Prashant Kumar
email: prashant.kumar@univ-amu.fr

Frédéric Topin
email: frederic.topin@univ-amu.fr

Different arrangements of simplified models to predict effective thermal conductivity of open-cell foams

Keywords: Series model, Parallel Model, Hashin-Shtrikman model, Effective medium theory, Reciprocity model

It is often desirable to predict the effective thermal conductivity (ETC) of a homogenous material like open-cell foams based on its composition, particularly when variations in composition are expected. A combination of five fundamental simplified thermal conductivity bounds and models (Series, Parallel, Hashin-Shtrikman, Effective Medium Theory, and Reciprocity models) is proposed to predict ETC of open-cell foams. Usually, these models use a parameter as the weighted mean to account the proportion of each bound arranged in arithmetic and geometric schemes. Based on the ETC data obtained on numerous virtual Kelvin-like foam samples, the dependence of this parameter has been deduced as a function of morphology and phase thermal conductivity ratio. Various effective thermal conductivity correlations based on material properties and foam structure valid for open-cell foams filled with any arbitrary working fluid over a solid conductivity of materials range ( # ! /# " =10-30000) and over a large porosity range (0.60< & # <0.95) are derived. Arrangement of series and parallel models together using the simplest models for both, arithmetic and geometric schemes, is found to predict excellent results among all the generic combinations.

Introduction

The thermal conductivity of porous materials, more particularly open-cell foams, plays an important role in many industrial processes. Heat conduction takes place through a solid skeleton and the fluid passing through the 3-D interconnected network of open-cells. In order to effectively utilize open-cell foams in heat transfer applications e.g. heat exchangers, volumetric solar receiver etc., accurate knowledge of their thermal transport properties are needed.

Many heat transfer mechanisms at pore scale could be included in equivalent thermal conductivity. The effective thermal conductivity refers to the case when only diffusion plays a role. The energy transport is then controlled by the effective thermal conductivity (e.g. [START_REF] Calmidi | The effective thermal conductivity of high porosity fibrous metal foams[END_REF][START_REF] Bhattacharya | Thermophysical properties of high porosity metal foams[END_REF][START_REF] Solorzano | An experimental study on the thermal conductivity of aluminium foams by using the transient plane source method[END_REF][START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF][START_REF] Yang | An analytical unit cell model for the effective thermal conductivity of high porosity open-cell metal foams[END_REF]). However, at high temperatures or for foam materials of extremely low conductivity, an equivalent thermal conductivity depends also on radiative heat transfer (e.g. [START_REF] Wang | Modeling and prediction of the effective thermal conductivity of random open-cell porous foams[END_REF][START_REF] Coquard | Conductive and radiative heat transfer in ceramic and metal foams at fire temperatures[END_REF][START_REF] Mendes | Detailed and simplified models for evaluation of effective thermal conductivity of open-cell porous foams at high temperatures in presence of thermal radiation[END_REF]) for which the expression or formulation is completely different.

The ETC of open-cell foams is not only dependent on the phases properties and porosity, but also on the structure of the materials (e.g. [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF][START_REF] Miettinen | Dependence of thermal conductivity on structural parameters in porous samples[END_REF][START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF]). Moreover, the intrinsic solid phase conductivity may be different from the parent/bulk material conductivity. Miettinen et al., [START_REF] Miettinen | Dependence of thermal conductivity on structural parameters in porous samples[END_REF] showed experimentally that there exists no simple relationship between intrinsic solid phase heat conductivity and porosity. Dietrich et al., [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF] measured the thermal conductivity of the strut materials and obtained a decrease in their intrinsic solid phase conductivity compared to the conductivity of pure/bulk material. Kumar et al., [START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF] showed a decrease of approximately 20-25% in the solid phase thermal conductivity of bulk material when transformed in foams from their calculated ETC data.

Apart from experiments, substantial efforts to estimate the effective thermal conductivity have been made in the recent years based on:

• numerical simulations using X-ray µCT images of actual samples (e.g. [START_REF] Vicente | Thermal conductivity of metallic foam: Simulation on real X-ray tomographied porous medium and photothermal experiments[END_REF][START_REF] Coquard | Numerical investigation of conductive heat transfer in highporosity foams[END_REF][START_REF] Bodla | Resistance network-based thermal conductivity model for metal foams[END_REF][START_REF] Randrianalisoa | Microscale direct calculation of solid phase conductivity of Voronoi's foams[END_REF][START_REF] Mendes | A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures[END_REF][START_REF] Mendes | An improved model for the effective thermal conductivity of open-cell porous foams[END_REF][START_REF] Ranut | High resolution X-ray microtomography-based CFD simulation for the characterization of flow permeability and effective thermal conductivity of aluminum metal foams[END_REF][START_REF] Wulf | Experimental and numerical determination of effective thermal conductivity of open cell FeCrAl-alloy metal foams[END_REF]),

• numerical simulations on idealized representation of open-cell foams (e.g. [START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF])

• empirical correlations by fitting experimental data (e.g. [1, 2 ,4,5,12, 23, 24] Despite the accuracy and precision of ETC results obtained from numerical simulations (or experiments); they are however considerably time and resource consuming. Therefore, their application is justified only when higher accuracy is required or, together with experimental measurements, for the validation and/or calibration of simpler models.

Moreover, ETC of a specific structure for a given working fluid has to be obtained individually on a case to case basis. Thus, an empirical correlation is a reasonable compromise between measurements and computational time for quick and accurate evaluation of ETC. It presents general applicability for a wide range of materials, porosities, and ratios of conductivities between solid and fluid phases.

Various simplified bounds and models exist in the literature to predict the ETC values for different porous media. The most common inputs for these models are material properties (e.g. solid and fluid conductivities) and material morphology (e.g. porosity). Nevertheless, prediction of ETC for different class of materials i.e. foam like structures is not straightforward. Consequently, various authors proposed empirical correlations using these bounds/models or combination between them to predict reasonable ETC values with or without an adjustable parameter depending on the ratio of constituent phases.

In open-cell foams, the empirical value of adjustable parameter in ETC correlations varies between employed models and used type of materials. Variations in adjustable parameter are due to the preferred choice of model/bound by authors and depend usually on the type of foam material used (metallic or ceramic) and thus, resulting in a massive gap to have access to complete range of conductivity ratios. Consequently, there is no simple model that can be used for all existing types of open-cell foams.

In this work, our intent is to develop generalized ETC correlations in accessible porosity range (0.60< & # <0.98) of foams resembling Kelvin-like cell structure and extend the conductivity ratios range (# ! /# " =10-30000) in order to fill this gap. In this view point, five basic arrangements of simplified models and their combinations including the series and parallel models (Leach,[START_REF] Leach | The thermal conductivity of foams. I: models for heat conduction[END_REF]), Hashin-Shtrikman upper and lower bounds [START_REF] Hashin | A variational approach to the theory of the effective magnetic permeability of multiphase materials[END_REF], Effective Medium Theory (EMT) (Landauer,[START_REF] Landauer | The electrical resistance of binary metallic mixtures[END_REF]), and, Reciprocity model (Keller,[START_REF] Keller | A theorem on the conductivity of a composite medium[END_REF]) are adapted as generic minimum and maximum bounds of thermal conductivity in both arithmetic and geometric schemes. A functional parameter is obtained by fitting ETC data obtained numerically from our previous works [START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF]. Obviously, this parameter cannot be obtained beforehand but it has been determined in the present work for the whole range of porosities and thermal conductivity ratios. Thus, it could be used as a function of the traditional input parameters (foam morphology and material property: conductivities) in order to predict ETC value for any Kelvin-like foam structure.

Literature study of ETC correlations

We discuss here ETC as an intrinsic property of conductive heat transfer and not an apparent property representing many other unspecified heat transfer phenomena (e.g. radiation, micro-convection, dispersion, evaporation-condensation etc.). Mixing heat transfer mode in a single apparent coefficient leads to a confusing situation and prohibit the comparison and use of obtained data. Nevertheless in several cases, the diffusive heat transfer is not the principal one for a given situation (e.g. [START_REF] Wang | Modeling and prediction of the effective thermal conductivity of random open-cell porous foams[END_REF]). This contribution could be calculated separately and added eventually.

Usually, commercially available real foams are close to periodic in nature. Pieper and Klein [START_REF] Pieper | Application of simple, periodic homogenization techniques to nonlinear heat conduction problems in non-periodic, porous media[END_REF] demonstrated that the periodic homogenization for real structures close to periodic gives very accurate results. This led researchers to derive ETC empirical correlations that are generally based either on asymptotic approaches or on a micro-structural approach in case of open-cell foams. It is widely known that the ETC strongly depends on porosity and the ratio of thermal conductivities of the constituent phases but also on a lesser extent to the distributions of the solid phase (e.g. between struts and lumps). that depends on the manufacturing process. Using analytical modelling, some authors (e.g. [1-2, 5, 14, 23, 30])

have considered lumps at the node junctions. On the other hand, others (e.g. [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF][START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF][START_REF] Singh | Computational aspects of effective thermal conductivity of highly porous metal foams[END_REF]) did not assume the presence of lumps of matter at the node junctions while deriving their empirical correlations.

For the validation of empirical correlations against ETC data, it is henceforth of major importance to measure the intrinsic solid phase thermal conductivity of the foam sample because different commercially available foams employ different manufacturing techniques; that lead to significant changes in the intrinsic solid phase thermal conductivity of foams, compared to the same parent/bulk material ( [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF], [START_REF] Miettinen | Dependence of thermal conductivity on structural parameters in porous samples[END_REF][START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF]).

The correlations reported in the literature can be classified into three major groups: [START_REF] Calmidi | The effective thermal conductivity of high porosity fibrous metal foams[END_REF] obtained from fitting experimental data using simplified asymptotic bounds models (e.g. [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF][START_REF] Singh | Computational aspects of effective thermal conductivity of highly porous metal foams[END_REF]), (2) obtained from fitting experimental data on micro-structural foam (e.g. [1-2, 5, 12, 23, 30, 31]), and, (3) fitting numerical simulations data on real and virtual foam structures: exploration of various geometries (e.g. [6-7,10-11, 14, 17, 18,,22]). Most commonly, one or two adjustable parameters that contain information about morphology are obtained by fitting experimental or numerical ETC data to derive correlations.

A first group of authors used only porosity and derived correlations based on series and bounds arranged in either geometric (e.g. [START_REF] Singh | Computational aspects of effective thermal conductivity of highly porous metal foams[END_REF]) or arithmetic (e.g. [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF]) scheme. Note that these models do not introduce additional morphological parameters to describe ETC; although this influence has already been shown to be significant (see. [START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF][START_REF] Mendes | A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures[END_REF][START_REF] Mendes | An improved model for the effective thermal conductivity of open-cell porous foams[END_REF]). Qu et al., [START_REF] Qu | A theoretical octet-truss lattice unit cell model for effective thermal conductivity of consolidated porous materials saturated with fluid[END_REF] used a so-called "morphological parameter" that was obtained by fitting the experimental data that is rather material dependent (thermal conductivity ratio) instead of foam morphology and is only valid for a few types of foam materials.

A second group of authors tried to introduce other morphological parameters in their correlation based on 2-D (e.g. [START_REF] Calmidi | The effective thermal conductivity of high porosity fibrous metal foams[END_REF][START_REF] Bhattacharya | Thermophysical properties of high porosity metal foams[END_REF]) or 3-D foam geometry (e.g. [START_REF] Yang | An analytical unit cell model for the effective thermal conductivity of high porosity open-cell metal foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF][START_REF] Boomsma | On the effective thermal conductivity of a threedimensionally structured fluid-saturated metal foam[END_REF][START_REF] Pieper | Application of simple, periodic homogenization techniques to nonlinear heat conduction problems in non-periodic, porous media[END_REF][START_REF] Qu | A theoretical octet-truss lattice unit cell model for effective thermal conductivity of consolidated porous materials saturated with fluid[END_REF]) and deduced an adjustable parameter from experimental data (e.g. [1-2, 23, 30-31).. Moreover, most of these correlations are limited to very high conductive materials (e.g. Al, Cu) with air or water as fluid phase and give a poor ETC estimate for foams of different materials.

For these ETC correlations for high-porosity open-cell metal foams with low conductive fluids where only parallel heat conduction through solid cell ligaments was considered (# ! $ /# " >400) it was concluded that ligament cross-sectional shape do not play a strong role (at constant porosity). For low conductive foam, [START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF] extended the resistor model of [START_REF] Singh | Computational aspects of effective thermal conductivity of highly porous metal foams[END_REF] to derive ETC correlation for ceramic foams and modified the Lemlich approach. Their model could be used together to predict the intrinsic solid phase conductivity of foams of different materials irrespective of fluid phase and porosity when ETC is known.

The third of group of authors calculated ETC form numerical solution of heat equation on idealized or reconstructed open and closed-cell foams structures (.e.g. [START_REF] Coquard | Conductive and radiative heat transfer in ceramic and metal foams at fire temperatures[END_REF][START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF][START_REF] Vicente | Thermal conductivity of metallic foam: Simulation on real X-ray tomographied porous medium and photothermal experiments[END_REF][START_REF] Coquard | Numerical investigation of conductive heat transfer in highporosity foams[END_REF][START_REF] Bodla | Resistance network-based thermal conductivity model for metal foams[END_REF][START_REF] Randrianalisoa | Microscale direct calculation of solid phase conductivity of Voronoi's foams[END_REF][START_REF] Mendes | A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures[END_REF][START_REF] Mendes | An improved model for the effective thermal conductivity of open-cell porous foams[END_REF][START_REF] Druma | Surface area and conductivity of open-cell carbon foams[END_REF]). Pore scale numerical simulations (using finite element finite volume, Lattice-Boltzmann method) allow solving heat equation within both solid and fluid phases on such foam structures while varying porosity, morphology and conductivity ratios. Some of these authors further derived ETC correlations by introducing non-dimensional morphological parameters. [START_REF] Druma | Surface area and conductivity of open-cell carbon foams[END_REF] concluded that some simplified models could not accurately predict the ETC of foams (approximated by spherical pores, homogeneously dispersed within a solid matrix) over the complete range of porosities. [START_REF] Coquard | Numerical investigation of conductive heat transfer in highporosity foams[END_REF] deduced that the realistic representations of foam structures (e.g. Cubic,

Tetrakaidecahedron and Weaire-Phelan unit cells) did not account for the commercial foam irregularities and imperfections. [START_REF] Mendes | A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures[END_REF] observed that using HS bounds to build correlation give the best results for most of the investigated structures. They also propose a complex arrangement of Series and Parallel models [START_REF] Mendes | An improved model for the effective thermal conductivity of open-cell porous foams[END_REF]. In both cases, the weighting factors can only be obtained from ETC data. [START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF] performed numerical simulations on Kelvin-like with convex triangular cross section ligaments to determine ETC over a wide range of solid to fluid conductivity ratios. The weighting factor appearing in their correlation is deduced from morphological parameters and ratio of solid to fluid phase thermal conductivities. This work was further extended to different strut cross-section [START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF]. These authors found that size of the cells struts cross section shape have negligible effects on ETC.

Various empirical ETC correlations have been developed for foams of different materials (metallic and ceramic). However, the majority of correlations concern metal foam i.e. very high conductive solid material (e.g. [1-2, 5, 24]) with high porosity (0.88< ε % <0.97). A few are dedicated to ceramic foams (e.g. [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF]) (0.75< ε % <0.97) i.e. low conductive solid material.

Different bounds, models, adjustable and morphological parameters were used to buid correlations. No common basis could be identified among the numerous and very different proposed correlation save they all give goods results on a given set of foams and that, it is usually difficult to evaluate beforehand their applicability to a given case.

In this work, we tested different combinations of simplified bounds and models in order to retrieve their feasibility in a wide porosity range (0.60-0.97), large range of conductivity ratios of constituent phases as well as open-cell foam samples of different materials/alloys obtained from different manufacturing processes.

Simplified ETC models of porous media

Effective thermal conductivity of open-cell foams is much higher than the granular porous media one. Since the solid has a higher conductivity, the interconnected microstructure virtually increases the pathway of thermal conduction (or energy flux) in the homogenous medium compared with the discrete granular microstructure, and therefore enhances the effective thermal conductivity of the foam material. Various effective thermal conductivity models are found in the literature; those incorporated correspond to two limiting cases for the effective conductivity, called upper and lower bounds arranged either in arithmetic or geometric schemes.

Two models of a two-phase material, which are not themselves realistic for a foam but which do provide useful results are the "parallel" and "series" models illustrated in Figure 1 (a) and (b), respectively and are often used as benchmarks for new model validations. The physical structures assumed in the derivations of the Series and Parallel models are layers of the components aligned either perpendicular or parallel to the heat flux as presented in Equations 1 and 2.

# &'('))*) = (1 -& # )# ! + & # # " (1)
# !*(+*! = # ! . # " (1 -& # )# " + & # # ! (2)
Hashin & Shtrikman [START_REF] Hashin | A variational approach to the theory of the effective magnetic permeability of multiphase materials[END_REF] derived effective conductivity bounds on the basis of a variational approach that were the best (i.e. narrowest) possible bounds for macroscopically homogeneous, isotropic, two-phase materials that could be derived from the components' volume fractions and conductivities. The bounds state that the ETC of any isotropic mixture of several isotropic conducting materials satisfies certain inequalities independently of the structure of a porous medium (see Figure 1c). The Hashin-Shtrikman (HS) bounds always lie within the Series-Parallel bounds, regardless of the components volume fractions or thermal conductivities and are given by Equations 3 and 4.

# ,-,/&&*( = # ! 1 2 # ! + # " -22 # ! -# " 3& # 2 # ! + # " + 2 # ! -# " 3& # 4 (3) # ,-,0#1*( = # " 1 2 # " + # ! -22 # " -# ! 3(1 -& # ) 2 # " + # ! + 2 # " -# ! 3(1 -& # ) 4 ( 4
)
The problem of electrical conduction in an inhomogeneous medium could be solved by the effective medium theory (EMT). Laudauer's EMT model [START_REF] Landauer | The electrical resistance of binary metallic mixtures[END_REF] consists of a twocomponent medium composed of two different materials, with neither phase being necessarily continuous or dispersed. The shape of each material element is assumed to be spherical. The main assumption in Landauer's theory is that the medium surrounding an element is considered homogenous, which has an effective conductivity that characterizes the overall properties of the mixture (see Figure 1d). Either component may form continuous heat conduction pathways, depending on the relative amounts of the components, therefore this structure is unbiased towards its components. Moreover, for porous media in which the solid phase forms continuous pathways, such as open-cell foams, the minimum ETC bound is expected to be given by the Effective Medium Theory (EMT) as given in Equation 5.

# 234 = 1 4 1(3& # -1) # " + {3(1 -& # ) -1} # ! + 9 :(3& # -1) # " + {3(1 -& # ) -1} # ! ; 5 + 8 # " # ! 4 (5)
The reciprocity model was based on the reciprocity theorem [START_REF] Keller | A theorem on the conductivity of a composite medium[END_REF] which assumes that a microstructure of two-component remains statistically equivalent when exchanging the volume fractions of the components (see Figure 1e) and is given by Equation 6. Reciprocity model is quite different than other models and predictions agreed well with many granular materials of spherical nature (Rio et al., [START_REF] Del Rio | Formula for the conductivity of a twocomponent material based on the reciprocity theorem[END_REF]).

# 63 # " = 1 + 2 > # " / # ! -13& # 1 + 2 > # ! / # " -13& # (6)

Comparison between experimentally and numerically calculated ETC data

Most of the ETC data were reported in the literature for the cases where # ! (or [START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF]). In the Figure 2, total porosity as well as ETC data obtained numerically were compared against experimental data and are in excellent agreement (see also Table 1). However, there are very minimal differences which could be attributed to the measurements uncertainties, numerical errors and morphological disparity. This agreement confirms quantitatively and qualitatively the previously published numerical dataset of ETC, which lends confidence to develop different ETC configurations based on different simplified models.

# ! $ ) ≫ # " (

Effective thermal conductivity empirical modelling

Depending on the availability of foam material (ceramic, alloys and pure metals), authors found out different values of adjustable parameters based on different arrangement of bounds.

These adjustable parameters possessed fixed (e.g. [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF][START_REF] Boomsma | On the effective thermal conductivity of a threedimensionally structured fluid-saturated metal foam[END_REF][START_REF] Dai | Correcting and extending the Boomsma-Poulikakos effective thermal conductivity model for three-dimensional, fluidsaturated metal foams[END_REF]) and variable (e.g. [START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF][START_REF] Singh | Computational aspects of effective thermal conductivity of highly porous metal foams[END_REF]) values in the most of the reported works. On the other hand, a few authors did not obtain any adjustable parameter in ETC determination for very high conductive material. The most common belief in this case is that heat conduction is mainly due to solid conductivity occurring mainly through the parallel layout while neglecting the influence of working fluid conductivity like air or water.

The present work investigates in predicting ETC values by determining a functional parameter in terms of the traditional input parameters valid for the complete range of porosities and accessible thermal conductivity ratios of Kelvin-like foams. This has been carried out by arranging various combinations of simplified bounds and models (see Equation 7and 8).

As a first step, a generalization of the simplified model for ETC ( similar to [START_REF] Mendes | A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures[END_REF]) is proposed and presented in Equatin 7. This approach is extended and applied to a geometric scheme, Equation 8(also proposed by a few authors e.g. [START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF][START_REF] Singh | Computational aspects of effective thermal conductivity of highly porous metal foams[END_REF]) . The two schemes are chosen in order to identify how simple models can be best arranged based on generic minimum and maximum bounds and could predict accurate ETC values of open-cell foams.

# *"" = K# ;+< + (1 -K)# ;'= (7)
# *"" = # ;'= > ! . # ;+< 98> ! (8)
where, # ;'= and # ;+< are generic maximum and minimum bounds for the ETC respectively, and K and K ? are the functional parameters to be determined from the fitting of various experimental/numerical data.

Mendes et al., [START_REF] Mendes | A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures[END_REF] used an adjustable parameter K that was an explicit function of the structure of porous media and obtained by calculating ETC of solid phase alone (i.e. under vacuum condition). K values were obtained for each case from numerical simulations by giving # *"",! and respective values for # ;+< and # ;'= under vacuum conditions.

A different strategy is applied in the current work to obtain the parameters, K and K ?

by solving Equations 7 and 8 in terms of # *"" , # ;+< and # ;'= as:

K = # ;'= -# *"" # ;'= -# ;+< (9)
K ? = NO2# *"" 3 -NO(# ;+< ) NO(# ;'= ) -NO(# ;+< ) (10)
where, 0< K(or K ? ) <1

Based on the proposed generic ETC model, given by Equations 9 and 10, and 2) by selecting different expressions for # ;'= and # ;+< . In the following sections, they are called as "model" for the sake of simplicity.

These models are presented in Table 2 where explicit expressions of K and K ? are provided. The dataset of 2000 numerically obtained values of ETC obtained on virtual and ideal isotropic foam structures was gathered from our previous work [START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF] to determine values of K and K ? in order to account for a wide range of porosities, different morphological parameters of foam structures as well as low to high ratios of solid to fluid conductivity ratios. Previously, porosity (& # ) of an idealized isotropic Kelvin-like cell was expressed as a function of dimensionless morphological parameters (T) of foams of any strut cross section [START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF] (Equation 23).

Same procedure was followed and K values for minimum and maximum bounds arranged in arithmetic scheme were plotted against T for Models 1-6 (see based on Equation 9. It can be seen in Figure 3a 9using numerical ETC values [START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF]) collapse very well and a unique curve is obtained as presented in Equations 24-25.

-&

# = T = 12UV *@ 5 W + 32 3 UV *@ 7 ( 23
) K 9 = 0.6249(1 -T) ( 24
) K 7 = 0.6213(1 -T) ( 25
)
where V *@ = X *@ /Y A (ratio of equivalent circular strut radius to node-to-node length) and W = Y ! /Y A (ratio of strut length to node-to-node length) respectively. This formulation (Equation 23) is presented in [START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF].

The RMSD (root-mean-square-difference) values (see Equation 26) are 0.77% and 1.93% for K values of Models 1 and 3.

XZ[\ = 10 63-(20CD) -1 ]^_ℎ aYbc = Nde(K) A')A -Nde (K) *=& (26)
In case of Model 2 (# ,-,0#1*( and # ,-,/&&*( bounds), data of K values fit very well (see Figure 3b). An intercept has been introduced in order to obtain the best fit of ETC data.

However, no explanation is yet found for the meaning of the intercept. The models 4-6 (see Figure 3d, 3e, 3f) clearly indicate that they follow same trend with porosity but do not form unique characteristics. However, these fittings are not coherent over the entire range of porosity and thus, they are not suitable to be arranged in arithmetic scheme to predict accurate values of ETC.

In the case of arrangements of simplified models in geometric scheme, the derivation of an empirical correlation is more difficult to obtain an excellent combination of dimensionless morphological parameters and ratio of constituent phases. The geometric scheme of series and parallel models using a weighted parameter (Model 7, Equation 17) has already been presented [START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF][START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF] and thus, this scheme is not shown here. Following their procedure, K ? values for Models 8-12 (see are calculated (obtained from Equation 10 using numerical ETC values [START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF]) and further efforts have been made to obtain an accurate relation between K ? and a combination of T and # ! /# " as presented in Figure 4.

However, we could not advance to obtain any relationship between K ? , T and # ! /# " for Models 9 and 10. The plots of δ

K F ? = -0.005 g′ 5 + 0.0876 g′ + 0.4914

K G ? = -0.0029 g 5 + 0.0458 g + 0.759

There is no physical reason to choose a quadratic polynomial function in Equations 27-29 and we do not claim any physical meaning to the curve fitting. The quadratic polynomial function is the simplest function that gives a good approximation of ETC data.

The development of functional parameter, K (or K ? ) is clearly a function of foam morphology (T) and constituent conductivities (# ! /# " ) depending the combination of bounds and models employed. For each combination, the functional parameter is unique. Figure 5 presents a simple algorithm to predict ETC from input parameters for foam structures.

Moreover, this algorithm can also be used to predict the solid phase conductivity and morphology from a known ETC value that could help in optimizing foam structure. The applicability of these functional parameters to predict ETC is validated in the next section.

Comparison and validation

ETC correlations developed in the section 4 for different arrangements and schemes are compared and validated against the experimental and numerical data reported in the literature data for foams of different materials (ceramic and metal).

With ceramic open-cell foams

Predicted ETC results are firstly compared and validated against experimental data reported by [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF] fro ceramic foams. In their work, data containing simultaneously morphological data, intrinsic solid phase thermal conductivity and effective thermal conductivity were measured for different foam samples (Al2O3, Mullite and OBSiC). Values of K and K ? of different models are calculated using empirical correlations according to different arrangements and schemes (Models 1, 3, 8, 11 and 12) and further applied them to estimate ETC. The analytical results are presented in Table 3. From Table 3, it can be clearly observed that all these models predict excellent ETC results (see also Figure 6-left).

With metal open-cell foams

A few authors (e.g. [START_REF] Miettinen | Dependence of thermal conductivity on structural parameters in porous samples[END_REF][START_REF] Kumar | Determination of effective thermal conductivity from geometrical properties: Application to open cell foams[END_REF][START_REF] Kumar | Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams[END_REF]) have already highlighted the problem of non-reporting of intrinsic solid phase conductivity of most of the foams. Most commonly, the correlations that have been reported in the literature were based on parent/bulk solid phase conductivity.

Different metal foam materials (or alloys) and their associated ETC values of various authors are presented in Table 4. However, no information regarding their intrinsic solid phase conductivity is known.

Consequently, the methodology proposed [START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF] has been used to determine intrinsic solid phase conductivity while using the experimental ETC data. These authors proposed the following formulation of modified Lemlich model that is valid for any arbitrary fluid as:

# *"" # " = 2 3 . # ! # " . (T) 9/J (30)
where, i = -0.0042NO (T 5 . # ! /# " )3

5

+ 0.0593 NO (T 5 . # ! /# " ) + 0.7144 A simple method has been described below to calculate the intrinsic solid phase conductivity (# ! ) for a given T, # *"" and # " .

Step I: Equation 30 can be rewritten as:

# *"" # " = 2 3 . # ! # " . (T) 9/K8L.LLMN)< (P " .Q # /Q $ )R " SL.LFT7 )< (P " .Q # /Q $ )SL.U9MMV (31)
Step II: Assign # ! /# " = j ! and # *"" /# " = j * and apply natural log functions to both sides:

!"(1.5. ' ! ) = !"(' " ) + 1 [" # {!"(-$ . ' " )} $ + " $ {!"(-$ . ' " )} + " % ] . !"(-) (323)
!"(1.5. ' ! ) -!"('

" ) - 1 [" # {!"(-$ . ' " )} $ + " $ {!"(-$ . ' " )} + " % ] . !"(-) = 0 (326)
where, O 9 =-0.004, O 5 =0.0593, O 7 =0.7144

Step III: Using iterative process, solve for j ! .

Intrinsic solid phase conductivities were calculated using Equations 31-32 for each foam material and these values were subsequently substituted in different models (Models 1, 3, 8, 11 and 12) to predict analytical ETC values (see Table 4). From Table 4, it can be prompted that the different models are consistent with each other and their uses can be combined to predict accurate ETC values (see also Figure 6-right). However, the predicted values underestimate the experimental ETC data which can be attributed to the formulation of [START_REF] Kumar | The geometric and thermohydraulic characterization of ceramic foams: An analytical approach[END_REF] by which the intrinsic solid phase conductivity of a different material is under-evaluated.

Conclusion

Different arrangements of simplified models in arithmetic and geometric schemes were Depending on the availability of morphological resources, any of these models can be easily used to determine effective thermal conductivity from morphology and thermal conductivity ratio. From the present results, it can be safely concluded that the proposed correlations are most suitable for evaluating the functional parameters of both schemes.

Finally, it may be emphasized that the most remarkable feature of the proposed models lies in the fact that they allow one to quite accurately predict the effective thermal conductivity of open-cell foam structures for any working fluid.

Acknowledgement

The authors would like to thank the ANR (Agence Nationale de la Recherche) for financial support in the framework of FOAM project and all project partners for their assistance. The ETC data of ceramic foams were taken from the works of Dietrich et al., [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF] while ETC data of metal foams were taken from Bhattacharya et al., [START_REF] Bhattacharya | Thermophysical properties of high porosity metal foams[END_REF], Solórzano et al., [START_REF] Solorzano | An experimental study on the thermal conductivity of aluminium foams by using the transient plane source method[END_REF], Bodla et al., [START_REF] Bodla | Resistance network-based thermal conductivity model for metal foams[END_REF]; Ranut et al., [START_REF] Ranut | High resolution X-ray microtomography-based CFD simulation for the characterization of flow permeability and effective thermal conductivity of aluminum metal foams[END_REF]; Wulf et al., [START_REF] Wulf | Experimental and numerical determination of effective thermal conductivity of open cell FeCrAl-alloy metal foams[END_REF]; Takegoshi et al., [START_REF] Takegoshi | A study on effective thermal conductivity of porous metals[END_REF] and Paek et al., [START_REF] Paek | Effective thermal conductivity and permeability of aluminium foam materials[END_REF].

Nomenclature

  '))*) and # ,-,/&&*( ) bounds, twelve different arrangements of these models are formed (Models 1-12, six for each scheme, Eqs. 11-22 in Table

  and 3c corresponding to the Model 1 (# !*(+*! and # &'('))*) bounds) and Model 3 (# 234 and # &'('))*) bounds), K values (obtained from Equation

G? 5 ?

 5 are represented against g = NO (T 5.5F # ! /# " ), g′ = NO (T 9.F # ! /# " ) and g = NO (T 5.5F # ! /# " ) in Figures 4a, 4b and 4c respectively. The dimensionless parameters g (and g′) are the best fitting parameters to estimate effective thermal conductivity (using Eq. 8). It can be easily observed that all the values of K ? in relation with g (and g′) collapsed on a single curve for all the different strut shapes. The RMSD values are 1.04%, 1.82% and 1.6% for Models 8, 11, and 12 respectively, obtained just by replacing K with K ? in Equation 26. From Figure 4, numerical approximation of K ? for Models 8, 11, and 12 are given by Equations 27-29 as: K = -0.0028 g 5 + 0.0395 g + 0.8226

  schemes predicts the most accurate effective thermal conductivity results. Moreover, the

Figure 1 .

 1 Figure 1. Five fundamental effective thermal conductivity structural models for two-component (solid and fluid phases) open-cell foam materials (assuming the heat flow is in the vertical direction).

Figure 2 .Figure 3 .

 23 Figure 2. Comparison and validation of experimental ETC data (taken from the works of Dietrich et al., [4]) against numerically calculated ETC data of ceramic foams.

Figure 4 .

 4 Figure 4. Plot of !′ (dimensionless) versus fitting parameters # and #′ (dimensionless) of different strut shapes. (a) Model 8-Plot of ! ! " versus # = %& () !.!$ * % /* & ). (b) Model 11-Plot of ! $ " versus #′ = %& () '.$ * % /* & ). (c) Model 12-Plot of ! ( " versus # = %& () !.!$ * % /* & ). The errors in these models are evenly distributed. Different colours of cubic data points represent ! values obtained for different porosities.

Figure 5 .

 5 Figure 5. Algorithm to predict effective thermal conductivity (ETC) by morphological and material properties characteristics of a foam matrix. This algorithm can be used in reciprocal way-from input to output and vice versa.

Figure 6 .

 6 Figure 6. Comparison and Validation of experimentally and analytically obtained effective thermal conductivity (! !"" ) data by using different models (Models 1, 3, 8, 11, 12) for ceramic foams (Left) and metal foams (Right). Note that, Ana. -Analytical, Exp. -Experimental. The fitting comparison between experimental and analytical ETC values is also presented (Right). The ETC data of ceramic foams were taken from the works of Dietrich et al.,[START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF] while ETC data of metal foams were taken from Bhattacharya et al.,[START_REF] Bhattacharya | Thermophysical properties of high porosity metal foams[END_REF], Solórzano et al.,[START_REF] Solorzano | An experimental study on the thermal conductivity of aluminium foams by using the transient plane source method[END_REF], Bodla et al.,[START_REF] Bodla | Resistance network-based thermal conductivity model for metal foams[END_REF]; Ranut et al.,[START_REF] Ranut | High resolution X-ray microtomography-based CFD simulation for the characterization of flow permeability and effective thermal conductivity of aluminum metal foams[END_REF]; Wulf et al.,[START_REF] Wulf | Experimental and numerical determination of effective thermal conductivity of open cell FeCrAl-alloy metal foams[END_REF]; Takegoshi et al.,[START_REF] Takegoshi | A study on effective thermal conductivity of porous metals[END_REF] and Paek et al.,[START_REF] Paek | Effective thermal conductivity and permeability of aluminium foam materials[END_REF].

  usually metal foams). In order to perform numerical simulations to calculate ETC of foam samples, it is necessary to provide solid and fluid conductivities as an input parameters on foam structures. As discussed in sections 1 and 2, intrinsic solid phase thermal conductivity of foam materials (#

! ) has not been generally measured in majority of the works. In this view point, we chose to perform numerical simulations using the experimentally measured intrinsic solid phase conductivities of Al 5 O 7 , Mullite and OBSiC ceramic foams (# ! =26, 4.4, 15 Wm 89 K 89 respectively) by Dietrich et al., [4]. The fluid conductivity (# " ) of air used was 0.03 Wm 89 K 89 . As ETC is mainly porosity driven, virtual open-cell foams were generated based on total porosity (& : ) reported by Dietrich et al., [4]. 3-D numerical simulations at pore scale were performed on these virtual open-cell foams with circular strut cross-section in LTE condition (for detailed description, see our previous work

Table 2 .

 2 Simplified models for effective thermal conductivity, based on the generic model given by Equations 9 and 10, obtained by selecting different thermal arrangements for ! !"# and ! !$% .

	Latin symbols
	k/X	Side length of strut shape/Radius of strut shape	ll
	Y A	Node-to-node length Strut length	ll
	i	Correlation factor (Eq. 30)	-
	X *@	Equivalent circular strut radius	ll
	Greek Symbols
	& #	Open porosity	-
	& :	Total porosity	-
	V *@	Ratio of equivalent circular strut radius to node-to-node length	-
	W	Ratio of strut length to node-to-node length	-
	K	Adjustable parameter in arithmetic scheme (Eq. 7)	-
	K ?	Adjustable parameter in geometric scheme (Eq. 8)	-
	T	Dimensionless geometrical parameter (Eq. 23)	-
	g	Dimensionless fitting parameter (Eqs. 27 and 29)	-
	g′	Dimensionless fitting parameter (Eq. 28)	-
	# !	Intrinsic solid phase conductivity of foam	m. l 89 . j 89
	# !	$	Solid/Bulk phase conductivity of foam material	m. l 89 . j 89
	# "	Fluid phase conductivity	m. l 89 . j 89
	# *""	Effective thermal conductivity	m. l 89 . j 89
	# &'('))*)	Effective parallel thermal conductivity (Eq. 1)	m. l 89 . j 89
	# !*(+*!	Effective series thermal conductivity (Eq. 2)	m. l 89 . j 89
	# ,-,/&&*(	HS upper bound thermal conductivity (Eq. 3)	m. l 89 . j 89
	# ,-,0#1*(	HS lower bound thermal conductivity (Eq. 4)	m. l 89 . j 89
	# 234	Effective medium theory thermal conductivity (Eq. 5)	m. l 89 . j 89
	# 63	Reciprocity model (Eq. 6)	m. l 89 . j 89
	Abbreviation
	µCT	Micro-computed tomography

*$)$++(+ ( -%&(! 678 ) 19 & : = %&'! (--( -%&(! 678 ) %&'! ./,4**() ( -%&(! 678 ) 20

Table 4 .

 4 Determination of intrinsic solid phase conductivity (! # ) and validation of analytical effective thermal conductivity (! !"" ) against experiments for foam samples of different pore sizes and porosities with air as interstitial fluid.

	Experiments

Table 3. Validation of analytically determined effective thermal conductivity (! !"" ) of Al2O3, Mullite and OBSiC ceramic foams (data taken from Dietrich et al., [START_REF] Dietrich | Determination of the thermal properties of ceramic sponges[END_REF]) of different pore sizes and porosities with air as an interstitial fluid.

Experiments


Vitor A C Pavinato

Stéphane De Mita

Jean-Michel Marin

Miguel De Navascués
email: miguel.navascues@inrae.fr

Joint inference of effective population size and genetic load from temporal population genomic data

Most classical population genetic inference methods assume that genome wide genetic diversity is mostly influenced by neutral processes (commonly named as "demography") while selective processes affecting few isolated loci. Thus, changes in allele frequencies through time for a large number of loci can be used to infer the amount of drift (i.e. the effective population size, N e ) and the presence of some selection is assumed to have a negligible impact on that estimate. Loci under selection can then be identified as they will present allele frequency changes larger or more often in the same direction, than expected under pure drift.

Reality is more complex

In some cases these assumptions might not hold. If the action of selection is widespread along the genome or recurrent in time, the high proportion of loci affected by selection and hitch-hiking could significantly bias the estimates of effective population size. Removing outlier loci would not solve this problem because only loci with large effects will be detected but many other loci under weaker selection (e.g. selection on polygenic characters) could be present. Our proposal: joint inference using ABC We propose to make population genetic inference using models that include both the neutral and adaptive processes on the genome scale. This will allow to estimate demographic and selection parameters taking into account their interaction. Because these models are difficult to address under a likelihood framework we recourse to approximate Bayesian computation via Random Forest (ABC-RF). ABC-RF uses simulations to generate a training data set from which RF can learn and make inferences from real data.

Our model

We applied this approach to the case of a single population of size N. Advantageous mutations arrive to the population at rate µ b with selection coefficients (s) taken from a gamma distribution. We simulate this model with SLiM and at each generation the effective population size (N e ) is calculated from the variance of reproductive success of individuals. We also calculate genetic load (L) and proportion of polymorphisms with Ns > 1 (P). Samples of individuals are taken at two different generations, separated by a period of time τ .

Results: Estimation of N e

The performance of the method was assessed using the simulations from the training data. For each of the the calculated N e during the simulation (true N e ) was compared to its estimate (out-of-bag, OOB, prediction from ABC-RF). An estimate of N e from temporal F ST was also calculated as comparison with a method that assumes all loci to be neutral.

Conclusions

Including both neutral and adaptive processes in the model allows:

Good estimate of demography in presence of pervasive selection Estimate of quantities describing the action of selection based exclusively on polymorphism data. Further work needed to evaluate more complex models

Figure :

 : Figure: A naive view of population genetics. Simulation of allele frequencies change through time in nine neutral loci (black) and one adaptive locus (red) due to drift (N e = 500) and selection (s = 0.4).

Figure : Figure :

 :: Figure: A proportion of neutral loci are afected by linked selection. Simulation of allele frequencies change through time in nine neutral loci (black), one adaptive locus (red) and five hitch-hiking neutral loci (blue) due to drift (N e = 500), selection (s = 0.4) and recombination (0.005 ≤ r ≤ 0.1).

Figure :

 : Figure: Performance of the N e estimation. Left: Estimates from ABC-RF. Right: Estimates from temporal F ST (assume no selection).

Figure : Figure :

 :: Figure:Good N e estimation despite pervasive selection. Error in N e estimate in function of the presence of selection (measured as θ b = 4N e µ b ). Estimates from F ST show a dramatic increase in error when selection is frequent in the population but ABRF estimates are only weakly affected.

Acknowledgements

This project has received funding from the LabEx AGRO (convention ANR-10-LABX-0001-01), CEMEB (convention ANR-10-LABX-0004) and NUMEV (convention ANR-10-LABX-20) through the AAP Inter-LabEx (ABCSelection). This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 791695 (TimeAdapt). S. De Mita was funded by INRAE (Projet Innovant EFPA).

Joint inference of demography and selection

Pavinato, De Mita, Marin, de Navascués


Spiros Kotopoulis
email: spiros.kotopoulis@googlemail.com

Anthony Delalande

Chantal Pichon

Michiel Postema

ON CELLS AND SOUND

Ultrasound as a therapeutic means is increasing in popularity, owing to its non-invasiveness and low cost. Biomedical applications include surgery, cyanobacteria eradication, and drug and gene delivery. Recent advances have been associated with the ultrasound-induced formation of transient porosities in cell membranes. These and other applications have been associated with the behaviour of medical microbubbles near cells. Common methods of studying the dynamic behaviour of microbubbles are high-speed photography and optical microscopy.

Here, we give a brief overview of our fields of study in biomedical ultrasonics, concentrating on bubble-cell interactions.

There are three standards for the safe use of biomedical sound. When introducing bubbles, these safety guidelines may be misleading.

Definition of ultrasound

Sound waves are a form of mechanical vibration. They correspond to particle displacements in matter. Unlike electromagnetic waves, which can propagate in vacuum, sound waves need matter to support their propagation: a solid, a liquid, or a gas. The ear is an excellent acoustic detector in air but its sensitivity is limited to an interval between 20 Hz and 20 kHz. Audiofrequency sound is essential in communication and entertainment. The acoustics of buildings, particularly concert halls, has been the subject of considerable study. Unwanted audiofrequency sound is called noise. The study of noise and noise control is an important part of engineering. Ultrasound refers to sounds and vibrations at frequencies above the upper audible limit of 20 kHz to values that can reach 1 GHz, as shown in Figure 1. Consequently, ultrasonics involve higher frequencies and smaller wavelengths than audio acoustics. The highest theoretical ultrasonic frequency that can be generated has been elegantly derived by Kuttruff. [START_REF] Kuttruff | Acoustics: An introduction[END_REF] Consider a crystal consisting of molecules separated by a distance !, through which a monotonous longitudinal wave with speed ! ! is travelling. The phase difference ∆! between two molecules is then

∆! = !" = 2! !" ! ! , ( 1
)
where ! is the sound frequency and ! is the wave number. If adjacent masses oscillate in opposite phase, the situation is that of a standing wave. Therefore, the highest theoretical ultrasonic frequency must be the frequency where

∆! = ! or ! = ! ! !! . (2)
Conversely, infrasound involves sounds and vibrations at low frequencies (below 20 Hz) and long wavelengths. Because the physiological sensation of sound has disappeared at these frequencies, our perceptions of infrasound and ultrasound are different. Ultrasonic waves in fluids and solids are used for non-destructive evaluation. The general principle is to excite and detect a wave at ultrasonic frequencies and to deduce information from the signals detected.

Example applications include the detection of flaws and inhomogeneities in solids, SONAR, SODAR, medical imaging, and acoustic microscopy. The medical ultrasound range is between 0.5-100 MHz. Ultrasound as a therapeutic means is increasing in popularity, owing to its non-invasiveness and low cost.

The device used for transmitting and receiving ultrasound is called a transducer. Traditional diagnostic transducers use lead zirconate titanate (PZT) materials to send and receive the ultrasound. The acoustic pressures generated by diagnostic transducers have an upper limit due to de-poling. For some therapeutic applications, where higher acoustic amplitudes are desirable different piezo-electric materials have been investigated, such as lithium niobate. [START_REF] Kotopoulis | Lithium niobate ultrasound transducer for high-resolution focused ultrasound surgery[END_REF] The density and compressibility parameters of blood cells hardly differ from those of plasma. Therefore, blood cells are poor scatterers in the clinical diagnostic frequency range. Since imaging blood flow and measuring organ perfusion is desirable for diagnostic purposes, markers should be added to the blood to differentiate between blood and other tissue types. Such markers must be acoustically active in the medical ultrasonic frequency range. The resonance frequencies of encapsulated microbubbles lie well within the clinical diagnostic range, i.e. 1-10µm. Based on their acoustic properties, microbubbles are well suited as an ultrasound contrast agent.

Cell-ultrasound interaction

It has been proven by numerous groups, that the cellular uptake of drugs and genes is increased, when the region of interest is under sonication, and even more so when an ultrasound contrast agent is present. [START_REF] Postema | Contrast-enhanced and targeted ultrasound[END_REF] This increased uptake has been attributed to the formation of transient porosities in the cell membrane, which are big enough for the transport of drugs into cells. The transient permeabilisation and resealing of a cell membrane in an ultrasound field is called sonoporation. [START_REF] Bao | Transfection of a reporter plasmid into cultured cells by sonoporation in vitro[END_REF] There are five non-exclusive hypotheses for the sonoporation phenomenon. [START_REF] Delalande | Sonoporation at a low mechanical index[END_REF] Cell death due to sonication has also been reported. This observation has been associated with inertial cavitation, the ultrasound-induced generation and subsequent collapse of cavities. [START_REF] Gerold | Laser-nucleated acoustic cavitation in focused ultrasound[END_REF] Structural damage to a cell by an ultrasound contrast agent is shown in Figure 2.

To study these rapid cell-ultrasound interactions, high-speed photography combined with optical microscopy is required.

Phenomena during a single ultrasound cycle need to be recorded at MHz frame rates whereas slower phenomena such as microbubble dissolution or microbubble translation still require kHz frame rates, as opposed to acoustic streaming and fluid flow.

Biomedical applications

Focused ultrasound surgery (FUS) is based on the application of high-intensity focussed ultrasound (HIFU) to heat tissue to a temperature that causes protein denaturation and coagulative necrosis. Typical lesions have been found to be ellipsoid in shape with typical lengths of 1 cm after 1-MHz ultrasound. FUS is used for treatment of malignant diseases in bone and breast. Smaller lesions are necessary for microsurgery. [START_REF] Kotopoulis | Lithium niobate ultrasound transducer for high-resolution focused ultrasound surgery[END_REF] Another use of biomedical ultrasound is cyanobacteria eradication. Cyanobacteria contain gas vesicles that can be disrupted with the aid of ultrasound in the clinical diagnostic frequency range. [START_REF] Kotopoulis | Sonic cracking of blue-green algae[END_REF] If therapeutic agents can be coupled to the shells of ultrasound contrast agents microbubbles or be incorporated in these microbubbles, and the bubbles could be directed to or into cells with the use of ultrasound, this could imply an economic revolution drug and gene delivery. [START_REF] Delalande | Sonoporation at a low mechanical index[END_REF]

Figure 1 :

 1 Figure 1: Typical applications of electromagnetic (top) and acoustic (bottom) frequencies.

Figure 2 :

 2 Figure 2: Cell destruction due to bubble-cell interaction at sonication time t. Dotted lines indicate the cell membrane.

Safety

There are three standards for the safe use of biomedical sound, two of which are represented in Figure 3. The NATO Undersea Research Centre (NURC) guidelines concern acoustic frequencies up to 250 kHz. [START_REF]Human diver and marine mammal risk mitigation rules and procedures[END_REF] The mechanical index (MI) has been split up into three regimes, of which MI<0.3 has been considered safe in pregnant women and neonatals. [START_REF]Guidelines for the safe use of diagnostic ultrasound equipment[END_REF] The thermal index (TI) refers to the transmitted power divided by the estimated power needed to raise the tissue temperature one degrees Celsius. The use of the TI has been disputed.

When introducing bubbles, these safety guidelines may be misleading.