Sepideh Mirrahimi
email: sepideh.mirrahimi@math.univ-toulouse.fr

Benoît Perthame
email: benoit.perthame@upmc.fr

Asymptotic analysis of a selection model with space

Keywords: Asymptotic concentration, Adaptive evolution, Tumor growth, Resistance to therapy, Mathematics Subject Classification 35B25, 45M05, 92C50, 92D15

Selection of a phenotypical trait can be described in mathematical terms by 'stage structured' equations which are usually written under the form of integral equations so as to express competition for resource between individuals whatever is their trait. The solutions exhibit a concentration effect (selection of the fittest); when a small parameter is introduced they converge to a Dirac mass.

An additional space variable can be considered in order to take into account local environmental conditions. Here we assume this environment is a single nutrient which diffuses in the domain. In this framework, we prove that the solution converges to a Dirac mass in the physiological trait which depends on time and on the location in space with Lipschitz continuity. The main difficulties come from the lack of compactness in time and trait variables. Strong convergence can be recovered from uniqueness in the limiting constrained equation after Hopf-Cole change of unknown.

Our analysis is motivated by a model of tumor growth introduced in [15] in order to explain emergence of resistance to therapy.

Setting the problem

In this paper, we are interested in the study of the evolutionary dynamics of populations structured by phenotypical traits and space. While our motivation comes from the study of tumor growth, we investigate the selection of the fittest individuals and the heterogeneity of the population. A population structured by a phenotypical trait can be modeled using integro-differential Lotka-Volterra equations. The solutions of such equations, when we consider small mutation steps and in long time, converge to Dirac masses (see [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF]); this property corresponds to the selection of the fittest traits. In this paper, we study such behavior considering a spatial structure for the population.

A simple way to describe the selection of the fittest individuals, when environmental conditions depend on space, was proposed in [START_REF] Lorz | Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors[END_REF] as a model for emergence of resistance to drug in cancer therapy. This model assumes the evolution of cells and is written as a coupled system of integro-differential equations structured by trait x and by a space variable y    ε∂ t n ε (y, x, t) = [r(x)c ε (t, y) -d(x)(1 + ̺ ε (y, t))] n ε (y, x, t), y ∈ R, 0 < x < 1, t ≥ 0, -∆ y c ε (y, t) + [̺ ε (y, t) + λ] c ε (y, t) = λc B , ̺ ε (y, t) = n ε (y, x, t)dx.

(

) 1
The first equation describes the dynamics of a cell population density n ε . The second equation describes a nutrient c ε (and a drug can be included in the same way) diffused within the tumor from a constant input concentration c B with rate λ. The term r(x) denotes the proliferation rate of cells expressing trait x due to the consumption of resource. The function d(x) models the death rate of cells with trait x due to the competition with other cells at the same position. The small parameter ε is introduced to consider the long time behavior of the cell population. Note that we do not consider mutations in this model, supposing that all traits are already present in the population, possibly at very small quantities.

Our goal is to show that, when ε vanishes, there is selection of a space and time dependent fittest trait X(y, t) in the cell population as numerically shown in [START_REF] Lorz | Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors[END_REF].

In order to get more complete results, and show better the difficulties when handling the time variable, we also study a related model where the integral equation for n ε is coupled to a parabolic equation for the nutrient.

ε∂ t n ε (y, x, t) = [r(x)c ε (t, y) -d(x)(1 + ̺ ε (y, t))] n ε (y, x, t), y ∈ R, 0 < x < 1, t ≥ 0, (2)
∂ ∂t c ε -∆ y c ε (y, t) + [̺ ε (y, t) + λ] c ε (y, t) = λc B , (3)
̺ ε (y, t) = n ε (y, x, t)dx. (4)
Recent technologic advances reveal evidence of heterogeneity within cancer tumors (see for instance [START_REF] Gerlinger | Intratumor heterogeneity and branched evolution revealed by multiregion sequencing[END_REF]). Taking into account this intratumor heterogeneity is crucial in the study of the tumor growth and the emergence of drug resistance (see [START_REF] Busch | Fluence rate-dependent intratumor heterogeneity in physiologic and cytotoxic responses to photofrin photodynamic therapy[END_REF][START_REF] Swanton | Intratumor heterogeneity: evolution through space and time[END_REF][START_REF] Gerlinger | Intratumor heterogeneity and branched evolution revealed by multiregion sequencing[END_REF]), and leads to important challenges in finding effective treatment strategies. The above model introduces a simple way to include spatial and phenotypical structure of the cell population together with the diffusion of the nutrient in the domain. Our study indicates that intratumor heterogeneity can emerge as an evolutionary process and provides a description of the space-dependent dominant traits.

The dynamics of phenotypically structured populations under the effect of mutations and competition between the traits has been studied widely during the last decade using stochastic methods and integro-differential equations (see for instance [START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF][START_REF] Champagnat | Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models[END_REF][START_REF] Desvillettes | On mutation-selection dynamics for continuous structured populations[END_REF][START_REF] Champagnat | Mathematical study of stochastic models of evolution belonging to the ecological theory of adaptive dynamics[END_REF][START_REF] Raoul | Etude qualitative et numérique d'équations aux dérivées partielles issues des sciences de la nature[END_REF][START_REF] Mirrahimi | Phénomènes de concentrations dans certaines EDPs issues de la biologie[END_REF] and the references therein). In particular an approach based on a WKB type ansatz, leading to Hamilton-Jacobi equations, provides an analysis of the asymptotic behavior of the populations structured by phenotypical traits (see [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF][START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF] and the references therein). It can be shown using this method that in long time and considering small mutation steps the population concentrates on a dominant trait that evolves in time. In other words, the population density tends to a Dirac mass in the phenotypical trait which depends on time.

Several refinements are necessary to consider problems which are more relevant biologically. In particular, one can include the interaction of the population with nutrients (see [START_REF] Mirrahimi | Evolution of species trait through resource competition[END_REF][START_REF] Champagnat | Adaptive dynamics in a stochastic multi-resources chemostat model[END_REF] for some results in this direction). The introduction of the resource leads to the study of integro-differential coupled systems. Moreover, most of the cited works neglect the spatial structure of the environment and consider a well-mixed population. However, as mentioned above the environmental heterogeneity is an important element to be considered. The study of models of populations structured jointly by space and trait has gained much attention recently and leads to several important difficulties to be overcome. These difficulties are mainly due to the integral term in one of the variables in the equations.

Most of the recent attempts to tackle these problems concentrate mostly on the spatial propagation of the population and less on the phenotypical selection (see [START_REF] Mirrahimi | Dynamics of sexual populations structured by a space variable and a phenotypical trait[END_REF][START_REF] Alfaro | Traveling waves in a nonlocal equation as a model for a population structured by a space variable and a phenotypical trait[END_REF][START_REF] Bouin | A Hamilton-Jacobi limit for a model of population structured by space and trait[END_REF][START_REF] Bouin | Travelling waves for the cane toads equation with bounded traits[END_REF], and also [START_REF] Champagnat | Invasion and adaptive evolution for individual-based spatially structured populations[END_REF][START_REF] Arnold | Existence of nontrivial steady states for populations structured with respect to space and a continuous trait[END_REF] for the stochastic derivation of such models and the study of steady states). In this paper, using the WKB approach mentioned above, we study models which take into account the spatial and phenotypical structure of the population and lead to the selection of a space and time dependent phenotypical trait.

Note that in our model the spatial heterogeneity is induced by a nutrient which diffuses in the domain.

We assume some conditions on the model parameters and on the initial data

̺ m ≤ ̺ 0 ε (y) ≤ ̺ M , ̺ m := c B λ λ + ̺ M min 0<x<1 r(x) d(x) -1 > 0, ̺ M := c B max 0<x<1 r(x) d(x) -1, (5)
c m < c 0 (y) < c B , c m := c B λ λ + ̺ M , c 0 ∈ W 1,∞ (R). (6)
Note that the non-extinction condition, ̺ m > 0, is equivalent to write

λc B min 0<x<1 r(x) d(x) ≥ λ + c B max 0<x<1 r(x) d(x) ,
a condition which is satisfied for parameters λ and c B sufficiently large.

For n ε , we assume initially a 'Gaussian type' concentration

           n 0 ε ⇀ ̺ 0 (y)δ(x -X 0 (y)), with the condition r(X 0 (y))c 0 (y) -d(X 0 (y))(1 + ̺ 0 (y)) = 0, n 0 ε = e u 0 ε (y,x)/ε , u 0 ε (y, x) -→ ε→0 u 0 (y, x) locally uniformly, | ∂ ∂x u 0 ε | + | ∂ 3 ∂x 3 u 0 ε | ≤ K 0 , ∂ 2 ∂x 2 u 0 ε ≤ -a < 0. (7)
In particular these conditions imply that

max 0≤x≤1 u 0 ε (y, x) = u 0 ε (y, X 0 ε (y)) -→ ε→0 0 = max 0≤x≤1 u 0 (y, x) = u 0 (y, X 0 (y)) with X 0 ε (y)) -→ ε→0 X 0 (y) locally uniformly.
Finally, we assume that c and d are smooth and that for some constant

K 0    |r ′ | + |d ′ | + |r ′′ | + |d ′′ | + |r ′′′ | + |d ′′′ | ≤ K 0 , r ′′ < 0, d ′′ > 0. ( 8
)
Theorem 1.1 (Parabolic case) With assumptions (5)-( 8), there is

X(y, t) ∈ W 1,∞ (R×R + ), ̺(y, t) ∈ C(R × R + ) and c(y, t) ∈ C(R × R + )
, such that solutions of (2)-( 4) satisfy, as ε → 0,

̺ ε → ̺(y, t), almost everywhere, (9)
c ε → c(y, t), locally uniformly, (10)
n ε (y, x, t) ⇀ ̺(y, t)δ x -X(y, t) , weakly in measures.

Moreover, we have

∂ ∂t c -∆ y c(y, t) + [ ̺(y, t) + λ ] c(y, t) = λc B , y ∈ R, t ≥ 0, (12)
and r X(y, t) c(y, t) -d X(y, t)

1 + ̺(y, t) = 0. ( 13
)
Finally, ̺(y, t) and c(y, t) are Hölder continuous in t and Lipschitz continuous in y.

The convergence of c ε can be derived from parabolic regularity, while more elaborate arguments are needed to obtain the limit of n ε . To obtain a priori bounds on n ε we first use a Hopf-Cole transformation to deal with bounded values. Next, we prove regularity estimates on variables x and t. However, we don't have a priori estimates on variable y due to the nonlocal term ̺ ε (t, y). To handle this difficulty we first pass to the weak limit, fixing the variable y, and next we recover pointwise and strong convergence from the uniqueness and the structure of the limit.

We postpone the statement and proof of a similar result in elliptic case, that is system (1) to the end of the paper (Section 5). We begin by giving general a priori bounds which hold both for the elliptic and parabolic case, in Section 2. With these at hand, we prove Theorem 1.1 in the next section. For the sake of completeness, we recall some Hölder continuity results for parabolic equations in Section 4. Some conclusions and perspectives are drawn in Section 6.

Preliminary estimates

Several bounds can be obtained from elementary manipulations of the equations (1) or equations ( 2)-( 4). Here we make the assumptions ( 5)-( 7) in the parabolic case and assume ( 5) and ( 7) in the elliptic case. These bounds are Lemma 2.1 The following estimates hold true:

(i) 0 ≤ c ε (y, t) ≤ c B , (ii) ̺ ε (y, t) ≤ c B max 0<x<1 r(x) d(x) -1 = ̺ M , (iii) c ε (y, t) ≥ c B λ λ+̺ M = c m , (iv) ̺ ε (y, t) ≥ c m min 0<x<1 r(x) d(x) -1 = ̺ m > 0.
Proof. We only give the proofs for the parabolic case ( 2)-( 4). The estimates can be proved for the elliptic case (1) following similar arguments. We first notice, from ( 2) and ( 7), that

n ε > 0, in R × R × R + . In particular, ̺ ε > 0 in R × R + .
Similarly, from (3), ( 6) and the comparison principle we obtain that c ε > 0 in R × R + .

(i) From ( 3), ( 6), ̺ ε > 0 and the comparison principle, we immediately deduce that

c ε ≤ c B in R × R + .
(ii) We integrate (2) with respect to x and use (i) to obtain

ε∂ t ̺ ε (y, t) ≤ c B max 0<x<1 r(x) d(x) -(1 + ̺ ε (y, t)) d(x)n ε (x, t)dx.
Using the above inequality together with the definitions (5), we obtain (ii).

(iii) The third inequality follows directly from (ii), ( 6) and the comparison principle for (3).

(iv) We integrate (2) with respect to x, use (iii) and obtain

ε∂ t ̺ ε ≥ c m min 0<x<1 r(x) d(x) -(1 + ̺ ε ) d(x)n ε (x, t)dx.
Using the above inequality together with the definitions (5), we obtain (iv).

The limiting problem

First step. Limits for ̺ ε (y, t) and c ε (y, t). From the uniform bound on ̺ ε (y, t), we define ̺(y, t) as the weak limit

̺ ε (y, t) ⇀ ̺(y, t) in L ∞ R × (0, ∞) -w- * . (14)
Then, we may pass to the limit in the equation (3) for c ε , and for that, we use that

c ε (y, t) -→ ε→0 c(y, t) locally uniformly in R × [0, ∞], (15)
(see Section 4 for a proof) and we find the equation for the limiting nutrient concentration

∂ ∂t c -∆ y c(y, t) + [ ̺(y, t) + λ ] c(y, t) = λc B , y ∈ R. ( 16
)
Its solution is C 1,α in y, with α ∈ (0, 1), by parabolic regularity (see also Section 4).

Second step. The WKB change of unknown. Rather than working on ̺ ε directly, we define as usual the function

u ε = ε ln(n ε ),
which satisfies

∂ t u ε (y, x, t) = r(x)c ε (t, y) -d(x)(1 + ̺ ε (y, t)), y ∈ R, 0 < x < 1. ( 17
)
It is standard to derive the bounds

   | ∂ ∂t u ε (x, y, t)| ≤ K(t), | ∂ ∂x u ε (x, y, t)| + | ∂ 3 ∂x 3 u ε (x, y, t)| + | ∂ 3 ∂txx u ε (x, y, t)| ≤ K(t), ∂ 2 ∂x 2 u ε (y, x, t) ≤ -a, u ε (y, x, t) ≤ o(1). (18)
Indeed, these estimates can be obtained by differentiating [START_REF] Mirrahimi | Phénomènes de concentrations dans certaines EDPs issues de la biologie[END_REF] and using ( 7), ( 8) and Lemma 2.1.

For our arguments below, we fix y and pass to the limit. Extracting several subsequences which a priori depend on y,

̺ ε (y, t) ⇀ ̺(y, t), L ∞ (0, ∞)-w- * ̺ m ≤ ̺(y, t) ≤ ̺ M , u ε (y, x, t) -→ u(y, x, t), uniformly in x, t ∈ [0, T ], ∀t > 0.
Notice that this value ̺(y, t) may differ from ̺(y, t) . Passing to the limit, we find, y by y, that

           u(y, x, t) = u 0 (y, x) + r(x) t 0 c(s, y)ds -d(x)t + d(x) t 0 ̺(y, s)ds, t ≥ 0, 0 < x < 1, max 0≤x≤1 u(y, x, t) = 0 = u(y, X(y, t), t), u(y, x, t = 0) = u 0 (y, x). ( 19
)
By concavity of u in y, the maximum point X(y, t) is unique.

Third step. ̺(y, t) = ̺(y, t). Because of the particular structure on the right hand side, we know there is a unique solution (u(y, t), t 0 ̺(y, s)ds) of ( 19) for each y (see also [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF] for general argument). Therefore the full families t 0 ̺ ε (y, s)ds and u ε (y, t) converge pointwise and not only subsequences, for each y and t. Also by continuous dependence upon the parameter y in the data for [START_REF] Mirrahimi | Dynamics of sexual populations structured by a space variable and a phenotypical trait[END_REF], both u(y, t) and t 0 ̺(y, s)ds also have continuous dependence on y. Consequently t 0 ̺(y, s)ds = t 0 ̺(y, s) ds and thus ̺(y, t) = ̺(y, t). However this does not imply strong convergence of ̺ ε (y, t) in the time variable.

Fourth step. The mapping t → X(y, t) is Lipschitz continuous in t, y by y. To prove this, let X ε be the unique maximum point of u ε and hence ∂ x u ε (y, X ε (y, t), t) = 0. We differentiate this equality with respect to t and find

∂ xt u ε (y, X ε (y, t), t) + ∂ xx u ε (y, X ε (y, t), t) d dt X ε (y, t) = 0.
It follows that

-∂ xx u ε (y, X ε (y, t), t) d dt X ε (y, t) = r ′ (x)c ε (t, y) -d ′ (x)(1 + ̺ ε (y, t)).
As a consequence, t → X ε (y, t) is Lipschitz continuous for all y. Therefore, we can pass to the limit as ε → 0, and X ε (y, t) converges uniformly locally to X(y, t) which is therefore Lipschitz continuous in t (because this value achieves the maximum of u(y, t) and is unique).

From the estimates on ∂ xxx u and ∂ txx u, we may pass to the strong limit in the term -∂ xx u ε y, X ε (y, t), t and obtain

Ẋ(y, t) = -∂ xx u y, X(y, t), t -1 r ′ X(y, t) c(t, y) -d ′ X(y, t) (1 + ̺(y, t)) , (20)
Using the arguments in [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF] we can also show, using [START_REF] Mirrahimi | Dynamics of sexual populations structured by a space variable and a phenotypical trait[END_REF], that at the Lebesgue points in t of ̺(y, t) we have r X(y, t) c(t, y) -d X(y, t)

(1 + ̺(y, t)) = 0. ( 21
)
Since ̺ ∈ L ∞ , we deduce that the above equality holds true for almost every t. This implies that ̺ is also Hölder continuous in t.

Fifth step. The mapping y → X(y, t) is Lipschitz continuous in y. We can use the value of 1 + ̺(y, t) given by formula [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF], and write the equation ( 21) for X under the form

Ẋ(y, t) = -∂ xx u y, X(y, t), t -1 r ′ X(y, t) - d ′ X(y, t) r X(y, t) d X(y, t) c(t, y).
This is an ordinary differential equation which inherits the regularity of the initial data and coefficients, c(y, t) and u(y, t). Therefore its solution X(y, t) is Lipschitz continuous in y.

Sixth step. Strong convergence of ̺ ε . With the steps above, the conclusion on the convergence of n ε is a direct consequence of the analysis of the convergence of u ε .

To conclude, we prove the strong convergence of ̺ ε following an argument in [START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF], equation (9.23). We divide equation (2) by d(x) and integrate. We obtain

ε ∂ ∂t 1 0 n ε (y, x, t) d(x) dx = c ε (t, y) 1 0 r(x)n ε (y, x, t) d(x) dx -̺ ε (y, t)(1 + ̺ ε (y, t)).
We pass to the limit and obtain, with n = ̺(y, t)δ x -X(y, t) , ̺(y, t) 2 ≤ c(t, y)

1 0 r(x)n(y, x, t) d(x) dx -̺(y, t) = ̺(y, t)c(t, y) r X(y, t) d X(y, t) -̺(y, t).
Comparing with [START_REF] Raoul | Etude qualitative et numérique d'équations aux dérivées partielles issues des sciences de la nature[END_REF], we conclude that this inequality is, in fact, an equality and thus the strong convergence.

The proof of Theorem 1.1 is complete.

4 Uniform estimates on c ε (parabolic case)

In the analysis of the limit of u ε and ̺ ε , we have used standard local uniform continuity for c(y, t). We recall the proof for the sake of completeness and show that, locally, c ε (y, t) is uniformly 1/4-Hölder continuous in t and is 1/2-Hölder continuous in y. Better regularity can be obtained using regularizing effects of parabolic equations with the available Lipschitz regularity of ρ(•, t); however, we have chosen to keep a simple complete proof and find a weaker result which is enough for our purpose.

First step. Localization method. We first indicate how to work in L 2 after localizing the problem.

Consider a smooth cut-off function χ with compact support. From equation (3) for c ε (which is uniformly bounded in L ∞ (R)), we find

∂ ∂t [χc ε ] -∆ y [χc ε (y, t)] + 2∇χ.∇c ε + c ε ∆χ = χF ε , y ∈ R, t ≥ 0, (22)
with

F ε = c B -c ε (λ + ̺ ε (y, t)) which is also uniformly bounded in L ∞ .
Therefore, multiplying by χc ε and integrating in y, we find after integrations by part 1 2

d dt R [χc ε ] 2 dy + R |∇(χc ε )| 2 dy ≤ R c 2 ε |∇χ| 2 dy + R c ε F ε χ 2 dy ≤ C.
From this estimate, we also control uniformly Next, we multiply equation ( 22) by 2∆ y [χc ε (y, t)] and integrate and obtain

d dt R |∇(χc ε )| 2 + 2 R |∆ y (χc ε (y, t))| 2 ≤ R |∆ y (χc ε (y, t))| 2 + R ε where R ε = R χF ε -c ε ∆χ -2∇χ.∇c ε 2 is uniformly controlled in L 1 loc thanks to the previous esti- mates on T 0 B |∇c ε | 2 dyds. As a consequence, for all 0 ≤ t ≤ T , R ∇[χc ε (y, t)] 2 dy ≤ C 1 (T ), T 0 R ∂ ∂t [χc ε ] 2 dydt ≤ C 2 (T ). (23)
Second step. Hölder regularity. We set v = χc ε and prove the following uniform estimate Proof. The space regularity is obvious since from the first bound (uniform in time) and the Cauchy Schwarz inequality

|v(y 2 , t) -v(y 1 , t)| ≤ (y 1 ,y 2 ) |∇v(y, t)|dy ≤ |y 2 -y 1 | 1/2 C 1 (T ) 1/2 .
Then, we estimate the time increments as follows (with

h = |t 2 -t 1 |) |v(y, t 2 ) -v(y, t 1 )| ≤ (t 1 ,t 2 ) | ∂ ∂t v(y, t)|dt ≤ h 1/2 T 0 | ∂ ∂t v(y, t)| 2 dt 1/2
and thus, being given y 0 and k > 0,

|y-y 0 |≤k |v(y, t 2 ) -v(y, t 1 )|dy ≤ h 1/2 |y-y 0 |≤k T 0 | ∂ ∂t v(y, t)| 2 dt 1/2 dy ≤ C 3 (T )h 1/2 k 1/2 .
Finally, we write for all y ∈ R,

|v(y 0 , t 2 ) -v(y 0 , t 1 )| ≤ |v(y, t 1 ) -v(y 0 , t 1 )| + |v(y 0 , t 2 ) -v(y, t 2 )| + |v(y, t 2 ) -v(y, t 1 )|
and we integrate in y for |y -y 0 | ≤ k (with k to be chosen later). We find

2k|v(y 0 , t 2 ) -v(y 0 , t 1 )| ≤ 8 3 C 1 (T ) 1 2 k k 1/2 + C 3 (T )h 1/2 k 1/2 .
We take k = h 1/2 and find the result.

Elliptic coupling

In case of elliptic coupling, that is of the system given by equations ( 1), an additional difficulty occurs because the regularizing effect in time for c ε and c cannot occur. Therefore, c ε (y, t) and ̺ ε (y, t) have the same regularity in t, that is we only handle L ∞ bounds and, consequently, weak limits. For that reason our result is weaker Theorem 5.1 (Elliptic case) With assumptions (5), ( 6)-( 8), there is

̺(y, t) ∈ L ∞ R × (0, ∞) , X(y, t) ∈ W 1,∞ R × [0, ∞) and c(y, t) ∈ L ∞ R × (0, ∞)
, such that solutions of (1) satisfy, as ε → 0,

̺ ε ⇀ ̺(y, t), c ε ⇀ c(y, t), L ∞ R × (0, ∞) -w- * , (24)
n ε (y, x, t) ⇀ ̺(y, t)δ x -X(y, t) , weakly in measures.

Moreover, we have, almost everywhere in t,

-∆ y c(y, t) + ̺ c (y, t) + λ c(y, t) = λc B , y ∈ R, t ≥ 0, (26) and r
X(y, t) c(y, t) -d X(y, t) 1 + ̺(y, t) = 0. (27)
Proof. We just indicate the modifications to the proof of Theorem 1.1.

With the uniform estimates of Lemma 2.1, we can follow the limiting procedure of Section 3. The limit in (15) is just a weak limit because of the time variable, but a.e. in t, c(y, t) belongs to W 2,∞ (R) and the equation ( 16) is replaced by equation (26). Then, the analysis of the limit u(y, t) can be performed as in Section 3 and both equations ( 20) and ( 21) hold a.e. in t. The y-regularity can be derived for ̺, but not t-Lipschitz regularity, because of the lack of time regularity in c. However X itself wins one degree of regularity and is indeed Lipschitz continuous. The Sixth step (strong convergence of ̺ ε ) also fails.

Conclusion and perspectives

The asymptotic problem we have handled is one of the simplest where both a trait variable x and a space variable y are used. The main difficulty is that the behaviors in these variables are very different because the solution concentrates as a Dirac mass in x and stays bounded in y. We do not know of methods adapted to prove compactness in this kind of situations. Indeed, a simple tool would be to prove a priori estimates in Sobolev spaces in the variables t and y for integrals in x as ρ(y, t) here; because of the concentration in the variable x we cannot expect such regularity except for the Hopf-Cole transform which however gives indirect information. Here we have been able to use uniqueness for the limit in order to recover compactness avoiding strong a priori estimates. This method is limited to the particular situation at hand. We cannot expect it to work in several other situations, for example more general (nonlinear) growth rates under the form R(y, c, ̺) or dispersion depending on the trait (as in [START_REF] Bouin | Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration[END_REF][START_REF] Bouin | Travelling waves for the cane toads equation with bounded traits[END_REF]).

T 0 B

 0 |∇c ε | 2 dyds on each ball B and for each T ∈ R + .

Lemma 4 . 1 A

 41 function v with compact support which satisfies (23) is 1/4-Hölder continuous in t and is 1/2-Hölder continuous in y.


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email: j-charles.de-hemptinne@ifpen.fr

Benoit Creton

Georgios M Kontogeorgis

Improving GC-PPC-SAFT equation of state for LLE of hydrocarbons and oxygenated compounds with water

Keywords: Group Contribution, GC-PPC-SAFT, binary interaction parameter, LLE, oxygenated compounds

The GC-PPC-SAFT model has been shown to be useful for predicting the liquidliquid phase split with water [Nguyen-Huynh et al. Ind. Eng. Chem. Res. 2011, 50, 7467-7483]. In order to extend the use of this model to oxygenated compounds for a large number of families (aliphatic ethers, aldehydes, ketones, formates, acetates, propionates/butyrates, n-aliphatic acids), it is proposed to consider cross-association in addition to a binary interaction parameter ij l on the combining rules for the crosssegment diameter between water and the investigated compound.

The binary interaction parameters ij l , αβ u , and αβ w are fitted on mutual solubilities of water and organic compounds. The regressed values which are obtained for each chemical family, are subsequently used for predicting infinite dilution activity coefficient in water and n-octanol/water partition coefficient.

In general, the results obtained are very much improved compared to the predictive approach discussed previously [Nguyen et al. Ind. Eng. Chem. Res. 2013, 52, 7014-7029]. The global deviation values on the decimal log scale for infinite dilution activity coefficient in water, water solubility and n-octanol/water partition coefficient are 0.377, 0.419, and 0.469, respectively.

Introduction

A reliable knowledge of the solubility of chemicals in various solvents, among which water has an important place, is of great importance for many fields of the chemical industry e.g. the areas of biorefining and environmental engineering. The liquidliquid equilibrium (LLE) properties serve also in the design and optimization of separation processes (e.g. liquid-liquid extraction, distillation). In this work, we have an extra focus on biorefining-related processes which are often faced with fluids covering a very large range of oxygenated compounds which are polar and hydrogen-bonding [START_REF] Huber | Synthesis of transportation fuels from biomass: Chemistry, catalysts, and engineering[END_REF][START_REF] Garcia-Perez | Characterization of bio-oils in chemical families[END_REF] As a consequence, the development of predictive approaches for phase equilibrium calculations is a major challenge, particularly for watercontaining mixtures with oxygenated compounds. Strongly associating and/or polar interactions between water and oxygen-bearing compounds (e.g. alcohols, acids, etc) are responsible for the extreme non-ideality of these mixtures. The modelling of these complex mixtures by applying cubic equations of state (EoS) with conventional mixing and combining rules is not appropriate [START_REF] Georgeton | Application of Cubic Equations of State to Polar Fluids and Fluid Mixtures[END_REF][START_REF] Voutsas | Equations of state with emphasis on Excess Gibbs Energy Mixing Rules[END_REF] Many research efforts have been made in order to overcome this problem. Among these efforts, it is worth mentioning the so-called EoS/G E models which were initially proposed by Huron and Vidal [START_REF] Huron | New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures[END_REF][START_REF] Vidal | Mixing Rules and Excess Properties in Cubic Equations of State[END_REF].

The EoS/G E models, based on the combination of an EoS and an activity coefficient model (e.g. 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 composition in thermodynamic excess functions for liquid mixtures[END_REF] UNIQUAC, [START_REF] Abrams | Statistical thermodynamics of mixtures: a new expression for the Excess Gibbs free energy of partly or completely miscible systems[END_REF] UNIFAC [START_REF] Fredenslund | Group contribution estimation of Activity coefficients in nonideal liquid mixtures[END_REF]), are mixing rules for the energy (and sometimes also the co-volume) parameters of an EoS. More details about this type of mixing and combining rules can be found in recent books [START_REF] Vidal | Thermodynamics: Applications in Chemical Engineering and the Petroleum Industry[END_REF][START_REF] Kontogeorgis | Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories[END_REF] and reviews. [START_REF] Voutsas | Equations of state with emphasis on Excess Gibbs Energy Mixing Rules[END_REF][START_REF] Kontogeorgis | Thirty Years with EoS/G E Models -What Have We Learned?[END_REF][START_REF] Wei | Equation of state for the calculation of fluid phase equilibria[END_REF] The application of the EoS/G E models for correlating water + chemicals (e.g. hydrocarbons, cyclohexanone, 1-butanol, surfactants, etc) LLE has been reported by some authors [START_REF] Escobedo-Alvarado | Study of EOS-G(ex) mixing rules for liquid-liquid equilibria[END_REF][START_REF] Rudolph | Phase behaviour of water+nonionic surfactant systems: experiments and modelling applying the Peng-Robinson equation of state with the Wong-Sandler mixing rules and the UNIQUAC gE-model[END_REF].

Another approach for modelling complex mixtures is to use EoS that account explicitly for associating and/or polar interactions between molecules. Many thermodynamic models have been proposed, such as APACT, [START_REF] Ikonomou | Thermodynamics of Hydrogen-bonded molecules: The associated Perturbed Anisotropic Chain Theory[END_REF] GCA-EoS, [START_REF] Gros | A group contribution equation of state for associating mixtures[END_REF] SAFT, [START_REF] Chapman | SAFT: Equation-of-state solution model for associating fluids[END_REF][START_REF] Chapman | New Reference equation of state for associating liquids[END_REF] CPA [START_REF] Kontogeorgis | An equation of state for associating fluids[END_REF] or NRHB [START_REF] Tsivintzelis | Evaluation of the nonrandom hydrogen bonding (NRHB) theory and the simplified perturbed-chain-statistical associating fluid theory (sPC-SAFT). 2. Liquid-liquid equilibria and prediction of monomer fraction in hydrogen bonding systems[END_REF]. The Group Contribution Association Eos has been used with some success by Pereda et al. [START_REF] Pereda | Solubility of hydrocarbons in water: Experimental measurements and modeling using a group contribution with association equation of state (GCA-EoS)[END_REF] [START_REF] Soria | Modeling alcohol+water+hydrocarbon mixtures with the group contribution with association equation of state GCA-EoS[END_REF] for describing phase equilibria of systems containing water, alkanes and alcohols. The most successful for the purpose of computing water + hydrocarbon LLE is CPA [START_REF] Yakoumis | Prediction of phase equilibria in binary aqueous systems containing alkanes, cycloalkanes, and alkenes with the cubic-plus-association equation of state[END_REF][START_REF] Oliveira | Mutual solubilities of hydrocarbons and water with the CPA EoS[END_REF]. It has been extended to water + bio-oil mixtures by Oliveira et al. [START_REF] Oliveira | Prediction of water solubility in biodiesel with the CPA equation of state[END_REF][START_REF] Oliveira | Another look at the water solubility in biodiesels: Further experimental measurements and prediction with the CPA EoS[END_REF].

Using the SAFT theory, the first such paper was by Galindo et al. [START_REF] Galindo | Predicting the highpressure phase equilibria of water plus n-alkanes using a simplified SAFT theory with transferable intermolecular interaction parameters[END_REF] who focused on the critical lines of water and n-alkanes. McCabe et al. [START_REF] Mccabe | Anomalies in the solubility of alkanes in near-critical water[END_REF] then showed that the infinite dilution properties could be adequately modelled as well with the same parameters. Using another version of SAFT, the group of Vega [START_REF] Vega | Capturing the solubility minima of n-alkanes in water by Soft-SAFT[END_REF], and Patel et al. [START_REF] Patel | Prediction of the Salting Out effect of Strong Electrolytes on Water+Alkane Solutions[END_REF] also proposed a discussion on this subject. NguyenHuynh et al. [START_REF] Nguyen-Huynh | Modeling Liquid-Liquid and Liquid-Vapor Equilibria of Binary Systems Containing Water with an Alkane, an Aromatic Hydrocarbon, an Alcohol or a Gas (Methane, Ethane, CO2 or H2S), Using Group Contribution Polar Perturbed-Chain Statistical Associating Fluid Theory[END_REF] used the group-contribution polar PC-SAFT (GC-PPC-SAFT) approach to propose a predictive scheme on systems containing water, alkanes, aromatics and alcohols. Some other authors have used SAFT with water and oxygenated compounds, as Garcia-Lisbona et al. [START_REF] Garcia-Lisbona | An Examination of the Cloud Curves of LiquidΓêAELiquid Immiscibility in Aqueous Solutions of Alkyl Polyoxyethylene Surfactants Using the SAFT-HS Approach with Transferable Parameters[END_REF] on polyoxyethylene compounds, and Pedrosa et al. on glycols and polyethylene glycols [START_REF] Pedrosa | Phase Equilibria of Ethylene Glycol Oligomers and Their Mixtures[END_REF][START_REF] Pedrosa | Modeling the Phase Equilibria of Poly(ethylene glycol) Binary Mixtures with soft-SAFT EoS[END_REF]. Up to our knowledge, no systematic investigation was carried so far on the liquidliquid phase split of water with all oxygenated families. Yet, in bio-oils, one finds not only hydrocarbons and alcohols, but also esters, ethers, ketones, aldehydes and acids.

The purpose of this work is therefore to extend the GC-PPC-SAFT approach for a large number of systems with transferable parameters in order to be able to extrapolate beyond the components considered.

There are many SAFT-type versions proposed e.g. the original SAFT [START_REF] Chapman | SAFT: Equation-of-state solution model for associating fluids[END_REF][START_REF] Chapman | New Reference equation of state for associating liquids[END_REF], CK-SAFT [START_REF] Huang | Equation of state for small, large, polydisperse and associating molecules[END_REF], the simplified SAFT, [START_REF] Fu | A simplified SAFT equation of state for associating compounds and mixtures[END_REF] LJ-SAFT, [START_REF] Kraska | Phase Equilibria calculations with a modified SAFT equation of state: 1. Pure alkanes, alcohols and water[END_REF][START_REF] Kraska | Phase Equilibria calculations with a modified SAFT equation of state: 2.Binary mixtures of n-alkanes, 1-alcohols and water[END_REF] soft-SAFT, [START_REF] Blas | Prediction of Binary and Ternary Diagrams Using the Statistical Associating Fluids Theory (SAFT) Equation of State[END_REF] SAFT-VR, [START_REF] Gil-Villegas | Statistical Associating Fluid Theory for Chain Molecules with Attractive potentials of Variable Range[END_REF] PC-SAFT [START_REF] Gross | Perturbed-Chain SAFT: An equation of state based on a perturbation theory for chain molecules[END_REF], the simplified PC-SAFT [START_REF] Solms | Computational and physical performance of a modified PC-SAFT equation of state for highly asymmetric and associating mixtures[END_REF], tPC-SAFT [START_REF] Karakatsani | Evaluation of the truncated perturbed chain-polar statistical associating fluid theory for complex mixture fluid phase equilibria[END_REF], and PPC-SAFT [START_REF] Nguyen-Huynh | Application of GC-SAFT EOS to polar systems using a segment approach[END_REF]. More details can be found in some references [START_REF] Kontogeorgis | Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories[END_REF][START_REF] De Hemptinne | Select thermodynamic models for process simulation: A practical guide using a three steps methodology[END_REF][START_REF] Tan | Recent Advances and Applications of Statistical Associating Fluid Theory[END_REF]. The main differences between these SAFT-type versions are mostly due to [START_REF] Huber | Synthesis of transportation fuels from biomass: Chemistry, catalysts, and engineering[END_REF] the choice of the reference system, and (2) the choice of various descriptions for the repulsive and attractive interactions. Some authors extended these equations by adding a polar term to take into consideration the dipolar interactions between molecules [START_REF] Kraska | Phase Equilibria calculations with a modified SAFT equation of state: 2.Binary mixtures of n-alkanes, 1-alcohols and water[END_REF][START_REF] Blas | Prediction of Binary and Ternary Diagrams Using the Statistical Associating Fluids Theory (SAFT) Equation of State[END_REF][START_REF] Nguyen-Huynh | Application of GC-SAFT EOS to polar systems using a segment approach[END_REF][START_REF] Chapman | Phase behavior applications of SAFT based equations of state -from associating fluids to polydisperse, polar copolymers[END_REF][START_REF] Tumakaka | Application of the Perturbed-Chain SAFT equation of state to polar systems[END_REF]. The resulting polar SAFT models are of great interest in the modelling of oxygenated compounds/water mixtures. Successful modelling of various thermodynamic properties of many complex fluids and fluid mixtures by using SAFT-type versions has been reviewed in some references [START_REF] Kontogeorgis | Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories[END_REF][START_REF] Economou | Statistical Associating Fluid Theory: A succesful model for the calculation of Thermodynamic and Phase equilibrium properties of complex mixtures[END_REF][START_REF] Muller | Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches[END_REF]. As discussed above, although several studies deal with the LLE behaviour as computed by SAFT-type models, particularly mutual solubilities, very few have investigated systematically several chemical families. This work, which is the follow-up of our previous study, [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF] proposes to use Group Contribution -Polar Perturbed Chain -SAFT (GC-PPC-SAFT) for computing LLE of water-containing mixtures with oxygenated compounds. This EoS, coupled with the GC concept of Tamouza et al., [START_REF] Tamouza | Group contribution method with SAFT EOS applied to vapor liquid equilibria of various hydrocarbon series[END_REF] is an extension of the PC-SAFT model [START_REF] Gross | Perturbed-Chain SAFT: An equation of state based on a perturbation theory for chain molecules[END_REF] for polar molecules [START_REF] Nguyen-Huynh | Application of GC-SAFT EOS to polar systems using a segment approach[END_REF]. More details will be given in the next section. In our previous work, [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF] it was observed that the predictive capability of this model for calculating water solubility is very challenging. In addition, a sensitivity analysis indicated that a very small modification in the segment diameter could lead to very significant changes in solubility. To be concrete, a 5% modification in the segment diameter caused an average absolute deviation of 314% in the prediction of the water solubility of oxygenated compounds. Because of the practical importance of this property, this paper will focus on an extension of the model that is based on the use of a binary interaction parameter on the cross-diameter parameter.

As for any thermodynamic model, mixing and combining rules are necessary for the GC-PPC-SAFT EoS when applied to mixtures. In SAFT models, as in many other EoS, a binary interaction parameter ij k is often used to correct the dispersion energy parameter ij ε , through the equation ( )

ij ij i j 1 k ε = - ε ε (1)
The use of this ij k parameter was also a part of our previous work [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF]. The physical meaning of ij k refers to the London's theory and several researchers have used this for proposing predictive methods for this parameter [START_REF] Schnabel | Unlike Lennard-Jones parameters for vaporliquid equilibria[END_REF][START_REF] Nguyen-Huynh | Modeling Phase Equilibria of Asymmetric Mixtures Using a Group-Contribution SAFT (GC-SAFT) with a k(ij) Correlation Method Based on London's Theory. 2. Application to Binary Mixtures Containing Aromatic Hydrocarbons, n-Alkanes, CO2, N2, and H2S[END_REF]. In general, these predictive approaches result in positive values for ij k . As explained in the very interesting discussion by Haslam et al [START_REF] Haslam | Prediction of binary intermolecular potential parameters for use in modelling fluid mixtures[END_REF], negative values for this parameter can be found in polar mixtures: the polarity results in stronger attractive interactions than those that can be justified by the London dispersion forces, resulting in ij could be proposed by NguyenHuynh [START_REF] Nguyen-Huynh | Modeling Phase Equilibria of Asymmetric Mixtures Using a Group-Contribution SAFT (GC-SAFT) with a k(ij) Correlation Method Based on London's Theory. 1. Application to CO2 + n-Alkane, Methane + n-Alkane, and Ethane + n-Alkane Systems[END_REF] in the PPC-SAFT framework. Yet, in some cases, it appears that still a negative value for this interaction parameter is unavoidable [START_REF] Rozmus | Application of GC-PPC-SAFT EoS to amine mixtures with a predictive approach[END_REF].

In this work, we have therefore investigated the path that was already used, but without any justification, by Held et al. [START_REF] Held | Measuring and modeling alcohol/salt systems[END_REF][START_REF] Nann | Liquid-Liquid Equilibria of 1-Butanol/Water/IL Systems[END_REF], which is the use of a correction parameter ij l on the cross-diameter parameter ij σ . Schnabel et al. [START_REF] Schnabel | Unlike Lennard-Jones parameters for vaporliquid equilibria[END_REF] propose a very interesting review of various combining rules and conclude that the so-called Lohrentz-Berthelot combining rule perform well on vapour-liquid equilibria and densities of a number of selected mixtures. Both Santos et al. [START_REF] Santos | Equation of state of nonadditive ddimensional hard-sphere mixtures[END_REF] and Paricaud [START_REF] Paricaud | Phase equilibria in polydisperse nonadditive hard-sphere systems[END_REF] propose an analysis of non-additive hard sphere and show that the use of this nonadditive character has a large effect on liquid-liquid phase split. This has motivated us to investigate the use of this type of binary interaction parameter, called ij l , with:

( ) i j ij ij 2 1 l + σ σ σ = - (2)
At this point, we should state that a few studies on the effect of ij l in the cross covolume parameter of cubic equations of state have been presented [START_REF] Kontogeorgis | Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories[END_REF]. However, the ij l in cubic EoS essentially affects the entropic part of the models that is actually not the case for SAFT models, as will become clear in sections 2.1 and 2.2 where the model equations are shown and this point is further discussed.

A second point that should be further investigated in this work is to revise the association scheme between water and oxygenated compounds. As shown in our previous work [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF], the predictive capability of the GC-PPC-SAFT EoS on the solubilities was not correct for water with some oxygenated chemical families (i.e. aliphatic ethers, aldehydes, ketones, formates, acetates, propionates/butyrates, naliphatic acids) whose association parameters (

A B α β ε , A B α β K
) were all taken from that of alcohols, without further justification, originating from the work of Nguyen-Huynh et al [START_REF] Nguyen-Huynh | Modeling Liquid-Liquid and Liquid-Vapor Equilibria of Binary Systems Containing Water with an Alkane, an Aromatic Hydrocarbon, an Alcohol or a Gas (Methane, Ethane, CO2 or H2S), Using Group Contribution Polar Perturbed-Chain Statistical Associating Fluid Theory[END_REF]. The thermodynamic modelling of aqueous solutions of oxygenated compounds is not easily accessible due to the effect of the temperature on the crossassociating interaction between water and chemicals, as illustrated in Figure 1. At low temperatures, the cross-association degree between water and oxygenated compounds is strong, resulting in relatively higher mutual solubilities. When the temperature increases, the cross-association diminished and the self-association of water will take over resulting in decreased mutual solubilities. Finally, at high temperatures all association phenomena fade away resulting in an increased solubility. This discussion was already presented by Patel et al. [START_REF] Patel | Prediction of the Salting Out effect of Strong Electrolytes on Water+Alkane Solutions[END_REF], although they acknowledged the fact that this balance between self-and cross-association could not be invoked for water + alkane mixtures which was the subject of their study. In our case, such cross-association can be used, because the oxygenated groups are electrondonors in hydrogen bonds. This is why a specific effort will be devoted to optimize the cross-association between water and oxygenated compounds with the objective to improve the temperature dependence of the model. This will be further discussed in section 4.

The paper is organized as follows. Firstly, the GC-PPC-SAFT EoS and the determination of its parameters are described. The collection and treatment of experimental data are presented in section 3. Also in this section, the regression results of ij l and the cross-association parameters αβ u & αβ w (defined below) are presented. In section 4, the model is used to evaluate infinite dilution activity coefficients in water and n-octanol/water partition coefficient. Our conclusions are given at the end.

GC-PPC-SAFT model 2.1 Model description

The GC-PPC-SAFT model, namely Group Contribution -Polar Perturbed Chain -Statistical Associating Fluid Theory, is the combination of the PPC-SAFT model [START_REF] Nguyen-Huynh | Application of GC-SAFT EOS to polar systems using a segment approach[END_REF] with the GC concept [START_REF] Tamouza | Application to binary mixtures of a group contribution SAFT EOS[END_REF] in order to determine three parameters ε/k , σ , and m . This model is written as a sum of intermolecular contributions to the residual Helmholtz energy for given temperature, volume and composition, as shown in eq [START_REF] Georgeton | Application of Cubic Equations of State to Polar Fluids and Fluid Mixtures[END_REF]. These contributions are independent of each other.

where hs A is the hard sphere term which accounts for repulsive interactions between hard spheres. This term is based on the hard-sphere theory developed by Boubli'k and Mansoori [START_REF] Boublik | Hard Sphere Equation of State[END_REF][START_REF] Mansoori | Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres[END_REF]. The most important parameter of this term is the segment diameter σ . The term ). The London-type attractive interactions between molecular chains are represented by the so-called dispersion term disp A . This term is constructed on the basis of the second-order perturbation theory of Barker and Henderson starting from a hard chain reference [START_REF] Barker | Perturbation Theory and Equation of State for Fluids: the Square well Potential[END_REF]. The characteristic parameter of this term is dispersive energy / k ε (where k is the Boltzmann constant) with units of temperature (in kelvin).

The contribution assoc A in eq (3) originating from the Wertheim's theory is used for representing the hydrogen-bonding association. Its expression can be found in literature [START_REF] Jackson | Phase equilibria of associating fluids: Spherical molecules with multiple bonding sites[END_REF]. It is characterized by an association energy parameter AB ε and an association volume parameter AB K (where A, B are the associating sites). The last term multi polar A - of eq (3) is used to take into consideration the polar interactions.

Regarding the PPC-SAFT model, this term is based on the theory of Gubbins and Twu [START_REF] Gubbins | Thermodynamics of Polyatomic Fluid Mixtures -1 Theory[END_REF] for spherical molecules and the segment approach of Jog and Chapman [START_REF] Jog | Application of Wertheim's thermodynamic Perturbation theory to dipolar hard sphere chains[END_REF].

The characteristic parameters of this term are the polar moments including the dipole μ and quadrupole Q moments and the corresponding polar fractions pμ mx and pQ mx .

Parameterization of the PPC-SAFT model a) Pure component parameters

By using the GC method of Tamouza et al., [START_REF] Tamouza | Application to binary mixtures of a group contribution SAFT EOS[END_REF] the three parameters of the PPC-SAFT model (ε/k , σ , and m ) are calculated, as shown in eqs ( 4), ( 5), and [START_REF] Vidal | Mixing Rules and Excess Properties in Cubic Equations of State[END_REF].

n groups n i n groups n i 1 i molecule i i 1           =   = ε ∑ ε = ∏ (4)
n groups

i i i 1 n molecule groups i i 1 n n = = σ σ = ∑ ∑ (5) n groups molecule i i i 1 m n R = = ∑ ( 6
)
where i ε , i σ represent the energy and diameter of the functional group i in the molecule composed of groups n different groups, respectively. i R is the contribution of functional group i to the chain length of the molecule. Its value depends on the position of functional group i in the molecule, as explained in reference [START_REF] De Hemptinne | Application to petroleum engineering of statistical thermodynamics -Based equations of state[END_REF].

The association and polarity parameters are family-specific parameters, or more exactly functional groups-specific parameters (e.g. hydroxyl group OH). Within the GC scheme of the GC-PPC-SAFT EoS, these specific parameters are transferable within the entire family of molecules.

The reader is referred to our previous publication for further details about the parameterization of this model [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF].

b) Mixture parameters

In order to apply the GC-PPC-SAFT model to mixtures, combining rules are required. Equations ( 1) and ( 2) are the so-called Lorentz-Berthelot combining rules for the dispersive energy ij ε and segment diameter ij σ , when considering two different molecules i and j. In this work, the binary interaction parameter ij k on the dispersive energy are taken as zero as observed in our previous work [START_REF] Nguyen | GC-PPC-SAFT equation of state for VLE and LLE of hydrocarbons and oxygenated compounds. Sensitivity analysis[END_REF]. We will focus on the use of the non-additivity of the hard sphere diameters. The ij l values will be regressed as explained in the next section.

It is worth noting that although the non-additivity on the hard-sphere diameters may be interpreted as an entropic effect, it has no effect at all on the so-called 'hardsphere' term that originates from Boubli'k [START_REF] Boublik | Hard Sphere Equation of State[END_REF] and Mansoori [START_REF] Mansoori | Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres[END_REF] and uses only a generalized definition of the density k ζ defined as:

( )

1 6 n k Av k i i ii i N x m d v π =   ζ =     ∑ . ( 7
)
Here k = 1, 2, 3, v is the molar volume, NAv is Avogadro's number and dii represents the temperature dependent hard-sphere diameter:

3 1 0.12 exp ii ii ii d kT ε σ     = - -         . ( 8
)
No unlike interaction is involved here.

The unlike segment diameter instead comes into the equation through the dispersion term that uses mixing rules for 2 3 m εσ and 2 2 3 m ε σ that appear in the first and second order contributions of the dispersion term:

n n ij 2 3 3 i j i j ij i j m x x m m kT ε   εσ = σ     ∑∑ (9) 2 n n ij 2 2 3 3 i j i j ij i j m x x m m kT ε   ε σ = σ     ∑∑ (10)
where i x , j x are the mole fraction of components i and j, respectively.

More details about the expressions can be found in the original paper [START_REF] Gross | Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules[END_REF].

This point essentially illustrates that ij l in the cross diameter combining rule of SAFT model is very different from the ij l in the cross co-volume of cubic EoS. First of all, the former refers to an interaction parameter in a segment-based combining rule while the latter is an interaction parameter in a molecular combining rule where covolume refers to the whole molecule. This may not even be the most significant difference. What is interesting is that the cross-diameter and subsequently the ij l enters in the SAFT equations only via eqs ( 9) & [START_REF] Fredenslund | Group contribution estimation of Activity coefficients in nonideal liquid mixtures[END_REF]. This means that ij l in SAFT is not an entropic-related interaction parameter (as is clearly the case in cubic EoS) and it does not affect the reference term of the model. Via eqs ( 9) & [START_REF] Fredenslund | Group contribution estimation of Activity coefficients in nonideal liquid mixtures[END_REF], it is clear that ij l affects the dispersion term of the SAFT model, more or less in the same way as ij k but possibly even more significantly, as the cross diameter enters in the third power in these equations. In this interpretation, it makes sense to only use one of the two interaction parameters, as they have almost the same physical meaning and evidently they will be highly inter-correlated the way they enter in eqs ( 9) & ( 10). Yet the choice of one rather than the other is here rather opportunistic as it is easier to justify a negative lij rather than a negative kij.

As proposed by Derawi et al. [START_REF] Derawi | Extension of the cubic-plus-association equation of state to glycol-water cross-associating systems[END_REF] and Wolbach and Sandler, [START_REF] Wolbach | Using molecular orbital calculations to describe the phase behavior of cross-associating mixtures[END_REF] equations [START_REF] Vidal | Thermodynamics: Applications in Chemical Engineering and the Petroleum Industry[END_REF] and ( 12) are combining rules for the association parameters in the case of crossassociating mixtures which contain two associating components (e.g. water and ketones).

( )

B A A B 1 w 2 β α α β αβ ε + ε ε = - (11)
and ( )

A B B A K K K 1 u α β β α αβ = - ( 12
)
where the superscripts A and B represent the associating sites on the different functional groups α or β . It should be noted that parameters are directly grouprelated parameters, and as such, they are fully transferable within a chemical family.

Additional interaction parameters ( u αβ and w αβ ) are also used to correct the given combining rules when the groups α and β belong to different chemical families.

Note that it would be more correct to use

α β A B
u and

α β A B
w , since the interactions are truly site-site and not group-group. Yet, in this work, all sites on the same group carry identical parameters, so we have simplified the notation. Except for (alcohols + water) and (n-alkylbenzenes + water) mixtures whose cross-association parameters were studied by Nguyen-Huynh et al. [START_REF] Nguyen-Huynh | Modeling Liquid-Liquid and Liquid-Vapor Equilibria of Binary Systems Containing Water with an Alkane, an Aromatic Hydrocarbon, an Alcohol or a Gas (Methane, Ethane, CO2 or H2S), Using Group Contribution Polar Perturbed-Chain Statistical Associating Fluid Theory[END_REF], we consider in this work cross-association parameters between water and other oxygenated compounds (the oxygen atom always contains a free electron pair that makes it possible to hydrogen bond with water). These families include aliphatic ethers, aldehydes, ketones, formates, acetates, propionates/butyrates, n-aliphatic acids.

Table 1 summarizes the PPC-SAFT parameters for different oxygenated families studied in this work. All details of the parameterization of this model (using group contribution approach) are provided in our previous paper [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF]. Also in Table 1, the PPC-SAFT parameters of two solvents (water and n-octanol) which are considered in this work are presented. These parameters are re-used from our previous work [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF].

3 Regression results

Experimental data

Experimental data for mutual solubilities of water/oxygenated compounds mixtures The experimental data of org i x , originating from the DETHERM database, [START_REF] Detherm | Thermophysical properties of pure substances and mixtures[END_REF] are gathered for some oxygenated chemical families i.e. acetates, formates, propionates/butyrates, aldehydes, aliphatic ethers, ketones, n-aliphatic acids. A summary of the data for this property is given in Table 3.

( aq i x , org i x ),
To make the use of these data easier, all properties considered in this work are correlated for each compound as a function of temperature, as shown in eq [START_REF] Kontogeorgis | Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories[END_REF]. X X X B logX = A + + C log(T) T [START_REF] Kontogeorgis | Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories[END_REF] where X is the property to be correlated (i.e.

Regression for ij l , αβ u , and αβ w

Our previous studies pointed out that the solubility in water was correctly calculated in a purely predictive way for hydrocarbons and alcohols, but the predictive capability of the GC-PPC-SAFT model was not correct for other oxygenated compounds [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF]. That's why this study is only concentrated on improving the predictive scheme for oxygenated families excluding n-alcohols, other aliphatic alcohols, and n-alkylbenzenes which were investigated by Nguyen-Huynh et al [START_REF] Nguyen-Huynh | Modeling Liquid-Liquid and Liquid-Vapor Equilibria of Binary Systems Containing Water with an Alkane, an Aromatic Hydrocarbon, an Alcohol or a Gas (Methane, Ethane, CO2 or H2S), Using Group Contribution Polar Perturbed-Chain Statistical Associating Fluid Theory[END_REF].

The ij l , αβ u , and αβ w values are fitted for each oxygenated chemical family, consisting of aliphatic ethers, aldehydes, ketones, formates, acetates, propionates/butyrates, n-aliphatic acids, to the mutual solubilities of water and oxygenated compounds by minimizing the following objective function:

= =     - - = +         ∑ ∑ 1 x x 1 x x OF n x n ' x (14)
where aq ,exp i x and aq ,calc i x are experimental and calculated solubilities of oxygenated compounds in water, respectively. org ,exp i x and org ,calc i x are experimental and calculated solubilities of water in organic phase, respectively. n , n ' are the number of experimental points created by using eq [START_REF] Kontogeorgis | Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories[END_REF]. The values of n , n ' equal 11 for each property in the allowed range of temperature. It should be noted that the calculation is performed without using the predictive ij k given by Nguyen-Huynh et al [START_REF] Nguyen-Huynh | Modeling Phase Equilibria of Asymmetric Mixtures Using a Group-Contribution SAFT (GC-SAFT) with a k(ij) Correlation Method Based on London's Theory. 1. Application to CO2 + n-Alkane, Methane + n-Alkane, and Ethane + n-Alkane Systems[END_REF][START_REF] Nguyen-Huynh | Modeling Phase Equilibria of Asymmetric Mixtures Using a Group-Contribution SAFT (GC-SAFT) with a k(ij) Correlation Method Based on London's Theory. 2. Application to Binary Mixtures Containing Aromatic Hydrocarbons, n-Alkanes, CO2, N2, and H2S[END_REF].

The estimated values of ij l , αβ u , and αβ w are presented in Table 4. Generally, the ij l values are estimated to be between -0.006 and -0.019 which correspond to an increase of the segment diameter ij σ in the range of 0.6% to 2%. Note that αβ u and αβ w are presented through the values of the cross-association parameters (

A B α β ε & A B α β K
) between the oxygenated families and water that are actually affected by these two correction parameters. The cross-association parameters of oxygenated family and water are calculated by using the combining rules as shown in eqs ( 11) & [START_REF] Orbey | Vapor-Liquid Equilibria: Cubic Equations of State and Their Mixing Rules[END_REF]. In order to evaluate the values of 4, we calculate the so-called association strength present in the association term of the PPC-SAFT EoS. Equation [START_REF] Wei | Equation of state for the calculation of fluid phase equilibria[END_REF] presents its expression as a function of temperature:

A B α β ε & A B α β K in Table
A B A B A B 3 hs ij ij D g K exp 1 kT α β α β α β     ε ∆ = -           (15)
where

A B α β
∆ is the association strength; k is Boltzmann factor; T is temperature; hs ij g is radial distribution function given according to Boubli'k [START_REF] Boublik | Hard Sphere Equation of State[END_REF] and Mansoori [START_REF] Mansoori | Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres[END_REF] as:

( ) ( ) ( ) 2 2 2 2 3 3 3 3 1 3 2 1 1 1 ij ij hs ij d d g ζ ζ ζ ζ ζ = + + - - - (16)
and ij D is an average diameter defined as

2 ii jj ij d d D + =
. Note that lij has no effect here because the average is performed on the temperature-dependent diameters.

In order to simplify the calculation (diameter and radial distribution function depend only slightly on volume), the association strength [START_REF] Kontogeorgis | Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories[END_REF] within an allowed temperature range. The deviation between the computed results and experimental data is expressed by using the standard error ( SAFT SE ) that was also used in our previous paper [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF].

n SAFT Corr 2 i i SAFT i=1 (logX -logX ) SE = n -1 ∑ (16)
where SAFT i

log X and

Corr i log X correspond to the values predicted by the GC-PPC-SAFT model and calculated from the correlations of the experimental data (eq (13)), respectively. Each property is evaluated at 11 different temperatures and therefore n = 11, except for OW K , for which data only exist at 298.15 K. In the latter case, the logarithmic absolute deviation between the experimental and predicted values is used. This definition is similar to the eq [START_REF] Escobedo-Alvarado | Study of EOS-G(ex) mixing rules for liquid-liquid equilibria[END_REF] with n = 1 and without the denominator.

We still call this deviation SAFT SE . The reason for using the logarithmic values in eq [START_REF] Escobedo-Alvarado | Study of EOS-G(ex) mixing rules for liquid-liquid equilibria[END_REF] is related to the fact that the data considered cover a very large range of orders of magnitude ranging from 10 0 to 10 In what follows, two different approaches are compared:

• the so-called "predictive" approach (called Case 1), which uses the group contribution pure component parameters, and ij k =0 (note that this is not the "predictive approach" of the original paper by Nguyen-Huynh [START_REF] Nguyen-Huynh | Modeling Liquid-Liquid and Liquid-Vapor Equilibria of Binary Systems Containing Water with an Alkane, an Aromatic Hydrocarbon, an Alcohol or a Gas (Methane, Ethane, CO2 or H2S), Using Group Contribution Polar Perturbed-Chain Statistical Associating Fluid Theory[END_REF][START_REF] Nguyen-Huynh | Modeling Phase Equilibria of Asymmetric Mixtures Using a Group-Contribution SAFT (GC-SAFT) with a k(ij) Correlation Method Based on London's Theory. 1. Application to CO2 + n-Alkane, Methane + n-Alkane, and Ethane + n-Alkane Systems[END_REF] that computes kij using pseudo-ionization energies, even though the results are almost identical). This approach was already presented in the previous paper [START_REF] Nguyen | GC-PPC-SAFT equation of state for VLE and LLE of hydrocarbons and oxygenated compounds. Sensitivity analysis[END_REF]. It should be noticed that in the "predictive" approach the crossassociation parameters ( • the " ij l , αβ u , and αβ w " approach (called Case 2), which uses the three parameters ij l , αβ u , and αβ w .

• In addition, whenever possible, these results are compared with the experimental uncertainties of the data (called Case 3). These have been obtained by evaluating the average deviation between the raw data and the correlation used in eq [START_REF] Kontogeorgis | Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories[END_REF]. More detail are available in our previous paper [START_REF] Nguyen | GC-PPC-SAFT equation of state for VLE and LLE of hydrocarbons and oxygenated compounds. Sensitivity analysis[END_REF].

Solubilities

Firstly, we present the evaluation results for solubilities which are also used in the parameter estimation of the set of ij l , αβ u , and αβ w (Case 2). Figure 3 and Figure 4 show Yet, these average values do not show the full picture, which can only be appreciated when considering the temperature dependence of these solubilities as shown in Figure 5. These figures show that the improved model almost always crosses the experimental trend. Yet, while according to the model both solubilities always increases with temperature, the experimental behaviour may be opposite. In the case of di-n-butyl ether, for example, while the solubility of water in the organic phase increases with temperature (this is the observed trend for all chemicals investigated), the solubility of the chemicals in water decreases with temperature. As discussed previously, the balance between cross-and self-association that is proposed by SAFT should make it possible to describe a solubility curve that exhibits a minimum. Yet, several phenomena make it difficult for SAFT to capture this behaviour. First, no such minimum is observed for the water solubility in the organic phase; second, the location of the minimum varies within a chemical family, probably because of experimental uncertainties. As a consequence, while the model predictive capacity has been clearly improved (it is capable of describing the correct order of magnitudes of both solubilities), a more fundamental investigation will be needed for improving the detailed shape of the curves.

Infinite dilution activity coefficient in water

On the basis of the regressed values, we now evaluate the predictive capability of the

n-octanol/water partition coefficient

Finally, we consider the n-octanol/water partition coefficient OW K , again using the same regressed values as discussed above (the values presented in Table 4), obtained from mutual solubilities. The results are presented in Figure 8 and in more detail in Table 6.

The best results are obtained when the previously regressed values are used. The global SAFT SE value is 0.469 (using the cross-association parameters), in comparison to 1.060 in the case of using no regressed values.

Figure 9 is the parity diagram for comparing measured and predicted OW K . The values are predicted by using " ij l + αβ u + αβ w " approach, excluding n-alcohols and nalkanes. The calculated values of n-alcohols and n-alkanes are taken directly from our previous work [START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF]. It should be noted that OW K of n-alcohols is calculated by using the cross-association of Nguyen-Huynh et al [START_REF] Nguyen-Huynh | Modeling Liquid-Liquid and Liquid-Vapor Equilibria of Binary Systems Containing Water with an Alkane, an Aromatic Hydrocarbon, an Alcohol or a Gas (Methane, Ethane, CO2 or H2S), Using Group Contribution Polar Perturbed-Chain Statistical Associating Fluid Theory[END_REF]. A good prediction is observed for almost all chemical families considered in this work. The results are also good for n-alcohols and n-alkanes having small molecular weight. However, larger deviations are observed for some heavy molecules, particularly n-aliphatic acids, alcohols and n-alkanes with more than 10 carbon atoms. Figure 10 shows the trends for these chemical families as a function of the number of carbon atoms, both according to the data and the model. It is expected that all trends converge as the number of carbon atoms increase because the functional group is then "diluted" in the aliphatic chain. This is observed for aliphatic acids and n-alkanes data, but not for the n-alcohols whose OW K is much smaller than the observations for n-aliphatic acids and n-alkanes.

In order to understand the observed deviations, the OW K behaviour can be analyzed using the definition of n-octanol/water partition coefficient, as shown in eq [START_REF] Ohta | Representation of liquid-liquid equilibria at low and high pressures using the EOS-GE mixing rules[END_REF].

∞ ∞ → γ = = = γ aq aq , aq Oct Oct i i i OW aq aq Oct Oct , Oct x 0 i i i i c x v v K lim c x v v (17)
Equation [START_REF] Ohta | Representation of liquid-liquid equilibria at low and high pressures using the EOS-GE mixing rules[END_REF] shows that several quantities affect directly the results:

♦ the infinite dilution activity coefficients in water aq , i ∞ γ , which we have analyzed above. The numerical values may be slightly because values used here originate from a liquid-liquid phase split calculation for a ternary mixture containing water, n-octanol and the chemical under investigation.

Hence, the water phase is not pure as before, it contains a small amount of n-octanol (molar fraction of n-octanol in water at 298.15 K = 2 ppm). The chemical is always considered at infinite dilution.

♦ the ratio of molar volumes This mistake will lead to an error of 21% on OW K , much less than that observed on Table 6. The calculations have been performed again with the experimental ratio, but no major change can be seen on the OW K results.

♦ the infinite dilution activity coefficient in n-octanol Oct , Figure 12 presents the comparison between the experimental, pseudoexperimental and calculated values for OW K of n-alkanes families.

It is found that the values calculated by the GC-PPC-SAFT model are very consistent with the pseudo-experimental data. There is a large deviation between the experimental data and the predicted values. On the basis of the results on Figure 10 and Figure 12, the large deviation observed on n-alkanes, n-aliphatic acids and alcohols seems to point to inaccuracy in the data. It should be recalled that OW K is often measured for evaluating the toxicity of a chemical in the environment ecosystems. The type of measurements and their conditions may be quite different from those considered in thermodynamic calculations. This may explain the observed disagreements, but more measurements will be needed in order to clarify the question.

Conclusion

The

(K) σ (Å) m (-) AB / k ε (K) AB K (-) sites n (-) charges (-) µ [D] p mx µ (-)
Water [START_REF] Nguyen-Huynh | Modeling Liquid-Liquid and Liquid-Vapor Equilibria of Binary Systems Containing Water with an Alkane, an Aromatic Hydrocarbon, an Alcohol or a Gas (Methane, Ethane, CO2 or H2S), Using Group Contribution Polar Perturbed-Chain Statistical Associating Fluid Theory[END_REF] 218. [START_REF] Gross | Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules[END_REF] 4), [START_REF] Huron | New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures[END_REF], and (6). c,d the values are calculated using the formulas proposed by Nguyen-Huynh et al. [START_REF] Nguyen-Huynh | Application of GC-SAFT EOS to polar systems using a segment approach[END_REF] e Cross-association parameters with water calculated by using eqs [START_REF] Vidal | Thermodynamics: Applications in Chemical Engineering and the Petroleum Industry[END_REF] and [START_REF] Orbey | Vapor-Liquid Equilibria: Cubic Equations of State and Their Mixing Rules[END_REF]. f charge zero means that the site may associate with any other site, be it zero, positive or negative.

Solubility

Temperature

Cross-association

No cross-association

Self-association

  the polar versions of SAFT, the polarity effect is explicitly taken into consideration. Hence, the ij ε parameter should represent only the London dispersion forces, which is why a Hudson & McCoubrey[65]-type theory (implying positive kij)

  of the molecular chain, developed with the assumption of infinite association strength. The parameter m representing the number of segments in one molecular chain is the parameter of this term. In the PC-SAFT EoS, the reference term is the sum of the first and the

  infinite dilution activity coefficient in water ( aq , i ∞ γ ), and n-octanol/water partition coefficient ( OW K ) are firstly collected for the purposes of regression and evaluation.

Figure 2

 2 shows the calculated cross-association strengths for 7 chemical families i.e. aliphatic ethers, aldehydes, ketones, formates, acetates, propionates/butyrates, and n-aliphatic acids. The values for n-alcohols, other aliphatic alcohols, n-alkylbenzenes are taken directly from the work of Nguyen-Huynh et al.[84] Figure 2 shows the trend of association energies as a function of the functional group: the order follows that of the expected association energies. The values of n-aliphatic acids are extremely large, testifying that the acids have a very strong interaction with water. Next are alcohols, aliphatic ethers, aldehydes, ketones and esters. The aliphatic ethers have a low interaction, probably because their lone electron pair is not well accessible, and n-alkylbenzenes have a very small interaction with water. OW K is carried out. The evaluation is performed by comparing the values computed by the model and the experimental values which are generated from eq

Figure 7 Figure 5 ,

 75 Figure7presents some examples for the prediction of aq , i

  solvents. In our calculation, this ratio is the result of the GC-PPC-SAFT calculation, which yields at

Figure 11 .

 11 of the predictions in this type of solvent, we have further investigated a number of alkane + n-octanol binary systems, as shown in It appears that the GC-PPC-SAFT results agree with experimental data. The deviations in logarithmic units are at most 0.1. We should note that the n-octanol phase contains a non-negligible amount of water ( 14% ≈), but we have tested the effect of this concentration on the alkane activity coefficient and found a very small effect through the reconstruction of "pseudo-experimental"

Figure 1 .

 1 Figure 1. Illustration of the mutual solubility behaviour of oxygenated compounds/water mixture.

Figure 2 .

 2 Figure 2. Association strength as a function of inverse temperature.

Figure 5 .Figure 6 .Figure 7 .StandardFigure 8 .

 5678 Figure 5. Some examples of the solubilities of water in organic phase org i x and that of compounds in

Figure 9 .

 9 Figure 9. Parity diagram of n-octanol/water partition coefficient OW K . The values are calculated by using " ij l + αβ u + αβ w " approach. The values of n-alcohols and n-alkanes are taken directly from our previous work[59].

Figure 10 .Figure 11 .

 1011 Figure 10. n-octanol/water partition coefficient OW K as a function of number of carbon atoms. Comparison of the experimental data (points) and the modelling results by the GC-PPC-SAFT model (lines) with the use of " ij l + αβ u + αβ w " approach for n-aliphatic acids.

Figure 12 .

 12 Figure 12. n-octanol/water partition coefficient OW K of n-alkanes as a function of number of carbon atoms. The experimental data are taken from references[93,97]. Notation: experimental data (triangles), pseudo-experimental data (solid squares), the calculated values without using " ij l + u αβ + w αβ " (solid line). To be able to calculate pseudo-experimental data

Table 2

 2 summarizes the number of molecules for which data of aq

	i x , aq , i γ , ∞

OW K are available. These data are re-used from our earlier work.

[START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF]

  11 . It should be noted that

					SE	SAFT	= 1 is equivalent to
	a factor of 10 in deviation between the measured and model calculated values. For
	small values,	SE	SAFT	values can be compared to the well-known relative deviations
	multiplying with 2.3 (e.g.	SE	SAFT	= 0.04 is equivalent with a relative deviation of 10%).

Table 1 .

 1 overall objective of this work is to evaluate the performance and especially the predictive capability of the GC-PPC-SAFT model for LLE-type properties ( org PPC-SAFT parameters for different chemical families a

	i x , aq i x ,

  correspond to dispersive energy, segment diameter, chain length, association energy, association volume, number of association sites, dipolar moment, dipolar fraction. b GC: the values are calculated by using eqs (

	formates	GC	GC	GC	1978.2 e	0.0175 e	1	-	1.9	1.20
	n-aliphatic acids	GC	GC	GC	6085.5	0.0010	1	0 f	1.5	1.00
	a More details can be found in the supplementary material (SAFTBANK.xls). Notations: / k ε , σ , m , AB / k ε	,
	AB K , sites n , µ ,
			3.3845	0.8096	1813	0.0356	4	++--	1.85	0.295
	n-octanol [59]	256.65	3.76	0.81	2143.3	0.0088	3	+--	1.7	0.5
	aliphatic ethers	GC b	GC	GC	1978.2 e	0.0175 e	1	-	1.2	1.20
	aldehydes	GC	GC	GC	1978.2 e	0.0175 e	1	-	c	0.60
	ketones	GC	GC	GC	1978.2 e	0.0175 e	1	-	2.7	0.57
	acetates	GC	GC	GC	1978.2 e	0.0175 e	1	-	d	1.15
	propionates/butyrates	GC	GC	GC	1978.2 e	0.0175 e	1	-	d	1.15

p mx µ

Table 2 . Summary of the number of molecules for which data are used in this study, per family. Data are taken from DETHERM database[82] and from other sources[85,92].

 2

	Chemical family	x	aq i	γ	aq, i	∞ ∞ ∞ ∞	K	OW
	acetates	13	13		5
	aldehydes	7		7		5
	aliphatic ethers	12	10		4
	formats	7		7		2
	ketones	18	18		11
	n-alcohols	13	14		11
	n-aliphatic acids	8		8		8
	other aliphatic alcohols	13	15		12
	propionates / butyrates	10	10		6
	Total	101	102		64

Table 3 . Summary of the experimental data for solubilities of water in oxygenated compounds. The data are taken from DETHERM database[82].

 3

	Chemical family	n° of molecules	n° of exp. points	range of carbon atoms	range of temperature
	Aliphatic ethers	11	127	4 -8	273.15 -368.15
	Aldehydes	7	74	3 -9	273.15 -363.15
	Ketones	14	186	5 -9	273.15 -453.15
	Formates	6	40	3 -9	273.15 -363.75
	Acetates	13	211	4 -10	273.15 -413.75
	Propionates/Butyrates	10	89	4 -9	273.15 -363.85
	n-aliphatic acids	4	23	5 -11	291.15 -545.65

Table 4

 4

	. Values of ij l and the cross-association parameters ( α β A B ε	and	K	α β A B	) for oxygenated
	compounds/water mixtures.
	Chemical family	ij (-) l	AB ε /k b [K]	AB K b (-)
	Aliphatic ethers	-0.01872	1978.15	0.0077
	Aldehydes	-0.00582	1905.90	0.0201
	Ketones	-0.01099	1699.50	0.0353
	Formates	-0.01348	1313.63	0.1066
	Acetates	-0.00943	1504.94	0.0744
	Propionates/Butyrates	-0.01722	1828.37	0.0208
	Aliphatic acids	-0.01886	4146.93	0.0034
	n-alcohols a	--	1978.15	0.0220
	Other aliphatic alcohols a	--	1978.15	0.0168
	n-alkylbenzenes a	--	1208.18	0.0189
	a the values of n-alcohols, other aliphatic alcohols, n-alkylbenzenes are
	taken, and hydrocarbons directly from the work of Nguyen-Huynh et
	al. .[37] b the values are calculated by using eqs (11) & (12). The
	regressed values of αβ u and αβ w are directly used in the calculation of
	AB ε /k and AB K .

Table 5 . Standard error ( SAFT SE ) per family for solubilities. of water in oxygenated organic phase org i x and solubilities of oxygenated compounds in water aq i x .

 5 The values are calculated by using the " αβ u + αβ w " values originating from the work of Nguyen-Huynh et al.[START_REF] Nguyen-Huynh | Modeling Liquid-Liquid and Liquid-Vapor Equilibria of Binary Systems Containing Water with an Alkane, an Aromatic Hydrocarbon, an Alcohol or a Gas (Methane, Ethane, CO2 or H2S), Using Group Contribution Polar Perturbed-Chain Statistical Associating Fluid Theory[END_REF] These calculated values were presented in our previous work[START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF].b The values are re-used from our previous work.[START_REF] Nguyen | GC-PPC-SAFT Equation of State for VLE and LLE of Hydrocarbons and Oxygenated Compounds. Sensitivity Analysis[END_REF] c The values are averaged without that of n-alcohols, n-alkanes, and n-alkylbenzenes.

	Family	Case 1	org i Case 2 x	Case 3	Case 1	aq i Case 2 x	Case 3
	Ethers	0.306	0.259	0.027	1.301	0.608	0.024
	Aldehydes	0.086	0.084	0.026	0.407	0.155	0.034
	Ketones	0.182	0.172	0.017	1.016	0.449	0.018
	Formates	0.110	0.143	0.013	0.911	0.371	0.033
	Acetates	0.156	0.116	0.020	1.154	0.617	0.015
	Propionates/Butyrates	0.132	0.118	0.018	1.141	0.228	0.009
	Aliphatic acids	0.405	0.248	0.033	1.646	0.257	0.043
	n-alcohols	--	--	--	0.529 a	--	0.100 b
	n-alkanes	--	--	--	0.229 a	--	0.063 b

a

Table 6 . Standard error ( SAFT SE ) per family for infinite dilution activity coefficient in water aq

 6

	γ	i	,	∞

c

The values are averaged without that of n-alcohols, n-alkanes, and nalkylbenzenes.

Standard error of solubilities of water in different oxygenated chemical families.

		0.45
		0.40
	Standard Error	0.15 0.20 0.25 0.30 0.35 -1.00	di-n-butyl ether
		0.10 -1.20
		0.05
		0.00 -1.40
		a l i p h a t i c e t h e r s -1.80 -1.60	a l d e h y d e s		k e t o n e s	f o r m a t e s	a c e t a t e s p r o p i o n a t e s / b u t y r a t e s n -a l i p h a t i c a c i d s
		-2.00
		270	290	310	330	350	370
	Figure 3. Notation: "predictive" approach (white column), " i j + αβ u + αβ w " approach (diagonal column), l Temperature (K)
	experimental data (black column).
		1.50
		1.20
	Standard Error	0.60 0.90
		0.30
		0.00
		a l i p h a t i c e t h e r s	a l d e h y d e s		k e t o n e s	f o r m a t e s	a c e t a t e s p r o p i o n a t e s / b u t y r a t e s	n -a l i p h a t i c a c i d s
	Figure 4. Standard error of solubility of oxygenated compounds in water. Notation: "predictive"
	approach (white column), ' ij l + αβ u + αβ w ' approach (diagonal column), experimental data (black
	column).

2 aq ,exp aq ,calc org ,exp org ,calc n n' i i i i aq ,exp org ,exp i 1 j 1 i i