Mourad Larbi

Riccardo Trinchero

Flavio G Canavero

Philippe Besnier

Madhavan Swaminathan

Analysis of Parameter Variability in Integrated Devices by Partial Least Squares Regression

Keywords: Machine learning, uncertainty quantification, surrogate model, PLS regression, sensitivity analysis, integrated voltage regulator (IVR)

This paper focuses on the application of the partial least squares (PLS) regression to the uncertainty quantification of the responses of complex stochastic systems. It considers the development of a surrogate model using a limited set of training samples in order to estimate statistical quantities of the system output with relatively low computational cost compared to the standard brute force Monte Carlo (MC) simulation. The performance and the strength of the proposed modeling scheme is investigated for an integrated voltage regulator (IVR) with 8 random variables. The results highlight the ability of the PLS regression to deals with complex nonlinear problems with very few principal components, also providing important insights about the input variables.

I. INTRODUCTION

Uncertainties associated with fabrication process, tolerances and unknown parameters in complex electronic systems may generated large and uncontrolled variations of the system outputs, which can lead to a potential failure of the system. In advanced electronic systems, the number of parameters impacting the variability of output signals is so large that the brute force Monte Carlo (MC) approach turns out to be inefficient.

For the above reason, in the last decades, various surrogate modeling techniques such as Polynomial Chaos (PC) expansion and its variant [START_REF] Stievano | Stochastic Analysis of Multiconductor Cables and Interconnects[END_REF]- [START_REF] Larbi | Variability Impact of Many Design Parameters: The Case of a Realistic Electronic Link[END_REF] and Machine Learning-based regressions [START_REF] Trinchero | Machine Learning and Uncertainty Quantification for Surrogate Models of Integrated Devices With a Large Number of Parameters[END_REF]- [START_REF] Houret | Controlled Stratification Based on Kriging Surrogate Model: An Algorithm for Determining Extreme Quantiles in Electromagnetic Compatibility Risk Analysis[END_REF] have been successfully proposed as viable and efficient alternatives to the plain MC simulation.

This paper investigate the accuracy and the strength of an alternative technique for the uncertainty quantification, namely the partial least squares (PLS) regression [START_REF] Frank | A statistical view of some chemometrics regression tools[END_REF]. Such technique allows building a compact surrogate model of the output of a generic stochastic system as a function of its uncertain parameters. Similarly to the principal component analysis (PCA) [START_REF] Wold | Principal component analysis[END_REF], PLS allows to reduce the problem dimensionality using a limited number of components while mapping the relationship between input and output variables. This feature allows to capture the most prominent input variables of the systems. Moreover, the surrogate model can then be used to compute statistical moments and the probability density function (PDF) of the output of interest. In Sec. III, the proposed PLS regression is adopted to predict the efficiency of an integrated voltage regulator (IVR) as a function of 8 stochastic parameters.

II. PARTIAL LEAST SQUARES (PLS)

Given a limited set of L training samples {(x i , y i )} L i=1 , with x i ⊆ R D and y i ⊆ R generated by the full-computational model M (i.e., y i = M(x i )), the PLS regression allows defining a reduced set of the input variables, called principal components, which better explain the behavior of the system response y as a function of the input x [START_REF] Frank | A statistical view of some chemometrics regression tools[END_REF].

First, the input matrix X = [x 1 , . . . , x L ] t of size (L × D) and the response vector y = [y 1 , . . . , y L ] t are centered and reduced in order to avoid a bias of the input variables having large values and strong variations. Then, the first principal component t (1) is computed by searching the best direction u (1) maximizing the squared covariance between t (1) = Xu (1) and y as:

u (1) = arg max u t u=1
u t X t yy t Xu.

(

) 1
The above optimization problem (1) is maximized when u (1) is the eigenvector of the matrix X t yy t X associated with the eigenvalue with the largest absolute value. The vector u (1) , called loading vector, corresponds to the X weights of the first component. Then, the residual matrix of X (0) = X and y (0) = y, which are denoted by X (1) and y (1) , are computed as:

X (1) = X (0) -t (1) b (1) ,

y (1) = y (0) -w 1 t (1) , (2)
where b (1) represents a vector of size (1 × D) including the regression coefficients of the local regression of X onto the first principal component t (1) of dimension (L × 1), and w 1 is the regression coefficient of the local regression of y related the first principal component t (1) . The equation (2) corresponds to the local regression of X and y onto the first principal component. The second principal component, which is orthogonal to t (1) , is computed via the PLS again by solving the problem w1 t (1) w2 t (2) y (1) y w 1 t (1) + w 2 t (2) Fig. 1. Top left represents the construction of two principal components in the initial X space. Top right and Bottom left illustrate the prediction of y (0) and y (1) , respectively. Bottom right shows the final prediction of the output y.

in [START_REF] Stievano | Stochastic Analysis of Multiconductor Cables and Interconnects[END_REF] where X and y are replaced by X (1) and y (1) , respectively. The same iterative scheme can be used to compute the rest of the components. A graphical interpretation in a three-dimensional example with two principal components is depicted in Fig. 1.

Usually, the number of principal components q are selected by retaining the one minimizing the root mean square error (RMSE) computed between the model predictions and the training samples used to build the model. The principal components, accounted for within the PLS regression, define a new coordinate system which corresponds to a rotation of the initial system in the parameter x = [x 1 , . . . , x D ] t . The r-th component of the new coordinate system, denoted t (r) with r = 1, . . . , q, is given by:

t (r) = X (r-1) u (r) = Xu (r) * . (3)
The relationship between the vectors u (r) and u (r) * are defined through the matrices U = u (1) , . . . , u (q) and U * = u

(1) * , . . . , u (q) * as follows:

U * = U B t U -1 , (4)
where B = b (1) t , . . . , b (q) t . The PLS regression can be suitably adopted to identify the importance of each input parameter via a variable selection method such as the variables importance projections (VIP) [START_REF] Chong | Performance of some variable selection methods when multicollinearity is present[END_REF]. The VIP score for the input variable j is defined as: where u (rj) is the weight of the j th variable in component r and R 2 (y, t (r) ) is the percentage of y explained by the component r.

V IP j = D • q r=1 R 2 (y, t (r) ) • (u (rj) / u (rj) ) 2 q r=1 R 2 (y, t (r) ) (5)
The VIP value is a weighted sum of squares of the PLS weights (u (rj) ), which takes into account the explained variance of each PLS dimension. The input variable with a VIP score greater than one is generally considered to have a significant impact on the response y [START_REF] Chong | Performance of some variable selection methods when multicollinearity is present[END_REF].

III. APPLICATION EXAMPLES

The integrated voltage regulator shown in Fig. 2 is considered with the objective of estimating its power conversion efficiency. The architecture consists of a system-in-package solution with an integrated inductor on an organic package with a solenoid structure with a Nickel-Zinc ferrite magnetic core [START_REF] Mueller | Design of high efficiency integrated voltage regulators with embedded magnetic core inductors[END_REF].

The goal of this study is to evaluate the impact of 8 stochastic geometrical parameters of the integrated inductor (see Table I) on the IVR efficiency for 5V:1V conversion ratio at the frequency of 100 MHz [START_REF] Trinchero | Machine Learning and Uncertainty Quantification for Surrogate Models of Integrated Devices With a Large Number of Parameters[END_REF], [START_REF] Trinchero | Statistical Analysis of the Efficiency of an Integrated Voltage Regulator by means of a Machine Learning Model Coupled with Kriging Regression[END_REF]. The efficiency is calculated via an extensive model that accounts for switching and conduction losses of power switches, DC, power delivery network (PDN) and AC losses of inductor and output capacitance by relaying on the results of the full-wave solver of Ansys HFSS [START_REF] Mueller | Design of high efficiency integrated voltage regulators with embedded magnetic core inductors[END_REF], [START_REF]Ansys HFSS ver[END_REF].

We now analyze the effect of uncertain input parameters on the IVR efficiency E f f . In order to do that, we construct a PLS regression model with [START_REF] Pedregosa | Scikit-learn: Machine Learning in Python[END_REF] using 200 realizations from Latin Hypercube sampling (LHS) and a number of principal components varying from 1 to 5. An evaluation of the performance of each PLS model is performed by comparison with the brute force MC simulation. Table II shows, from 10000 realizations, a comparison of various PLS models with an increasing number of components and MC simulation, by computing the RMSE, the mean value μ and the standard deviation σ. We see that the PLS model with one principal component provides a RMSE equal to 0.162, while the PLS models with 2 to 5 principal components provide a RMSE of 0.158. As the PLS surrogate model with 2 components achieves the lowest RMSE, i.e. 0.158, the minimum number of components is retained. It is worth noting that the accuracy of this PLS model is very close to LS-SVM surrogate model with a RBF kernel, i.e. with a RMSE of 0.155, and better than the sparse PC which has a RMSE of 0.170 as presented in [START_REF] Trinchero | Machine Learning and Uncertainty Quantification for Surrogate Models of Integrated Devices With a Large Number of Parameters[END_REF]. This result points out to the ability of the PLS surrogate model to estimate the main variability of a complex output while reducing the dimensionality of the problem, i.e. from 8 original variables to 2 principal components. This feature of the method is very interesting and will be exploited in the following to hierarchize the effect of the uncertain parameters on the IVR efficiency.

For the purpose of illustration, we compare in Fig. 3 the predictions of the PLS regression model with 2 principal components with 10000 MC realizations. As the scatter plot is very well aligned along the dashed line, we then deduce that there is a good agreement between the PLS model and MC simulation. As far as the variability of the IVR efficiency is concerned, a representation of the PDFs estimated by the PLS surrogate model and by MC simulation with 10000 realizations is given in Fig. 4. It can be observed that the PLS surrogate model reproduces well the central tendency of the output estimated by MC simulation, although some discrepancy exists for the left tails of the PDFs. In terms of computational cost, 10000 MC simulations took about 7 days while the PLS surrogate model required less than 1 s to carry out the predictions. It is important to mention that this comparison does not include the computational cost related to the generation of the training data set (200 simulations), which needed about 3 h 27 min as shown in Table II.

In addition to the variability of the output, the PLS model provides a sensitivity analysis of the model response. The histogram in Fig. 5 illustrates the VIP scores calculated via the proposed PLS-based surrogate. The VIP scores indicate the impact of the input random variables on the efficiency E f f . We see that the main variability of E f f is related to 3 variables, i.e., the copper thickness bottom t c,b , the copper thickness top t c,t and the copper trace width w v of the inductors, as their VIP scores are greater than or equal to the threshold of 1, as suggested by [START_REF] Chong | Performance of some variable selection methods when multicollinearity is present[END_REF]. Other variables, such as the size of via s v and the dielectric thickness t d , have less impact while the dielectric width w d , the gap between windings g and the magnetic core width offset ∆w m of the inductors have an insignificant effect on the variability of the response E f f . Furthermore, at 100 MHz, total losses of the IVR system are dominated by switching and conduction losses of power switches and the resistive loss due to the PDN [START_REF] Mueller | Design of high efficiency integrated voltage regulators with embedded magnetic core inductors[END_REF]. Since the DC resistance of the inductor impacts the finite resistance of the PDN, its AC losses become less important. Hence, the parameters of the inductor directly influencing its DC resistance become more important and have more effect on the IVR efficiency. This confirms that the sensitivity analysis illustrated in Fig. 5 is coherent since it presents a more significant impact of the parameters that directly affect the DC resistance than the other parameters related to the magnetic material.

Based on the projection provided by the PLS surrogate model, we can also display the variables on the 2 components. Figure 6 shows the correlation circle to highlight the correlation between the input variables and the output. This graph is obtained by computing the correlation between the principal components (i.e., the 2 axes of the graph) and the original variables (input variables and output). We can see that the distance between the output E f f and the input variables such as the copper thickness bottom t c,b , the copper thickness top t c,t and the copper trace width w v of the inductors is very short. This means that they are highly positively correlated. In other words, large values of those input variables will lead to an increase of the IVR efficiency E f f . Concerning the other variables, the size of vias s v is positively correlated with the output E f f but with less impact, while other variables seem to have a negligible effect.

IV. CONCLUSIONS

This paper presents a PLS surrogate model, with a very low computational cost compared to MC simulation, for the uncertainty quantification in high-dimensional space. This PLS surrogate model allowed to efficiently estimate the efficiency of an IVR characterized by 8 uncertain input variables.

Additionally to a good estimation of the first two statistical moments, the PLS surrogate model also provides a sensitivity analysis of the model response at a negligible computational cost. Using this sensitivity analysis, the designer may identify the uncertain input variables affecting the most the variability of the output of the system. Also, the designer may determine the configurations of input variables allowing to improve the performance of the system during the design stage.

Fig. 2 .

 2 Fig. 2. Illustration of the two-chip SiP IVR architecture [11].

Fig. 3 .

 3 Fig. 3. Scatter plot of the IVR efficiency computed by the PLS surrogate model and by the real numerical model with 10000 MC realizations. The dashed-line illustrates a good correlation between the models.

Fig. 4 .

 4 Fig. 4. PDFs of the IVR efficiency estimated by the PLS surrogate model (solid curve) and by MC simulation (histogram) from 10000 MC realizations.

Fig. 5 .

 5 Fig. 5. Variable Importance in Projection (VIP) scores of the IVR efficiency E f f . Input variables having a VIP score greater than or equal to 1 have an important impact on the variability of the IVR efficiency E f f .

Fig. 6 .

 6 Fig.6. Correlation circle representing the projection of the input variables and the output variable (IVR efficiency E f f ). The variables are represented through their projections onto the plane defined by the first two PLS components. A positive correlation is highlighted between the input variables sv, wv, tc,t, t c,b and the output variable E f f as they are projected in the same direction and at a large distance from the origin.

TABLE I UNCERTAIN

 I GEOMETRICAL PARAMETERS OF THE SOLENOIDAL INDUCTOR USED FOR THE IVR.

	Uniform random variables		Unit U [M in; M ax]
	Gap between windings	g	mil	U [4; 6]
	Size of via	sv	µm	U [80; 120]
	Copper Trace Width	wv	mil	U [9; 11]
	Copper Thickness Bottom	t c,b	µm	U [64; 96]
	Copper Thickness Top	tc,t	µm	U [64; 96]
	Dielectric Thickness	t d	µm	U [180; 220]
	Dielectric Width	w d	mil	U [59.4; 60.6]
	Magnetic Core Width offset ∆wm	mil	U [9; 11]

TABLE II EVALUATION

 II OF THE ACCURACY OF THE PLS REGRESSION MODEL W.R.T THE NUMBER OF PRINCIPAL COMPONENTS.

	Method	Number of RMSE Components	μ	σ	t model	*	tcost
	MC	-	-	67.01 0.31	-	7 days
		1	0.162 67.01 0.28 <1s		<1s
		2	0.158 67.01 0.28 <1s		<1s
	PLS	3	0.158 67.01 0.28 <1s		<1s
		4	0.158 67.01 0.28 <1s		<1s
		5	0.158 67.01 0.28 <1s		<1s
	* In addition to 200 LHS simulations which took 3 h 27 min.


Armand Roucher

Véronique Schmitt

Jean-Luc Blin

Rénal Backov
email: backov@crpp-bordeaux.cnrs.fr

Sol-gel process and complex fluids: sculpting porous matter at various lengths scales toward the Si(HIPE), Si(PHIPE) and SBA15-Si(HIPE) series

Keywords:

    Sol-gel process and complex fluids: sculpting porous matter at various lengths scales towards the Si(HIPE), Si(PHIPE), and SBA-15-Si(HIPE) series

Introduction

Today societal need is requestingthe generation of smart materials being more and more complex, multi-scales, multifunctional, responsive toward external fields or even developing a certain degree of autonomy, taking inspiration from Mother Nature. Sol-gel chemistry is indeed the perfect chemical tool to generate such sophisticated materials. As occurring at low temperature this "Chimie Douce" [START_REF] Livage | Vers une chimie écologique. Quand l'air et l'eau remplacent le pétrole[END_REF][START_REF] Brinker | The Physics and Chemistry of Sol-Gel Processing[END_REF][START_REF] Livage | Sol-gel chemistry of transition metal oxides[END_REF] will not disrupt the organization of various templates occurring at various length scales. These templates might be supramolecular entities like micelles, but we can also include the whole field of the physical chemistry of complex fluids (air-liquid foams, biliquid foams, liquid crystals, hydrodynamic instabilities and so forth). To these shaping modes we can associate the application of external fields (electro-spinning, extrusion, spin-coating, spray-drying, 3D printing and so forth), in one word "technology". Of course, in order to specify the final application, sol-gel chemistry is also ideal toward the generation of hybrid organic-inorganic materials or even living bio-materials. We can already sense that the chemical path to generate upto-date advance functional materials is evidently complex, certainly interdisciplinary and possibly bio-inspired [START_REF] Mann | Synthesis of inorganic materials with complex form[END_REF]. For instance, when considering the diatom frustule of the Figure 1a-h we can see that their silica exoskeletons are bearing a hierarchically organized porosity. If Mother Nature is constructing such sophisticated scaffolds, this is because at each level of porosity is occurring a specific functionality. This hierarchical porosity (Figure 1i) is at first glance important for many applications but is imperative when dealing with heterogeneous catalysis where exchange surface is involved. Considering the IUPAC nomenclature microporosity is addressing pores bearing diameters below 2 nm, mesoporous ones between 2 nm and 50 nm while all diameters above 50 At the microscopic length scale, the fluid motion is driven only by diffusion and convection is negligible (molecular reactivity will occur at this length scale). At the macroscopic length scale, the fluid hydrodynamics is based only on convection with a basically Poiseuille flow (when fluids are Newtonian, this is to say that their viscosity remains constant under shear) and diffusion is now negligible considering the whole mass transport. At the mesoscopic length scale, the hydrodynamics is hybrid where the overall scenario can be regarded as a fluid moving through convection in which molecules are diffusing, this behavior is known as a dispersion flow. This is the reason why the hierarchical porosity from macroporous (optimizing the mass transport) through microporous (where molecular reactivity occurs) is at first glance of importance for catalysis or energy conversion applications. Coming back to the diatom exoskeleton of Figure 1ah we can notice a nano-porosity, this nano-porosity being somewhat organized and not totally randomly dispersed. Again if Mother Nature is doing so, this is not for the contemplative issue, but because the organization of the porosity at the nanoscale favors selective photonic behavior, where low energetic wavelengths (above 500 nm) will penetrate the skeleton while higher energetic wavelengths will not, protecting thus the diatom bioorganic counterpart (Fig 1i) for nondesirable energetic UV-light. Overall we can see that Mother Nature has optimized complex architectures where both, molecular reactivity and both mass and light transport are optimized as far as it can be. It is thus important, and challenging, to try to replicate such complex architectures in the Lab, way of thinking being bio-inspired, leading to the notion of chemistry of shapes.

Inhere we focus on the coupling of sol-gel chemistry, direct concentrated emulsion employed as macroscopic molds and lyotopic mesophases as the mesoscale ones. We show how it is possible to conceive 3D-macrocellular monolithic foams bearing hierarchical porosities, through a fine coupling between the sol-gel process and the physical chemistry of complex fluids. In this minireview we do not treat final applications on purpose, we stay at the chemistry of shapes level, while disregarding the Integrative Chemistry [START_REF] Backov | Combining soft matter and soft chemistry: integrative chemistry towards designing novel and complex multiscale architectures[END_REF][START_REF] Faustini | Integrative Sol-gel Chemistry": A Nanofoundry for Materials Science[END_REF] concept dedicated toward a systemic approach of generating advanced functional materials.

2. Solid silica foams obtained by combining sol-gel process and emulsion templating 2.1. Solid Foams, emulsion and lyotropic mesophase templates at a glance Foams are materials containing gaseous voids surrounded by a denser matrix, usually a liquid or a solid. Due to this specific structure they are widely used in applications like insulation (thermal and acoustic), absorbents and weight-bearing structures [START_REF] Klemper | Handbook of polymeric foams and foam technology[END_REF]. They are also employed to take benefit of their high specific surface in applications like heterogeneous catalysis and water cleaning [START_REF][END_REF].

Depending on their cell sizes, foams can be classified as macrocellular (>100 µm), microcellular Polymers foams are the most common but their applications are limited by their inferior mechanical strength, poor surface quality and low thermal and dimensional stability. They are usually prepared by chemical or physical foaming [START_REF] Lee | Polymeric Foams: Mechanisms and Materials 2 nd Ed[END_REF], but the control of the cell size and morphology is difficult as well as the preparation of fully open-cell structures. Therefore, the emulsion-template approach to prepare foams may appear as an outstanding path for both mineral [START_REF] Esquera | [END_REF] or organic foams [START_REF] Ceglia | Formulation and mechanical properties of emulsion-based model polymer foams[END_REF]13] due to the possibility of better controlling the emulsions properties.

In this vein, High Internal Phase Emulsions (HIPE) are a class of emulsions characterized by an internal phase volume fraction exceeding 0.74, which corresponds to the most compact arrangement of uniform, undistorted spherical droplets (for a random packing the compact fraction is equal to 0.64). At higher volume fractions, their structure consists of deformed (polyhedral) and/or polydispersed droplets separated by a thin film of continuous phase, a structure resembling gas-liquid foams. Emulsions are classified into two categories depending on the nature of both the dispersed and continuous phases: water-in-oil (W/O) also called reverse emulsions and oil-inwater (O/W) also called direct emulsions (Scheme 1). Scheme 1. Schematic representation of: a) Oil-in-water concentrated emulsion where oil droplets are dispersed within a continuous aqueous phase, b) water-in-oil concentrated emulsion where water droplets are dispersed within a continuous oily phase. The hydrophilic or hydrophobic continuous phase will contain respectively direct or reverse micelles (as soon as the surfactant concentration becomes larger than the critical micellar concentration (CMC)), a micelle being a supramolecular self-assembly of surfactant entities. For very high surfactant concentrations, these micelles can self-organize into lyotropic mesophases like a hexagonal phase in the present example. RSC Copyright 2011.

Emulsions are thermodynamically meta-stable systems, meaning that over time the emulsion will evolve until the dispersed and continuous phases macroscopically separate. To enhance their kinetic stability, surfactants are used to stabilize the oil/water interfaces. As shown in Scheme 1 those surfactant molecules constituted of a polar head group and a hydrophobic chain will adsorb at the oil/water interface to minimize the interfacial energy. The choice of the surfactant is important and the Bancroft rule states that the phase in which the surfactant is preferentially soluble becomes the continuous phase [START_REF]The Theory of Emulsification[END_REF]. Also, micelles will be swollen with a small amount of the disperse phase, creating a microemulsion (nanometer length scale) which is thermodynamically stable contrary to a macro-emulsion (micrometer length scale). Considering their use as soft templates, these microemulsions will promote a second porosity, called mesoporosity, within the final materials after calcination. If the inorganic skeleton is amorphous as it the case for silica or polymer, final foams will bear intrinsic microporosity, leading overall to a hierarchical porosity.

Solid Foams, emulsion and lyotropic mesophase templates: The Si(HIPE) series

Materials synthesized from concentrated emulsions were labeled poly-HIPE (polymerized High Internal Phase Emulsion) and discovered by the industry [15] and very nice recent review articles have been proposed recently on this domain [START_REF] Silverstein | PolyHIPEs: recent advances in emulsion-templated porous polymers[END_REF][START_REF] Silverstein | Emulsion-templated polymers: Contemporary contemplations[END_REF]. Such materials are most often obtained from the polymerization of a reverse concentrated emulsion, the continuous phase being composed of the organic polymerizable species [START_REF] Silverstein | PolyHIPEs: recent advances in emulsion-templated porous polymers[END_REF][START_REF] Silverstein | Emulsion-templated polymers: Contemporary contemplations[END_REF]. Contrary to what is often admitted, the first inorganic-HIPE has also been pioneered by the industry [START_REF] Araya | Hydrophobic, highly porous, three-dimensional inorganic structures[END_REF]. Eight years later Imhof and Pine [START_REF] Imhof | Ordered macroporous materials by emulsion templating[END_REF] published a paper exhibiting the advantage of working with monodisperse non aqueous emulsions. Our group has shown how the macroscopic voids can be tuned on demand either playing with the starting emulsion oil volume fraction, in order to make a parallel with the organic foams, we labeled those materials Si(HIPE) [START_REF] Carn | Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates[END_REF]. The first step is the hydrolysis of TEOS toward Si(OH)4, the silicic acid being the inorganic polymer precursor. Thereby, as the precursors are hydrophilic, oil droplet are dispersed within the continuous aqueous one, so that we are now dealing with direct emulsions (dispersion of oil droplets in water), see scheme 1. The Si-(HIPE) foams possess very high porosity and very low bulk density. The void size is usually situated in the microcellular range (1-100 µm). These materials will be used as supports for a wide range of applications. In order to tailor the macroporous solids for different applications, it is important to be able to modify the macroscopic void space diameters (Fig. 2). Varying the starting oil volume fraction (ϕo) of an oil-in-water concentrated emulsion [START_REF] Silverstein | PolyHIPEs: recent advances in emulsion-templated porous polymers[END_REF] is one way to achieve that. Several O/W emulsions with increasing oil volume fractions (1Si-HIPE, ϕo = 0.70; 2Si-HIPE, ϕo = 0.73; and 3Si-HIPE, ϕo = 0.78) have been prepared and the resulting solid foams analyzed. Whatever the oil volume fraction, the general texture resembles aggregated hollow spheres. The macrocellular void sizes of the resulting material diminish drastically because the viscosity of concentrated oil-inwater emulsions increases dramatically when the oil volume fraction reaches values above 0.64 [START_REF] Carn | Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates[END_REF]. The enhanced viscosity of the emulsions increases the shear applied to the droplets, thus inducing smaller macrocellular voids within the silica replica. These porous materials possess a secondary micro-mesoporosity, due to the lyotropic mesophase as explained previously. This secondary porosity results in specific surface areas (BET) values around 800 m 2 g -1 addressing only 100 m 2 g -1 of mesoporosity (BJH). It is possible to align the macropores by using a hydrophobic ferrofluid instead of oil and applying an external magnetic field during the condensation process [START_REF] Imhof | Ordered macroporous materials by emulsion templating[END_REF]. As observed within Figure 2, the macroscopic cell diameters are rather polydisperse in size, we can express a strong monodispersity of the macroscopic cells while switching from surfactantstabilized emulsion toward particles-stabilized ones. These Si(HIPE) materials can be easily hybridized with organically modified silanes (OROMOSILs) to generated hybrid Organo-Si(HIPE) series fostering applications toward, photonic [START_REF] Brun | Eu3+@Organo-Si(HIPE) Macro-Mesocellular Hybrid Foams Generation: Syntheses, Characterizations, and Photonic Properties[END_REF], heterogeneous catalysis whatever metallic [START_REF] Ungureanu | Palladium nanoparticles heterogeneous nucleation within organically grafted silica foams and their use as catalyst supports toward the Suzuki-Miyaura and Mizoroki-Heck coupling reactions[END_REF], enzymatic [START_REF] Brun | Enzyme-based biohybrid foams designed for continuous flow heterogeneous catalysis and biodiesel production[END_REF] or bacteriologic [START_REF] Roucher | From Compartmentalization of Bacteria within Inorganic Macrocellular Beads to the Assembly of Microbial Consortia[END_REF]. These Si(HIPE) materials exhibited a low mesoporosity (around 100 m 2 g -1 BJH surface area). Overall on a Chemistry of Shapes point of view two main challenges remained. The first one was to increase the monodisperse character of the macroscopic voids, and the second one was to enhance the Si(HIPE) mesoscopic surface area.

Solid Foams, Pickering emulsion templates: The Si(PHIPE) series

Under this new configuration we are dealing with Pickering-based emulsions, that is to say emulsions stabilized by particles [START_REF] Ramsden | Separation of Solids in the Surface-Layers of Solutions and 'Suspensions' (Observations on Surface-Membranes, Bubbles, Emulsions, and Mechanical Coagulation).-Preliminary Account[END_REF][START_REF] Pickering | CXCVI.-Emulsions[END_REF]. Thanks to the limited coalescence process occurring in Pickering emulsions, the oil droplets size may be easily tuned by the amount of particles [START_REF] Arditty | Some general features of limited coalescence in solid-stabilized emulsions[END_REF]. This brings the advantage of an independent control over the initial oil volume fraction and the drop size. The sol-gel process is equivalent to the one described earlier and the resulting materials are noted PHIPE for Pickering-based HIPEs. For example, keeping the oil volume fraction fixed but varying the amount of nanoparticles in use, the drop size was varied and so did the resulting voids, ranging from 20 to 800 µm, a macrocellular void size domain never reached before (Figure 3) with the Si-PHIPE series [START_REF] Destribats | Tailored Silica Macrocellular Foams: Combining Limited Coalescence-Based Pickering Emulsion and Sol-Gel Process[END_REF]. In addition to the void size, the size of the junction windows that connect adjacent macroscopic voids is of utmost importance as it governs the convectional mass transport within the solid-state foams. The junction size can be triggered at constant initial drop size, that is to say at constant nanoparticle-to-oil weight ratio, by tuning the dispersed phase volume fraction (Fig. 4). Additionally, both parameters, void diameters and connecting windows can be tuned independently, offering thus a strong versatility toward the foams morphology. These Si PHIPE based foams can be further employed as hard templates for the design of carbonaceous foams, for biofuel cell application, with the advantage of tuning on demand the macroscopic mass transport. The nomenclature is the same as the one adopted for Fig. 3. Copyright Wiley-VCH 2012.

Solid Foams, concentrated direct emulsions and P123 lyotropic mesophase templates: The SBA15-(PHIPE) series

Ordered mesoporous silicates [START_REF] Beck | A new family of mesoporous molecular sieves prepared with liquid crystal templates[END_REF][START_REF] Renzo | A 28-year-old synthesis of micelle-templated mesoporous silica[END_REF] have generated fascination in materials science and heterogeneous catalysis, as witness by thousands of publications in this domain. The interested reader can for example refer to [START_REF] Taguchi | Ordered mesoporous materials in catalysis[END_REF][START_REF] Davis | Ordered porous materials for emerging applications[END_REF][START_REF] Balas | Confinement and Controlled Release of Bisphosphonates on Ordered Mesoporous Silica-Based Materials[END_REF][START_REF] Trewin | Synthesis and Functionalization of a Mesoporous Silica Nanoparticle Based on the Sol-Gel Process and Applications in Controlled Release[END_REF]. These mesoporous materials were stated as outstanding candidates for extending heterogeneous catalysis or adsorbents for use in domains where pore sizes larger than those of zeolites were expected. Among the synthesized materials the most renowned are certainly the MCM-41 (Mobil Crystalline Materials) [START_REF] Kresge | Ordered mesoporous molecular sieves synthesized by a liquid-crystal template mechanism[END_REF] using cetyltrimethylammonium as templating agent, SBA-15 (Santa Barbara Amorphous) [START_REF] Zhao | Triblock copolymer syntheses of mesoporous silica with periodic 50 to 300 angstrom pores[END_REF] using triblock copolymer Pluronic 123 as meso-scale template, MSU (Michigan State University) [START_REF] Bagshaw | Templating of mesoporous molecular sieves by nonionic polyethylene oxide surfactants[END_REF] obtained from non-ionic polyoxyethylene alkyl ethers and HMS (Hexagonal Molecular Sieves) [START_REF] Tanev | A Neutral Templating Route to Mesoporous Molecular Sieves[END_REF][START_REF] Boissière | Mechanisms of Pore Size Control in MSU-X Mesoporous Silica[END_REF] where dodecylamine is employed as a mesoscale templating agent. The drawback is that obtaining selfstanding monoliths and not only powders bearing hierarchical porosity is everything but an easy task to reach, in that vein we have to underline the Nakanishi and co-workers' highly efficient path to address monolith generation obtained by coupling sol-gel process, swelling lyotropic mesophases and phase separation instead of concentrated direct emulsions [START_REF] Amatani | Monolithic Periodic Mesoporous Silica with Well-Defined Macropores[END_REF].

We have recently extended the SBA-15 mesopoporous materials synthesis toward the generation of macro-mesocellular Si(HIPE) that we have labelled SBA15x-Si(HIPE) (Figure 5) where x represents the molar salt concentration. In our case we have employed a salt effect to promote efficient silica polycondensation while using (NH4)2SO4. We have demonstrated that this salt effect is not based only on the salt ionic strength but also on his chaotrope-kosmotrope character. Apart from electrostatic screening, ions are known for their ability to provoke order (kosmotrope) or disorder (chaotrope) following their position in the Hofmeister series [START_REF] Hofmeister | Zur Lehre von der Wirkung der Salze[END_REF]. For the anions the series follows : CO3 2-> SO4 2-> S2O3 2-> H2PO4 -> F -> Cl -> Br -≈ NO3 -> I -> SCN -and for the cations : (CH3)4N + > NH4 + > K + > Na + > Li + > Mg 2+ > Ca 2+ [START_REF] Zhao | Triblock copolymer syntheses of mesoporous silica with periodic 50 to 300 angstrom pores[END_REF]. As seen on Figure 5 the salt concentration has a strong effect over both the macroscopic voids diameters and monolith selfstanding properties. First, at the macroscale level, ammonium sulfate decreases the kinetic stability of the oilin-water emulsion by changing the solubility of the P123 surfactant that precipitates inducing a decrease the surfactant density at the oil/water interface and the enhancement of coalescence. This phenomenon involves an increase on the cell diameter with the salt concentration. Second, at the

a) b) c) d) e) f) g) h)
mesoscale level sulfate anions strengthen the hydrophobic interactions (salting out effect of sulfate anions) leading to a mesopore ordering as depicted in Figure 6. The pore sizes distributions have been calculated through the B.J.H desorption curves.

These pore sizes distributions reveal rather well-defined profile of the mesoscopic voids centered at the values ranging between 5 nm-9 nm. The BJH surface areas provide the mesoscopic surface with values ranging from 250 to 350 m 2 g -1 , far above the ones obtained with the traditional Si(HIPE) series while employing TTAB (tetradecyltrimethylammonium bromide) as a mesoscopic templating agent [START_REF] Carn | Soft matter, sol-gel process and external magnetic field to design macrocellular silica scaffolds[END_REF]. Beyond the fact that we obtained self-standing macrocellular foams bearing organized 2D-hexagonal mesoporosity, it was the first time that the "kosmotrope-chaotrope" balance was introduced into the sol-gel process, beyond the well-known screening and ionic strength salt effects [START_REF] Roucher | First Macro-Mesocellular Silica SBA-15-Si(HIPE) Monoliths: Conditions for Obtaining Self-Standing Materials[END_REF].

Conclusions

With this mini review we have emphasized the possibility to couple with a certain success sol-gel chemistry and the physical chemistry of complex fluid to chisel condensed matter at various length scales. Using various examples as traditional direct emulsions, Pickering emulsions and surfactant self-assemblies, we have shown that it is possible to combine all these systems to obtain materials perfectly controlled at all scales. Moreover these materials are modular according to the targeted applications. First, only classical direct emulsions were used for the synthesis of the Si(HIPE) series employing to tune the macroscopic void diameters and cationic surfactant molecules to both stabilize the emulsion oil/water interface at the macroscopic length scale and create vermicular poorly organized porosity at the mesoscopic length scale. The use of Pickering emulsions allowed us to reach macroporous size that have never been achieved before thanks to very large controllable droplets playing with the limited coalescence process. Controlling the initial drop size and dispersed phase volume fraction of surfactant-stabilized or particle-stabilized emulsion leads to independently tuning macroscopic cell diameters and cell junctions. Finally taking the synergetic effect of nonionic surfactants and a (NH4)2SO4 kosmotrope salt effect, we have shown that it is possible to obtained self-standing Si(HIPE) bearing high organized mesoporosity while addressing the SBA15-Si(HIPE) series. When considering the diatom exoskeleton of the Figure 1 and comparing the morphologies with the Si(HIPE), Si(PHIPE) and SBA15-Si(HIPE) series, we understand very easily that the next step is to combine very high monodisperse void and cell junction diameters at the macroscopic length scales with a very high organization of the mesoscopic porosity. In brief the SBA15-Si(PHIPE) series is still missing where all the porosity will be organized on demand at various length scales, proposing potentially the first synthetic silica-diatom-exoskeleton. Up to now, when operating this complex synthetic approaches, we were completely driven by the thermodynamic equilibrium in term of silica synthesis (emulsion preparation being governed by kinetics). Maybe, sooner or later it would be interesting to perform sol-gel route out of this thermodynamic dead end while promoting continuous chemical reactions driven by kinetics rather than thermodynamics, feature that Mother Nature is mastering since a certain time scale.

Figure 1 . 1

 11 Figure 1. 1 Structures of diatom cell walls. Electron microscopy images of isolated cell walls from different diatom species: a) Cylindrotheca fusiformis; b, c) Coscinodiscus asteromphalus; d, e) Thalassiosira pseudonana. RSC, Copyright 2004, f), Cell wall morphology of Coscinodiscus wailesii with valve and girdle patterns h), image taken by optical microscopy g) and details by scanning electron microscopy (f, h), i) scheme representing both the specific mass and light transport occurring at various length scales.

( 1 -

 1 100 µm), ultramicrocellular (0.1-1 µm) and nanocellular (0.1-100 nm). Those cells can be close or open: in closed cells foams, the voids are isolated from each other and cavities are surrounded by complete cell walls whereas in open cell foams the structure consists mainly of ribs and struts.

Figure 2 .

 2 Figure 2. SEM visualization of the inorganic monolith-type material macrostructure. a) and b) 1Si-HIPE, c) and d) 2Si-HIPE, e) and f) 3Si-HIPE. RSC Copyright 2004.

Figure 3 .

 3 Figure 3. SEM images of solid foams synthesized from 64% PDMS-in-water emulsions: a) and b) 7.3Si(PHIPE)64, c) and d) 3.7Si(PHIPE)64, e) and f) 2.5Si(PHIPE)64. Materials are observed either after drying (a, c and e) or after thermal treatment (b, d and f). The emulsions are labeled X (PHIPE) YZ where X is the amount of particles used to stabilize the emulsion with respect to the total oil weight (mg g 1 ), Y is the oil volume fraction (%) with respect to total sample volume Wiley Copyright 2012.

Figure 4 .

 4 Figure 4. SEM images of sintered solid foams synthesized from hexadecane-in-water emulsions containing an increasing internal phase volume fraction: a) 2.6Si(PHIPE) 64, b) 2.6Si(PHIPE) 75, c) 2.6Si(PHIPE)85 and d) 2.3Si(PHIPE)90The nomenclature is the same as the one adopted for Fig.3. Copyright Wiley-VCH 2012.

Figure 5 .

 5 Figure 5. Overall appearance of the as-synthesized materials when varying the (NH4)2SO4 salt concentration. SEM visualization at different magnifications a-b) SBA-150.1Si(HIPE), c-d) SBA-150.5Si(HIPE), e-f) SBA-151.0Si(HIPE), g-h) SBA-152.0Si(HIPE). Inserts are photos of the materials showing whether they are self-standing or not. Copyright ACS 2018.

Figure 6 .

 6 Figure 6. TEM observations. a, b) SBA-150.1Si(HIPE), c, d) SBA-150.5-Si(HIPE), e, f) SBA-151.0Si(HIPE) and g, h) SBA-152.0Si(HIPE). ACS Copyright 2018.

Acknowledgments

The work was financed by the ANR project n°ANR-15-CE07-0023: « Intensified & Sustainable Enzymatic Acylation Processes on Innovative Macroporous/Mesoporous Materials ».