Daniel Borgis
email: daniel.borgis@ens.psl.eu.fr

Sohvi Luukkonen

Luc Belloni

Guillaume Jeanmairet
email: guillaume.jeanmairet@sorbonne-universite.fr

Simple parameter-free bridge functionals for molecular density functional theory. Application to hydrophobic solvation

Keywords:

Computer simulations have been substantial in understanding the fine details of hydrophobic solvation and hydrophobic interactions. Alternative approaches based on liquid-state theories have been proposed, but are not yet at the same degree of completeness and accuracy. In this vein, a classical, molecular density functional theory approach to hydrophobic solvation is introduced. The lowest, second-order approximation of the theory, equivalent to the hypernetted chain approximation in integral equations, fails in describing correctly cavitation free-energies. It is corrected here by two simple, angular-independent, so-called bridge functionals; they are parameter-free in the sense that all variables can be fixed unambiguously from the water bulk properties, including pressure, isothermal compressibility, and liquid-gas surface tension. A hard-sphere bridge functional, based on the known functional of a reference hard fluid system, turns out to face strong limitations for water. A simpler weighted density approximation is shown to properly reproduce the solvation free energy of hydrophobes of various sizes, from microscopic ones to the nanoscale, and predicting the solvation free-energy of a dataset of more than 600 model hydrophobic molecules having a variety of shapes and sizes with an accuracy of a quarter of k B T compared to Monte-Carlo simulations values. It constitutes an excellent starting point for a general functional describing accurately both hydrophobic and hydrophilic solvation, and making it possible to study non-idealized hydrophobic interactions.

Introduction

Much of the physical chemistry of hydrophobic solvation has been understood from early statistical mechanics theories such as scaled-particle theory, and from the advent of molecular simulations in the 1980's [START_REF] Lee | The structure of liquid water at an extended hydrophobic surface[END_REF][START_REF] Zichi | The equilibrium solvation structure for the solvent-separated hydrophobic bond[END_REF][START_REF] Zichi | Solvent molecular dynamics in regions of hydrophobic hydration[END_REF] , and its subsequent developments [START_REF] Lee | A comparison of the structure and dynamics of liquid water at Hydrophobic and hydrophilic surfaces -A molecular dynamics simulation study[END_REF][START_REF] Cheng | Surface topography dependence of biomolecular hydrophobic hydration[END_REF][START_REF] Giovambattista | Hydration behavior under confinement by nanoscale surfaces with patterned hydrophobicity and hydrophilicity[END_REF][START_REF] Giovambattista | Hydrophobicity of protein surfaces: Separating geometry from chemistry[END_REF][START_REF] Giovambattista | Effect of temperature on the structure and phase behavior of water confined by hydrophobic, hydrophilic, and heterogeneous surfaces[END_REF][START_REF] Matysiak | Dissecting the energetics of hydrophobic hydration of polypeptides[END_REF][START_REF] Giovambattista | Computational Studies of Pressure, Temperature, and Surface Effects on the Structure and Thermodynamics of Confined Water[END_REF] . In particular, those simulations have contributed to show that for small hydrophobic solutes, hydration is characterized by a clathrate-like geometry for the solvent, driven by the large enthalpic penalties which would follow from a less ordered solvation layer [START_REF] Zichi | The equilibrium solvation structure for the solvent-separated hydrophobic bond[END_REF] . The molecular structure of liquid water around larger, nanoscale hydrophobic entities appears governed by the same principle that determines the hydration structure of small hydrophobic solutes, namely the optimization of hydrogen bonding interactions. Substantial structural reorganization occurs at the surface, where the sacrifice of possible H-bonds is required to maximize the total interaction.

The resulting orientational structure is inverted from that found for small solutes and the solvation free energy is dominated by enthalpic rather than entropic effects. [START_REF] Lee | The structure of liquid water at an extended hydrophobic surface[END_REF] It was shown further that the structure and free energy of hydrophobic hydration is strongly influenced by the detailed topography of the biomolecule surfaces that contain convex patches, deep or shallow concave grooves and roughly planar area. [START_REF] Lee | A comparison of the structure and dynamics of liquid water at Hydrophobic and hydrophilic surfaces -A molecular dynamics simulation study[END_REF][START_REF] Cheng | Surface topography dependence of biomolecular hydrophobic hydration[END_REF] Indeed, it has been thought for long that the hydrophobic effect plays a key role in the stability of protein structures, in the protein folding process, or in the aggregation of self-assemblies. The proteins contain hydrophobic regions which associate due to the favorable solvent-mediated free energy of aggregation of nonpolar moieties in an aqueous environment. This globally accepted picture has nevertheless given rise to hot debates at the turn of the 21 st century and during its first decade as to ascertain the details of the process. Be it a concerted scenario in which water is gradually reduced between the associating regions when the moieties approach each other, or a sequential one where a thermodynamic liquid-gas phase transition occurs in the interstitial region, leading to instantaneous hydrophobic collapse to contact [START_REF] Ten Wolde | Drying-induced hydrophobic polymer collapse[END_REF] . Simulations of water next or in between realistic hydrophobic surfaces patched with hydrophilic sites were substantial to show that the conditions for an ideal dewetting transition scenario is hard to achieve in real systems and that water density next to nanoscale hydrophobic patches can be influenced substantially by nearby hydrophilic motifs. [START_REF] Giovambattista | Hydration behavior under confinement by nanoscale surfaces with patterned hydrophobicity and hydrophilicity[END_REF][START_REF] Giovambattista | Hydrophobicity of protein surfaces: Separating geometry from chemistry[END_REF][START_REF] Giovambattista | Effect of temperature on the structure and phase behavior of water confined by hydrophobic, hydrophilic, and heterogeneous surfaces[END_REF] On the theoretical side, the Gaussian field theory of hydrophobicity initiated by the 1999 paper of Lum, Weeks and Chandler [START_REF] Lum | Hydrophobicity at Small and Large Length Scales[END_REF] , and further developed in subsequent articles [START_REF] Ten Wolde | Drying-induced hydrophobic polymer collapse[END_REF][START_REF] Huang | Cavity formation and the drying transition in the Lennard-Jones fluid[END_REF][START_REF] Ten Wolde | Model of a fluid at small and large length scales and the hydrophobic effect[END_REF][START_REF] Varilly | An improved coarse-grained model of solvation and the hydrophobic effect[END_REF] , has driven the view that nanoscale hydrophobic attraction is induced by a local gas-liquid phase transition. This theory is based on the fluctuations of the density field n(r) and, to our knowledge, was never fully developed to include its coupling to the electrostatic polarisation field [START_REF] Song | Gaussian Field Model of Dielectric Solvation Dynamics[END_REF] . A unified field theoretical approach of this sort would be able to describe both hydrophobic and hydrophilic interactions and could be applied to the hydrophobic/hydrophilic patchy surfaces described above.

Another possible theoretical approach is 3D-RISM, a method that has emerged from a consistent formulation of integral equation theory for molecular liquids and solutions [START_REF] Hirata | An extended rism equation for molecular polar fluids[END_REF][START_REF] Hirata | Application of an extended RISM equation to dipolar and quadrupolar fluids[END_REF][START_REF] Pettitt | Integral equation predictions of liquid state structure for waterlike intermolecular potentials[END_REF][START_REF] Hirata | Application of an extended RISM equation to dipolar and quadrupolar fluids[END_REF][START_REF] Cortis | A three-dimensional reduction of the Ornstein-Zernicke equation for molecular liquids[END_REF] .

That approach is gaining nowadays considerable momentum to describe the solvation properties of complex molecular entities at a numerical cost far below that of molecular simulations, giving access to properties that are as hardly accessible by direct simulations [START_REF] Cortis | A three-dimensional reduction of the Ornstein-Zernicke equation for molecular liquids[END_REF][START_REF] Beglov | An Integral Equation to Describe the Solvation of Polar Molecules in Liquid Water[END_REF][START_REF] Kovalenko | Three-dimensional density profiles of water in contact with a solute of arbitrary shape; a RISM approach[END_REF][START_REF] Kovalenko | Hydration free energy of hydrophobic solutes studied by a reference interaction site model with a repulsive bridge correction and a thermodynamic perturbation method[END_REF][START_REF] Chuev | Improved estimates for hydration free energy obtained by the reference interaction site model[END_REF][START_REF] Palmer | Toward a universal model to calculate the solvation thermodynamics of druglike molecules: The importance of new experimental databases[END_REF][START_REF] Truchon | A cavity corrected 3D-RISM functional for accurate solvation free energies[END_REF][START_REF] Misin | Predicting Solvation Free Energies Using Parameter-Free Solvent Models[END_REF][START_REF] Nguyen | A molecular reconstruction approach to site-based 3D-RISM and comparison to GIST hydration thermodynamic maps in an enzyme active site[END_REF] .

The method, however, was shown recently to face strong limitations when describing the hydration properties of nanosized hydrophobic solutes [START_REF] Fujita | Assessing the accuracy of integral equation theories for nanosized hydrophobic solutes in water[END_REF] .

The purpose of this work is to propose an alternative approach to either 3D-RISM [START_REF] Cortis | A three-dimensional reduction of the Ornstein-Zernicke equation for molecular liquids[END_REF][START_REF] Beglov | An Integral Equation to Describe the Solvation of Polar Molecules in Liquid Water[END_REF][START_REF] Kovalenko | Three-dimensional density profiles of water in contact with a solute of arbitrary shape; a RISM approach[END_REF][START_REF] Kovalenko | Hydration free energy of hydrophobic solutes studied by a reference interaction site model with a repulsive bridge correction and a thermodynamic perturbation method[END_REF][START_REF] Chuev | Improved estimates for hydration free energy obtained by the reference interaction site model[END_REF][START_REF] Palmer | Toward a universal model to calculate the solvation thermodynamics of druglike molecules: The importance of new experimental databases[END_REF][START_REF] Truchon | A cavity corrected 3D-RISM functional for accurate solvation free energies[END_REF][START_REF] Misin | Predicting Solvation Free Energies Using Parameter-Free Solvent Models[END_REF][START_REF] Nguyen | A molecular reconstruction approach to site-based 3D-RISM and comparison to GIST hydration thermodynamic maps in an enzyme active site[END_REF] , or Gaussian field theory, namely classical density functional theory in its molecular version, MDFT [START_REF] Gendre | Classical density functional theory of solvation in molecular solvents: Angular grid implementation[END_REF][START_REF] Borgis | Molecular density functional theory: Application to solvation and electron-transfer thermodynamics in polar solvents[END_REF][START_REF] Zhao | Molecular density functional theory of solvation: From polar solvents to water[END_REF][START_REF] Jeanmairet | Molecular Density Functional Theory of Water[END_REF][START_REF] Ding | Efficient molecular density functional theory using generalized spherical harmonics expansions[END_REF][START_REF] Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF] . Shortly, MDFT allows to compute the solvation free-energy and the equilibrium solvent structure around a given solute by minimizing a functional of the solvent density ρ(r, ω). In the most general case, the solvent density depends on the space coordinates r and on the orientational ones ω. In this paper, we consider only pure hydrophobes, described by Lennard-Jones potentials and no partial charges, so that it is possible to stay at the level of the current Lum-Weeks-Chandler theory and its extensions and consider the orientationally averaged density field n(r) = dω ρ(r, ω). We know how to extend the theory to the polarisation field, and in fact to the whole orientational density field ρ(r, ω) in a numerically efficient way [START_REF] Ding | Efficient molecular density functional theory using generalized spherical harmonics expansions[END_REF] , but this will be put in a future work.

The question raised here is how to go beyond the so-called HNC approximation which we have studied thoroughly in a recent article [START_REF] Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF] . The weaknesses of the HNC approximation have been known for long [START_REF] Rickayzen | Integral equations and the pressure at the liquid-solid interface[END_REF] : it suffers from the original sin of being a quadratic theory around the liquid bulk density. The homogeneous free energy as a function of density presents a single instead of a double well; it cannot accommodate for the liquid-gas transition and consequently it largely overestimates the cost of creation of an empty cavity (a gas bubble in the bulk). This has to be cured by pushing the theory beyond second order and introducing the so-called bridge functional which, by definition, starts at cubic order in density perturbation [START_REF] Levesque | Scalar fundamental measure theory for hard spheres in three dimensions: Application to hydrophobic solvation[END_REF][START_REF] Jeanmairet | Molecular density functional theory of water describing hydrophobicity at short and long length scales[END_REF][START_REF] Jeanmairet | Molecular density functional theory for water with liquid-gas coexistence and correct pressure[END_REF][START_REF] Gageat | Bridge functional for the molecular density functional theory with consistent pressure and surface tension and its importance for solvation in water[END_REF] . We will explore here what we refer to as parameter-free approaches, in the sense that all the parameters appearing in the expression of the bridge functional are determined unambiguously from the properties of the bulk solvent (pressure, isothermal compressibility, liquid-gas surface tension). The resulting functional is approximate but universal, that is independent of the solute; no parameter has to be subsequently fitted on the dataset or part of the dataset that is studied.

The outline of the paper is as follows. Section 2 exposes the theoretical background.

In section 2.1 we start by a general consideration about MDFT and the properties of the bridge functional which need to be fulfilled. In section 2.2, we examine a hard-sphere bridge (HSB) functional that has been proposed as a "universal" bridge functional in the past. In section 2.3, inspired by our previous works, we propose a simple bridge functional based on a weighted density approximation (WDA) which appears more flexible than HSB or than a previous WDA version of ours [START_REF] Jeanmairet | Molecular density functional theory of water describing hydrophobicity at short and long length scales[END_REF] . In section 3, as application, we depart from idealized spherical solutes and we compute the hydration free energies of a database of 642 hydrophobic molecules of arbitrary shape using the WDA bridge.

Theory

General considerations

We recall briefly the well-established results of molecular density functional theory: the free energy of solvation of a molecule can be written as

∆G solv = min (F id [ρ] + F ext [ρ] + F exc [ρ]) , (1)
where F id , F ext , and F exc are the ideal, external and excess functionals of the solvent position and orientation density, ρ(r, ω). The notation ω stands for the three Euler angles that describe the orientation of a rigid body in three dimensions. The whole MDFT procedure can be summerized as how to find the spatial and angular density that minimizes the sum of the three contributions. Knowing the functionals, the density at minimum, by virtue of the theorems derived by Mermin [START_REF] Mermin | Thermal Properties of the Inhomogeneous Electron Gas[END_REF] and Evans [START_REF] Evans | The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids[END_REF] , is also the thermodynamic equilibrium density of the solvent around the solute ρ(r, ω). As far as the various contributions are concerned, we have (i) the ideal contribution, that is known exactly, (ii) the external contribution that is the direct cost of the interaction of the solute with the solvent density. The interaction energy φ (r, ω) is typically given by classical force fields like SPC/E water for the solvent and, e.g., OPLS for the solute. (iii) the excess contribution which, as usual in liquid state theories, is known analytically as an infinite diagrammatic resummation of virial diagrams, but is not tractable numerically. It can be approximated by a Taylor expansion around the homogeneous bulk solvent density, ρ b ≡ n b /8π 2 with n b the number density. Truncation at second order in ∆ρ(r, ω) = ρ(r, ω) -ρ b yields the so-called hyper-netted chain (HNC) approximation and higher orders are gently put into the so-called bridge functional, F B .

More explicitly:

F id [ρ] = k B T drdω ρ (r, ω) log ρ (r, ω) ρ 0 -ρ (r, ω) + ρ 0 , (2)
F ext [ρ] = drdωρ (r, ω) φ (r, ω) , (3)
F exc [ρ] = F HN C [ρ] + F B [ρ] (4) = - k B T 2 drdω∆ρ (r, ω) dr dω c (|r -r |, ω, ω ) ∆ρ (r , ω ) + F B [ρ], (5)
where k B is the Boltzmann constant, T is the temperature. c is the direct correlation function of the bulk solvent, which depends upon the relative position of two solvent molecules and their respective orientations. It is an input of the theory that needs to be calculated once per solvent and thermodynamic condition. We use the ones obtained by Belloni for either the transferable intermolecular potential 3-points (TIP3P) or the extended simple point charge (SPC/E 44 ) models of water, using a mixture of MC calculations and integral equations to invert the simulation data and make the asymptotic behaviours well-defined. [START_REF] Belloni | Exact molecular direct, cavity, and bridge functions in water system[END_REF] Even in its simplest HNC form, the efficient calculation of the excess term is a numerical challenge because of the spatial and angular convolution. For this we use expansions of both the density and the direct correlation function onto generalized spherical harmonics; all technicalities are described in Ref. [START_REF] Ding | Efficient molecular density functional theory using generalized spherical harmonics expansions[END_REF] F B [ρ] involves by definition terms of order ∆ρ 3 and higher.

The purpose of this paper is to explore the simplest, conceivable approximations for the bridge functional, which amount to neglect the angular dependence and suppose that F B depends only on the number density n(r) = dω ρ(r, ω), that is to say

F B [ρ] = F B [n].
Various bridge functionals of this sort have been proposed in the past, built in particular from the fundamental measure theory of hard-sphere fluids [START_REF] Rosenfeld | Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing[END_REF][START_REF] Kierlik | Free-energy density functional for the inhomogeneous hard-sphere fluid: Application to interfacial adsorption[END_REF][START_REF] Roth | Fundamental measure theory for hard-sphere mixtures revisited: the White Bear version[END_REF][START_REF] Yu | Structures of hard-sphere fluids from a modified fundamental-measure theory[END_REF] The criteria that make it challenging to build a bridge functional for molecular fluids are: (i) to predict correct solvation free energies and structures, and (ii) to remain thermodynamically consistent, i.e., describe correctly the solvation thermodynamic properties of the solvent, in particular its pressure acting on solutes, its compressibility, and its liquid-gas surface tension. As of today, no bridge functional designed for water, including ours [START_REF] Levesque | Scalar fundamental measure theory for hard spheres in three dimensions: Application to hydrophobic solvation[END_REF][START_REF] Jeanmairet | Molecular density functional theory of water describing hydrophobicity at short and long length scales[END_REF][START_REF] Jeanmairet | Molecular density functional theory for water with liquid-gas coexistence and correct pressure[END_REF][START_REF] Gageat | Bridge functional for the molecular density functional theory with consistent pressure and surface tension and its importance for solvation in water[END_REF] , has proved to fulfil thoroughly and convincingly those two criteria.

We will focus in this paper on hydrophobic solutes described as a distribution of atomic sites with no partial charges, so that for water models such as TIP3P or SPC/E, built with a single Lennard-Jones center on oxygen, the angular dependence of F ext in eqn. 3 disappears.

We will furthermore neglect the orientational couplings in the excess functional and consider a simplified angular-independent functional

F[n] = k B T dr n(r) log n(r) n b -n(r) + n b + drΦ LJ (r) n(r) - k B T 2 drdr ∆n(r) c s (|r -r |)∆n(r ) + F B [n(r)], (6)
where c s (r) is the spherically-symmetric direct correlation function. We define that function from the bulk water-water isotropic pair-correlation function, h s (r), or equivalently from the density-density structure factor χ s (r) by the following relation in Fourier space:

1 -n b ĉs (q) = (1 + n b ĥs (q)) -1 = χs (q) -1 . (7)
For an homogeneous fluid of volume V with no external constraints, this functional becomes a function of the bulk density n which can be written, per molecule, as

βF (n)/N = f (x) = f HN C (x) + f B (x) (8)
= x log x -x + 1 - 1 2 n b ĉs (0)∆x 2 + f B (x), (9)
with x = n/n b and ĉs (0) the zeroth-order spherical Hankel transform of c s (r) taken at q = 0.

f B (x) is now a dimensionless bridge function of order at least ∆x 3 . As we noted several times, this simple expression already imposes strong restrictions on the possible forms of f B .

Indeed for an empty macroscopic volume V , n = x = 0, one should have F [n] = P V , P the pressure of the fluid, 1 atm at ambient conditions, i.e., virtually P = 0 for since P V 0 for volumes below micrometer sizes. The HNC approximation, corresponding to a f (x) that is an almost quadratic function centered at x = 1 as depicted in Fig. 1-a, yields a largely overestimated value at x = 0, related to the pressure

βP HN C /n b = f HN C (0) = 1 - 1 2 n b ĉs (0). ( 10
)
This overestimation is largely responsible for the deficiencies of HNC in predicting solvation free energies, and lead to the definition of so-called a posteriori pressure corrections [START_REF] Misin | Predicting Solvation Free Energies Using Parameter-Free Solvent Models[END_REF][START_REF] Sergiievskyi | Fast Computation of Solvation Free Energies with Molecular Density Functional Theory: Thermodynamic-Ensemble Partial Molar Volume Corrections[END_REF][START_REF] Sergiievskyi | Solvation free-energy pressure corrections in the three dimensional reference interaction site model[END_REF] .

This deficiency has to be compensated by the bridge term, through the condition

f B (0) = -f HN C (0) = -1 + 1 2 n b ĉs (0). (11)
This leads to an overall function f (x) depicted in Fig. 1-a and presenting a double-well, with equal minima at x = 0 and x = 1. Such picture is indeed at the heart of the Gaussian Field theory of hydrophobicity developed along the years by Chandler and collaborators [START_REF] Lum | Hydrophobicity at Small and Large Length Scales[END_REF][START_REF] Huang | Cavity formation and the drying transition in the Lennard-Jones fluid[END_REF][START_REF] Varilly | An improved coarse-grained model of solvation and the hydrophobic effect[END_REF] .

Note that the condition above can be related to the value of bulk liquid isothermal com- [START_REF] Hansen | Theory of Simple Liquids: With Applications to Soft Matter[END_REF] , we get

pressibility χ T . Using n b k B T χ T = 1/(1 -n b ĉs (0))
f B (0) = - 1 2 (1 + 1/n b k B T χ T ) . (12)
With these conditions in hand, we turn now to the examination of several suggestions reported in the literature for approximating F B [∆n], including our own proposals. The different approximations will be tested below for three systems for which reference simulation results are available: 1) The solvation of a hydrophobic sphere of increasing radius going from microscopic to macroscopic dimensions; this is the paradigmatic system for either the standard scaled particle theory [START_REF] Ashbaugh | Colloquium: Scaled particle theory and the length scales of hydrophobicity[END_REF][START_REF] Floris | Modeling the cavitation free energy[END_REF] or the more recent advances in the theory of hydrophobicity and hydrophobic interactions [START_REF] Huang | Scaling of hydrophobic solvation free energies[END_REF][START_REF] Huang | The hydrophobic effect and the influence of solute-solvent attractions[END_REF] .

2) The solvation of a spherical Lennard-Jones particle of increasing diameter up to nanoscale sizes.

3) The hydration free energies of a dataset of more than 600 realistic although virtual hydrophobic molecules of various shapes and sizes, going up to 40 atomic sites.

Hard-sphere bridge functional

Following the recommendation of Rosenfeld that the fundamental measure theory of hardsphere fluids should provide a "universal" hard-sphere bridge (HSB) functional applicable for general fluids [START_REF] Rosenfeld | Theory of simple classical fluids: Universality in the short-range structure[END_REF][START_REF] Rosenfeld | Free energy model for inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas[END_REF][START_REF] Kahl | Structure and thermodynamics of binary liquid mixtures: Universality of the bridge functional[END_REF] , Wu and collaborators [START_REF] Zhao | New Theoretical Method for Rapid Prediction of Solvation Free Energy in Water[END_REF][START_REF] Zhao | Corrections to New Theoretical Method for Rapid Prediction of Solvation Free Energy in Water[END_REF][START_REF] Liu | A Site Density Functional Theory for Water: Application to Solvation of Amino Acid Side Chains[END_REF][START_REF] Liu | High-throughput prediction of the hydration free energies of small molecules from a classical density functional theory[END_REF][START_REF] Li | A Multiscale Procedure for Predicting the Hydration Free Energies of Polycyclic Aromatic Hydrocarbons[END_REF] , and later ourselves [START_REF] Levesque | Scalar fundamental measure theory for hard spheres in three dimensions: Application to hydrophobic solvation[END_REF] , proposed to use such bridge functional for water. It has the form

F B [n] = F HS exc [n] -F HS exc [n b ] -µ HS exc (n b ) dr∆n(r) + k B T 2 drdr ∆n(r) ∆n(r ) c HS s (|r -r |; n b ). (13)
The first two term correspond to the fundamental measure theory (FMT) hard sphere functional taken at the inhomogeneous density n(r) and homogeneous density n b , respectively. We considered the scalar version of FMT derived by Kierlik and Rosinberg leaving only terms of order ∆n(r) 3 and higher. The unknown bridge is thus mimicked by that of an equivalent hard-sphere system at the same density. There is one single adjustable parameter to play with: the hard-sphere radius r hs . According to our discussion above, it has to be determined by the condition in eq.11, which leads, according to our criteria, to a parameter-free theory.

Writing the equivalent of eq. 13 for the homogeneous fluid, and using the textbook

results 52 f HS exc (x) = log(1 -η) + 3η 1 -η + 3η 2 2(1 -η) 2 (PY) (14)
= η(4 -3η) (1 -η) 2 (CS) (15)
for f HS exc (x) = βF HS exc (n)/N in either the PY or CS approximation (here η = η b x and η b = 4πr 3 hs n b /3), it is a matter of simple algebra to derive the expression of f HS B (x) from eqn. 13, and get the resulting f (x), eq. 9. One can then plot the dimensionless pressure βP/n b = f (0)

as a function of the reference hard-sphere fluid radius in order to identify the value of r hs for which the pressure vanishes. This is represented in Fig. 2 We computed the solvation free-energy of hard-spheres of different radius R using the HS bridge functional. For microscopic spheres, of radii up to ≈ 5 Å, results are compared to MC simulations obtained by Hummer et al when they tested their information theory of hydrophobicity 67 as displayed in Figure 3. In Figure 4, results are compared to MD simulations of Huang and Chandler 55 for spheres up to R = 14 Å, reaching the nanoscale.

For the value r hs = 1.4576 Å that yields correct pressure, the hydrophobic solvation freeenergy appears considerably underestimated on the whole microscopic to nanoscopic range.

In Fig. 4 it can be seen that the solvation free energy per surface area, ∆G solv /4πR 2 , correctly reaches a plateau for large R's, giving a value for the liquid-vapor surface tension γ (since ∆G solv = γA + P V in the macroscopic limit, the volumetric term being negligible for P = 1 atm and volumes of nanoscale sizes). The measured value is around 22.5 mJ/m 2 , thus far below the value of 63.6 mJ/m 2 quoted for SPC/E water [START_REF] Vega | Surface tension of the most popular models of water by using the test-area simulation method[END_REF] . One can indeed fit the correct solvation free energy at small R by taking a slightly smaller parameter r hs = 1.42 Å; see Fig. 3. But doing so one gets a considerable overestimation at larger radii above 5 Å, and, as can be seen in Fig. 4, the solvation free energy per surface area does not reach a plateau, as it should. It behaves asymptotically as P R/3 with a pressure that is already far from 0, as can be estimated from Fig. 2. Figure 3: Hydration free-energy of a hard-sphere solute as a function of its radius R. The bullets points are the simulation results of Hummer et al [START_REF] Hummer | An information theory model of hydrophobic interactions[END_REF] . The MDFT results concerns the hard-sphere bridge functional of eqn. 13 in its parameter-free version with r hs = 1.4576 Å, or with a slightly smaller radius r hs = 1.42 Å that matches the simulation results at small R, or the parameter-free WDA bridge functional of eqn. 16 adjusted on pressure, isothermal compressibility, and liquid-gas surface tension of the bulk solvent.

0 1 2 3 4 5 6

R(Å)

This deficiency observed for hard cavities extends to solute presenting attractive forces. In Fig. 5, we compare the MDFT-HSB results to the simulation results obtained by Fujita et al [START_REF] Fujita | Assessing the accuracy of integral equation theories for nanosized hydrophobic solutes in water[END_REF] for a Lennard-Jones sphere of diameter σ increasing up to σ = 18 Å ( is constant and taken equal to 0.5 kJ/mol). Again the hard-sphere parameter r hs that yields the correct pressure leads to a strong underestimation over the whole range of σ, whereas the smaller parameter r hs = 1.42 Å determined above does improve the results for small spheres, but leads to a diverging curve afterwards. Beside the fact that playing with the hard-sphere radius r hs contradicts the parameter-free strategy announced in the title, we found no convincing results when trying to fix r hs between the two preceding bounds.

We conclude that section by the ascertainment that a hard-sphere bridge functional, although being quite appropriate in the case of Lennard-Jones fluids, has limitations when extended to water. This is at least the case in our MDFT approach, here limited to an angular independent functional, and where the second order is fixed by the exact direct correlation function of the bulk fluid, yielding the correct isothermal compressibility. We thus have to turn to another approach.

Weighted density bridge functional

The well-founded fundamental measure theory of hard-sphere fluids is rooted in the socalled weighted density approximations (WDA) that go back to the foundations of classical DFT [START_REF] Nordholm | Generalized van der Waals theory. III. The prediction of hard sphere structure[END_REF][START_REF] Tarazona | Free-energy density functional for hard spheres[END_REF][START_REF] Curtin | Weighted-density-functional theory of inhomogeneous liquids and the freezing transition[END_REF] . In a recent contribution [START_REF] Gageat | Bridge functional for the molecular density functional theory with consistent pressure and surface tension and its importance for solvation in water[END_REF] , we have proposed to use a more empirical, but more flexible version of WDA than FMT that is able to fulfil more physical conditions than only equation 11. and we have introduced a bridge functional in the form where n(r) is a weighted density using a Gaussian weighting function

F B [n] = k B T n b drf B (n(r)/n b ), (16)
n(r) = dr w(|r -r |) n(r ) (17)
w(r) = (2πσ 2 w ) -3/2 exp -r 2 /2σ 2 w . (18)
The bridge function f B is defined again as a function of x = n/n b and ∆x = x -1

f B (x) = a ∆x 3 + b x 2 ∆x 4 0 < x < 1 = a ∆x 3 x > 1, (19)
where the second condition prevents spurious behaviours of the function at large x. Note that the preceding formulas are riminescent of those employed by Chandler and collaborators in their Gaussian field approach of hydrophobicity. They are also of similar essence as a WDA approach that we proposed previously, based on a more complex, three-body expression of [START_REF] Jeanmairet | Molecular density functional theory for water with liquid-gas coexistence and correct pressure[END_REF] . The approach here has the merit of simplicity. In the expression of f B (x), the value of a is fixed by the condition of eqn. 11 at x = 0, yielding a = 1 -1 2 n b ĉs (0). The role of the parameter b is to control the height of the double-well barrier of the total free energy function f (x) (eq. 9) without altering the well-depth equality condition. This is depicted in Fig. 1-b. There are thus two parameters to be fixed, the coarse-graining length σ w and b.

F B [n]
In regard to the hard-sphere bridge parametrisation above, σ w should roughly correspond to a coarse-graining over the size of a water molecule, with an associated radius r hs 1.4 Å.

Equating Gaussian and hard sphere volumes leads to σ w (2/3 √ 2π) 1/3 r hs = 0.9 Å. The second parameter b can then be determined to yield the correct liquid-vapor surface tension for the selected water model. This is done by adjusting the asymptotic value obtained for the solvation free energy per surface area of a hard sphere to the correct surface tension, i.e. Note that we have 2 parameters, b and σ w to reproduce a sole physical property, the surface tension γ. This leaves some flexibility that somehow contradicts the parameterfree claim of the title. Actually, we find that varying σ w around the above value while readjusting b to reproduce the correct surface tension has very little effect on the computed solvent structure and only small consequences on the hydration free energies. It cannot be denied, however, that even with this procedure, σ w remains as a fine adjustment cursor.

As seen in Figs. 3 and4 for SPC/E, not only the asymptotic, macroscopic behaviour of the HS solvation is reproduced properly this way, but also the whole microscopic to nanoscopic range where the 'exact' simulation results of Hummer et al [START_REF] Hummer | An information theory model of hydrophobic interactions[END_REF] and Huang and Chandler 55 apply. This excellent and almost miraculous agreement extends to the solvation of the Lennard-Jones sphere of microscopic to nanoscopic size in Fig. 5. Other important features of hydrophobic solvation have been shown to be correctly reproduced, be it with our previous WDA version in Ref. [START_REF] Jeanmairet | Molecular density functional theory for water with liquid-gas coexistence and correct pressure[END_REF] or with the present one in Ref. [START_REF] Gageat | Bridge functional for the molecular density functional theory with consistent pressure and surface tension and its importance for solvation in water[END_REF] This applies to the solvent pair-distribution function around spherical hydrophobic solutes or to the decreasing contact values of the pair distribution function around hydrophobic spheres as a function of sphere radius, leading to depletion at very large R. For those two cases, the present weighted density bridge functional yields equivalently good results and does considerably improve the HNC ones.

3 Results Same in the HNC approximation with an a-posteriori pressure correction; Results taken from Ref. [START_REF] Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF] The mean absolute error is 0.35, 0.25, and 0.7 k B T in panels (a) to (c), respectively.

We depart here from idealized spherical entities and turn to a more realistic problem: the hydration properties of hydrophobic solutes of arbitrary 3D-shapes. As a model to this, we consider the calculation of the non-electrostatic contribution to the hydration free-energy of drug-like molecules, a quantity that is systematically computed in free-energy perturbation methods (growing slowly the Lennard-Jones potential before introducing the charges), and that is a main source of error when using MDFT-HNC. We refer to the Mobley's dataset, containing 642 drug-like molecules for which the experimental solvation free-energies, and more importantly here, the solvation free-energies computed by simulation and decomposed into their non-electrostatic and electrostatic contributions, are available [START_REF] Mobley | Small Molecule Hydration Free Energies in Explicit Solvent: An Extensive Test of Fixed-Charge Atomistic Simulations[END_REF][START_REF] Matos | Approaches for Calculating Solvation Free Energies and Enthalpies Demonstrated with an Update of the FreeSolv Database[END_REF] . We have generated ourselves our own reference data with the same force-field parameters as in Mobley's FEP calculations but with fixed solute geometries in order to compare directly with MDFT.

This was done using an original MC/4D-hydrid method and associated code developed by one of us [START_REF] Belloni | Non-equilibrium hybrid insertion/extraction through the 4th dimension in grand-canonical simulation[END_REF] . This amounts to test our approach on a dataset of virtual hydrophobic molecular solutes with a large variety of shapes and sizes. In Fig. 6-a we compare the MC results obtained for the non-electrostatic contribution to the solvation free-energy (all partial charges put to zero) to the same quantities obtained with MDFT in its angular-independent version, eqn. 6, and using the WDA bridge functional defined above. Anticipating future developments, we have also tested the angular-dependent version, eqs. 2-5, which accounts for the density/polarisation couplings that exists even for non-polar solutes, even though they are small. We have used an angular grid of 84 orientations per spatial point, corresponding to a maximum order m max = 3 in a generalized spherical harmonics expansion [START_REF] Ding | Efficient molecular density functional theory using generalized spherical harmonics expansions[END_REF] . A single minimization using a box length of 24 Å and a resolution of 0.3 Å takes a few seconds on a single CPU for the angular-independent case, m max = 0, and a few minutes for m max = 3. The correlation between simulation and MDFT is already found excellent in the angular-independent case with a mean slope very close to 1 and an overall mean absolute error (MAE) of 0.35 k B T (Fig. 6-a). The MAE is decreased to 0.25 k B T when using the full angular-dependent functional (with a slight re-adjustment of the parameter b to stick to the correct value of the surface tension γ); see Fig. 6-b. This is much better than that was obtained previously using the HNC approximation and the best so-called pressure corrections to date. As seen in Fig. 6-c, this lead to a mean slope of 0.55 and an overall MAE of 0.7 k B T , which is already satisfactory compared to the non-corrected HNC approximation [START_REF] Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF] . The pressure correction just mentioned amounts to correct the HNC free energies by an optimized term -0.86P HN C ∆V , where P HN C is the pressure defined in eqn. 10 and ∆V is the measured partial molar volume of the solute. This correction turns out to be very close to the so-called parameter-free PC+ correction which we have introduced some time ago for MDFT 50 and 3D-RISM [START_REF] Sergiievskyi | Solvation free-energy pressure corrections in the three dimensional reference interaction site model[END_REF] , and which is now used routinely in 3D-RISM free-energy calculations [START_REF] Misin | Predicting Solvation Free Energies Using Parameter-Free Solvent Models[END_REF][START_REF] Roy | Performance of 3D-RISM-KH in Predicting Hydration Free Energy[END_REF] . It should be noted that such simple correction formula applies only in a limited range of ∆V values and cannot be true at the nanoscale, where the solvation free energy should go as the surface and not as the volume [START_REF] Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF] . In any case, it is rewarding to find that, at least in the hydrophobic case, a parameter-free bridge functional approach bypasses the need for such a-posteriori corrections while yielding more accurate results.

Conclusions

Going further than above, i.e. computing also the electrostatic contribution when the charges are switched on, will require to consider the angular-dependent functional of eqs. 2-5. This bypasses the goal of the present paper which was intended to show that even a simpler and computationally much more efficient angular-independent functional as in eqn. 6, combined with the proper angular-independent bridge functional F B [n], is able to capture the main physical features of hydrophobic solvation, at the same time as it provides estimates of hydrophobic solvation free energies with chemical accuracy, a fraction of k B T , compared to state-of-the-art simulation methods. This is made possible using a rather simple weighted density bridge functional yielding liquid-gas coexistence and built from bulk solvent properties such as isothermal compressibility and liquid-gas surface tension. We found that the reputed hard sphere bridge functional proposed by Rosenfeld for general fluids [START_REF] Rosenfeld | Free energy model for inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas[END_REF][START_REF] Kahl | Structure and thermodynamics of binary liquid mixtures: Universality of the bridge functional[END_REF] turns out to be a dead end in our case. The value of the hard-sphere parameter which matches the correct pressure and liquid-gas coexistence yields a poor value of the surface tension, and systematically underestimates the solvation free energy of hydrophobes at all length scales.

Adjusting this value to reproduce the solvation free-energies of microscopic solutes, as we did in the past [START_REF] Levesque | Scalar fundamental measure theory for hard spheres in three dimensions: Application to hydrophobic solvation[END_REF] , leads to an overestimated pressure and diverging values of the solvation free-energy at the nanoscale. The model is thus lacking here some flexibility. It proved per-fectly legitimate in other approaches with different formalism and different water properties requirements [START_REF] Zhao | New Theoretical Method for Rapid Prediction of Solvation Free Energy in Water[END_REF][START_REF] Zhao | Corrections to New Theoretical Method for Rapid Prediction of Solvation Free Energy in Water[END_REF][START_REF] Liu | A Site Density Functional Theory for Water: Application to Solvation of Amino Acid Side Chains[END_REF][START_REF] Liu | High-throughput prediction of the hydration free energies of small molecules from a classical density functional theory[END_REF][START_REF] Li | A Multiscale Procedure for Predicting the Hydration Free Energies of Polycyclic Aromatic Hydrocarbons[END_REF] Extending the proposed angular-independent bridge functional to the angular-dependent functional of eqs. 2-5, in order to ascertain all the results obtained here and systematically improve the HNC results of Ref. [START_REF] Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF] by circumventing a-posteriori pressure/surface tension corrections, will be the subject of a following work. We take our new bridge functional as an important step towards a general density functional for water, describing accurately both hydrophobic and hydrophilic interactions, and making it possible to study hydrophobic interactions in biologically relevant, non-idealized systems, such as those characterized along the years using statistical theories and simulations. [START_REF] Lee | The structure of liquid water at an extended hydrophobic surface[END_REF][START_REF] Zichi | The equilibrium solvation structure for the solvent-separated hydrophobic bond[END_REF][START_REF] Zichi | Solvent molecular dynamics in regions of hydrophobic hydration[END_REF][START_REF] Lee | A comparison of the structure and dynamics of liquid water at Hydrophobic and hydrophilic surfaces -A molecular dynamics simulation study[END_REF][START_REF] Cheng | Surface topography dependence of biomolecular hydrophobic hydration[END_REF][START_REF] Giovambattista | Hydration behavior under confinement by nanoscale surfaces with patterned hydrophobicity and hydrophilicity[END_REF][START_REF] Giovambattista | Hydrophobicity of protein surfaces: Separating geometry from chemistry[END_REF][START_REF] Giovambattista | Effect of temperature on the structure and phase behavior of water confined by hydrophobic, hydrophilic, and heterogeneous surfaces[END_REF][START_REF] Matysiak | Dissecting the energetics of hydrophobic hydration of polypeptides[END_REF][START_REF] Giovambattista | Computational Studies of Pressure, Temperature, and Surface Effects on the Structure and Thermodynamics of Confined Water[END_REF]

Figure 1 :

 1 Figure1: a) Free-energy per particle for the homogeneous solvent, f (x) = βF (n)/N , as function of the dimensionless density x = n/n b , in the second-order HNC approximation or with a bridge function correcting the pressure and restoring the near coexistence of gas and liquid according to eqn 11. (b) The function f (x) obtained for SPC/E water using the hard-sphere bridge functional of eqn. 13 with r hs = 1.4576 Å, or using the WDA expression of eqn. 19 with different values of the barrier control parameter b.

  [START_REF] Kierlik | Free-energy density functional for the inhomogeneous hard-sphere fluid: Application to interfacial adsorption[END_REF] , which was shown to be equivalent to the earlier vectorial version of Rosenfeld 46,65 , but presents supplementary numerical advantages[START_REF] Levesque | Scalar fundamental measure theory for hard spheres in three dimensions: Application to hydrophobic solvation[END_REF] . We use it in either its Percus-Yevick (PY) or Carnahan-Starling (CS) formulation, relying on the PY or CS hard-sphere equation of state.Other authors use the modified FMT version[START_REF] Roth | Fundamental measure theory for hard-sphere mixtures revisited: the White Bear version[END_REF] , or the white-bear versions[START_REF] Yu | Structures of hard-sphere fluids from a modified fundamental-measure theory[END_REF] . µ HS exc (n b ) and c HS s (|r -r |; n b ) represent the corresponding chemical potential and direct correlation functions of the HS fluid at the homogeneous density n b , by definition minus the first and second derivative of F HS exc [n(r)] at n(r) = n b . This expression amounts to substract from the exact hard-sphere functional all terms of order 0, 1, 2 in the expansion in ∆n(r) around n(r) = n b ,

Figure 2 :

 2 Figure 2: Dimensionless pressure as a function of reference hard-sphere fluid radius r hs taken in the hard-sphere functional of eqn 11, either in the Percus-Yevick or Carnahan-Starling approiximation. The red line indicates a pressure of 1 atm, virtually 0 at the scale that matters in the figure.

  hs = 1.4576 HSB, r hs = 1.42 Hummer et al.

  hs = 1.4576 HSB, r hs = 1.42 Huang et al.

Figure 4 :

 4 Figure 4: Same as Fig. 3 for the hydration free-energy per unit area, ∆G solv /4πR 2 , as a function of the hard-sphere radius radius R . Here the bullet points are the simulation results of Huang and Chandler 55 .

Figure 5 :

 5 Figure5: Same as Fig.3for the hydration free energy of a Lennard-Jones solute of increasing diameter σ and constant well-depth = 0.5 kJ/mol. The bullet points are the simulation results of Fujita et al[START_REF] Fujita | Assessing the accuracy of integral equation theories for nanosized hydrophobic solutes in water[END_REF] . Both the simulation and MD results have been corrected by the 1/r 6 long-range contributions beyond cut-off distance.

γ

  =63.6 mJ/m 2 for SPC/E water, and γ =52.3 mJ/m 2 for TIP3P; see Fig. 4. This lead us to σ w = 0.91 Å and b = 25 and b = 12 for SPC/E and TIP3P respectively; those models differ by the value of the compressibility and surface tension, hence requiring different parameter adjustments.

Figure 6 :

 6 Figure6: Comparison of the hydrophobic hydration free energies obtained by Monte-Carlo or MDFT (in kJ/mol) for the whole Mobley's dataset of drug-like molecules (all charges set to zero). (a) Angular-independent free-energy functional of eqn. 6 (m max = 0), complemented by the parameters-free WDA functional defined in the text. (b) Same using the full angulardependent functional of eqs. 2-5 with 84 orientations per spatial grid point (m max = 3). (c) Same in the HNC approximation with an a-posteriori pressure correction; Results taken from Ref.[START_REF] Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF] The mean absolute error is 0.35, 0.25, and 0.7 k B T in panels (a) to (c), respectively.

Acknowledgement

This work has been supported by the Agence Nationale de la Recherche, projet ANR BRIDGE AAP CE29. There are obviously two meetings of the present work with pioneering contributions of Peter Rossky in the field of theoretical physical chemistry and biochemistry.

The first one concerns his early and more recent elucidation by computer simulation of hydrophobic solvation and hydrophobic interactions. The second one is his invention in the 1980's of a consistent formulation of integral equation theories for molecular liquids and solutions, the extended reference interaction site model, which has given rise later to 3D-RISM. This contribution is faithfully dedicated to him.


Z Hammache

S Bensaadi

Y Berbar

N Audebrand

A Szymczyk
email: anthony.szymczyk@univ-rennes1.fr

M Amara

Recovery of rare earth elements from electronic waste by diffusion dialysis

Keywords: Rare earths, polymer membranes, dialysis, Nd-Fe-B magnet, recovery

Recycling rare earth elements (REEs) contained in electronic waste is of utmost importance due to their ever-increasing use in the high-tech sector. In this context, the use of non-polluting techniques represents a real challenge for scientists. In this work, we demonstrated that a clean and energy-efficient process, diffusion dialysis, has great potential for the recovery of neodymium (Nd) and praseodymium (Pr), contained in the Nd-Fe-B magnets of end-of-life computer hard disk drives (HDDs). Four kinds of polymer membranes were prepared by blending cellulose triacetate (CTA) and polyethylenimine (PEI) with the addition of di-(2ethylhexyl) phosphoric acid (D2EHPA), tridodecylamine (TDDA), trioctylamine (TOA) or trioctylphosphine oxide (TOPO). The membranes were first characterized by several techniques such as water uptake, contact angle, ATR-FTIR, SEM, XRD and zeta potential. They were further implemented in diffusion dialysis experiments in which diluted HDDs leachates were used as feed solutions. The relevance of the diffusion dialysis process for the recovery of REEs was demonstrated as it was possible to extract up to 15% of the boron by spontaneous diffusion through the CTA / PEI / TOA membrane in 6 hours with a lab-scale cell operating in batch mode. All membranes were positively charged under the operating conditions and then, REEs (present mainly in the form of trivalent cations Nd 3+ and Pr 3+ ) were strongly rejected. A fairly good correlation was found between the membrane water uptake and boron transfer, with boric acid molecules passing more easily through the more hydrated membranes. The selectivity factor between boron and REEs resulted from the interplay between the membrane structure, its water uptake ability and surface charge. It was also found to be dependent on the leachate composition. The CTA / PEI / TDDA membrane was found to exhibit the greatest B / REE selectivity factors, with values reaching up to 3706 and 140 for Nd and Pr, respectively. The lower selectivity towards Pr was explained by the weaker Gibbs energy of hydration of Pr 3+ cations compared with Nd 3+ .

Introduction

Global demand for rare earth elements (REEs) continues to increase due to their numerous applications in the high-tech sector [START_REF] Humphries | Rare earth elements: The global supply chain[END_REF]. Conventional ores cannot satisfy the staggering demand for these strategic elements. REEs are found in computers hard-disk drives (HDDs) [START_REF] Hurst | China's Rare Earth Elements Industry: What Can the West Learn? Institute for the Analysis of Global Security (IAGS)[END_REF], wind turbines, Ni -metal hydride batteries, luminophores, acoustic transducers, magnetic separators, hybrid cars, etc [START_REF] Omodara | Recycling and substitution of light rare earth elements, cerium, lanthanum, neodymium, and praseodymium from end-of-life applications -A review[END_REF][START_REF] Auerbach | Critical raw materials -Advanced recycling technologies and processes: Recycling of rare earth metals out of end of life magnets by bioleaching with various bacteria as an example of an intelligent recycling strategy[END_REF]. Recycling of neodymium (Nd) from end-of-life consumer products has attracted growing interest [START_REF] Yin | An Efficient Process for Recycling Nd-Fe-B Sludge as High-Performance Sintered Magnets[END_REF]. In this regard, several studies focused on the recovery of neodymium and praseodymium (Pr) from Nd-Fe-B permanent magnets contained in HDDs [START_REF] Yoon | Solvent extraction, separation and recovery of dysprosium (Dy) and neodymium (Nd) from aqueous solutions: Waste recycling strategies for permanent magnet processing[END_REF][START_REF] Balaram | Rare earth elements: A review of applications, occurrence, exploration, analysis, recycling, and environmental impact[END_REF]. Nd-Fe-B magnets typically contain ∼30 wt. % REEs. Besides Nd, several other REEs, such as Pr, Dy, Tb, Gd and Ho, may be found in Nd-Fe-B magnets [START_REF] Skotnicova | Structural and magnetic engineering of (Nd, Pr, Dy, Tb)-Fe-B sintered magnets with Tb3Co0.6Cu0.4H composition in the powder mixture[END_REF][START_REF] Sasaki | Structure and chemical compositions of the grain boundary phase in Nd-Fe-B sintered magnets[END_REF].

Several REEs recovery techniques have been used, such as flotation [START_REF] Houot | Recovery of Rare Earth Minerals, with Emphasis on Flotation Process | Scientific.Net[END_REF], magnetic separation [START_REF] Jordens | Processing a rare earth mineral deposit using gravity and magnetic separation[END_REF], complexation [START_REF] Tang | Ligand extraction of rare earth elements from aquifer sediments: Implications for rare earth element complexation with organic matter in natural waters[END_REF] or solvent extraction [START_REF] Safarbali | Solvent extraction-separation of La(III), Eu(III) and Er(III) ions from aqueous chloride medium using carbamoyl-carboxylic acid extractants[END_REF][START_REF] Xia | Recovery of rare earths from acid leach solutions of spent nickel-metal hydride batteries using solvent extraction[END_REF][START_REF] Parhi | Liquid-liquid extraction and separation of total rare earth (RE) metals from polymetallic manganese nodule leaching solution[END_REF]. In this context, membrane processes have also been considered in recent years. Some studies dealing with the extraction and separation of REEs used liquid membranes [START_REF] Chen | Extraction of rare earth ions from phosphate leach solution using emulsion liquid membrane in concentrated nitric acid medium[END_REF]. A bubbling organic liquid membrane extraction using primary amine N1923 was used to extract and enrich low concentration REEs from leaching solutions of rare-earth ores [START_REF] Liu | Enrichment of Low Concentration Rare Earths from Leach Solutions of Ion-Adsorption Ores by Bubbling Organic Liquid Membrane Extraction Using N1923[END_REF]. Low-concentration REEs were also extracted using a bubbling organic liquid membrane with un-saponified P507 [START_REF] Liu | Extraction of rare earths using bubbling organic liquid membrane with un-saponified P507[END_REF]. Nanofiltration membranes [START_REF] López | Integration of nanofiltration membranes in recovery options of rare earth elements from acidic mine waters[END_REF][START_REF] López | Application of nanofiltration for acidic waters containing rare earth elements: Influence of transition elements, acidity and membrane stability[END_REF], dispersion micro-extractor membranes [START_REF] Chen | Efficient extraction and stripping of Nd(III), Eu(III) and Er(III) by membrane dispersion micro-extractors[END_REF][START_REF] Chen | Efficient enrichment and recovery of rare earth elements with low concentration by membrane dispersion microextractors[END_REF] or hollow-fiber membrane extractor [START_REF] Hango | Extraction Kinetics of Rare Earth Metals with 2-Ethylhexyl Phosphonic Acid Mono-2-ethylhexyl Ester Using a Hollow Fiber Membrane Extractor[END_REF] were also considered in recovery options of REEs.

Diffusion dialysis is a well-established process for the separation of chemical species in solution [START_REF] Luo | Diffusion dialysis-concept, principle and applications[END_REF]. It is based on the application of a concentration difference between two compartments separated by a membrane. It is a clean technique that does not require the addition of chemicals and it is energy-efficient as no hydraulic-pressure difference is applied through the membrane.

Both the membrane features (e.g. pore size, surface charge density, nature of surface functional groups, etc.) and the feed solution composition impact the separation efficiency [START_REF] Lahiri | Separation of iron and cobalt using 59Fe and 60Co by dialysis of polyvinylpyrrolidone-metal complexes: A greener approach[END_REF][START_REF] Bensaadi | Dialysis and photo-electrodialysis processes using new synthesized polymeric membranes for the selective removal of bivalent cations[END_REF].

Dialysis polymer membranes synthesized from mixtures of cellulose triacetate (CTA) and polyethyleneimine (PEI) were shown to be low-cost materials with good separation performance [START_REF] Kebiche-Senhadji | Facilitated Cd(II) transport across CTA polymer inclusion membrane using anion (Aliquat 336) and cation (D2EHPA) metal carriers[END_REF][START_REF] Bensaadi | Evaluating molecular weight of PVP on characteristics of CTA membrane dialysis[END_REF]. The addition of PEI has been widely used as it allows to form a charged layer on the membrane surface [START_REF] Bagheri | Preparation of a positively charged NF membrane by evaporation deposition and the reaction of PEI on the surface of the C-PES/PES blend UF membrane[END_REF]. Furthermore, PEI can form complexes with heavy metals [START_REF] Amara | Modification of cation-exchange membrane properties by electro-adsorption of polyethyleneimine[END_REF]. The addition of carriers, such as di-(2-ethylhexyl) phosphoric acid (D2EHPA), tridodecylamine (TDDA), trioctylamine (TOA) and trioctylphosphine oxide (TOPO), in liquid membranes and polymer inclusion membranes has been successfully applied to selective extraction of metal ions [START_REF] Gherrou | Removal of silver and copper ions from acidic thiourea solutions with a supported liquid membrane containing D2EHPA as carrier[END_REF][START_REF] Bayou | Elaboration and characterisation of a plasticized cellulose triacetate membrane containing trioctylphosphine oxyde (TOPO): Application to the transport of uranium and molybdenum ions[END_REF]. It is worth mentioning that D2EHPA and TOPO have also been found efficient to extract REEs by liquid / liquid extraction [START_REF] Arous | Cadmium (II) and lead (II) transport in a polymer inclusion membrane using tributyl phosphate as mobile carrier and CuFeO2 as a polarized photo electrode[END_REF][START_REF] Wu | Simultaneous recovery of rare earths and uranium from wet process phosphoric acid using solvent extraction with D2EHPA[END_REF][START_REF] Jha | Review on hydrometallurgical recovery of rare earth metals[END_REF]. Although D2EHPA, TDDA, TOA and TOPO have been used as carriers in liquid membranes and polymer inclusion membranes, their chemical structure with a hydrophobic alkyl backbone and one or several polar groups give them also plasticizing properties [START_REF] Nghiem | Extraction and transport of metal ions and small organic compounds using polymer inclusion membranes (PIMs)[END_REF][START_REF] Gherasim | Removal of lead(II) from aqueous solutions by a polyvinyl-chloride inclusion membrane without added plasticizer[END_REF], which allows improving the membrane mechanical properties.

In this study, we investigated the possibility of recovering REEs (Nd and Pr) from electronic waste (Nd-Fe-B permanent magnets) using diffusion dialysis as a clean alternative process to liquid / liquid extraction [START_REF] Rodrigues | Liquid-liquid extraction of metal ions without use of organic solvent[END_REF]. Four dialysis CTA / PEI membranes containing D2EHPA, TDDA, TOA or TOPO were first synthesized. The membranes were further characterized by water uptake, water contact angle, Fourier-transform infrared spectroscopy in attenuated total reflectance mode (ATR-FTIR), X-ray diffraction analysis (XRD), scanning electronic microscopy (SEM) and zeta potential. Eventually, the four membranes were implemented in dialysis experiments and their ability to separate boron from REEs was evaluated.

Materials and methods

Chemicals

Cellulose triacetate (CTA), polyethyleneimine (PEI) (50%), di-(2-ethylhexyl) phosphoric acid (D2EHPA) (97%), tridodecylamine (TDDA) (97%), trioctylamine (TOA) (98%), trioctylphosphine oxide (TOPO) (98.5%), nitric acid (HNO3) (65%), hydrochloric acid (HCl) (37%), potassium hydroxide and chloroform (99.8%) were purchased from Sigma Aldrich.

HDD magnet leaching

Magnets contained in the HDD of end-of-life desktop and laptop computers (Table 1) were first collected and demagnetized in a muffle furnace at 250 °C [START_REF] Zhang | Squareness factors of demagnetization curves for multi-main-phase Nd-Ce-Fe-B magnets with different Ce contents[END_REF]. After crushing and fine grinding, the powder obtained from each magnet was weighed with an analytical balance before leaching in 3N HCl and HNO3 solutions. The leachates composition was determined by inductively coupled plasma mass spectrometry (ICP-MS; Thermo Fisher, model ICAP RQ). Ten replicas were performed.

Membrane preparation

The membranes were synthesized by the solvent-evaporation method [START_REF] Sugiura | Carrier-mediated transport of rare earth ions through cellulose triacetate membranes[END_REF][START_REF] Sugiura | Transport of lanthanide ions through cellulose triacetate membranes containing hinokitiol and flavonol as carriers[END_REF][START_REF] Soroko | The effect of membrane formation parameters on performance of polyimide membranes for organic solvent nanofiltration (OSN). Part B: Analysis of evaporation step and the role of a cosolvent[END_REF][START_REF] Prihatiningtyas | Effect of solvent on the morphology and performance of cellulose triacetate membrane/cellulose nanocrystal nanocomposite pervaporation desalination membranes[END_REF]. CTA and PEI were dissolved separately in CHCl3 with vigorous stirring for 2 h at room temperature. The two polymer solutions were then mixed and left under stirring for 3 h before adding D2EHPA, TDDA, TOA or TOPO. Stirring was pursued for another 2 h until a homogeneous solution was obtained. The latter was poured into a glass Petri dish (130 mm diameter) and left to evaporate the solvent for 24 h. Membranes in the form of homogeneous films were obtained and detached from the bottom of the Petri dish by gently adding a little water. It was observed that the addition of D2EHPA, TDDA, TOA or TOPO to the CTA / PEI led to much less brittle membranes, which confirmed that these compounds can act as plasticizers [START_REF] Nghiem | Extraction and transport of metal ions and small organic compounds using polymer inclusion membranes (PIMs)[END_REF][START_REF] Gherasim | Removal of lead(II) from aqueous solutions by a polyvinyl-chloride inclusion membrane without added plasticizer[END_REF]. Details of the amount of solvent as well as polymer and additive concentrations used for each membrane (labelled M1, M2, M3 and M4) are presented in Table 2. The chemical structures of CTA, PEI and additives used for membrane preparation are shown in Figure 1.

Membrane characterization

Water uptake

A dry membrane sample (surface area: 4 cm 2 ) of known mass was soaked in deionized (DI) water for 24 h under gentle stirring. The membrane sample was then removed, wiped with absorbent paper and weighed [START_REF] Bensaadi | Evaluating molecular weight of PVP on characteristics of CTA membrane dialysis[END_REF]. The water content T (%) was determined by the following equation:

T = (𝑚𝑚 𝑤𝑤 -𝑚𝑚 𝑑𝑑 ) 𝑚𝑚 𝑑𝑑 × 100 (1)
where 𝑚𝑚 𝑤𝑤 and 𝑚𝑚 𝑑𝑑 are the mass of the wet and dry membrane sample, respectively.

Water contact angle

Contact angles were determined by the sessile-drop method using a DIGIDROP GBX-DS apparatus. A syringe was used to deposit DI water droplets with controlled size. Contact angles were determined by means of a video capture system and the Windrop++ software. The reported contact angles are the average of 10 measurements performed at different locations on the membrane surface. Measurements were performed at room temperature. Membrane samples were vacuum-dried for 48 h prior to measurements.

Attenuated Total Reflectance-Fourier Transform Infrared spectroscopy (ATR-FTIR)

The ATR-FTIR spectra of membranes were recorded using a PerkinElmer Spectrum™ 100 FTIR spectrometer equipped with a single-reflection diamond crystal (incidence angle: 45°).

Each spectrum was averaged from 20 scans in the range 600-3700 cm -1 at a resolution of 2 cm -1 . Membrane samples were vacuum-dried for 48 h prior to analysis.

Scanning electronic microscopy (SEM)

Membrane surface morphology and membrane cross-section were characterized by SEM using a JEOL (JSM-7100 FEG EDS EBSD Oxford) microscope. For cross-section imaging, membrane samples were fractured after soaking in liquid nitrogen. Membrane samples were coated with an Au/Pd alloy prior to analysis.

X-ray diffraction analysis (XRD)

XRD analysis was performed with a Bruker D8 Advance diffractometer with Cu Kα1 radiation (λ= 1.5406 Å) selected with an incident beam Ge (111) monochromator and equipped with a LynxEye detector in the 2θ range 10-80°. An adhesive tape was used to properly fix the membranes on the sample holder. An additional in-depth structural analysis was carried out in the 2θ range 3-45° with both the M4 membrane and TOPO powder.

Zeta potential

A SurPASS electrokinetic analyzer (Anton Paar GmbH) was used to determine the surface zeta potential of the various membranes from tangential streaming current measurements [START_REF] Hanafi | Electrokinetic analysis of PES/PVP membranes aged by sodium hypochlorite solutions at different pH[END_REF]. The streaming current technique was preferred over the streaming potential to avoid any potential contribution of the electrical conduction through the membrane thickness [START_REF] Fievet | A new tangential streaming potential setup for the electrokinetic characterization of tubular membranes[END_REF]. Measurements

were carried out at room temperature with a 1 mM KCl background solution in the pH range 3-9 (the pH was set by addition of 0.05M HCl and KOH solutions). Data were analyzed with the Visiolab software.

Diffusion dialysis

A lab-made diffusion-dialysis cell was used. The membrane (effective surface area: 15 cm 2 )

was sandwiched between two compartments of 85 mL each. The stripping and feed compartments were filled with DI water and diluted leachate, respectively. The leachates were diluted until a pH of ∼4 was reached so as to limit the hydrolysis of the acetyl groups of CTA (the hydrolysis rate is at a minimum in the pH range of 4-6) [START_REF] Farooque | Degradation study of cellulose triacetate hollow fine-fiber SWRO membranes[END_REF][START_REF] Geise | Water purification by membranes: The role of polymer science[END_REF] while keeping the membrane surface positively charged (see below). The increase in pH resulting from leachate dilution led to the precipitation of iron(III), which was further removed by filtration.

Dialysis experiments were performed in a batch-configuration for 6 h at room temperature. The feed and stripping solutions were stirred at 400 rpm to limit establishment of boundary layers.

The solution composition was determined by ICP-MS by regularly withdrawing 0.2 mL volumes from both compartments. The transport yield of species i (ηi) was calculated by the following equation:

𝜂𝜂 𝑖𝑖 = (𝐶𝐶 𝑖𝑖,𝐹𝐹 (0)-𝐶𝐶 𝑖𝑖,𝑆𝑆 ) 𝐶𝐶 𝑖𝑖,𝐹𝐹 (0) × 100 (2)
where C i,F (0) is the initial concentration of species i in the feed compartment and C i,S is the concentration of i transferred to the stripping compartment. The volume change is the cell compartments did not exceed 0.2 mL and was then neglected in the calculation of the transport yield.

The selectivity factor (SFB/REE) [START_REF] Yan | Separation of NaOH and NaAl(OH)4 in alumina alkaline solution through diffusion dialysis and electrodialysis[END_REF], calculated by equations (3), was used to quantify the membrane efficiency for the selective separation of boron (B) and REEs (Nd and Pr):

𝑆𝑆𝑆𝑆 𝐵𝐵/𝑅𝑅𝑅𝑅𝑅𝑅 = 𝐶𝐶 𝐵𝐵,𝑆𝑆 /𝐶𝐶 𝐵𝐵,𝐹𝐹 𝐶𝐶 𝑅𝑅𝑅𝑅𝑅𝑅,𝑆𝑆 /𝐶𝐶 𝑅𝑅𝑅𝑅𝑅𝑅,𝐹𝐹 (3)
where SFB/REE is the selectivity factor between boron and an REE (Nd or Pr), CB,S and CB,F are the concentration of boron in the stripping and feed compartments, respectively, CREE,S and CREE,F are the concentration of REE (Nd or Pr) in the stripping and feed compartments, respectively.

Results and discussion

Composition of Nd-Fe-B permanent magnets

The mass of the Nd-Fe-B magnet was 17.66 g and 2.88 g for the desktop-computer and laptopcomputer HDDs, respectively. The composition of leachates obtained after leaching of Nd-Fe-B magnets is given in Table 3. The main component found in HDDs leachates was iron (∼60-65 wt. %), followed by Nd (∼20-28 wt. %). A significant difference in Pr (this element is always found with Nd) was detected in the two types of HDDs, with a much larger amount found in the laptop HDD (16.06 wt. % vs. 5.31 wt. %). The amount of boron (the third essential component in Nd-Fe-B magnets) was found between ∼1 and 4 wt. %. Only a few traces of the light REEs, La, Ce, Sm and Eu, have been quantified in HDDs leachates. Results collected in Table 3 are in line with data reported in the literature [START_REF] Skotnicova | Structural and magnetic engineering of (Nd, Pr, Dy, Tb)-Fe-B sintered magnets with Tb3Co0.6Cu0.4H composition in the powder mixture[END_REF][START_REF] Ikram | The sintering mechanism of fully dense and highly coercive Nd-Fe-B magnets from the recycled HDDR powders reprocessed by spark plasma sintering[END_REF].

Membrane characterization

The water uptake and contact angle of the various membranes are reported in Table 4. The membranes incorporating phosphorus-based additives (M1 and M4) exhibited much less water uptake than their counterparts with amino additives (M2 and M3). M3 and M4 membranes exhibited the greatest and lowest water uptake, respectively. These results were correlated with water contact angles as M3 and M4 were found to be the most and least hydrophilic membranes, respectively. However, no such correlation was observed with M1 and M2 membranes, for which quite different water uptakes but similar contact angles were obtained.

Figure 2 shows the ATR-FTIR spectra of the various membranes. M1, M2 and M3 membranes exhibit the characteristic bands of CTA / PEI blend membranes. The band at 1747 cm -1 is associated with stretching of the C=O bond of CTA ester groups [START_REF] Dias | Structure of water in asymmetric cellulose ester membranes -An ATR-FTIR study[END_REF]. The characteristic bands of PEI are located around 1635 cm -1 (NH2 bending from primary amines) and 1035 cm -1

(stretching of the C-N band) [START_REF] Kotte | A facile route to the preparation of mixed matrix polyvinylidene fluoride membranes with in-situ generated polyethyleneimine particles[END_REF][START_REF] Zarei | Preparation of thin film composite nano-filtration membranes for brackish water softening based on the reaction between functionalized UF membranes and polyethyleneimine[END_REF]. The band around 1240 cm -1 results from stretching contributions of both C-O single bonds (CTA) and C-N from tertiary amines (PEI) [START_REF] Dias | Structure of water in asymmetric cellulose ester membranes -An ATR-FTIR study[END_REF][START_REF] Doan-Nguyen | Structure determination and modeling of monoclinic trioctylphosphine oxide[END_REF]. Strikingly, the ATR-FTIR spectrum of the M4 membrane did not show any of the abovementioned characteristic bands of CTA and PEI. On the other hand, the spectrum showed the characteristic bands of TOPO. Indeed, the bands at 1154 and 1476 cm -1 correspond to the stretching vibrations of the P=O and P-C bonds of TOPO, respectively [START_REF] Doan-Nguyen | Structure determination and modeling of monoclinic trioctylphosphine oxide[END_REF]. It is also worth noting the greater intensity of the bands associated with stretching of C(sp 3 )-H (∼2850-3000 cm -1 ) compared with the other membranes. These results indicate that TOPO was not homogeneously distributed in the bulk of the membrane and that it was preferentially located on the membrane surface, forming a layer thicker than the penetration depth of the IR beam, which is in the µm range. It was most likely the result of the bad affinity between TOPO and CHCl3, which led to migration of TOPO towards the membrane surface during the slow solvent evaporation process.

Conclusions drawn from ATR-FTIR were confirmed by SEM. Figure 3 shows the cross section of the M1 and M4 membranes. Unlike the M1 membrane, which showed a relatively homogeneous (sponge-like) structure, a several micrometer-thick layer can be seen on the M4 membrane surface. The TOPO layer formed on the surface may also explain why the M4 membrane had the highest water contact angle of the four membranes (Table 4). The surface of all membranes was found to be dense without pores (see for instance the SEM images of the surface of M2 and M3 membranes in Figure 4). The membrane XRD patterns in the 2θ range 10-80° are reported in Figure 5. As expected, the M1, M2 and M3 membranes were found fully amorphous (the peaks observed were associated

M4 M1

M2 M3

with the adhesive tape used to properly fix the membrane on the sample holder). Still, the results of the M4 membrane were found to be different from those of the other membranes as several peaks belonging to the M4 membrane were detected. These results are in line with ATR-FTIR and SEM analyses as the observed crystallinity can be associated with the TOPO layer formed on the surface of M4 membrane, which was confirmed by comparison with the XRD pattern of the TOPO powder used for membrane preparation (see Figure 6). The results of the membrane electrokinetic characterization are displayed in Figure 7. The reported isoelectric points for CTA membranes are usually below pH 4 [START_REF] Nguyen | Preparation of cellulose triacetate/cellulose acetate (CTA/CA)-based membranes for forward osmosis[END_REF][START_REF] Nguyen | The chlorination mechanism of integrally asymmetric cellulose triacetate (CTA)-based and thin film composite polyamide-based forward osmosis membrane[END_REF]. The M2, M3 and M4 membranes were found to have isoelectric points around 4.6, 6.1 and 8.6, respectively, while the M1 membrane was positively charged in the whole pH range. The significant decrease in the surface acidity of the various membranes, compared CTA membranes [START_REF] Nguyen | Preparation of cellulose triacetate/cellulose acetate (CTA/CA)-based membranes for forward osmosis[END_REF][START_REF] Nguyen | The chlorination mechanism of integrally asymmetric cellulose triacetate (CTA)-based and thin film composite polyamide-based forward osmosis membrane[END_REF], results from the addition of PEI in the membrane preparation and the resulting formation of positively charged quaternary ammonium groups [START_REF] Xu | Positive charged PEI-TMC composite nanofiltration membrane for separation of Li+ and Mg2+ from brine with high Mg2+/Li+ ratio[END_REF][START_REF] Bagheri | Preparation of a positively charged NF membrane by evaporation deposition and the reaction of PEI on the surface of the C-PES/PES blend UF membrane[END_REF]. It can be noticed that the addition of phosphorus-based additives (D2EHPA and TOPO) led to more basic surfaces (M1 and M4 membranes) compared with membranes containing TDDA and TOA (M2 and M3 membranes, respectively).

Membrane separation performance

M1-M4 membranes were employed in dialysis experiments to assess their ability to separate boron from REEs (Nd and Pr). The composition of the diluted leachates (see section 2.5) is given in Table 5 (the solutions resulting from leaching of desktop-computer and laptopcomputer HDDs are referred to as S1 and S2, respectively). As stated in section 2.5, HDD leachates were diluted to attain a pH of ∼4 so as to limit the hydrolysis of the acetyl groups of CTA. Furthermore, as shown in Figure 7, this pH guaranteed that all membranes were positively charged, which was expected to be beneficial to prevent transfer of REEs due to strong electrostatic repulsion between the membranes surface and Nd 3+

and Pr 3+ ions. Interestingly, they were also the less hydrated membranes with a water uptake in the range ∼6-9 wt. % (see Table 4). As all membranes had a relatively dense structure, it is unlikely that solutes could penetrate the membrane phase while keeping their entire hydration shell [START_REF] Ding | On the structure and rejection of ions by a polyamide membrane in pressure-driven molecular dynamics simulations[END_REF]. The associated energetic penalty associated with solute partial dehydration could be partly overcome if the solute could interact favorably with water molecules inside the membrane phase. That is why M2 (CTA / PEI / TDDA) and M3 (CTA / PEI / TOA) membranes, which were the most hydrated membranes with a water uptake of 20.3 and 23.1 wt%, respectively (see Unlike boron, which was present in the form of uncharged boric acid molecules (B(OH)3 (pKa = 9.24 at 25 °C), REEs were mainly in the form of trivalent cations (Nd 3+ and Pr 3+ ) under the conditions of our study [START_REF] Migdisov | A spectrophotometric study of neodymium(III) complexation in chloride solutions[END_REF][START_REF] Liu | Effect of Solvation? Complexation of Neodymium(III) with Nitrate in an Ionic Liquid (BumimTf2N) in Comparison with Water[END_REF]. The latter were then strongly rejected by the four positively charged membranes. Iron was also strongly rejected by all membranes. As indicated in section 2.5, the increase of pH up to 4 as a result of leachate dilution led to the precipitation of Fe 3+ ions. It was therefore assumed that the residual iron detected in the leachates after dilution and filtration was in the form of Fe 2+ ions. As expected, the most strongly charged membrane, M1, exhibited very high REEs rejection. Although the M2 membrane was only slightly positively charged at pH 4 (see Figure 7), it was able to reject REEs at a higher level than the more positively charged M3 and M4 membranes. These results suggest that the M2 membrane had a bit denser structure than M3 and M4. In general, Pr 3+ cations were transferred slightly more easily across membranes than Nd 3+ , which may be due to the slightly higher Gibbs energy of hydration of Nd 3+ (-3280 and -3245 kJ / mol for Nd 3+ and Pr 3+ , respectively).

It is noteworthy that D2EHPA and TOPO are known to be efficient to extract REEs when they are used in the liquid / liquid extraction process, in which they are "free" to form coordination complexes with REEs [START_REF] Wu | Simultaneous recovery of rare earths and uranium from wet process phosphoric acid using solvent extraction with D2EHPA[END_REF][START_REF] Radhika | Liquid-liquid extraction and separation possibilities of heavy and light rare-earths from phosphoric acid solutions with acidic organophosphorus reagents[END_REF][START_REF] Battsengel | Recovery of light and heavy rare earth elements from apatite ore using sulphuric acid leaching, solvent extraction and precipitation[END_REF]. However, when embedded into the CTA / PEI polymer matrix, they were found unable to bind with REEs as confirmed by mass balance performed in the feed and stripping compartments of the dialysis cell. It is most probably due to the favorable interaction between the positively charged PEI and the phosphoryl and phosphonyl groups of TOPO and D2EHPA, respectively. Furthermore, the positive membrane charge repelled the like-charged REEs from the membrane surface, thus preventing transport of REEs through the membrane.

M1 and M2 membranes exhibited the highest rejection of REEs but the M1 also strongly rejected boron. As a result, better selectivity factors between B and REEs were reached with the M2 membrane, as reported in Figures 9. Notably, the M2 membrane exhibited an impressive selectivity factor B / Nd in the range 559 -3706, depending on the leachate composition, which makes it very promising for the recovery of REEs from end-of-life computer hard drives.

Conclusion

Various polymer membranes were synthesized from cellulose triacetate (CTA) and polyethylenimine (PEI) by the solvent evaporation method. Amino-based additives, tridodecylamine (TDDA) and trioctylamine (TOA), and phosphorus-based additives, di-(2ethylhexyl) phosphoric acid (D2EHPA) and trioctylphosphine oxide (TOPO), were added to the CTA / PEI blends in order to obtain dense membranes with appropriate mechanical properties. Water uptake, contact angle, ATR-FTIR, SEM, XRD and zeta potential techniques were used to characterize the membranes. Unlike TDDA, TOA and D2EHPA, TOPO was found to preferentially migrate towards the membrane surface during the membrane formation process, thus forming a crystalline layer on the membrane surface, as revealed by ATR-FTIR, SEM and XRD analyses.

Nd-Fe-B magnets contained in end-of-life hard disk drives were demagnetized, crushed and grinded before being leached by strong acids. The leachates were further diluted until a pH of ∼4 was reached. Setting the pH at this value (i) enabled to limit the hydrolysis of the acetyl groups of CTA, (ii) ensured a positive surface charge for all membranes and (iii) made it possible to remove most of the iron by precipitation of iron(III) hydroxide.

The diluted leachates were then used as feed solutions in diffusion-dialysis experiments and the ability of the various membranes to separate boron from REEs was evaluated. Owing to the positive membrane charge, the transfer of REEs (present in the form of Nd 3+ and Pr 3+ cations) was found very small through all membranes. Boron, which was present in the form of neutral boric acid molecules under the operating conditions, was transferred to a greater extent, especially through membranes with high water uptake (i.e. membranes containing an aminobased additive, TDDA or TOA). The CTA / PEI / TOA membrane was able to extract ∼15% of the boron after 6-hour dialysis. It should be stressed that this performance could be readily improved as the dialysis experiments were carried out in batch-mode with a lab-scale cell. The CTA / PEI / TDDA membrane had the highest selectivity between boron and neodymium with a B / Nd selectivity factor up to 3706. For all membranes, the selectivity between boron and praseodymium was less than for neodymium (max. 140) because the membranes rejected less strongly Pr 3+ than Nd 3+ , which can be understood through their different Gibbs energy of hydration.

This work demonstrated that a clean and energy-efficient process like diffusion dialysis is an attractive alternative method to e.g. liquid / liquid extraction for the recovery of REEs from electronic waste.

Figure 1 .

 1 Figure 1. Chemical structures of CTA, PEI, D2EHPA, TDDA, TOA and TOPO.

Figure 2 .

 2 Figure 2. ATR-FTIR spectra of the various membranes (M1-M4) in the range 600-3700 cm -1 .

Figure. 3 .

 3 Figure. 3. SEM images of the M1 (left) and M4 (right) membranes cross-sections (scale bar: 10 µm).

Figure 4 .

 4 Figure 4. SEM images of the M2 (left) and M3 (right) membranes surfaces (scale bar: 1 µm).

Figure 5 .

 5 Figure 5. XRD patterns of the various membranes (M1-M4) in the range in the 2θ range 10-80°. The XRD pattern of the adhesive tape (T) used to fix the membranes on the sample holder is also shown for comparison.

Figure 6 .

 6 Figure 6. XRD patterns of the M4 membrane and TOPO powder. The dotted vertical lines indicate the peaks associated with the adhesive tape used to fix the membrane.

Figure 7 .

 7 Figure 7. Variation of the zeta potential of the various membranes (M1-M4) with the pH of a 10 -3 M KCl solution.

Figures 8a and 8b

  Figures 8a and 8b show the transport yield of the various species (B, Fe, Nd and Pr) contained in solutions S1 and S2, respectively. As a batch cell was used in this study, no steady state could have been reached and values reported in Figures 8 correspond to the membrane performance after a 6-hour dialysis experiment. Although transport yields were found to be dependent on the composition of the feed solution, similar qualitative conclusions could be drawn with the two solutions S1 and S2. Notably, all membranes showed preferential transport of boron over REEs and residual iron. The membranes incorporating phosphorus-based compounds, M1 (CTA / PEI / D2EHPA) and M4 (CTA / PEI / TOPO), were the ones that rejected boron the most strongly.

Figure 8 .

 8 Figure 8. Transport yield of B, Fe, Nd and Pr through the membranes for (a) S1 feed solution and (b) S2 feed solution.

Figure 9 .

 9 Figure 9. Selectivity factor B / Nd and B / Pr of the various membranes for (a) S1 feed solution and (b) S2 feed solution.

Table 1 .

 1 HDDs features.

	HDDs Company	Model	Reference S/N Country
	Desktop	Western Digital	WD200 20gb	WD200EB-00BHF0	Malaysia
	Laptop	Seagate	Momentus 5400,6 160 GB	5VC5SKY3	China

Table 2 .

 2 Amount of solvent, polymers and additives used in membrane preparation.

Membrane ID CHCl3 (mL) CTA (wt. %) PEI (wt. %) D2EHPA (wt. %) TDDA (wt. %) TOA (wt. %) TOPO (wt. %)

	M1	20	33.34	22.22	44.44	/	/	/
	M2	20	33.34	22.22	/	44.44	/	/
	M3	20	33.34	22.22	/	/	44.44	/
	M4	20	33.34	22.22	/	/	/	44.44

Table 3 .

 3 Composition of leachates (wt. %) obtained after leaching of Nd-Fe-B magnets.

	Confidence level: 95 %.
		B	Fe	La	Ce	Pr	Nd	Sm	Eu
	Desktop	1.30	65.39	0.01	0.01	5.31	27.97	0.01	n.d.
	Laptop	3.95	59.63	0.14	0.01	16.06	20.19	0.01	0.01
	n.d.: not detected.

Table 4 .

 4 Water uptake and contact angle of the various membranes.

	M1	9.1	67.8 ± 1.5
	M2	20.3	67.6 ± 1.3
	M3	23.1	59.7 ± 3.8
	M4	6.4	75.5 ± 0.8

Membrane ID Water uptake (wt. %) Contact angle (°)

Table 5 .

 5 Composition of leachates after dilution. S1 and S2 stand for diluted leachates after desktop-computer-HDD and laptop-computer-HDD lixiviation, respectively.

			Confidence level: 95 %.
		B (mg/L)	Fe (mg/L)	Pr (mg/L)	Nd (mg/L)
	S1	8.724	18.578	35.711	119.141
	S2	12.630	129.476	48.142	944.721

Table 4 )

 4 , were found to exhibit the highest transport yields for boron. The latter reached almost

	15% with the M3 membrane after 6-hour dialysis. It is worth reminding that a batch-cell
	configuration was used in this work, thus the reported performance could be easily
	outperformed by optimizing the dialysis cell. These results highlight the great interest of
	diffusion dialysis for REEs recovery as it is (i) a clean process and (ii) it operates without the
	energy-intensive application of hydraulic pressure difference or electric field through the
	membrane but only by spontaneous diffusion, thus making it very attractive in terms of energy
	savings.

Acknowledgements

Francis Gouttefangeas is acknowledged for SEM analyses (UMS 2001 ScanMAT).