Denis A Ivanov

Olga Klimchuk

Isabelle Billard

Nikolai Kh

Petrov

N Petrov

The microheterogeneous structure of

Introduction

Room temperature ionic liquids (ILs) have unusual solvent properties and in particular a negligible low vapor pressure. Although a large viscosity inherent to ILs might be a limitation of their applications, they remain a very active field of theoretical and experimental studies as novel reaction media (see a recent review [START_REF] Castner | Spotlight on ionic liquids[END_REF]).

In order to tune a reaction environment, binary solvent mixtures have long been in use.

An important feature of such media, which is often ignored, is that such mixtures have microdomains, whose properties are different from that of the bulk. Organic solvent mixtures that comprise components of significantly different polarity, i.e. toluene and DMSO, provide micro-heterogeneous media for a charged solute, for example cationic cyanine dye [START_REF] Kh | Solvation-Shell Effect on the Cyanine-Dye Fluorescence in Binary Liquid Mixtures[END_REF].

Components of binary mixtures are not necessarily to be conventional solvents. Hapiot and coworkers [START_REF] Zigah | Diffusion of Molecules in Ionic Liquids/Organic Solvent Mixtures. Example of the Reversible Reduction of O 2 to Superoxide[END_REF] found a significant increase in diffusion coefficients of O 2 and superoxide that increase exponentially with increasing the dimethylformamide volume fraction in the mixture with triethyl(butyl)ammonium bis((trifluoromethyl)sulfonyl) imide. Although such a IL singularity is not still completely understandable, one can suggest that the observed effect is mainly caused by structural heterogeneities.

It is worth noting that neat IL by itself tends to produce structural heterogeneities. For example, Wakasa and coworkers [START_REF] Hamasaki | Anomalous Magnetic Field Effects on Photochemical Reactions in Ionic Liquid under Ultrahigh Fields of up to 28 T[END_REF] observed an unusually large influence of external magnetic field on the photoinduced hydrogen-abstraction reaction between benzophenone and thiophenol in ionic liquid, N,N,N-trimethyl-N-propylammonium bis(trifluoromethanesulfonyl) imide. Such magnetic field effects require a large cage effect in reaction media that is associated with micellar solutions rather than with homogeneous liquids. The comparison of experimental results with a reasonable cage model showed that in the ionic liquid there exist domains of 20-40 Å size Thiacarbocyanine dye, employed earlier in studying binary mixtures of conventional solvents [START_REF] Kh | Solvation-Shell Effect on the Cyanine-Dye Fluorescence in Binary Liquid Mixtures[END_REF][START_REF] Kh | Photophysical Properties of 3,3'-Diethylthiacarbocyanine Iodide in Binary Mixtures[END_REF], was used as a fluorescent probe.

Experimental

The dye, 3,3'-diethylthiacarbocyanine iodide (DTCI), was purchased from Fluka and used without further purification. Ammonium-based ionic liquid, trimethyl(butyl)ammonium bis((trifluoromethyl)sulfonyl)imide ([N(4)111] [Tf 2 N]) and imidazolium-based ionic liquid, 1butyl-3methylimidazolium bis((trifluoromethyl)sulfonyl)imide, [bmim] [Tf 2 N], were purchased from Solvionic, France. Their water content was checked to be less than 100 ppm. The structural formulas of cyanine dye and ionic liquids are shown in Scheme 1. Dimethylsulfoxide (DMSO) was of spectrograde (Uvasol, Merck). Scheme 1. The structural formulas of cyanine dye (DTCI) and ions of which the used ionic liquids consist.

Absorption spectra were recorded by a UV-3101PC spectrophotometer (Shimadzu) and steady-state fluorescence emission spectra of solutions, by a Fluorolog-3τ spectrofluorometer (Jobin Yvon). Steady-state fluorescence depolarization experiments were carried out by a Fluorolog-3τ spectrofluorometer using automatically operated polarizers (the DataMax software) with excitation at the peaks of the absorption spectra, i.e., about 560 nm, and emission at the peaks of the DTCI-fluorescence spectra about 580 nm. By definition, the polarization of fluorescence is

    H V H V GI I GI I P   
where I V and I H are the fluorescence components parallel and perpendicular to the plane of polarization of the excitation beam, respectively, and G is the correction factor that was taken into account in the course of experiment (the DataMax software). All measurements were carried out with 10 mm cells at room temperature.

Results and discussion

When the IL volume fraction increases in DMSO/ [N(4)111] [Tf 2 N] mixtures up to 100 vol %, the absorption peak of DTCI solution slightly (about 7 nm) shifts to the shorter wavelengths from 565 nm in neat DMSO (see Figure 1). The time-resolved fluorescence measurements have shown that upon excitation in a 560 nm spectral band, the fluorescence-decay dynamics recorded at 580 nm depends on the mixture composition (see Table 1). So that fluorescence decays were fitted to a bi-exponential function

                      2 2 1 1 exp exp     t t t I , (1)
where τ 1 and τ 2 are the lifetimes, α 1 and α 2 are their relative fractions ( 1

2 1     ).
For neat DMSO and DMSO-rich mixtures (volume fraction of IL is less than 40 vol %), the (where n is an average value), can be described as follows [START_REF] Landau | Statistical Physics[END_REF]:

  T P n RT n , 2            , ( 2
)
where μ is the chemical potential of DMSO, R is the universal gas constant, and T is the absolute temperature. Introducing the molar ratio of a DMSO component, x 1, one can rewrite Eq(2)

 

T P x RT x , 1 2 1              . ( 3
)
According to the Hildebrand theory [START_REF] Barton | Solubility Parameters[END_REF], the chemical potential of a DMSO component is

  2 2 1 1 2 2 1 0 ln          V x RT , (4)
where 2  is the volume fraction of IL, 1 V is the molar volume of DMSO, 1  and 2  are Hildebrand's solubility parameters for DMSO and IL respectively. Substituting Eq.( 4) into Eq.( 3)

and taking into account that 1 2

V V  , one can obtain for small values of x 1 :

    RT x V x x 2 1 2 2 1 1 1 2 1 4 1       . ( 5
)
On increasing   2 1

x  , the mixture becomes more fluctuating. For 2 1    , Eq.( 5) reduces to

  1 2 1 x x  
that corresponds to the stochastic level of the ideal mixture. Eq(5) is physically meaningful if the denominator is positive; on being so the closer a value of

  RT x V 2 1 2 2 1 1 4   
to unity, the stronger should be concentration fluctuations of the mixture.

Thus, in this approximation the conditions for strongly fluctuating mixtures in the range of 2

1 1  x is   RT V   2 2 1 1   . ( 6
)
Taking 1 V =71.3 cm 3 and 1  =12.0 cal RT ≈600 cal, so that condition ( 6) is pretty well fulfilled for [N( 4)111] [Tf 2 N] that is confirmed by experiment. This is in fairly good agreement with the results observed, considering the empirical nature of Hildebrand's theory and quite contradictory data on solubility parameters of ILs available in the literature. It is worth noting that similar results were obtained in the case of binary mixtures of molecular solvents [START_REF] Kh | Study of Preferential Solvation in Binary Solvent Mixtures by the Fluorescence-detected Magnetic Field Effect[END_REF][START_REF] Kh | A Fluorescence Spectroscopy Study of Preferential Solvation in Binary Solvents[END_REF]. 2), we can estimate the distribution of cyanine-dye molecules between these two types of clusters in the framework of the pseudo-phase model that was successively used in studying water-in-oil microemulsions [START_REF] Fletcher | Partitioning of p-nitroaniline between pseudophases within a water-in-oil microemulsion. Effects of temperature and microwave irradiation[END_REF].

By analogy with microemulsions, a binary mixture of DMSO with IL may be divided into two pseudo-phases and treated as if they are separated phases in equilibrium. The pseudo-phases introduced in this fashion are a continuous DMSO solvent and a phase of ionic liquid.

P      , (7)
where P 1 and P 2 are polarization in neat DMSO and neat IL respectively; N 1 and N 2 are the number of DTCI molecules which are comprised in DMSO-and IL-domains respectively; N 0 =N 1 +N 2 , where N 0 is the total number of dye molecules in samples and that was kept the same for all experiments.

For a system of two pseudo-phases the partition ratio is defined as follows

2 1 C C D  , (8)
where 1 C and 2 C are the concentration of the dye in DMSO and IL pseudo-phases respectively.

This definition of partition ratio, that neglects possible non-ideality effects, seems reasonable because of a very low total concentration of the dye in mixtures. It follows form Eq.( 8) that

            1 1 1 1 2 0 2 D N N . ( 9
)
Substituting Eq.( 9) into Eq.( 7), one can easily obtain . This formula allows the estimation of the DTCI partition ratio on the basis of the measured values of polarization for various compositions of the mixture (see Table 2).

               1 1

Conclusion

In summary, DMSO in the presence of ILs tends to produce clusters, instead of being completely mixed. These clusters become large and long-lived under certain conditions so that a fluorescence probe can "see" coarse-grain structure of the binary mixture. This accounts for the The ability of DMSO and ILs to segregate at the molecular level has a dynamic character and may be considered as concentration fluctuations of the mixture. In the framework of the empirical Hildebrand's theory of solubility, the condition for the observation such micro -

segregation is   RT V   2 2 1 1  
, where 1 V is the molar volume of DMSO, R is the universal gas constant, T is the absolute temperature, 1  and 2  are solubility parameters for DMSO and IL respectively.

  and 1-2 cP viscosity. Although there are voluminous publications on the IL topic, relatively few studies of IL mixtures with organic solvents are available. So that the purpose of this work is to study the microstructure of binary solvent mixtures, one component of which is IL, by means of fluorescence spectroscopy. As a representative example, we have chosen ionic liquids, [N(4)111][NTf 2 ] and [bmim][Tf2N], for studying the effect of the addition of organic solvent dimethylsulfoxide (DMSO) into these ILs on the microheterogeneous structure of mixtures.

  Viscosity of mixtures of ILs with DMSO were measured by a cone-and plate ARES viscosimeter (Rheometric scientific) at (25 ±0.1) C  and ambient pressure.

Figure 1 .

 1 Figure 1. DTCI absorption spectra in DMSO/ [N(4)111] [Tf 2 N] mixtures for various volume

Figure 2 .

 2 Figure 2. Fluorescence spectra of DTCI in DMSO/[N(4)111] [Tf 2 N] mixtures for various volume

  decay can be fitted to a single exponent (α 2 =0) with the lifetime component of about 330 ps. For IL-rich mixtures, a longer component is found which increases from about 670 to 1120 ps when a the volume fraction of IL becomes larger than 40 vol.%. In the case of [bmim] [Tf 2 N], only a single-exponential fluorescence decay has been observed for all mixture compositions.The appearance of a bi-exponential character of fluorescence decay seems to reflect microstructure of binary mixtures. Such features can be regarded as spatial fluctuations of mixture's composition that are created and destroyed by random diffusional motion of solvent molecules of both components. In greatly fluctuating mixtures, the fluorophore molecule always can find the long-lived (as compared with the life time of the fluorophore fluorescence) mode of spatial fluctuation, i.e. the molecular cluster mainly consisting of similar particles either of components.The ability of the binary mixture to fluctuate can determine the observed distinction in photophysical properties of DTCI in the mixtures that otherwise are quite similar in physical sense.

Figure 3 .

 3 Figure 3. The viscosity of the [N(4)111] [Tf 2 N] /DMSO mixture as a function the volume

  observed distinction between two liquid binary mixtures, DMSO/ [N(4)111] [Tf2N] and DMSO/[bmim] [Tf2N].

Table 1 .

 1 The lifetimes and their fractions of TDCI fluorescence in mixtures of DMSO with ILs as a function of volume fraction of DMSO.

						[N(4)111] [Tf 2 N]	[bmim] [Tf 2 N]
	DMSO, vol. %
		τ 1 (α 1 ), ps	τ 2 (α 2 ), ps	τ, ps
	100	326 (1.0)	--	326
	90		328(1.0)	--	--
	80	331 (1.0)	--	--
	75			--		--	385
	60	346 (0.93)	676 (0.07)	419
	40	333 (0.2)	451 (0.8)	512
	20	361 (0.1)	601 (0.9)	642
	0.4	736 (0.53)	1159 (0.47)	870
	Thermally generated fluctuations of the number of DMSO molecules, n, in a given
	nano-sized volume, i.e.	n 		n		n

Table 2 .

 2 The polarization of DTCI fluorescence, P, and the DTCI partition ratio, D,

	in mixtures

Acknowledgements

The authors are indebted to Dr. Christophe Melart for his help with the viscosity measurements and to Mr. Pavel Kozlov for assistance in fluorescence measurements.


Vilas Shinde
email: vilas.shinde@polytechnique.edu

Elisabeth Longatte

Franck Baj

Yannick Hoarau

Marianna Braza

Galerkin-free model reduction for fluid-structure interaction using proper orthogonal decomposition

Keywords: Reduced-order modeling, fluid-structure interaction, proper orthogonal decomposition, Navier-Stokes equations, panel flutter

A Galerkin-free model reduction approach for fluid-structure interaction (FSI) is presented in this article.

The reduced order model (ROM) is based on proper orthogonal decomposition (POD), where a reduced basis is formed using energy dominant POD modes. The reduced basis also consists of characteristics POD time modes that are derived from the POD time modes (coefficients) by using their periodicity. In addition to flow variables, the solution state vector comprises the mesh deformation, taking into account the structural deformation in FSI. A ROM solution is obtained by performing a linear interpolation of the reduced basis for changing operating/control parameters. The proposed Galerkin-free POD-ROM approach is demonstrated in terms of two test cases: a canonical case study of vortex-induced vibration (VIV) of a cylinder at Reynolds number Re = 100, where simulations are performed for various structural-to-fluid mass ratios; and a shock wave boundary layer induced panel flutter. For the second case, we use previously computed high-fidelity simulations, considering only the effect of panel thickness on the aeroelastic coupling between the flexible panel and shock wave boundary layer interaction (SWBLI); the inflow is at Mach 2 and Reynolds number based on panel length Re a = 50000. The presented Galerking-free ROM procedure is clean and robust for large mesh deformations, in addition to a significantly lower cost of computation compared to the FSI high-fidelity simulations.

Introduction

Computational fluid dynamics (CFD) can be a costly endeavor, particularly for the simulations involving large scale real world problems [START_REF] Lucia | Reduced-order modeling: new approaches for computational physics[END_REF]. Reduced-order models (ROM) can provide computationally inexpensive possibility to perform same computations with a minimum complexity, keeping the essential features of the system intact. Although ROM were initially developed in the context of dynamic systems and control theory, now they are essential in many disciplines. In the context of systems and control theory, the method of balanced truncation [START_REF] Moore | Principal component analysis in linear systems: Controllability, observability, and model reduction. Automatic Control[END_REF][START_REF] Li | Etbr: Extended truncated balanced realization method for on-chip power grid network analysis[END_REF] is based on the controllability and observability of a control system, while the Hankel-norm reduction [START_REF] Glover | All optimal hankel-norm approximations of linear multivariable systems and their l ∞ -error bounds[END_REF] method is applied for the approximation of transfer functions. The proper orthogonal decomposition (POD) or Karhunen-Loève expansion is commonly used in fluid dynamics [START_REF] Lumley | The Structure of Inhomogeneous Turbulent Flows[END_REF][START_REF] Sirovich | Turbulence and the dynamics of coherent structures. part i: Coherent structures[END_REF] in order to derive ROM as well as to investigate the flow structures and stability. The POD basis provides as an optimal choice, in terms of minimization of the quadratic errors, in order to form a reduced basis [START_REF] Holmes | Turbulence, dynamical systems and the unreasonable effectiveness of empirical eigenfunctions[END_REF],

which is used for the dimensionality reduction. A traditional procedure to derive a ROM is the Galerkin projection of Navier-Stokes equations (NSE) on the POD time-invariant reduced basis, leading to a reduced system of ordinary differential equations (ODE). The reduced set of ODE is then time integrated to obtain the space-invariant POD modes (also called POD time coefficients). An approximate solution is readily obtained using the POD reduced basis functions. In addition to the numerical issues associated with the Galerkin ROM, the dynamically moving mesh is a major difficulty in constructing a ROM for fluid-structure interaction (FSI) problems In general for FSI, two separate ROM are derived, one for the fluid domain and other for the structural domain. The two ROM are then coupled to form an FSI-ROM [START_REF] Dowell | Modeling of fluid-structure interaction[END_REF][START_REF] Beran | Reduced-order modelling of limit-cycle oscillation for aeroelastic systems[END_REF][START_REF] Vierendeels | Implicit coupling of partitioned fluid-structure interaction problems with reduced order models[END_REF][START_REF] Kalashnikova | A stable galerkin reduced order model for coupled fluid-structure interaction problems[END_REF]. This procedure has been beneficial in terms of reducing the number of parameters and also keeping the two ROM separate. In several other cases, the interface between the fluid and solid domains is also modeled [START_REF] Vierendeels | Implicit coupling of partitioned fluid-structure interaction problems with reduced order models[END_REF].

The FSI-ROM are prominently developed for the aero-elastic flutter [START_REF] Amsallem | Real-time solution of linear computational problems using databases of parametric reduced-order models with arbitrary underlying meshes[END_REF] and haemodynamics applications [START_REF] Vierendeels | Implicit coupling of partitioned fluid-structure interaction problems with reduced order models[END_REF]. The aeroelastic flutter is also considered as a weakly coupled FSI problem, and it has been studied using FSI-ROM in Silva and Bartels [START_REF] Silva | Development of reduced-order models for aeroelastic analysis and flutter prediction using the cfl3dv6. 0 code[END_REF], Beran et al. [START_REF] Beran | Reduced-order modelling of limit-cycle oscillation for aeroelastic systems[END_REF] and Raveh [START_REF] Raveh | Computational-fluid-dynamics-based aeroelastic analysis and structural design opimization -a researcher's perspective[END_REF], to name a few. The recent advancements of ROM for aerodynamic applications are discussed in Lucia et al. [START_REF] Lucia | Reduced-order modeling: new approaches for computational physics[END_REF] and Raveh [START_REF] Raveh | Computational-fluid-dynamics-based aeroelastic analysis and structural design opimization -a researcher's perspective[END_REF], they include mainly the ROM based on POD, Volterra series and method of Harmonic balance. Contrary to the weekly coupled FSI, the blood flow through arteries and heart represents a strongly coupled FSI problem.

where the ROM are expected to provide simplistic models [START_REF] Colciago | Comparisons between reduced order models and full 3d models for fluid-structure interaction problems in haemodynamics[END_REF][START_REF] Gerbeau | A quasi-newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows[END_REF][START_REF] Vierendeels | Implicit coupling of partitioned fluid-structure interaction problems with reduced order models[END_REF].

The influence of mesh deformation on the POD modes has been studied in Anttonen et al. [START_REF] Anttonen | Pod-based reduced-order models with deforming grids[END_REF], where a multi-POD technique has been adopted in order to select the POD reduced basis depending on the grid displacement; furthermore, the multi-POD was applied to a flow around a pitching and plunging airfoil [START_REF] Anttonen | Applications of multi-pod to a pitching and plunging airfoil[END_REF].

The blended POD/ROM based on a forced deformation analysis resulted in an accurate and effective FSI-ROM. A Galerkin POD-ROM for an FSI is developed in Bourguet et al. [START_REF] Bourguet | Reduced-order modeling of transonic flows around an airfoil submitted to small deformations[END_REF] for small imposed domain deformations in order to capture the transitional features of a compressible transonic flow. Iemma and Gennaretti [START_REF] Iemma | Reduced-order modeling for linearized aeroelasticity of fixed wings in transonic flight[END_REF] developed a ROM using a transonic full-potential aerodynamic model along with its transfer functions and a structural dynamic operator for the fluid-structure coupling. A general framework for constructing an optimization oriented ROM is presented in Bui-Thanh et al. [START_REF] Bui-Thanh | Goal-oriented, model-constrained optimization for reduction of large-scale systems[END_REF], with an example of a subsonic blade row. An FSI-ROM for a steady FSI problem has been presented in Lassila and Rozza [START_REF] Lassila | Reduced formulation of a steady fluid-structure interaction problem with parametric coupling[END_REF],

where a free-boundary problem has been reduced to a low-dimensional parameter space.

The present work builds on Shinde et al. [START_REF] Shinde | A galerkin-free model reduction approach for the navier-stokes equations[END_REF], where a Galerkin-free model reduction approach was proposed for Navier-Stokes equations. The Galerkin-free ROM approach treats the solution state variables separately, thus it can be conveniently used for multi-physics problems. In this effort, we extend the Galerkin-free ROM approach for FSI problems, treating large mesh deformations. The presented model reduction technique is demonstrated using two case studies: a two-dimensional vortex induced vibration in a cylinder and a three-dimensional shock wave boundary layer induced panel flutter. Section 2 describes the mathematical formulation of the Galerkin-free ROM procedure including a posteriori error estimate. The results and discussion of the test cases are presented in Sec. 3; while some concluding remarks are provided in Sec. 4.

Mathematical formulation

The computational fluid dynamics (CFD) simulations are considered as the high-fidelity model (HFM), where a solution state vector comprises the mesh deformation in addition to the solution of Navier-Stokes equations. Let the solution state vector s s s = s s s(χ χ χ, t; η) be spanned on the a time dependent space χ χ χ ∈ Ω, which is mapped by using a mapping function ξ ξ ξ on a time independent space x x x ∈ Ω, where Ω ⊂ R 3 . The time is denoted by t ∈ T , where T ⊂ R. The variable η stands for the controlling parameter. Let H be a Hilbert space and a state variable s i (x x x, t; η) ∈ H with i = 1, 2, • • • , r(s s s). r(s s s) is the number of state variables.

The standard inner product of a state variable s i (between two time instances, s i (x x x, t 1 ; η) and s i (x x x, t 2 ; η),) is given by, (s i (x x x, t 1 ; η), s i (x x x, t 2 ; η)

) Ω = Ω s i (x x x, t 1 ; η) • s i (x x x, t 2 ; η)dx x x (1)
The induced norm and time averaging (over the time T ) of the state variable is defined as,

||s i || Ω = (s i , s i ) Ω and si = 1 T T s i dt = s i T . (2)
The controlling/operating parameter η is an important parameter of ROM. For a change in η, the ROM must predict the solution state vector s s s(x x x, t; η), which can also be performed on the fly. The flow parameters such as the Reynolds number, Mach number and angle of attack can serve as the controlling parameter. In fluid-structure interaction problems, structural parameters such as the mass ratio, non-dimensional stiffness and geometric parameters are commonly considered, depending on the objective of reduced-order modeling.

The controlling parameter is also referred as control or objective function.

The Snapshot POD

Lumley [START_REF] Lumley | The Structure of Inhomogeneous Turbulent Flows[END_REF] introduced the method of proper orthogonal decomposition (POD) to fluid mechanics in order to analyze coherent turbulent flow structures. The direct POD method involves, generally, a large number of degrees-of-freedom when applied to a CFD simulation. The method of snapshot POD, proposed by Sirovich [START_REF] Sirovich | Turbulence and the dynamics of coherent structures. part i: Coherent structures[END_REF], results in significant reduction of the number of degrees-of-freedom. In an arbitrary Lagrangian Eulerian (ALE) formulation for fluid-structure coupling the computational mesh is re-positioned in accordance with the moving solid boundaries. A simple numerical approach to handle the mesh is to solve the Laplace equation for the mesh deformation by retaining the mesh topology. Let the solution state vector be s s s(x x x, t; η), which includes the deforming mesh in addition to the flow state variables. The solution state vector can be written as,

s s s(x x x, t; η) =         χ i (x x x, t; η) ρ(x x x, t; η) v i (x x x, t; η) p(x x x, t; η)         with i = 1, 2, 3, (3)
where the variables ρ, v i and p are the density, components of velocity vector and pressure respectively. A number of solution states or snapshots of an accurate high-fidelity (HF) CFD simulation are gathered as it advances in time. The state vector can be decomposed by using the Reynolds decomposition as,

s i (x x x, t; η) = s i (x x x; η) + s i (x x x, t; η) (4)
= s i (x x x; η) + ∞ j=1 φ si j (x x x; η)a si j (t; η), (5)
where s i (x x x, t; η) represents the unsteady part of the Reynolds decomposition. The flow solution is assumed to be statistically stationary. The unsteady part of the state vector in Eq. 4 is equivalently represented by means of the time-invariant POD space modes (topos) and space-invariant POD time modes (chronos) (Eq. 5). The solution state vector can be approximated by using a reduced number of energy dominant POD modes (φ si j ) and corresponding time coefficients (a si j ) as shown in Eq.6.

s i (x x x, t; η) ≈ s i (x x x; η) + N s i r j=1 φ si j (x x x; η)a si j (t; η) t 1 ≤ t ≤ t sn (6)
A reduced-order solution can be obtained for the time interval of snapshots collection i.e. for [t 1 , t sn ], where N si r is a reduced number of POD modes for s i ; this number can be different for each variable of the solution state vector. Here t 1 and t sn represent time of the first and last snapshots respectively. Let R R R(η) be a two point time-correlation function, given by,

R R R(η) =            R χ1 (η) (ti×tj ) . . . R ρ (η) (ti×tj ) . . . R p (η) (ti×tj )            = 1 N t            (χ 1 (x x x, t i ), χ 1 (x x x, t j )) Ω . . . (ρ(x x x, t i ), ρ(x x x, t j )) Ω . . . (p(x x x, t i ), p(x x x, t j )) Ω            , (7)
where N t stands for the number of snapshots used to estimate the time-correlation tensor. The subscript (t i × t j ) stands for the dimensions of each element of the correlation tensor. The correlation tensor R R R(η) is solved for an eigenvalue problem, as in Eq. 8.

R si (η)a si j (t; η) = λ si j a si j (t; η),

where λ si j are the eigenvalues. The eigenfunctions a si j (t; η) satisfy,

a si j (t; η), a si k (t; η) Tsn = δ jk , (9)
where, δ jk is the Kronecker delta. The POD modes are arranged in descending order of their energy content (eigenvalues), i.e λ si 1 ≥ λ si 2 ≥ ............... ≥ λ si N pod > 0. The orthonormal POD space modes are obtained using Eq. 10, such that φ si j (x x x; η), φ si k (x x x; η) Ω = δ jk .

φ si j (x x x; η) = 1 N t λ si j s i (x x x, t; η), a si j (t; η) Tsn (10)
The corresponding POD time coefficients are given by,

a si j (t; η) = φ si j (x x x; η), s i (x x x, t; η) Ω = N t λ si j a si j (t; η) (11)
Generally, the number of reduced POD modes (N si r ) is much smaller compared to the total POD modes (N si r << N si pod ), containing most of the energy E c . The relative energy of reduced system is given as,

%Ec si = N s i r j=1 λ si j N s i pod j=1 λ si j × 100. ( 12
)
The total energy of the system can be estimated by Eq. 13, which also provides an approximate estimate of the energy in terms of POD time coefficient (a si j ).

E si (η) = 1 2 Ω s i (x x x, t; η) 2 Tsn dx x x ≈ 1 2 N s i r j=1 λ si j = 1 2 N s i r j=1 a si j (t; η) 2 Tη , (13)
where T sn is the time interval of snapshots collection. An approximate value of total energy can be obtained for a minimum value of time T η , which is the time-period of one limit cycle oscillation (LCO) of the first POD mode, a 1 (t; η), for an operating parameter η. Equation 14 provides characteristic time coefficients, where t a is an arbitrary time in

t a = [0, (t sn -T η )]. ãsi i (t; η) = a si i (t; η) f or t ∈ [t a , t a + T η ] (14)
Equation 6 can be reframed as in terms of the characteristic POD time modes as:

s i (x x x, t; η) ≈ s i (x x x; η) + N s i r j=1 φ si j (x x x; η)ã si j (t; η) t ≥ 0, (15)
leading to a ROM that can time march outside the snapshots' time interval ([t 1 , t sn ]).

Linear interpolation

The time average of state vector ( s s s ) and the POD modes (φ si i , ãsi i ) are interpolated linearly for any change in the operating parameters such as Reynolds number, mass ratio (mass of solid/mass of displaced fluid medium) in fluid-structure interaction systems by using Eq. 16,

Γ(β β β; η) = Γ(β β β; η 1 ) + (Γ(β β β; η 2 ) -Γ(β β β; η 1 )) (η 2 -η 1 ) (η -η 1 ), (16)
where η 1 , η 2 are the values of controlling parameter for two high-fidelity reference cases. Γ is a dummy variable for s i , φ si j and ãsi j , whereas β β β is replaced by x x x for the state variable or space modes and by t for the characteristic POD time modes. A priori the standard inner product in Eq. 17 is satisfied, so that the interpolated quantities (terms on the right hand side of Eq.15) are in minimum phase difference with η 1 reference case.

(Γ(β β β; η 1 ), Γ(β β β; η 2 )) β β β ≥ 0 ( 17
)
The linear interpolation can be easily extended for a linear extrapolation using the reference state solutions, in order to broaden the range of operating conditions. Furthermore, a smooth transition from an operating condition to another can be achieved by using a linear transition, as discussed in Morzynski et al.

[24] and Shinde et al. [START_REF] Shinde | A galerkin-free model reduction approach for the navier-stokes equations[END_REF].

Error estimate

An error estimate based on the quadratic energy terms is considered, which provides an upper bound of the error for a Galerkin-POD ROM [START_REF] Balajewicz | Stabilization of projection-based reduced order models of the navier-stokes[END_REF][START_REF] Demmel | Applied Numerical Linear Algebra[END_REF]. The normalized error ( (t; η)) can be given by,

(t; η) = r(s s s) i=1 (E si (t; η) pod -E si (t; η)) σ i 2 (η) 2 = r(s s s) i=1   N pod j=1 ãsi j (t; η) 2 pod - Nr j=1 ãsi j (t; η) 2 σ i 2 (η)   2 , (18)
where the subscript pod stands for the energy of POD reconstructed solution. The variance is computed using HF model as,

σ i 2 (η) = Ω s i (x x x, t; η) 2 hf T dx x x (19)

Results and discussion

At first, we present a test case of vortex-induced vibration of cylinder at Reynolds number based on cylinder diameter Re D = 100. The computations are performed in two-dimensional configuration for various structural-to-fluid mass ratios, and a ROM is constructed for the predictions at off-reference values of the mass ratio. Secondly, a test case of panel flutter induced by shock wave boundary layer interaction is considered in order to demonstrate the applicability of the proposed method for three-dimensional and highly non-linear fluid-structure interaction problems. For this case, we consider an existing high-fidelity simulation database from Shinde et al. [START_REF] Shinde | Effect of structural parameters on shock wave boundary layer induced panel flutter[END_REF], where an incoming laminar boundary layer at Mach 2 and

Reynolds number base on panel length Re a = 50000 interacts with an incident shock at the shock angle of 35 • and shock strength of 1.8 over a flexible panel. The study [START_REF] Shinde | Effect of structural parameters on shock wave boundary layer induced panel flutter[END_REF] comprises effect of several structural parameters on the panel flutter, however we consider only the effect of panel thickness, and construct a ROM solution for the flutter response and wall pressure of the panel. ∂ ∂t

(w w w) + ∂ ∂x (f f f -f f f ν ) + ∂ ∂y (g g g -g g g ν ) = 0, (20)
where

w w w =         ρ ρu ρv ρE         , f f f =         ρu ρu 2 + p ρuv u(ρE + p)         , g g g =         ρv ρvu ρv 2 + p v(ρE + p)         f f f ν =         0 τ xx τ xy [τ τ τ , v v v] x -q x         , g g g ν =         0 τ yx τ yy [τ τ τ , v v v] y -q y         .
Here w w w is the state vector. f f f , g g g are the convective fluxes, while f f f ν , g g g ν are the viscous fluxes. The flow velocity components in x and y directions are denoted by u and v respectively. p stands for the pressure, while E represents the specific energy density. The components of shear stress tensor τ τ τ in the viscous fluxes are given by Eq. 21.

τ xx = 2 3 µ 2 ∂u ∂x - ∂v ∂y , τ yy = 2 3 µ - ∂u ∂x + 2 ∂v ∂y τ xy = τ yx = µ ∂u ∂y + ∂v ∂x (21)
The heat flux is calculated using Fourier's law as,

q x = -k θ ∂θ ∂x , q y = -k θ ∂θ ∂y with k θ = µC p /P r, (22)
where k θ , θ are the thermal conductivity, temperature respectively. The Prandtl number (P r) is taken 0.72 (for air).

The second order fully implicit LU-SGS (Lower-Upper Symmetric Gauss-Seidel) backward A-stable scheme is used for solving the discretized equations, as discussed in Vos et al. [START_REF] Vos | Recent advances in aerodynamics inside the nsmb (navier-stokes multi block) consortium[END_REF]. The space derivatives are discretized by using a forth order central finite volume scheme in a skew-symmetric form. For the incompressibility, the preconditioning method proposed in Turkel et al. [START_REF] Turkel | Preconditioning methods for low-speed flows[END_REF] is used for the flows with low Mach numbers.

Fluid-structure coupling

The cylinder's motion is modeled as the mass on a spring system, as shown in Fig. 2, where the cylinder oscillates only in the lift direction (Y direction). The equation of cylinder motion can be written as:

+ ky = F y , ( 23
)
where y is the cylinder displacement in the lift direction. c and k denote the structural damping and stiffness coefficients respectively. The fluid force F y , per unit length of the cylinder, drives the motion of cylinder.

The second order differential equation is solved using Hilber-Hughes-Taylor method (HHT) Newmark algorithm [START_REF] Erlicher | The analysis of the generalized-α method for non-linear dynamic problems[END_REF]. An arbitrary Lagrangian Eulerian (ALE) method is used to simulate the fluid-structure coupling [START_REF] Hoarau | Recent developments of the navier stokes multi block (nsmb) cfd solver[END_REF], where the displacement (y) and velocity ( dy dt ) of the cylinder are used to deform the mesh and update the flow velocities respectively.

The non-dimensional flow and structure parameters are listed in Table 1, where u * = u/(f n D) is the non-dimensional reduced velocity. The structural non-dimensional frequency f * n = f n D/u ∞ , leading to f * n = 1/u * at the far field. The damping coefficient ζ and natural frequency f n are respectively expressed as:

ζ = c 2 √ km and f n = 1 2π k m . (24)
The high value of the damping ratio is selected in order to control the amplitude of the cylinder oscillations.

The flow past a cylinder for Reynolds number Re D = 100 results in a Strouhal number, St = f sh D/u ∞ , of St ≈ 0.17, where f sh is the vortex shedding frequency. Thus, the cylinder frequency f * n = 0.17 leads to the VIV.

Re D u * f * n ζ (%)
100 5.88 0.17 20

Table 1: Simulation parameters

As noted before, the mass ratio m * is considered as the controlling parameter (η = m * ) for the VIV. The High-fidelity simulations are performed for a range of mass ratio values, such that the oscillation amplitudes of the cylinder vary approximately between y ≈ 0 and y ≈ 1D.

POD analysis

The solution state vector s s s(x x x, t; η) of Eq. 3 becomes

s s s(x x x, t; m * ) =         y(x x x, t; m * ) u(x x x, t; m * ) v(x x x, t; m * ) p(x x x, t; m * )         (25)
for the VIV of cylinder case study. A set of snapshots of the solution vector are recorded for the snapshot POD analysis, where a total of 576 snapshots are collected using a non-dimensional time step of ∆t = 0.05.

A time-correlation tensor is formed by using the unsteady part (s s s ) of the state vector, which is then solved for the eigenvalue problem as per the procedure detailed in Sec. 2. The % energy (λ i / i λ i ×100) associated with the POD modes of the different state variables is shown in Fig. 3 for the mass ratio of m * = 2.50. The POD modal energies are arranged in decreasing order for all variables The energy associated with the first POD mode of mesh deformation is ≈ 100% (Fig. 3 a). Thus, only one POD mode (N y r = 1) is sufficient for the reconstruction of mesh movement. On the other hand, first 10 POD modes of the velocity (u, v) and pressure comprise ≈ 99% of energy, thus only 10 out of 576 modes are retained for the ROM. Furthermore, the corresponding temporal modes exhibit stable limit cycle oscillations, ensuring the periodicity required in order to extract the characteristic time modes of Eq. 14. The pair of first POD modes for the mesh deformation is shown in Fig. 4, displaying the space mode (φ y 1 (x x x; m * )) in Fig. 4(a) and corresponding characteristic temporal mode (ã y 1 (t; m * )) in Fig. 4(b), where the period of LCO is T m * = 5.79622.

ROM solution

The vortex-induced vibration in a circular cylinder is extensively studied in the past [START_REF] Williamson | Vortex-induced vibrations[END_REF]. There exists a critical minimum value of the mass ratio, as discussed in Williamson and Govardhan [START_REF] Williamson | Vortex-induced vibrations[END_REF], which is m * ≈ 0.4241 ; this value of critical minimum mass ratio changes with other system parameters. In order to obtain the desired higher amplitudes of cylinder oscillations, the very high value (20%) of the damping ratio is used, varying the mass-damping parameter (mδ/ρD2 [START_REF] Shinde | A theoretical model of fluidelastic instability in tube arrays[END_REF], where δ ≈ 2πζ) from 0.2 to 4. The physics of the problem at small and very small mass-damping parameter is studied in Khalak and Williamson [START_REF] Khalak | Fluid forces and dynamics of a hydroelastic structure with very low mass and damping[END_REF] and Khalak and Williamson [START_REF] Khalak | Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping[END_REF].

The reduced number of POD modes (N si r ) with high energy content are selected to form the POD subspace. There are two main problems while constructing the ROM for this FSI system: first, the traditional Galerkin projection of the Navier-Stokes equations on the POD time invariant modes (or space modes) when the computation domain is time dependent, which is discussed in a great detail by Tadmor et al. [START_REF] Tadmor | Galerkin models enhancements for flow control[END_REF]. Second, the validity of a POD-based ROM is generally limited to a small range of controlling parameter, since the POD reduced basis non-linearly depends on the controlling parameter. Bui-Thanh et al. [START_REF] Bui-Thanh | Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics[END_REF] proposed a gappy POD procedure to construct the off-reference ROM solutions, thereby dealing with the change of controlling parameters. The method uses the POD coupled with an interpolation method, which avoids the Galerkin projection of the governing equations. In Lieu et al. [START_REF] Lieu | Reduced-order fluid/structure modeling of a complete aircraft configuration[END_REF] a ROM for a complete aircraft is formulated based on a Mach-adaptation strategy, where the angle between POD subspaces is interpolated in order to deal with the changes in controlling parameter. The interpolation of the reduced basis for the change in controlling parameter is performed in a tangent space to a Grassmann manifold in Amsallem and Farhat [START_REF] Amsallem | Interpolation method for adapting reduced-order models and application to aeroelasticity[END_REF] and further on matrix manifolds in Amsallem and Farhat [START_REF] Amsallem | An online method for interpolating linear parametric reduced-order models[END_REF]. In the present work, the periodic reconstruction of the POD time modes as per Eq. 14 is used to circumvent the Galerkin projection; and the solution, including mesh deformation, is readily built using Eq. 15. A direct linear interpolation of the POD space as well as time modes is performed in order to predict an off-reference solution state by using the pre-simulated reference cases.

The linear interpolation of the energy dominant (N si r ) POD modes is performed for any value of the mass ratio in the range of reference cases listed in Table 2. A comparison between the interpolated POD modes and high-fidelity POD modes for an off-reference case (mass ratio m * = 2.75) is shown in Fig. 5. The figure compares first three time-invariant POD modes (topos) of the streamwise velocity (u). Figure 5 (a), (c) and (e) show the first three energy dominant POD modes (φ u 1 , φ u 2 and φ u 3 ) respectively, which are obtained by using the HF simulation. In comparison, the POD modes in Fig. 5 (b),(d) and (f) are the corresponding linearly interpolated POD modes, which are obtained by using the reference state POD reduced sub-spaces at m * = 2.50 and m * = 3.00. A similar configuration at Reynolds number Re = 1690 is treated in Liberge and Hamdouni [START_REF] Liberge | Reduced order modelling method via proper orthogonal decomposition (pod) for flow around an oscillating cylinder[END_REF], where a non-linear FSI ROM based on a multiphase formulation of the Navier-Stokes equations and its extension in the solid domain is presented; in their study, Reynolds number is considered as a controlling parameter.

In addition to the interpolation of the POD space modes, the temporal coefficients are also linearly interpolated in phase space. The interpolation of the first three temporal coefficients in phase space is shown in Fig. 6. The interpolated chronos (in blue colour, Fig. 6)) coincide with the high-fidelity modes beneath (in green colour, Fig. 6), indicating a high accuracy of the presented FSI-ROM.

Figure 7 shows a flow chart of the operating principle of the presented reduced-order approach. Once the reference high-fidelity simulations are performed, one can build the reference POD basis. The POD reduced bases are formed separately for each state variable, including the mesh deformation. For a given Similar to most of the existing POD-based ROM, this method is also a posteriori formulation, which implies that the high-fidelity simulations need to be performed in order to obtain the reference cases. The formulation is based on the assumption of statistical stationarity of the coupled fluid-structure system.

Although this test case is a periodic flow, the method is not restricted to periodic flows but the stationary flows. This aspect of the ROM is already discussed in the previous work [START_REF] Shinde | A galerkin-free model reduction approach for the navier-stokes equations[END_REF][START_REF] Shinde | Fluidelastic instability in heat exchanger tube arrays and a galerkin-free model reduction of multiphysics systems[END_REF] based on the periodicity of the POD time coefficients in a stationary flow. For instance, Rempfer and Fasel [START_REF] Rempfer | Dynamics of three-dimensional coherent structures in a flat-plate boundary layer[END_REF] have presented the POD time coefficients of the vorticity for a flat plate boundary layer turbulent flow. Although the flow is not periodic, the POD time coefficients are periodic. A large difference between two reference states may lead to a higher interpolation error, due to the linear interpolation as well as due to the order/rank change of the POD modes.

A shock wave boundary layer induced panel flutter

The The simulations in Shinde et al. [START_REF] Shinde | Effect of structural parameters on shock wave boundary layer induced panel flutter[END_REF] were performed by using a well validated aeroelastic solver [START_REF] Gaitonde | Padé-type higher-order boundary filters for the navier-stokes equations[END_REF][START_REF] Visbal | High-order-accurate methods for complex unsteady subsonic flows[END_REF][START_REF] Visbal | On the use of higher-order finite-difference schemes on curvilinear and deforming meshes[END_REF][START_REF] Visbal | Numerical simulation of the interaction of a transitional boundary layer with a 2-d flexible panel in the subsonic regime[END_REF]. A 2nd order implicit time marching scheme alongside a 6th order compact difference scheme was used for solving Navier-Stokes equation in a full three-dimensional curvilinear formulation [START_REF] Shinde | Panel flutter induced by transitional shock wave boundary layer interaction[END_REF][START_REF] Shinde | Control of transitional shock boundary layer interaction using surface morphing[END_REF]. The structural dynamics of the flexible panel was modeled by using von-Karman plate theory. The spatial derivatives in the structural equations were solved using standard 2nd order central finite difference scheme, whereas the equations were time-marched by using a second order unconditionally stable Newmarks β algorithm. Further numerical details on the fluid-structure coupling are elaborated in Gordnier and Visbal [START_REF] Gordnier | Development of a three-dimensional viscous aeroelastic solver for nonlinear panel flutter[END_REF].

We consider the effect of panel thickness h * on panel flutter in order to construct the proposed Galerkinfree POD-ROM, thus the controlling parameter of ROM is the panel thickness (η = h * ). Furthermore, we consider only a computational plane of the geometry for the ROM that includes the panel, and a solution state vector comprising only the panel deflection and wall pressure, which are important variables for panel flutter. Table 3

POD analysis

The solution state vector of Eq. 3 comprises the panel deflection y and the wall pressure p w , given as:

s s s(x x x, t; h * ) =   y(x x x, t; h * ) p(x x x, t; h * )   . (26)
The simulations in Shinde et al. [START_REF] Shinde | Effect of structural parameters on shock wave boundary layer induced panel flutter[END_REF] were performed for sufficiently long non-dimensional time, ensuring the statistical stationarity of the flutter response. A total of 1000 snapshots are collected using a time interval of ∆t = 0.01 in order to perform the snapshot POD analysis as detailed in Sec. 2. The snapshot POD is performed for both the reference cases, while the results for panel thickness h/l = 0.008 are presented in this section.

The POD modal energy curves in Fig. 11 1000 snapshots; however, due to optimality of the POD [START_REF] Holmes | Turbulence, dynamical systems and the unreasonable effectiveness of empirical eigenfunctions[END_REF], only few modes comprise most of the energy.

The figure (Fig. 11(a)) shows first 100 modes ordered based on their energy content for both the variables.

Clearly, the first two modes of panel deflection and first ten modes of wall pressure contain ≈ 99% of the total energy. Thus, we retain only these modes, in addition to the time-averages of variables, for ROM predictions. The first two energy dominant POD modes for the panel deflection, Fig. 12 (a), indicate higher modes of panel vibration [START_REF] Shinde | Panel flutter induced by transitional shock wave boundary layer interaction[END_REF], where the first mode constitutes most energy; which is ≈ 97.43% as opposed to ) and (ã pw 5 -ãpw 6 ) etc., where the modes from a pair show a phase shift with respect to the first POD mode ãpw 1 for the same value of time period. The presence of POD mode pairs indicates organized flow structures of the transitional SWBLI, as the boundary layer is not developed to a fully turbulent state.

ROM solution

As noted before, only first two modes of the panel deflection and first ten modes of wall pressure constitute ≈ 99% of the total energy, forming a reduced order POD basis. The POD reduced basis comprise the reduced number of POD modes for all reference cases listed in Table 3, and an off-reference case solution can be readily obtained by the ROM procedure shown in Fig. 7. We consider an off-reference case with the panel thickness of h * = 0.010. In order to perform the linear interpolation defined in Sec. 2.2, two reference cases that encompasses the off-reference case are selected, which include: h * = 0.008 and h * = 0.016.

The interpolation leads to a reduced POD basis for h * = 0.010, which includes the POD space and time modes as well as the time-averaged variables. Figure 14 shows the interpolated POD modes in comparison with the high-fidelity POD modes at the same value of panel thickness (h * = 0.010). The first two energy dominant POD modes for the panel deflection, φ y 1 and φ y 2 , are shown are in Fig. 14(a) and Fig. 14(b), where the interpolated modes are almost identical to the high-fidelity modes. The interpolated modes follow the sign convention of any one reference case, here they follow h * = 0.008. Although a large difference in the reference cases can introduce non-linearities and the linear interpolation may result in higher numerical error, the dominant POD modes appear to be consistent over the considered range of panel thickness. Xie et al. [START_REF] Xie | Projection-free proper orthogonal decomposition method for a cantilever plate in supersonic flow[END_REF] made similar observation while developing a ROM for a nonlinear aeroelastic flutter based on their prior study [START_REF] Xie | Proper orthogonal decomposition reduced-order model for nonlinear aeroelastic oscillations[END_REF], where a set of POD modes was used for various cases with different flutter parameter The space-invariant POD reduced basis, which consists of the characteristics POD time modes, is also interpolated by using the procedure in Sec. 2.2. The linear interpolation of the first two and first ten POD modes for the panel deflection and wall pressure is performed for the off-reference case of h * = 0.010 by using the two reference cases: h * = 0.008 and h * = 0.016, similar to the POD space modes and time averages. flutter response at a probe location in Fig. 16. The POD modal energy contribution, Fig. 16 (a), for the ROM is in good agreement with the high-fidelity modes for both the panel deflection and wall pressure, where the reduced basis comprise ≈ 99% of the total energy. The error in ROM prediction based on Eq. 18 is shown in Fig. 16(a), where the time evolving error exhibits LCO with a maximum value of ≈ 0.17 in a cycle. As discussed in Shinde et al. [START_REF] Shinde | A galerkin-free model reduction approach for the navier-stokes equations[END_REF], the Galerkin-free POD ROM approach is unconditionally stable, where the error (t) remains in bounds for values of t. The panel flutter response in terms of the panel deflection at the 3/4 chord length is compared in Fig. 16(c), where the ROM predictions closely follow the high-fidelity solution. The time-averaged panel deflection and its standard deviation are y m /h = 0.9535 and y SD /h = 0.0564 against the high-fidelity simulation values of y m /h = 0.9331 and y SD /h = 0.0523 respectively.

A high-fidelity simulation costs ≈ 7500 hours of processor time in order to obtain a statistically stationary solution. The presented Galerkin-free procedure predicts the reduced order solution in ≈ 10 seconds, substantially reducing the computational cost; which is also due to the partial selection of the computational domain and solution state variables while constructing the ROM.

Conclusion

A Galerkin-free model reduction of fluid-structure interactions based on proper orthogonal decomposition is presented. The computational mesh deformation is considered as a part of the solution state vector while constructing the reduced POD basis. The approach is demonstrated by using two case studies: a vortexinduced vibration of a cylinder at low Reynolds number and a shock boundary layer induced panel flutter, which comprise the two-dimensional laminar and three-dimensional transitional flows respectively. The energy dominant POD time modes must exhibit a periodic response in order to extract the characteristic POD time modes. The periodicity assumption holds for both the two dimensional VIV of cylinder and threedimensional panel flutter cases. The presented ROM procedure is clean and robust in terms of handling large mesh deformations, in addition to a significant reduction of computing cost, which is enticing for the flow/flutter control applications.

3. 1 .

 1 Vortex-induced vibration of a cylinder at Re D = 100 for various mass ratios The two-dimensional vortex-induced vibration (VIV) of a circular is at Reynolds number Re D = 100 based on the cylinder diameter D, streamwise inflow velocity u ∞ , fluid density ρ and fluid dynamic viscosity µ. The structural parameters include the mass ratio m * = m/(ρD 2 ), cylinder natural frequency f n (in vacuum) and damping coefficient ζ, where m represents the cylinder mass per unit length. Figure 1 shows the computational domain (χ χ χ) and instantaneous streamwise velocity field (u(χ χ χ, t; η)), where the control parameter η = m * . The simulations are performed by using standard Dirichlet and far field boundary conditions, leading to the mean flow in +X direction. The cylinder surface is a no-slip adiabatic wall.(a) Geometry and mesh (b) Instantaneous streamwise velocity

Figure 1 :

 1 Figure 1: Computational domain and instantaneous flow field at Re = 100 and mass ratio m * = 2.50

Figure 2 :

 2 Figure 2: Coupling between fluid and the cylinder

Figure 3 :

 3 Figure 3: % energy associated with the POD modes at Re D = 100, m * = 2.50

Figure 4 :

 4 Figure 4: The pair of first POD modes for mesh deformation in y direction for Re D = 100 and m * = 2.50.

  (a) φ u 1 (x x x; m * ) (high-fidelity mode) (b) φ u 1 (x x x; m * ) (interpolated mode) (c) φ u 2 (x x x; m * ) (high-fidelity mode) (d) φ u 2 (x x x; m * ) (interpolated mode) (e) φ u 3 (x x x; m * ) (high-fidelity mode) (f) φ u 3 (x x x; m * ) (interpolated mode)

Figure 5 :Figure 6 :

 56 Figure 5: Comparison of the POD modes (HF simulations) with interpolated modes at Re = 100, m * = 2.75

Figure 7 :

 7 Figure 7: Working principle of the presented ROM

Figure 8 :

 8 Figure 8: Comparison between the ROM and HF CFD solutions at Re D = 100, m * = 2.75

Figure 8 (

 8 Figure 8(b) shows the normalized energy based error ( (t)) as defined in Eq. 8, limiting the errors to 2%. The ROM solution is obtained within ≈ 20s using a single processor by following the presented ROM model. On the contrary, the corresponding HF CFD simulation takes more than 24 hours on 8 processors to provide the solution. Figure9shows the instantaneous streamwise velocity curve compared between the

Figure 9 :

 9 Figure 9: Comparison of streamwise velocity (u) at t = 10 for m * = 2.75

  flow configuration of a shock wave boundary layer induced panel flutter consists of an oblique incident shock at the shock angle of 35 • and shock strength of p 3 /p 1 = 1.8. The schematic of flow configuration is shown in Fig. 10(a). The incident shock interacts with a laminar boundary layer that evolves from a Blasius profile, where the inflow boundary thickness is δ in /l = 0.0083, inflow Mach number M ∞ = 2.0 and Reynolds number based on panel length Re l = 50000. The shock impinges at the mid-chord location of the panel, where the panel length (l) to width (b) ratio is b/l = 1. The pressure inside the cavity that is beneath the flexible panel is p c /p 1 = 1.4. Shinde et al. [27] investigated the panel flutter for various structural parameters that include: panel aspect ratio b/l, panel thickness h * = h/l, flutter parameter and mass ratio.Figure 10(b) shows a three-dimensional flow field in terms of the Q criterion isosurface colored with velocity magnitude, shadowgraph (Laplacian of the density) and panel deflection contours, exhibiting a transitional shock wave boundary layer interaction (SWBLI) over a flexible panel, for a set of structural parameters. The thick lined square on the bottom surface indicates the panel edges (Fig. 10 b).

Figure 10 :

 10 Figure 10: Flow configuration. (a) schematic of the computational domain (b) an instantaneous three-dimensional fluidstructure interaction state. y axis is scaled by factor 4 for clarity.

  (a) show the percentage energy contained in each mode for the panel deflection and wall pressure. The snapshot procedure leads to a total of 1000 POD modes for 10 POD modal energy (b) time-mean panel deflection (c) time-mean wall pressure

Figure 11 :

 11 Figure 11: POD modal energy content and the time-averages of the panel deflection and wall pressure (a) Energy contribution of POD modes (b) time-averaged panel deflection (b) time-averaged wall pressure.

Figure 11 (Figure 12 :

 1112 Figure 11(b) and (c) show the time-averaged panel deflection and wall pressure profiles respectively, where the profiles are normalized by the panel thickness and free-stream pressure (p ∞ ) respectively. The timemean panel deflection profile exhibits a sinusoidal response (which corresponds to the mode 2,1 of panel deformation) due to the impinging shock, the wall pressure on the other hand (Fig. 11 c) indicates the typical increase across a shock.

≈ 1 .

 1 Fig.13(b) in terms of phase portraits, where LCO of the first seven modes are displayed. Furthermore, the phase portraits in Fig.13(b) exhibit pairs of POD modes: (ã pw 1 -ãpw 2 ), (ã pw 3 -ãpw 4 ) and (ã pw 5 -ãpw 6 ) etc., where the modes from a pair show a phase shift with respect to the first POD mode ãpw 1 for the same value of time period. The presence of POD mode pairs indicates organized flow structures of the transitional

1 (Figure 13 :

 113 Figure 13: Energy dominant POD space and characteristic time modes of wall pressure. (a) POD space modes φ pw i (b) POD

Figure 14 :

 14 Figure 14: Comparison between the high-fidelity and interpolated POD space modes from reduced POD basis for h * = 0.010 (a) high-fidelity modes for panel deflection (b) interpolated modes for panel deflection (c) high-fidelity modes for wall pressure (d) interpolated modes for wall pressure. The arrows indicate flow direction.

Figure 15 shows

 15 Figure15shows linearly interpolated characteristic POD time modes for the panel deflection and wall pressure for h * = 0.010 in comparison with the high-fidelity solutions. As opposed to the POD space modes, the modes exhibit non-linearity in their amplitudes and phase values for the change in controlling parameter, leading to higher differences between the high-fidelity solution and ROM interpolation. The phase portrait of the first two POD modes of the panel deflection is shown in Fig.15(a), comparing the high-fidelity and ROM LCO; where the interpolated modes, to some degree, under predict the amplitudes.The interpolation of the second, third and eighth modes against the first mode of the wall pressure is presented in Fig.15(a), (b) and (c) respectively, indicating a good match even for the higher mode ãpw 8 . The ROM results are compared with high-fidelity solution in terms of the POD modal energy and panel

Figure 15 :

 15 Figure 15: Interpolation of characteristic POD time modes h * = 0.010. (a) phase portrait of first and second modes of panel deflection (b) phase portrait of first and second modes of wall pressure (c) phase portrait of first and third modes of wall pressure (d) phase portrait of first and eighth modes of wall pressure

Figure 16 :

 16 Figure 16: ROM predictions and error h * = 0.010. (a) POD modal energy of reduced POD basis (b) time evolution of total error (refer Eq. 18) (c) comparison of flutter response between the HF and ROM in term of panel deflection at the 3/4 chord length location

Table 2 :

 2 Table 2 shows the reference cases simulated using different mass ratio values and the corresponding response of the cylinder in terms of a root mean squared displacement y * rms = y rms /D. Reference case simulations

	m *	1.5 1.75	2	2.5	3	3.5	5	7	10	20
	y * rms	0.69 0.64 0.59 0.51 0.44 0.38 0.25 0.21 0.10 0.04

Table 3 :

 3 Reference case simulations

In general, the mass ratio is defined as the structural mass to displaced fluid mass: m * = 4m/(ρπD

). Then the minimum critical value is ≈ 0.54.

Acknowledgement

The authors acknowledge Centre National de la Recherche Scientifique (CNRS) for facilitating the work via Agence Nationale de la Recherche (ANR) project Baresafe. The authors acknowledge Prof. Datta Gaitonde for providing access to the high-fidelity database of shock boundary layer induced panel flutter.


Thomas Thiebault

Jocelyne Brendlé

Grégoire Augé

Lionel Limousy

Zwitterionic-surfactant modified Laponites® for removal of ions (Cs + ; Sr 2+ and Co 2+ ) from aqueous solution as a sustainable recovery of radionuclides from aqueous wastes

The synthesis of zwitterionic-surfactant modified Laponites ® (LAP-CBs) was investigated as a function of the surfactant loading in order to obtain suitable adsorbent for the removal of Co 2+ , Sr 2+ and Cs + ions from aqueous solutions. The proper adsorption of surfactant was characterized by several techniques allowing to emphasize the modification of the surfactant distribution between the interlayer space and the surface of Laponite ® (LAP), with increasing the load of surfactant. The organophilic character of the obtained LAP-CBs strongly affected its affinity with the targeted contaminants. The adsorption of Sr 2+ was lowered onto LAP-CBs in comparison to LAP, while the adsorption of Co 2+ and Cs + were strongly improved. The competition between Sr 2+ and Cs + was important onto LAP-CBs, whereas no competition was found when using LAP, indicating different adsorption mechanisms depending on the adsorbent. The desorption of contaminants was performed in various solutions and confirmed these mechanisms. Whereas Cs + and Sr 2+ were adsorbed through cation exchange onto LAP-CBs, as highlighted by a significant desorption in saline solution, the desorption of Co 2+ from LAP-CBs was very poor whatever the composition of the solution indicating rather a complexation. Finally, the increase of the load of surfactant decreased the adsorption capacity of all the investigated contaminants. The arrangement of the surfactant, within the interlayer space or on the surface of LAP, was therefore of high concern when considering the potential of this material to durably remove radionuclides from low-level radioactive wastewaters.

Introduction

The production of electricity by nuclear power plants generates significant amounts of radioactive wastes. [START_REF] Gershey | Low-level radioactive waste[END_REF] These wastes are currently managed by containment from their environment in various conditions depending on the level and the half-life of the radionuclides. [START_REF] Ewing | [END_REF] The production of low-level radioactive wastes (LLRW) is the most important in volume due to the constant discharge of contaminated effluents by nuclear power plants. 3 The management of LLRW represents an important challenge due to the hardness for treating such complex effluents with economically viable productions. However, reaching a suitable solution for the decontamination of LLRW is of high concern in order to avoid the environmental dispersion of hazardous radionuclides. 4 With this aim, adsorption could represent a potential way to adsorb radioactive inorganic cations if the selected adsorbent presents both appropriate adsorption properties and reasonable costs. 5,6 Several materials and processes have been tested in order to adsorb radionuclides, such as zeolites, 7,8 oxides, 9 membranes, 10 or clay-based adsorbents. [11][12][13] Clay minerals are among the most abundant and cheapest materials, and they are especially recognized for the adsorption of cationic contaminants given their cation exchange capacity (CEC). 5,14,15 This structural property, associated with a very low hydraulic conductivity, 16,17 results in the promotion of this material for the long-term geological storage of high-level radioactive wastes. 18,19 Nevertheless, the use of natural clays could represent a risk considering the possible chemical variations and the presence of undesired substitutions in the extracted clay minerals. [START_REF] Bruno | Processes and features affecting the near field hydrochemistry. Groundwater-bentonite interaction[END_REF][START_REF] Wersin | [END_REF] Synthetic clay minerals could therefore represent an added-value, resulting from their stable chemical composition. One of the main limitation of clay minerals is their poor selectivity depending on the targeted contaminant. Liquid effluents originating from nuclear power plants are complex solutions, as they contain various inorganic cations (e.g. radionuclides) as well as organic matter or surfactants that could hinder the adsorption properties of clay minerals. 22 Therefore, raw clay minerals could be inappropriate for an efficient fixation of such contaminants. In order to improve the selectivity and/or the affinity of claybased adsorbents in relation to specific ions, several papers have demonstrated that their organo-modification could be beneficial. [23][24][25][26][27] The organo-modification of clay minerals can be carried out with various type of organic compounds such as surfactants. Each type of surfactant (i.e. cationic, anionic, nonionic and zwitterionic) allows the organophilisation of the interlayer space of the clay minerals. However, only cationic and zwitterionic surfactants lead to a strong affinity with the clay mineral. Their intercalation within the layers of the clay minerals is performed through cation exchange ensuring a stable and energetic adsorption of the surfactant. Yet, only zwitterionic surfactants could maintain the cation exchange capacity of the resulting organo-modified clay minerals, due to its zwitterionic charge. 28,29 Conversely, the intercalation of cationic surfactants could hinder the cation exchange capacity by compensating the negative surface charge of clay minerals. 30,31 The innovative approach of this research stands in the use of Cocamidopropyl Betaine (CB), a widely-spreaded low cost zwitterionic surfactant used as intercalated agent. CB is currently employed in various cosmetic products due to its safety and biodegradability, 32 and it has also been proposed for the treatment of wastewaters, due to its chemical properties. 33 A commercially available synthetic hectorite (Laponite ® RD, LAP) was selected as host materials to prepare a series of hybrids having different surfactant load, the goal of this study being to prepare adsorbent by an environmental friendly way, i.e. by using water as solvent and heating at low temperature (40°C at maximum). The adsorption of 133 Cs, 59 Co, 88 Sr, three stable radionuclides isotopes commonly detected (i.e. of 137 Cs, 60 Co and 90 Sr, respectively) in LLRW was then investigated on LAP and on the CB modified LAP (LAP-CBs). The influence of the surfactant loading on the adsorption properties of Co 2+ , Sr 2+ and Cs + of LAP-CBs was deciphered as well as the competitive adsorption and desorption of the contaminants in various type of solutions. These latter experiments underlined that the developed adsorbents may be suitable to replace the complex matrixes used for the removal of radionuclides from aqueous solution.

Materials and Methods

Chemical reagents

Raw Laponite® RD (LAP) was purchased to BYK company and was used without further modification. Zwitterionic surfactant, Cocamidopropyl Betaine (CB) was supplied by APC Pure Chemical Company assuming a minimal surfactant content of 30%. General information as well as the distribution of CB species as function of pH are presented in

Synthesis of zwitterionic-surfactant modified Laponite®

The protocol for the modification of raw LAP was the following. Fourier transform infrared (FTIR) measurements in the range 400-4000 cm -1 , were recorded using a Bruker 55 FT-IR spectrometer equipped with Nd: YAG laser operating at 1064 nm and a Ge detector. The analyses were performed in transmission mode and each spectrum was the average of 256 scans collected at 2 cm -1 resolution. Thermogravimetric (TG) analyses were performed on a TG Mettler Toledo STARe apparatus under air flow with a heating rate of 2°C min -1 from 30 to 900°C. All samples were air-dried during 48 hours in order to obtain comparable hydration state. Zeta potential (ZP) measurements were realized using a Malvern Nanosizer instrument. 20 mg of material were dispersed in 50 mL of a 10 -3 M NaCl solutions. The pH of the suspension was adjusted using 0.1 M NaOH and/or HCl solutions and the suspension was stirred for 24 h until ionic equilibrium was reached. The concentrations of the different cations (Cs + , Co 2+ and Sr 2+ ) were quantified with atomic absorption spectroscopy (Varian AA 240 FS), using the absorption mode and acetylene as fuel.

Adsorption experiments

Adsorption experiments were carried out both in single-solute and in competitive solutions. Typically, the adsorbent mass was 100 mg in 100 mL of solution spiked with different amounts of contaminant solution. The concentration of each contaminant ranged from 6 to 600 µM for Cs + , 4.2 to 420 µM for Co 2+ and 3.7 to 370 µM for Sr 2+ in both single-solute and competitive solutions, in pure-water solution spiked at 100 µM of NaCl. The pH value of each isotherm solution was let free and measured at the beginning and at the end of interaction (i.e. 6-6.5). Solutions were then centrifuged at 8,000 rpm during 5 minutes and the contaminant concentration in supernatant was analyzed by atomic absorption spectroscopy.

Please do not adjust margins

Please do not adjust margins

The impact of pH on the adsorption capacity of single-solute solutions of Cs + , Co 2+ and Sr 2+ was carried out with the same protocol. The only difference was that the pH of the solution was modified with 1M HCl and/or 1M NaOH solutions in order to obtain a range of pH values between 2 and 12.

Desorption experiments

After interaction with contaminants, the adsorbents were recovered by centrifugation at 8,000 rpm during 10 minutes. The recovered material was then dried at 60°C during 72 hours. 20 mg of dried adsorbent were then mixed with 20 mL of three different solutions; Pure water, 1 mM of CaCl 2 and 2 mM of NaCl. The solution was then gently stirred during one week at room temperature. Solutions were then centrifuged at 8,000 rpm during 5 minutes and the contaminant concentrations in supernatant were analyzed by atomic absorption spectroscopy.

Sorption modeling

The fitting of the resulting adsorption isotherms by using Langmuir, Freundlich and Dubinin-Radushkevich (DR) equation models drives to thermodynamic parameters allowing one to precisely quantify the affinity of contaminants with the sorbent. Briefly, Langmuir model is expressed by the following equation: 11,14 𝑞 𝑎𝑑𝑠 = 𝑞 𝑚𝑎𝑥 𝐾 𝐿 𝐶 𝑒𝑞

+ 𝐾 𝐿 𝐶 𝑒𝑞

Where q ads is the equilibrium amount adsorbed on sorbent (mmol.g -1 ), C eq the equilibrium concentration in the supernatant (mmol.L -1 ), q max the adsorption capacity of the sorbents (mol.g - 1 ) and K L is the Langmuir adsorption constant (L.mol -1 ) which is related to the free energy (G°) of adsorption. The linear Freundlich model equation is written as follow: 14,34 𝑙𝑛𝑞 𝑎𝑑𝑠 = 𝑙𝑛𝐾 𝐹 + 1

𝑛 𝑙𝑛𝐶 𝑒𝑞

where K F (mg.g -1 )/(mg.L -1 ) n and n are the Freundlich constants indicating the extent of the adsorption and the degree of nonlinearity between contaminants and the adsorbent respectively. The linear expression of the DR model used to adjust our experimental data is expressed as follow: 23,35 𝑙𝑛𝑞 𝑎𝑑𝑠 = 𝑙𝑛𝑞 𝑚 + 𝛽𝜀 2

where corresponds to the Polanyi potential of which relation 𝜀 including C eq the equilibrium concentration could be found elsewhere. The constant corresponds to the activity 𝛽 coefficient associated to the mean free energy E (kJ.mol -1 ) described by the following equation: 23,35

𝐸 = (2𝛽) -1/2
This later parameter gives information whether the adsorption mechanism involves a cation exchange or physical adsorption. Indeed, if the magnitude of E is below 8 kJ mol -1 , physisorption is envisaged, while for E > 8 kJ mol -1 the adsorption process follows chemisorption. 11,14

Results and Discussion

Characterization of LAP and zwitterionic-surfactant modified Laponite®

The LAP and zwitterionic-surfactant modified LAPs (LAP-CBs) were characterized by XRD, FTIR and TG/DTG analyses. The 00l diffraction patterns of LAP-CBs shift to lower angular values in comparison with LAP, underlining the intercalation of surfactant within the interlayer space of the starting clay mineral. LAP exhibits a 00l reflection at 7.0° (2θ) leading to an interlayer spacing of 13.7 Å (Figure 1). As expected, with a CB load of 4 times the CEC of LAP, the interlayer space increases, reaching 22.3 Å. This value matches with an organization of the surfactant in paraffin monolayer structure. 36,37 Considering the basal spacings of LAP-CBs, the arrangement of the surfactant is expected to be a monolayer at 0.5 and 1 CEC, and a pseudo-trilayer at 2 CEC. 36 FTIR spectra of LAP-CBs show typical features characteristics of the organic compounds: the absorption bands at 2840-2920 cm -1 are relative to the symmetric and antisymmetric CH 2 stretching vibrations of the CB aliphatic chain (Figure S2). The presence of these bands testifies the proper adsorption of CB onto LAP. Three main domains of weight losses can be expected from TG analyses of LAP-CBs. Below 150°C, the loss of weight is due to the evaporation of free and adsorbed water; between 150 and 600°C occurs the devolatilization and the thermal oxidation of organic matter (OM), and beyond this temperature, dehydroxylation of LAP is expected. 38 TG analyses of LAP and LAP-CBs exhibit two distinct patterns (Figure S3). With the increase of the CB load, a decrease of the weight loss associated to water content (i.e. T = 38-46 °C) is observed. Conversely, the loss of weight associated with the organic content strongly increases with the CB load (i.e. T= 230-319°C). The DTG curves illustrate these qualitative information too, but also indicates the doubling of the peak associated with the thermal decomposition of organic matter (i.e. CB, Figure 2). This doubling can be explained by the distinction between CB adsorbed onto the surface of the clay mineral (i.e. lower temperature) and intercalated CB (i.e. higher temperature). 28,39 Quantitative data can be derived from these curves and are expressed in Table S2. As observed in Figure 2 and Table S2 the water content decreases with increasing CB amount from 13.7% for LAP to 4.0% for LAP-CB-4CEC respectively. The weight loss associated to CB thermal oxidation can be calculated for a global weight loss considering the total amount of adsorbed CB (i.e. 150-550°C), or split between surfactant adsorbed onto the surface or intercalated (Figure 2). The total OM content ranges from 4.9 % at 0.5CEC of surfactant load and 29.1% at 4CEC of surfactant load. These values allow calculating the adsorbed amount of CB that represents 0.21, 0.39, 0.84 and 1.10 times the CEC of the LAP at 0.5, 1, 2 and 4CEC of surfactant load respectively (Figure 3). The amount of adsorbed CB is therefore higher for higher initial loads of CB. It should be noted that the amount of adsorbed CB slightly exceeds the CEC of LAP for LAP-CB-4CEC, exhibiting the potential recombination between the surfactant Please do not adjust margins Please do not adjust margins molecules. This increase of CB content is not equally distributed on LAP-CBs. As showed in Figure 3, there is a sudden increase of the surface adsorption between 1 and 2 CEC of CB loads. The amount of CB adsorbed onto the surface is even equal to the intercalated amount at 4 CEC. The intercalated amount of CB exhibits a weak increase between 2 and 4 CEC, with such a saturation around 0.5 CEC of LAP (Table S2). As a result, the CEC is not the only limiting factor for the intercalation of CB within the layers. The possible impact of a steric hindrance could be highlighted. This result was previously observed with other zwitterionic surfactants intercalated within montmorillonites, for which the maximal intercalated amount reached 0.6 CEC. 39 Zeta potential (ZP) values of LAP, LAP-CB-0.5CEC and LAP-CB-4CEC were evaluated as a function of the pH. ZP values of LAP are very close to 0 in a wide range of pH values (i.e. between pH = 2 and pH = 9), and become negative under alkaline pH conditions (Figure 4). This variation for alkaline pH values is generated by the deprotonation of the edge-sites of LAP. 40 LAP-CB-0.5CEC and LAP-CB-4CEC exhibit three ranges of ZP values. ZP values are positive for acidic pH values, close to 0 between pH ≈ 3.5 and pH ≈ 7.5, and are negative for alkaline pH values (Figure 4). The speciation of CB (Fig S1) is in accordance with these variations. Therefore, the variation of the surface charge of LAP-CBs is solely generated by the speciation of CB for the pH values < 10, and the deprotonotation of edge-sites is only significant for strong alkaline pH values. 41,42 It should be mentioned that the variation of ZP values are more important for LAP-CB-4CEC in comparison with LAP-CB-0.5CEC due to the higher amount of CB adsorbed onto the surface for the highest CB load (Figure 3).

Adsorption experiments

Impact of pH. The impact of pH on the single-solute adsorption of Cs + , Co 2+ and Sr 2+ onto LAP and LAP-CB-4CEC is presented Figure 5. The adsorbed amount of Sr 2+ onto LAP is important and increases from 58 to 98 % at pH = 2.3 and pH = 12.2 respectively. The same pattern is observed onto LAP-CB-4CEC with an increase of the adsorbed percentage from 19 to 95 % at pH = 2.4 and pH = 11.8 respectively. In the same way, the adsorption of Cs + onto LAP-CB-4CEC strongly increases under alkaline pH conditions, from 5 to 35% at pH=2.3 and pH=11.9, respectively. Conversely, the adsorption of Cs + onto LAP is very limited (i.e. between 5.5 and 9.9 %) and the increase of adsorption percentage under alkaline conditions is not significant.

The adsorption of Co 2+ onto LAP is very poor and due to the precipitation of Co 2+ when pH > 7.5, 11 it can be considered that the pH value has a weak impact on the adsorption extent of Co 2+ onto LAP.

In the same way the adsorption of Co 2+ onto LAP-CB-4CEC is not pH-dependent with high adsorption percentage (i.e. between 80 and 99 %) whatever the pH value, except for the most acidic solution, for which the adsorption of Co 2+ is very low (i.e. < 2%).

As previously observed with the variation of ZP values of LAP and two LAP-CBs, pH values play a key role on the speciation of the surfactant and in a less important manner on the deprotonation of edge-sites of LAP (Figure 4). The modification of the surfactant charge state is particularly important for pH > 3.2, due its zwitterionic form since this pH value. Cations such as Sr 2+ and Cs + are obviously sensible to this modification of speciation. However, it is not the case for Co 2+ , indicating that the adsorption of Co 2+ onto adsorbents is less sensible to the amount of available negative charges even if Co 2+ is obviously cationic. 11 Single-solute interactions. The single-solute adsorption of each selected contaminant has been investigated onto LAP and LAP-CBs.

The adsorption isotherms of Cs + are presented Figure 6.

Regardless the adsorbent, the adsorption of Cs + is not favorable with significant equilibrium concentrations even for the lower initial concentrations tested. However, the adsorption capacity significantly varies between each adsorbent. The lowest adsorption capacity is observed for LAP (i.e. 0.055 mmol.g -1 ), whereas the adsorption capacities onto LAP-CBs are much higher.

The increase of the surfactant load significantly decreases the adsorption capacity with maximal adsorbed concentration of 0.20 and 0.14 mmol g -1 at 0.5 and 4 CEC respectively. The best fit of experimental data is performed with Freundlich equation (Table 1). The increase of K F values between LAP and LAP-CB-0.5CEC from 0.08 to 0.32 (mg.g -1 )/(mg.L -1 ) n points out the better affinity of Cs + to LAP-CBs than LAP. The same trend can be observed for the single-solute adsorption of Co 2+ (Figure 7). Thus, the lowest adsorption capacity with unfavorable adsorption behavior is displayed onto LAP with a maximal adsorption capacity of 0.09 mmol g -1 (Figure 7). Conversely, onto LAP-CBs, the adsorption is more favorable with low equilibrium concentrations before reaching a saturation plateau at different levels for each CB load. At 0.5 and 1 CEC of CB, the adsorption capacity is equal to 0.37 and 0.34 mmol g -1 respectively, and at 2 and 4 CEC, the adsorption capacity is equal to 0.24 and 0.27 mmol g -1 . Hence, the modification of the surfactant arrangement (Figure 1) impacts the adsorption capacity of Co 2+ . This variation between LAP and LAP-CBs is also expressed in the model fits. The adsorption onto LAP is well-fitted by the Freundlich model whereas the adsorption onto LAP-CBs is properly fitted by the Langmuir equation (Table 1). It should be mentioned that the mean free energy (E) is below 8 kJ.mol -1 only for the adsorption of Co 2+ onto LAP. Therefore, the organo-modification of LAP strongly modifies the affinity between Co 2+ and the sorbent. The adsorption of Sr 2+ onto the different adsorbents show two distinct regimes with one displaying a gradual growth of adsorbed Sr 2+ with equilibrium Sr 2+ concentrations, whereas the second one points out a steady state for the highest initial concentrations (Figure 8). However, the saturation of each adsorbent occurs at different adsorbed amounts. The highest adsorption capacity of Sr 2+ is displayed onto LAP with a value equal to 0.26 mmol g -1 (i.e. corresponding to 70% of the CEC of LAP); whereas onto LAP-CBs, the adsorption capacity decreases with increasing the surfactant load from 0.23 to 0.16 mmol.g -1 at 0.5 and 4 CEC of CB respectively. The adsorption of Sr 2+ is well-fitted by the Langmuir equation exhibiting chemisorption

Please do not adjust margins Please do not adjust margins (E > 8kJ mol -1 ) and spontaneous adsorption (ΔG° < 0) regardless the sorbent (Table 1).

Competitive adsorption. The competitive adsorption was carried out with ternary solutions. The competitive adsorption isotherms of Co 2+ , Sr 2+ and Cs + are presented Figures S4-6. In order to assess the impact of competition on the adsorption of each contaminant, Figure 9 summarizes the adsorption isotherms of each contaminant in both single-solute and ternary solutions onto LAP and LAP-CB-0.5CEC. A competitive effect is observed in the same extent whatever the surfactant load onto LAP-CB (Figures S4-6, Table 1).

The adsorption of Co 2+ on both LAP and LAP-CBs is not impacted by the competition with Cs + and Sr 2+ in the used concentration range. Indeed, both the adsorption capacity and the adsorption behavior are identical for mono-molecular and ternary solutions whatever the adsorbent (Figure 9). It could indicate different adsorption sites of Co 2+ , or a strong selectivity of the adsorbent. 43 Conversely, the adsorption of Cs + onto LAP-CBs is affected by the competition. An important drop in the adsorption capacity is observed, from 0.2 to 0.1 mmol g -1 in single-solute and ternary solutions respectively (Figure 9). The affinity between Cs + and the adsorbent remains close but a competitive effect can be assumed. In the same way, the adsorption of Sr 2+ onto LAP-CBs is impacted by the competition. The adsorption capacity decreases from 0.25 to 0.12 mmol g -1 in single-solute and ternary solution experiments respectively. However, this trend is not visible for the adsorption of Sr 2+ onto LAP. On LAP-CBs, it exists a competitive effect between Cs + and Sr 2+ , and the adsorption capacities reach a saturation around 0.25 mmol g -1 for Sr 2+ or Cs + alone, or in competition. It is interesting to note that despite their different valence, the adsorption capacity remains the same. Therefore, it seems that the limiting factor is more the availability of adsorption sites rather than global charge. The sorption mechanisms remains identical whatever the composition of the solution, even if the adsorption capacities is modified (Table 1), as expressed by the proper fit with the same model for both single-solute and competitive adsorption tests.

Desorption experiments

Desorption experiments have been performed with adsorbent samples that were previously in interaction with the highest starting concentration of Co 2+ , Cs + and Sr 2+ in single-solute experiments.

After drying, sorbent samples were put in three different types of solutions in order to evaluate the impact of inorganic salts and their valence on desorption behavior. The results are presented in Figure 10.

The desorption behavior of each contaminant varies both with the adsorbent (i.e. LAP or LAP-CBs) and the initial composition of the solution.

Langmuir Freundlich Dubinin-Raduskevich Q max K L ΔG°r² n K F r² Q m E r² mmol g -1 L mmol -1 kJ mol -1
(mg.g -1 )/(mg.L -1 ) n mmol.g -1 kJ.mol Please do not adjust margins Please do not adjust margins For example, desorption of Co 2+ is important for the LAP samples, with desorbed percentages between 15 and 42%, although its desorption from LAP-CBs is below 1.5 % whatever the surfactant load or the initial composition of the solution. A slight (but non-significant) increase tendency of the desorbed amount is observed with increasing the CB load (Figure 10). The desorption behavior of Sr 2+ is less impacted by the CB load than the composition of the solution. For a same adsorbent, the highest desorbed amount is systematically observed in CaCl 2 solution, and the lowest in pure water. The desorption in NaCl solution is intermediate. This indicates that the valence of the inorganic cation plays a key role in the desorption of Sr 2+ . This trend can also be described on the desorption behavior of Cs + . Systematically, the desorption percentage is the highest in NaCl solution, but the desorption percentage is also very high in pure water and CaCl 2 solution. The Cs + cation is therefore highly desorbed whatever the release solution tested, even higher values are noticed for inorganic salt with the same valence (Figure 10).

Sorption mechanisms

Based on the adsorption and desorption results, the sorption mechanisms of each contaminants can be discussed. The adsorption of Sr 2+ onto LAP is favorable, properly fitted by the Langmuir model and the adsorption capacity corresponds to 70% of the CEC of LAP (Figure 8). Moreover, the desorption results exhibit that even if the desorbed amount of Sr 2+ is very weak in pure water or NaCl solution, the presence of Ca 2+ strongly increases the desorption of Sr 2+ (Figure 10). As a consequence, desorption increases with the presence of an inorganic cation with the same valence than the adsorbed contaminant. All these results demonstrate that Sr 2+ is adsorbed through cation exchange onto LAP. The same results were observed onto LAP-CBs (Figure 8). However, we don't have the information about the CEC of these adsorbents, and their adsorption capacities of Sr 2+ were lower than that of LAP. It indicates the impact of the arrangement of the surfactant within the LAP layer, which could hinder the cation exchange mechanism. However, Sr 2+ is also adsorbed through cation exchange onto LAP-CBs, and may be used as a probe of the CEC of each adsorbent, due to its selective adsorption instead of Na+, the compensation ion of LAP. 44 The adsorption of Co 2+ onto LAP is not favorable and seems to be dependent on the solid/water partition. As a result, none competitive effect has been observed as well as high desorption percentages whatever the composition of the solution (Figure 10). Co 2+ is therefore adsorbed through weak electrostatic bindings onto LAP. The organo-modification of LAP strongly modifies the affinity of the materials with Co 2+ . Co 2+ is indeed efficiently adsorbed onto LAP-CBs in a wide range of concentration with very low equilibrium concentrations. The desorption percentages of Co 2+ are also very low whatever the composition of the solution or the CB load. However, a decrease of the adsorption capacity is noticed for higher CB loads (Figures 7 andS6). As previously explained for the adsorption of Sr 2+ , high surfactant load could diminish the availability of adsorption sites. Yet, the addition of CB within the layers dramatically changes the affinity of the material with Co 2+ . We can therefore assume that Co 2+ is adsorbed through chelation mechanism with a specific chemical function of the surfactant. 45,46 Finally, the adsorption of Cs + onto the selected adsorbents is more difficult to understand. The adsorption of Cs + onto LAP is poorly performed (Figures 6 andS4) as already demonstrated on pure clay minerals. 47 Cs + is generally adsorbed onto surface defaults of clay minerals. 48 However, LAP is a synthetic material which presents only few structural defaults. The high desorption of Cs + from LAP whatever the releasing solution indicates that the adsorption is probably performed through physisorption (Figure 10). Then, the desorption percentage is systematically the highest in NaCl saline solutions, especially for LAP-CBs. Moreover, the competition with Sr 2+ is very important onto these adsorbents. The adsorption of Cs + is therefore partially performed through cation exchange, especially onto LAP-CBs, as revealed by the decreasing adsorption capacity with increasing the CB load (i.e. as displayed by Sr 2+ adsorption capacity). But a significant fraction of Cs + is also adsorbed through weak electrostatic interaction, 49 as exhibited by the adsorption isotherms of Cs + , which are moderately favorable.

Conclusions

In this work, the interest of the organo-modification of a synthetic clay mineral was demonstrated in order to develop green, safe and multi-skilled adsorbents for the removal of radionuclides. Indeed, LAP-CBs exhibit favorable adsorption properties for Co 2+ , Sr 2+ and Cs + in comparison with the raw LAP which only allows a significant adsorption of Sr 2+ . Moreover, this adsorption seems to be specific and highly stable especially for Co 2+ , given the strong binding with CB. However, our results demonstrate that the CB load, which impacts the CB arrangement within the layers of LAP, is of high concern for the availability of adsorption sites in one hand, and for the conservation of the cation exchange properties on the other hand. It appears that the lowest surfactant load presents the highest adsorption capacities of Co 2+ and Cs + . This minimal CB load seems to avoid the recombination of surfactant molecules and also the steric hindrance in the interlayer space. The monolayer arrangement also ensures a good further stability of the hybrid materials. Oppositely, the important amount of surfactant adsorbed onto the surface for higher CB load raises hard questions about the long-term stability of the organo-clays, especially in saline solutions.

In the tested ternary solutions, a competitive effect between Cs + and Sr 2+ was observed for the adsorption onto LAP-CBs. As these two contaminants were adsorbed through cation exchange, this competitive impact should be taken into consideration for the evaluation of the suitability of the developed adsorbent for the treatment of complex solutions. Thus, the selectivity of Sr 2+ and Cs + adsorption in comparison with other inorganic cations would be of high concern.

The proposed LAP-CB, especially LAP-CB-0.5CEC, could therefore represent a cheap, green and easy to prepare adsorbent for the removal of radionuclides from LLRW.

Please do not adjust margins Please do not adjust margins However, further tests remain necessary, especially on the adsorption capacities in complex solutions (i.e. real effluents) and on the aging of the adsorbent during the storage, especially when exposed to radioactivity. Indeed, the use of this green adsorbent for the further storage of radionuclides may significantly reduce the environmental footprint of nuclear waste packages.

ARTICLE

Please do not adjust margins Please do not adjust margins

Figure captions

4 |

 4 J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx

Figure 1 :Figure 2 : 1 Figure 3 :

 1213 Figure 1: X-ray diffraction patterns of LAP and LAP-CBs for different surfactant loads (from 0.5 to 4 CEC)

Figure 4 :Figure 5 :Figure 6 :

 456 Figure 4: ZP values as a function of the pH for LAP, LAP-CB-0.5CEC and LAP-CB-4CEC, the arrows at the bottom of the figure marks the speciation of CB

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: Single-solute adsorption isotherms of Co 2+ onto LAP and LAP-CBs, dashed dark lines represent the Langmuir fits

Figure 10 :

 10 Figure 10: Desorption percentage of Co 2+ (dark rods), Sr 2+ (gray rods) and Cs + (light gray rods) as a function of the adsorbent and the release solutions.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :

 12345678910 Figure 1: X-ray diffraction patterns of LAP and LAP-CBs for different surfactant loads (from 0.5 to 4 CEC) 122x74mm (300 x 300 DPI)

Table S1

 S1

	and Figure
	S1 respectively.
	Cesium chloride (CsCl, >99.99%), Strontium hexahydrated
	Chloride (SrCl 2 .6H 2 O, >99%) and Cobalt hexahydrated chloride
	(CoCl 2 .6H 2 O, >98%) were supplied by Carl Roth company.
	Chemical reagents such as HCl, NaOH, NaCl and CaCl 2 were
	supplied by Sigma-Aldrich company at analytical grade and
	were used without further modifications.

Table 1 :

 1 Single-solute (Mono) and competitive (Comp) sorption model parameters for sorption of Co 2+, Cs + and Sr 2+ onto LAP and LAP-CBs

Published on 19 August 2019. Downloaded by Sorbonne Université on 8/26/2019 8:56:35 AM. View Article Online DOI: 10.1039/C9GC02243K

Acknowledgements

The ONET Technologies company is gratefully thanked for funding this work.

Conflicts of interest

There are no conflicts to declare. Please do not adjust margins Please do not adjust margins

ARTICLE