. The partition coefficient (K AC ), also known as absolute volatility, describes the distribution of an aroma compound (AC)

between the liquid and vapor phases at equilibrium, as defined by:

The relative volatility of an aroma compound in relation to ethanol (α AC/Et ) can inform on the behavior of the compound during distillation. It is defined as the ratio of the absolute volatilities of the aroma compound and ethanol, which can be expressed as:

Calculation of activity coefficients from thermodynamic models

Three activity coefficient models are employed in this work, including semi-empirical models NRTL [START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF] and UNIQUAC [START_REF] Abrams | Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems[END_REF] and predictive model UNIFAC [START_REF] Fredenslund | Group-contribution estimation of activity coefficients in nonideal liquid mixtures[END_REF].

Semi-empirical models. Non-Random Two-Liquid (NRTL) and Universal Quasi-Chemical (UNIQUAC) are local composition models based on the concept introduced by Wilson [START_REF] Wilson | Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing[END_REF].

Both models are valid at low pressures (<1000 kPa) and well-adapted to ternary and 10 multicomponent mixtures involving highly diluted aroma compounds in hydroalcoholic mixtures [START_REF] Valderrama | Advances on modeling and simulation of alcoholic distillation. Part 1: Thermodynamic modeling[END_REF].

NRTL has two sets of adjustable parameters, including and , which are fitted to experimental VLE data, and non-randomness parameter , which is set to 0.30 as recommended for systems containing water and polar non-associated substances [START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF].

In the UNIQUAC model, the combinatorial part is constituted by pure-component properties including coordination number Z, lattice parameter , segment fraction Φi and area fractions θi and θ'i. The residual term comprises two sets of binary interaction parameters that are determined from experimental VLE data. In this work, Z is set to 10 and the segment and area fractions of the studied system are calculated from pure-component constants ( , and ) whose values are listed in Table 3. Predictive methods. UNIFAC is a group-contribution tool based on the UNIQUAC equation [START_REF] Abrams | Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems[END_REF] and on Wilson's solution-of-groups concept [START_REF] Wilson | Activity coefficients and molecular structure: Activity coefficients in changing environments'solutions of groups[END_REF]. In the modified (Dortmund)

version applied in this study, the residual term includes pure-component parameters and , which are calculated from the sum of group area parameters and group volume parameters , respectively, whereas the group interaction parameter is calculated from interaction parameters , and . Group decomposition for α-terpineol, ethanol and water, as well as , and interaction parameters between the corresponding functional groups are listed in Tables 4 and5. a Data from the DDBST Dortmund Data Bank [48].

12

RESULTS AND DISCUSSION

This section is divided into two parts: (1) validation of the experimental methodology using the ebulliometer and the VLE results for α-terpineol highly diluted in ethanol-water mixtures;

(2) correlation of the experimental data by semi-empirical thermodynamic models NRTL and UNIQUAC and predictions by the UNIFAC model.

Experimental VLE data

Validation of the apparatus and experimental protocol

The accuracy of the experimental device used for the VLE measurements and the suitability of the experimental protocol were validated by remeasuring the VLE of the binary system ethanol + water and applying thermodynamic consistency tests to the experimental data.

This validation procedure can be extended to VLE measurements of our ternary mixtures due to the low concentrations of α-terpineol in the hydroalcoholic mixtures (in the order of 10 -4 in mole fractions), which allow to infer that the thermodynamic behavior of the mixture is governed by the ethanol-water binary system [START_REF] Valderrama | Advances on modeling and simulation of alcoholic distillation. Part 1: Thermodynamic modeling[END_REF][START_REF] Alessi | Significance of dilute regions[END_REF]. This hypothesis was verified by plotting the densities at 25 °C measured for the liquid and vapor phases as a function of temperature for the binary system and comparing with those measured for the samples containing α-terpineol.

A total of 30 VLE points were measured for the binary ethanol-water. The experimental bubble and dew points are plotted in Figure 2 together with literature data [START_REF] Kojima | Determination of vapor-liquid equilibriums from boiling point curves[END_REF][START_REF] Hughes | The application of radioactive tracers to diffusional operations. Binary and ternary equilibrium data[END_REF] that are deemed consistent according to Jaubert et al. [START_REF] Jaubert | Benchmark Database Containing Binary-System-High-Quality-Certified Data for Cross-Comparing Thermodynamic Models and Assessing Their Accuracy[END_REF], and the corresponding VLE data are listed in Appendix B. Figure 2 also presents bubble and dew curves calculated using the NRTL model [START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF] with interaction parameters reported in previous works from our team [START_REF] Puentes | Vapor-Liquid Equilibrium of Ethyl Lactate Highly Diluted in Ethanol-Water Mixtures at 101.3 kPa. Experimental Measurements and Thermodynamic Modeling Using Semiempirical Models[END_REF][START_REF] Puentes | Review and Thermodynamic Modeling with NRTL Model of Vapor-Liquid Equilibria (VLE) of Aroma Compounds Highly Diluted in Ethanol-Water Mixtures at 101.3 kPa[END_REF].

These interaction parameters were adjusted to consistent, high-quality isothermal data measured by a static method [START_REF] Pemberton | Thermodynamic properties of aqueous non-electrolyte mixtures II. Vapour pressures and excess Gibbs energies for water + ethanol at 303[END_REF][START_REF] Kolbe | Thermodynamic properties of ethanol + water. I. Vapour-38 liquid equilibria measurements from 90 to 150°C by the static method[END_REF]. As a result, the NRTL model is used here in a

Vapor-liquid equilibrium for α-terpineol highly diluted in hydroalcoholic mixtures

The vapor-liquid equilibrium at 101.3 kPa of the system highly diluted α-terpineol + ethanol + water was measured as described in section 2.2. The mole compositions of the liquid and vapor phases as determined by gas chromatography and density measurements are listed in Table 6. Table 7 depicts the temperature dependency of pure component vapor pressure of αterpineol ( ), along with its partition coefficients (Kα T ), relative volatilities (αα T/Et ) and infinite-dilution activity coefficients ( ). The different properties were calculated from the experimental VLE data using eqs 1-3. The experimental values for Kα T and αα T/Et as a function of ethanol mole composition are plotted in Figure 3.

The high values of the infinite dilution activity coefficient of α-terpineol ( ) depicted in Table 7 for the dilute region (x Et < 0.04) are related to the high hydrophobicity of α-terpineol reported in Table 1 as log K ow (2.98). This phenomenon has been reported in the literature for other hydrophobic aroma compounds such as long chain esters (isopentyl acetate, ethyl octanoate and ethyl decanoate), for which is very large at high water content and decreases by several orders of magnitude when the proportion of ethanol is increased [START_REF] Athès | Vapour-liquid equilibria of aroma compounds in hydroalcoholic solutions: Measurements with a recirculation method and modelling with the NRTL and COSMO-SAC approaches[END_REF]. As observed in Figures 3a and3b, the absolute volatilities of α-terpineol in the hydroalcoholic mixtures are lower than 1 for ethanol mole fractions above 0.15, a region at which the aroma compound is more concentrated in the liquid phase. In the water-rich region (x Et < 0.15), Kα T increases exponentially, and the compound can be up to 30 times more abundant in the vapor phase. Inversely, in the ethanol-rich region (x Et > 0.50), αterpineol concentration can be over 10 times higher in the liquid phase (Kα T < 0.1). As per the relative volatility, Figures 3c and3d show that α-terpineol is more volatile than ethanol only for ethanol mole fractions below 0.04.

A similar behavior has been observed in other studies for the VLE of aroma compounds highly diluted in ethanol-water mixtures [START_REF] Athès | Vapour-liquid equilibria of aroma compounds in hydroalcoholic solutions: Measurements with a recirculation method and modelling with the NRTL and COSMO-SAC approaches[END_REF][START_REF] Deterre | Vapor-liquid equilibria measurements of bitter orange aroma compounds highly diluted in boiling hydro-alcoholic solutions at 101.3 kPa[END_REF][START_REF] Puentes | Vapor-Liquid Equilibrium of Ethyl Lactate Highly Diluted in Ethanol-Water Mixtures at 101.3 kPa. Experimental Measurements and Thermodynamic Modeling Using Semiempirical Models[END_REF][START_REF] Athès | Comparison of Experimental Methods for Measuring Infinite Dilution Volatilities of Aroma Compounds in Water/Ethanol Mixtures[END_REF][START_REF] Tsachaki | Dynamics of flavour release from ethanolic solutions[END_REF]] and is mainly associated with two 16 factors: (1) the decrease in equilibrium temperature with consequent decrease of pure component vapor pressures [START_REF] Ickes | Effect of ethanol on flavor perception of Rum[END_REF][START_REF] Conner | Release of distillate flavour compounds in Scotch malt whisky[END_REF] and (2) the increase in solubility of aroma compounds with increasing ethanol concentration due to a reduction in the free energy of the mixture [START_REF] Conner | Headspace concentrations of ethyl esters at different alcoholic strengths[END_REF]. The latter phenomenon is well correlated with the high log K ow value (i.e. high hydrophobicity) of α-terpineol reported in Table 1. Another factor potentially driving the accentuated decrease in volatility observed for α-terpineol until x Et < 0.25 is the change in molecular structure of the hydroalcoholic solution. In aqueous solutions containing small amounts of ethanol (x Et < 0.05), alcohol molecules are monodispersed in the matrix.

Increasing ethanol mole concentration up to about x Et = 0.25 results in progressive aggregation of ethanol molecules and the formation of pseudo-micelles, which are liable to entrap hydrophobic aroma molecules, thus hampering their release into the vapor phase.

Once a second critical value is reached with ethanol concentrations above x Et ≈ 0.25 the aggregates dissociate and water becomes monodispersed in the ethanolic solution instead [START_REF] Conner | Headspace concentrations of ethyl esters at different alcoholic strengths[END_REF][START_REF] Angelo | Self-Association Behaviour of Alcohols in Diluted Aqueous Solutions (*)[END_REF][START_REF] Ickes | Effects of Ethanol on Flavor Perception in Alcoholic Beverages[END_REF][START_REF] Nose | Hydrogen bonding of water-ethanol in alcoholic beverages[END_REF]. However, had α-terpineol been incorporated into ethanol pseudo-micelles, an increase in the volatility of the terpene would be expected once ethanol concentration in the mixtures surpassed the second critical point, which is not the case in this study.

Repeatability of VLE assays was lower in the more aqueous region, where the volatility of αterpineol is more pronounced and slight variations in temperature between experimental runs are likely to result in large differences in composition between the three independent runs. Temperature fluctuations are passible of occurring in dynamic ebulliometers operating at low pressures due to an overheating of the liquid in the boiling chamber [START_REF] Raal | Measurement of limiting activity coefficients: Nonanalytical tools[END_REF], a phenomenon that is more accentuated in the aqueous region.

Thermodynamic modeling and validation

The experimental VLE data for the system highly diluted α-terpineol + ethanol + water at 101.3 kPa were correlated using semi-empirical models NRTL and UNIQUAC. The results were compared to those obtained from the predictive method UNIFAC. The ideal gas law was used to model the vapor phase, whereas for the liquid phase activity coefficients were calculated by fitting the experimental data to the excess Gibbs energy models using the software Simulis Thermodynamics (v. 2.0.34, ProSim) [START_REF] Prosim | Simulis Thermodynamics[END_REF].

For the NRTL model, the non-randomness parameter was fixed at 0.30, as recommended in the literature for VLE of mixtures of polar liquids such as hydroalcoholic mixtures [START_REF] Renon | Local compositions in thermodynamic excess functions for liquid mixtures[END_REF]. For the sub-systems α-terpineol (1) + ethanol (2) and α-terpineol (1) + water

(3), the temperature-dependent parameters /J•mol -1 •K -1 and /J•mol -1 •K -1 are set to zero, since the influence of the highly diluted α-terpineol on the equilibrium temperature is deemed negligible [START_REF] Valderrama | Advances on modeling and simulation of alcoholic distillation. Part 1: Thermodynamic modeling[END_REF][START_REF] Alessi | Significance of dilute regions[END_REF] and the influence of temperature on measurements is incorporated in the variations in boiling temperatures as a function of ethanol composition. As per the binary sub-system ethanol (2) + water (3), the major compounds governing thermal equilibrium, binary interaction parameters were set to values from the literature [START_REF] Puentes | Review and Thermodynamic Modeling with NRTL Model of Vapor-Liquid Equilibria (VLE) of Aroma Compounds Highly Diluted in Ethanol-Water Mixtures at 101.3 kPa[END_REF] reported in Table 8, and parameters /J•mol -1 •K -1 and /J•mol -1 •K -1 are taken into consideration.

These ethanol-water interaction parameters have been consistently used by our team for all systems aroma compound + ethanol + water for the simulation of spirit distillation processes. They were thus not fitted to our experimental data.

Given these considerations, the correlation of the experimental data for the system αterpineol (1) + ethanol (2) + water (3) with NRTL and UNIQUAC models is narrowed down to a regression of four binary interaction parameters: /J•mol -1 and /J•mol -1 , for the binary α-terpineol (1) + ethanol (2); /J•mol -1 and /J•mol -1 , for the binary αterpineol (1) + water (3). The parameters were estimated through the Microsoft Excel Solver add-in by minimizing the objective function for the partition coefficient of α-terpineol, KαT:

in which is the experimental partition coefficient calculated from eq 2 and is

the partition coefficient calculated with the thermodynamic models. The calculation of was performed using the bubble temperature algorithm of the software Simulis Thermodynamics, which computes saturated liquid temperature (T) and vapor phase composition (y) at equilibrium from experimental pressure (P) and liquid phase composition (x).

18

The binary interaction parameters obtained for the two semi-empirical models are listed in Table 8. They were used to obtain values for , , and g E /RT for the entire composition range, which were then plotted with the experimental data. Figure 3 depicts the evolution of the partition coefficient (3a) and the relative volatility (3c) of α-terpineol as a function of ethanol mole composition in the liquid phase. A closeup view of the two figures is presented in Figures 3b and3d to increase visibility of the data in the region of lower volatility.

Table 8. Binary interaction parameters for the system highly diluted α-terpineol (1) + ethanol (2) + water [START_REF] Paravisini | Interactions between aroma compounds and food matrix[END_REF] from experimental data fitted by NRTL and UNIQUAC models to minimize the objective function (eq 4).

Model Binary (i-j) A good agreement was found between the NRTL model and the experimental data, with slight discrepancies for at ethanol mole fractions below 0.1 (Figure 3c). The UNIQUAC model underestimates both and in the region below 0.05 and above 0.25 ethanol mole fraction and thus fails to fit experimental data in the region nearing pure water. Since in spirit distillation the region of interest is specifically that of low ethanol concentration, UNIQUAC might not be well suited to explain the behavior of α-terpineol during distillation, in which case NRTL would be a more adequate model choice. For comparative purposes, VLE data for the system was additionally predicted by the modified UNIFAC (Dortmund) model, with the structural and interaction parameters listed in Tables 4 and5. As illustrated in Figure 3, values of absolute and relative volatility predicted by the UNIFAC model are lower than the experimental values for the entire ethanol composition range. Although the behavior of α-terpineol predicted by the model follows the same tendencies as that from the experimental data, the model was unable to accurately predict VLE data for the system studied. Since group interaction parameters used in the UNIFAC model derive from a reduction of binary experimental data over a large composition range, it is not surprising that the model is not adapted for highly diluted compounds as is the case in this work.

/J•mol -1 /J•mol -1 /J•mol -1 •K -1 /J•mol -1 •K -1 αij Source NRTL ( 1

20

The evolution of the infinite dilution activity coefficient of α-terpineol ( ) as a function of temperature and ethanol composition is depicted in Figure 4. The increase in values of with temperature or decreasing ethanol concentration indicate that the non-ideality of the liquid phase is more pronounced in more dilute regions, where hydrogen bonding between water molecules is stronger and partial pressure of α-terpineol is lower, as discussed in subsection 4.1. Such marked variations in activity coefficients at lower ethanol mole fractions make experimental VLE measurements quite challenging, increasing the chance of imprecisions [START_REF] Eckert | Measurement and application of limiting activity coefficients[END_REF]. Figure 4 highlights that NRTL was the best suited model amongst the three models tested to adequately represent the evolution of the activity coefficient with temperature and ethanol composition, followed by UNIQUAC. Once again, UNIFAC provides underestimated predictions over the entire composition range. The fitting quality of the thermodynamic models was evaluated by the root-mean-square deviation (RMSD) and the average absolute relative deviation (AAD%) according to eqs 5-6, in which N is the number of experimental points and E is equilibrium variable , or . The data calculated from NRTL, UNIQUAC and UNIFAC are reported in Table 9.

The residual values are higher than the tolerance stipulated in the literature [START_REF] Faúndez | Thermodynamic consistency test for binary gas + water equilibrium data at low and high pressures[END_REF], especially for UNIQUAC and UNIFAC. Nonetheless, given the difficulties to accurately measure and quantify highly diluted compounds, the RMSD and AAD% values for the NRTL model 21 remain acceptable. The relative deviations between the experimental data and the values calculated with the different models for are presented in Figure 5.

22

The results show that the semi-empirical models fitted to experimental data are better adapted than the group-contribution predictive method for the VLE of the system highly diluted α-terpineol + ethanol + water at atmospheric pressure. Similar results have been observed in other studies for systems consisting of aroma compounds highly diluted in hydroalcoholic mixtures [START_REF] Faúndez | Predictive models to describe VLE in ternary mixtures water + ethanol + congener for wine distillation[END_REF][START_REF] Faúndez | Phase equilibrium modeling in binary mixtures found in wine and must distillation[END_REF].

CONCLUSIONS

Vapor-liquid equilibrium of the system highly diluted α-terpineol + ethanol + water at 101.3 kPa has been measured from (354.99 to 369.93) K. Results show that α-terpineol is more volatile than ethanol in highly dilute regions with ethanol mole fractions below 0.04, its volatility decreasing when ethanol concentration in the liquid phase is increased. The experimental data was modeled using two semi-empirical models, NRTL and UNIQUAC, and compared to the data predicted by the UNIFAC model. The NRTL model provided the best correlation for the system, with average relative deviations of 12.9 and 11.8% for Kα T and , respectively. The binary interaction parameters obtained in this work will prove useful to represent the behavior of aroma compounds in distillation processes for alcoholic beverages using computer simulation models.

APPENDIX A. Verification of the density measurements and the relationship between density and ethanol concentration of hydroalcoholic solutions

The accuracy of the density measurements was verified by comparing experimental measurements for ethanol-water mixtures of known composition with data from the literature (Figure A-1). The mixtures were prepared by weight using a precision scale with a resolution of ± 0.0001 g. To limit evaporation during mixture preparation, both compounds , González et al. [START_REF] González | Density, dynamic viscosity, and derived properties of binary mixtures of methanol or ethanol with water, ethyl acetate, and methyl acetate at T =[END_REF]. The solid green line represents the evolution of density with ethanol composition from the OIML alcoholometric tables [START_REF]International alcoholometric tables, OIML R22[END_REF]. [START_REF] Kamihama | Isobaric Vapor-Liquid Equilibria for Ethanol + Water + Ethylene Glycol and Its Constituent Three Binary Systems[END_REF]; , experimental γ w ; , γ w from Kamihama et al. [START_REF] Kamihama | Isobaric Vapor-Liquid Equilibria for Ethanol + Water + Ethylene Glycol and Its Constituent Three Binary Systems[END_REF]; , g E /RT from this work; ∆, g E /RT from Kamihama et al. [START_REF] Kamihama | Isobaric Vapor-Liquid Equilibria for Ethanol + Water + Ethylene Glycol and Its Constituent Three Binary Systems[END_REF]; +, g E /RT from Arce et al. [START_REF] Arce | VLE for water + ethanol + 1-octanol mixtures. Experimental measurements and correlations[END_REF]. The solid blue lines represent the curves calculated using the NRTL model with the parameters from Puentes et al. ( 2018) [START_REF] Puentes | Review and Thermodynamic Modeling with NRTL Model of Vapor-Liquid Equilibria (VLE) of Aroma Compounds Highly Diluted in Ethanol-Water Mixtures at 101.3 kPa[END_REF] listed in Table 8.

The expanded combined uncertainty associated with the determination of ethanol composition from density measurements was calculated according to the law of propagation of uncertainty and is estimated to be ±0.015 (k=2). Sources of uncertainty used for the calculation and their respective estimates and probability distributions are listed in Table B Redlich-Kister area test [START_REF] Redlich | Algebraic Representation of Thermodynamic Properties and the Classification of Solutions[END_REF] The Redlich-Kister test is based on the Gibbs-Duhem equation. For experimental data measured under isobaric conditions, the following equality can be written:

where h E is the excess enthalpy of the mixture at the bubble point. The right-hand side of eq C1 was estimated using an empirical correlation proposed by Larkin [START_REF] Larkin | Thermodynamic properties of aqueous non-electrolyte mixtures I. Excess enthalpy for water + ethanol at 298[END_REF] and based on his experimental measurements. The estimated numerical value was found to be very small and thus can be neglected.

When ln( Et / w ) is plotted against x Et , the areas A+ and A-between the resulting curve and the x-axis must be the same (Figure C-1). Theoretically, perfect data would result in A+ = A-. In practice, however, experimental errors are inevitable, whether from systematic errors from the instruments, from the presence of impurities in the chemical components used or from manipulation errors. In light of this, experimental data is evaluated according to eq C2 and can be declared consistent if D < 2 [START_REF] Wisniak | A fresh look at the thermodynamic consistency of vapour-liquid equilibria data[END_REF].

In our case, experimental data successfully passed the area test with D ≈ 1.23. •, experimental data from this work.

Fredenslund's test (point test) [START_REF] Fredenslund | Vapor-liquid Equilibria Using Unifac[END_REF] The Van Ness test [START_REF] Van Ness | Vapor-Liquid Equilibrium: Part I. An Appraisal of Data Reduction Methods[END_REF] described by Fredenslund et al. [START_REF] Fredenslund | Vapor-liquid Equilibria Using Unifac[END_REF] and Gmehling et al. [START_REF] Gmehling | Chemical Thermodynamics for Process Simulation[END_REF] has also been used to test the consistency of the data. The test is described in depth in several works in the literature, being often referred to as the point-to-point test. Briefly, according to the phase rule, for a binary system at VLE variance equals 2. The principle of the test is thus to use only two variables of the reported data (e.g. T and x Et exp ) to calculate the third one (y Et ).

From the fitted parameters, it is possible to calculate the activity coefficients and finally the vapor phase compositions (y i calc ), given that the experimental values of y i were not used for fitting the parameters. Experimental values ln γ Et , ln γ w , and g E /(RTx Et x w ) plotted against x Et are shown in Figure C-2, along with pertinent curves calculated using a Legendre polynomial with five coefficients fitted to the experimental data.

To do so, (g E /RT) exp data are used to fit the parameters of a Redlich-Kister equation or a series of Legendre polynomials. A set of n data points is considered thermodynamically consistent by the point-to-point test [START_REF] Fredenslund | Vapor-liquid Equilibria Using Unifac[END_REF][START_REF] Gmehling | Chemical Thermodynamics for Process Simulation[END_REF]

A value of 0.0061 is obtained for our data, suggesting that the data are consistent. However, when the residuals in mole fraction are considered, larger values are obtained for the experimental points at small ethanol concentrations. Moreover, in the same dilute region of the system, there is a slight lack of randomness in the distribution of the residuals of vapor phase composition. Such observations are common for this kind of phase diagram [START_REF] Kolbe | Thermodynamic properties of ethanol + water II. Potentials and limits of GE models[END_REF].

Indeed, in this dilute region of the phase diagram (small ethanol mole fraction) the slope of the bubble curve is very high, so that a small error in experimental composition (or in the effective pressure of the experimental device) is likely to entail a large error in temperature.

The L-W test of Wisniak[78]

The thermodynamic consistency of our data was additionally evaluated using the L-W test developed by Wisniak[78], which is now routinely employed by many authors [START_REF] Zhang | Experimental isobaric vapour-liquid equilibrium data for binary systems of {sec-butyl acetate (SBAC) + acetamide} and {sec-butyl alcohol (SBA) + acetamide} and the ternary system of (SBAC + SBA + acetamide) at 101.3 kPa[END_REF][START_REF] Gomis | Phase equilibria of the water + 1-butanol + 2-pentanol ternary system at 101.3 kPa[END_REF][START_REF] Rubio-Pérez | Isobaric vapor-liquid equilibrium, density and speed of sound of binary mixtures 2,2,4trimethylpentane + 1-butanol or dibutyl ether (DBE) at 101.3 kPa[END_REF]. The test can be described by the following expressions: 31 in which, for a pure component i, /K is the boiling temperature, /J•K -1 is the entropy of vaporization, with , and /J is the vaporization enthalpy. The values of and used in this work originate from Poling et al. [START_REF] Poling | The Properties of Gases and Liquids[END_REF].

An experimental point passes the test if: For binary systems, the test can also be employed as an area test with:

Experimental data are declared consistent if D < 3 [START_REF] Wisniak | A fresh look at the thermodynamic consistency of vapour-liquid equilibria data[END_REF].

Our experimental data resulted in D = 1.10, with L = 7.81 and W = 7.99.

Results obtained using the different consistency tests described above evidence that the VLE data measured in this work are thermodynamically consistent.

Figure 1 .

 1 Figure 1. Schematic diagram of the Labodest VLE 602 (iFischer) system: (1) mixing chamber, (2) boiler, (3)

  All samples were adjusted to an ethanol mass fraction of wEt = 0.6 before analysis to overcome matrix effects and minimize measurement variability between the different samples. The mass compositions of α-terpineol were determined from calibration curves of the pure component in an ethanol-water mixture at the same ethanol mass fraction of w Et = 0.6. Seven calibration points were established with α-terpineol mass fractions ranging from 8 wαT = (2.2•10 -4 to 4.1•10 -2

  predictive way. As observed in the figure, the isobaric diagram obtained from our experimental points is in good agreement with the literature data and our experimental data are well described by the NRTL model with the parameters from Puentes et al. (2018)[54]. 13 Indeed, the absolute average deviation for vapor composition (|y Et |) was 0.007 in mole fraction, with a maximum value of 0.021 in the water-rich region of the phase diagram.

Figure 2 .

 2 Figure 2. Isobaric phase diagram for the binary system ethanol (Et) + water (w) at 101.3 kPa.•, experimental bubble and dew points; , Hughes and Maloney[START_REF] Hughes | The application of radioactive tracers to diffusional operations. Binary and ternary equilibrium data[END_REF]; ×, Kojima et al.[START_REF] Kojima | Determination of vapor-liquid equilibriums from boiling point curves[END_REF]; +, Kamihama et al.[START_REF] Kamihama | Isobaric Vapor-Liquid Equilibria for Ethanol + Water + Ethylene Glycol and Its Constituent Three Binary Systems[END_REF]; ∆, Arce et al.[START_REF] Arce | VLE for water + ethanol + 1-octanol mixtures. Experimental measurements and correlations[END_REF]. The solid blue lines represent the bubble and dew curves calculated using the NRTL model with the parameters from Puentes et al. (2018)[START_REF] Puentes | Review and Thermodynamic Modeling with NRTL Model of Vapor-Liquid Equilibria (VLE) of Aroma Compounds Highly Diluted in Ethanol-Water Mixtures at 101.3 kPa[END_REF] listed in Table8.

a

  Standard uncertainties are u(T) = 0.2 K, u(P) = 0.5 kPa, u(ρL) = u(ρCV) = 0.0002 g•cm -3 , u(Tρ_meas) = 0.03 K, u(x Et ) = u(y Et ) = 0.007. b T/K, equilibrium temperature; ρ/g•cm -3 , density of the liquid (L) and condensed vapor (CV) phases at 298.15 K;Tρ_meas/K, temperature of the density measurement; x and y, mole compositions of the liquid and vapor phase, respectively.

Figure 3 .

 3 Figure 3. Evolution of (a, b) the partition coefficient (Kα T ) and (c,d) the relative volatility of α-terpineol (αα T/Et ) with ethanol mole fraction in the liquid phase (x Et ): ∆, experimental data from this work; ---(solid blue line), NRTL model; (dash-dot red line), UNIQUAC model; (dashed green line), UNIFAC prediction.

Figure 4 .

 4 Figure 4. Evolution of log of infinite dilution activity coefficient of α-terpineol ( ) with (a) temperature and with (b) ethanol mole fraction (x Et ). o, experimental data from this work; ---(solid blue line), NRTL model; (dash-dot red line), UNIQUAC model; (dashed green line), UNIFAC prediction.

a

  Kα T , partition coefficient of α-terpineol; αα T/Et , relative volatility of α-terpineol in relation to ethanol; , infinite dilution activity coefficient of α-terpineol; RMSD, root-mean-square deviation; AAD%, average absolute relative deviation.

Figure 5 .

 5 Figure 5. Relative deviations of experimental partition coefficients ( ) from values calculated using the different models ( ) for alpha-terpineol highly diluted in ethanol + water mixtures. ,NRTL; ,UNIQUAC; ,UNIFAC.

  were kept at low temperatures and the least volatile compound was weighed first. The experimental composition and density measurements at 298.15 K and atmospheric pressure for nineteen mixtures are listed in Table A-1. Furthermore, Figure A-2 depicts a parity plot between the ethanol mole fractions from the weighed values and those calculated from density using the mathematical model from Wagenbreth and Blanke[43], as described in subsection 2.3. A good correspondence is observed between the two values, with average absolute and relative deviations of 0.0006 and 0.5%, respectively.

Figure A- 1 .

 1 Figure A-1. Density measurements for ethanol-water mixtures as a function of ethanol mole fraction (x Et ) at T = 298.15 K and atmospheric pressure. , this work; ∆, Arce et al. [70]; +, Hoga et al. [71]; ×, Zarei et al. [72];

a

  Standard uncertainties are u(x Et exp ) = 0.0001, u(ρ) = 0.0002 g•cm -3 and u(Tρ_meas) = 0.03 K.

Figure A- 2 .

 2 Figure A-2. Parity plot for ethanol mole fraction of ethanol-water mixtures determined experimentally (x Et exp ) and calculated from density (x Et calc ) at T = 298.15 K and atmospheric pressure.

  2 K, u(P) = 0.5 kPa, u(ρL) = u(ρCV) = 0.0002 g•cm -3 , u(Tρ_meas) = 0.03 K, and u(x Et ) = u(y Et ) = 0.007. b T/K, equilibrium temperature; ρ/g•cm -3 , density of the liquid (L) and condensed vapor (CV) phases at 298.15 K; Tρ_meas/K, temperature of the density measurement; x Et , liquid phase ethanol mole fraction; y Et , vapor phase ethanol mole fraction; γ i , activity coefficient of component i; g E /RT, Gibbs dimensionless function.

Figure B- 1

 1 Figure B-1 presents the activity coefficients and g E /RT obtained from the experimental data as a function of the mixture composition. Activity coefficients and g E /RT data for the same system from Kamihama et al.[START_REF] Kamihama | Isobaric Vapor-Liquid Equilibria for Ethanol + Water + Ethylene Glycol and Its Constituent Three Binary Systems[END_REF] and Arce et al.[START_REF] Arce | VLE for water + ethanol + 1-octanol mixtures. Experimental measurements and correlations[END_REF] have been added for comparison.

Figure B- 1 .

 1 Figure B-1. Plot of the experimental activity coefficients and excess Gibbs energy as a function of ethanol mole fraction for the binary system ethanol (Et) + water (w) at 101.3 kPa. , experimental γ Et ; , γ Et from Kamihama et al.[33]; , experimental γ w ; , γ w from Kamihama et al.[33]; , g E /RT from this work; ∆, g E /RT

Figure C- 1 .

 1 Figure C-1. Redlich-Kister consistency test for the binary system ethanol (Et) + water (w) at 101.3 kPa.

Figure C- 2 .

 2 Figure C-2. Activity coefficient-composition relationships for the consistency test of Fredenslund[77]. +, experimental ln γ Et ; ○, experimental ln γ w ; •, experimental g E /(RTx Et x w ). Solid and dashed lines represent the calculated curves by using Legendre polynomials.

Figure C- 3 .

 3 Figure C-3. Application of the Wisniak test to the experimental points.

Figure C- 3

 3 Figure C-3 indicates that our experimental data pass the Wisniak test, as confirmed by the resulting value of .

Table 1 .

 1 Riedel equation coefficients a for calculating the vapor pressures of each component in the mixture at a given temperature.

	Compound	A i	B i	C i	D i	E i	T min /K	T max /K
	α-terpineol	79.319	-9967.8	-7.6780	3.87•10 -18	6	309.65	675.00
	ethanol	73.304	-7122.3	-7.1424	2.89•10 -6	2	159.05	514.00
	water	73.649	-7258.2	-7.3037	4.17•10 -6	2	273.16	647.10

Table 3 .

 3 Values of size and surface parameters a for α-terpineol, ethanol and water

	Component	r i	q i	q' i
	α-terpineol	6.7897	5.2000
	ethanol	2.1055	1.9720	0.9600
	water	0.9200	1.4000	1.0000

a Data made available in the Simulis Thermodynamics package (ProSim)

[START_REF] Prosim | Simulis Thermodynamics[END_REF]

Table 4 .

 4 Modified UNIFAC (Dortmund) decomposition for α-terpineol, ethanol and water and volume and surface area parameters a for the different functional groups.

	Group	Subgroup	α-terpineol	ethanol	water	R	Q
	CH 2	CH 3	3	1		0.6325	1.0608
		CH 2		1		0.6325	0.7081
		C	1			0.6325	0.0000
	C=C	CH=C	1			1.2832	0.8962
	OH	OH(p)		1		1.2302	0.8927
		OH(t)	1			0.6895	0.8345
	H 2 O	H 2 O			1	1.7334	2.4561
	CY-CH 2	c-CH 2	3			0.7136	0.8635
		c-CH	1			0.3479	0.1071

a Data from the DDBST Dortmund Data Bank

[48]

, made available in the Simulis Thermodynamics package (ProSim)

[START_REF] Prosim | Simulis Thermodynamics[END_REF]

.

Table 5 .

 5 Modified UNIFAC (Dortmund) interaction parameters a between the subgroups in this study.

	m	n

Table 6 .

 6 Experimental VLE data for the system highly diluted α-terpineol (αT) + ethanol (Et) + water (w)

	at P = 101.3 kPa a,b
	T/K	ρ L /

g•cm -3 ρ CV /g•cm -3 xα T u(xα T ) x Et yα T u(yα T ) yEt

	369.93	0.9922	0.9565	4.10•10 -5	0.07•10 -5 0.010 83.95•10 -5	1.25•10 -5	0.125
	369.78	0.9918	0.9575	4.56•10 -5	0.08•10 -5 0.011 103.20•10 -5	1.55•10 -5	0.121
	369.60	0.9915	0.9543	4.78•10 -5	0.08•10 -5 0.012 112.38•10 -5	1.66•10 -5	0.132
	368.30	0.9894	0.9397	6.74•10 -5	0.12•10 -5 0.017 104.12•10 -5	1.45•10 -5	0.181
	368.31	0.9889	0.9403	6.28•10 -5	0.11•10 -5 0.018 92.50•10 -5	1.29•10 -5	0.179
	368.02	0.9885	0.9371	6.74•10 -5	0.12•10 -5 0.019 94.20•10 -5	1.30•10 -5	0.189
	364.35	0.9798	0.8950	7.61•10 -5	0.13•10 -5 0.044 58.86•10 -5	0.69•10 -5	0.339
	364.11	0.9799	0.8956	8.57•10 -5	0.14•10 -5 0.043 67.97•10 -5	0.80•10 -5	0.336
	364.08	0.9801	0.8955	8.96•10 -5	0.15•10 -5 0.042 68.97•10 -5	0.81•10 -5	0.337
	360.48	0.9673	0.8698	9.95•10 -5	0.16•10 -5 0.086 27.34•10 -5	0.29•10 -5	0.447
	360.46	0.9701	0.8699	9.54•10 -5	0.15•10 -5 0.076 24.58•10 -5	0.26•10 -5	0.446
	359.94	0.9690	0.8675	9.80•10 -5	0.16•10 -5 0.080 22.40•10 -5	0.23•10 -5	0.458
	356.63	0.9490	0.8528	10.72•10 -5 0.16•10 -5 0.149	8.02•10 -5	0.08•10 -5	0.532
	356.50	0.9423	0.8497	10.72•10 -5 0.15•10 -5 0.171	5.82•10 -5	0.06•10 -5	0.549
	356.60	0.9423	0.8511	10.48•10 -5 0.15•10 -5 0.171	5.95•10 -5	0.06•10 -5	0.541
	355.05	0.9097	0.8431	12.83•10 -5 0.16•10 -5 0.281	4.31•10 -5	0.04•10 -5	0.586
	355.01	0.9073	0.8413	12.43•10 -5 0.15•10 -5 0.290	4.20•10 -5	0.04•10 -5	0.596
	354.99	0.9092	0.8426	12.86•10 -5 0.16•10 -5 0.

Table 7 .

 7 Experimental VLE variables describing the behavior of α-terpineol (αT) highly diluted in hydroalcoholic mixtures at P = 101.3 kPa a,b .

	T/K	x Et	/kPa	Kα T	KEt	αα T/Et
	369.93	0.010	1.08	20.46	12.03	1.70	1925.6
	369.78	0.011	1.07	22.65	10.56	2.14	2148.9
	369.60	0.012	1.06	23.51	10.95	2.15	2250.9
	368.30	0.017	0.99	15.44	10.55	1.46	1582.8
	368.31	0.018	0.99	14.72	9.81	1.50	1508.3
	368.02	0.019	0.97	13.98	9.76	1.43	1454.5
	364.35	0.044	0.80	7.73	7.77	1.00	979.4
	364.11	0.043	0.79	7.93	7.76	1.02	1017.0
	364.08	0.042	0.79	7.70	7.92	0.97	989.3
	360.48	0.086	0.65	2.75	5.22	0.53	430.2
	360.46	0.076	0.65	2.58	5.87	0.44	403.8
	359.94	0.080	0.63	2.29	5.74	0.40	368.8
	356.63	0.149	0.52	0.75	3.57	0.21	145.4
	356.50	0.171	0.52	0.54	3.20	0.17	106.3
	356.60	0.171	0.52	0.57	3.16	0.18	110.5
	355.05	0.281	0.48	0.34	2.08	0.16	71.6
	355.01	0.290	0.47	0.34	2.05	0.16	72.1
	354.99	0.283	0.47	0.32	2.08	0.16	68.9

a Standard uncertainties are u(T) = 0.2 K, u(P) = 0.5 kPa, u(x Et ) = 0.007. b T/K, equilibrium temperature; x Et , ethanol mole fractions in the liquid phase;

, vapor pressure calculated with eq 1; Kα T and KEt, partition coefficient of α-terpineol and ethanol, respectively; αα T/Et , relative volatility of α-terpineol in relation to ethanol; , infinite dilution activity coefficient of α-terpineol.

Table 9 .

 9 Fitting quality statistics a for NRTL, UNIQUAC and UNIFAC models for the system highly diluted α-

	terpineol + ethanol + water.
			NRTL	UNIQUAC	UNIFAC
	Kα T	RMSD	1.1	2.9	4.6
		AAD%	12.9%	23.5%	35.3%
	αα T/Et	RMSD	0.2	0.2	0.4
		AAD%	15.1%	23.3%	31.8%
		RMSD	116.0	286.3	463.0
		AAD%	11.8%	22.6%	36.2%

Table A - 1 .

 A1 Density measurements at 298.15 K and atmospheric pressure (P = 0.1 MPa) and experimental (x Et exp ) ethanol mole fractions for ethanol-water mixtures of known compositions a .

	x Et exp	ρ/g.cm -3
	0.0000	0.9971
	0.0107	0.9920
	0.0200	0.9880
	0.0409	0.9805
	0.0502	0.9776
	0.0601	0.9747
	0.0801	0.9687
	0.1002	0.9630
	0.1501	0.9488
	0.2500	0.9187
	0.3507	0.8917
	0.4494	0.8693
	0.5504	0.8496
	0.6500	0.8325
	0.7491	0.8179
	0.8502	0.8042
	0.9252	0.7947
	0.9744	0.7884
	1.0000	0.7854

Table B - 1 .

 B1 Experimental isobaric VLE data at 101.3 kPa for the binary system ethanol (Et) + water (w) a .

	T/K	ρ L /g•

cm -3 ρ CV /g•cm -3 x Et y Et

						 Et	 w	g E /RT
	373.15	0.9971		0.0000	0.0000		1.000	0.000
	369.80	0.9912	0.9567	0.0128	0.1231	4.867	1.003	0.023
	368.66	0.9881	0.9435	0.0202	0.1672	4.366	1.000	0.030
	367.54	0.9859	0.9329	0.0261	0.2019	4.236	1.005	0.043
	366.07	0.9824	0.9183	0.0358	0.2510	4.056	1.006	0.056
	364.79	0.9803	0.9063	0.0420	0.2938	4.237	1.001	0.062
	362.00	0.9732	0.8844	0.0651	0.3805	3.912	1.001	0.089
	359.27	0.9633	0.8664	0.0999	0.4619	3.425	1.003	0.126
	357.06	0.9474	0.8574	0.1542	0.5071	2.649	1.066	0.204
	356.39	0.9363	0.8519	0.1909	0.5365	2.322	1.076	0.220
	355.78	0.9283	0.8483	0.2173	0.5566	2.165	1.090	0.236
	355.12	0.9140	0.8445	0.2660	0.5778	1.883	1.136	0.262
	354.81	0.9066	0.8435	0.2925	0.5835	1.751	1.177	0.279
	354.62	0.9014	0.8418	0.3120	0.5937	1.682	1.190	0.282
	354.16	0.8903	0.8388	0.3557	0.6112	1.546	1.239	0.293
	353.83	0.8807	0.8366	0.3963	0.6246	1.437	1.294	0.299
	353.40	0.8697	0.8338	0.4465	0.6422	1.333	1.369	0.302
	353.20	0.8631	0.8326	0.4783	0.6497	1.269	1.433	0.302
	353.03	0.8594	0.8307	0.4968	0.6616	1.252	1.445	0.297
	352.74	0.8516	0.8286	0.5381	0.6757	1.194	1.527	0.291
	352.65	0.8496	0.8274	0.5489	0.6834	1.188	1.532	0.287
	352.39	0.8414	0.8247	0.5960	0.7010	1.134	1.633	0.273
	352.18	0.8361	0.8221	0.6282	0.7186	1.112	1.684	0.260
	351.97	0.8290	0.8191	0.6730	0.7395	1.077	1.788	0.240
	351.73	0.8245	0.8163	0.7026	0.7591	1.069	1.836	0.227
	351.62	0.8158	0.8115	0.7627	0.7937	1.034	1.978	0.187
	351.42	0.8051	0.8038	0.8407	0.8504	1.013	2.156	0.133
	351.37	0.7977	0.7976	0.8977	0.8984	1.004	2.285	0.088
	351.37	0.7939	0.7941	0.9281	0.9263	1.001	2.359	0.063
	351.38	0.7912	0.7916	0.9494	0.9458	0.999	2.464	0.045
	351.41	0.7889	0.7893	0.9682	0.9650	0.998	2.526	0.028
	351.45	0.7854		1.0000	1.0000	1.000		0.000

a Standard uncertainties are u(T) = 0.

- 2 .

 2

Table B - 2 .

 B2 Standard uncertainties used for estimating the combined uncertainty related to the determination of ethanol composition from density measurements.

	Source of uncertainty	Estimate	Distribution
	Density (g•cm -3 )	0.0002	rectangular
	Repeatability of density	0.00006	gaussian
	Temperature (K)	0.03	rectangular
	Density equation	0.0003	gaussian
	Ethanol purity	0.001	rectangular
	APPENDIX C.

Thermodynamic consistency of the measured VLE data for the binary system ethanol (Et) + water (w)

  In order to validate our apparatus and the experimental method employed for the measurements, consistency tests were performed on the experimental binary VLE data using 28 three different tests: the Redlich-Kister area test[START_REF] Redlich | Algebraic Representation of Thermodynamic Properties and the Classification of Solutions[END_REF], as described by Wisniak et al.[START_REF] Wisniak | A fresh look at the thermodynamic consistency of vapour-liquid equilibria data[END_REF], the Van Ness[START_REF] Van Ness | Vapor-Liquid Equilibrium: Part I. An Appraisal of Data Reduction Methods[END_REF] point test modified by Fredenslund[START_REF] Fredenslund | Vapor-liquid Equilibria Using Unifac[END_REF] and the Wisniak test (also called the L-W test)[78] combining a point test and an area test.

Acknowledgements

The authors are thankful to Nicolas Descharles for the valuable assistance with the analytical measurements.

Funding

This work was supported by the foundation Jean Poupelain.

APPENDIX B. Phase behavior study of the binary system ethanol + water

Table B-1 reports experimental VLE data of T, x Et and y Et for the binary system ethanol + water at P = 101.3 kPa, in addition to the calculated activity coefficients and the excess Gibbs function in its dimensionless form. The non-ideality of the vapor phase was neglected, and activity coefficients and excess Gibbs energy (g E ) were calculated for the liquid phase.

The values reported for temperature and composition comprise all significant digits measured, even if those are greater than the uncertainties associated with the measurements. This choice was made considering the azeotropic region of the binary system, in which the bubble and dew curves are nearly flattened-out and the temperature range is extremely narrow. Despite the experimental uncertainty, the reported values around the azeotrope are logical in that they follow the expected trend, the ethanol mole fraction in the liquid phase being slightly superior to that in the vapor phase for the three points measured between the azeotrope and pure ethanol (at x Et > 0.9).

The azeotropic point of the system at 101.3 kPa can be estimated using our experimental data and a simple graphical method. The azeotropic point was found to be at x Et az ≈ 0.905, in mole fraction, and T az ≈ 351.37 K (78.22 °C). These values are in good agreement with previously reported azeotropic data, which range from x Et az = (0.893 to 0.905) and T az = (351.25 to 351.45) K [START_REF] Lai | Isobaric (vapor + liquid) equilibria for the ternary system of (ethanol + water + 1,3-propanediol) and three constituent binary systems at P = 101[END_REF][START_REF] Pemberton | Thermodynamic properties of aqueous non-electrolyte mixtures II. Vapour pressures and excess Gibbs energies for water + ethanol at 303[END_REF].

AUTHOR INFORMATION

Corresponding author

*Gabriela Zanghelini. E-mail: gabriela.zanghelini@agroparistech.fr ORCID Gabriela Zanghelini: 0000-0001-6993-3878 Violaine Athès: 0000-0002-2194-8517 Martine Esteban-Decloux: 0000-0002-8084-2528 Pierre Giampaoli: 0000-0002-6756-0403 Stéphane Vitu: 0000-0002-5442-1279

Notes

The authors declare no competing financial interest.