Louis-Pierre Bergé
email: louis-pierre.berge@irit.fr

Emmanuel Dubois
email: emmanuel.dubois@irit.fr

Minica Houry-Panchetti

Mathieu Raynal

Cédric Sanza

Towards an engineering approach for advanced interaction techniques in 3D environments

Keywords: 3D User Interaction, Design, Analytical Framework, 3D Interaction Techniques, Multimodal Interaction. ACM Classification D.2.2 [Software engineering]: Design Tools and Techniques -User interfaces, H.5.2 [Information interfaces and presentation]: User Interfaces -Theory and methods, User-centered design, I.3.6 [Computer Graphics]: Methodology and Techniques -Interaction techniques

In recent years, Virtual Environments have appeared in new areas such as mass-market, web or mobile situations. In parallel, advanced forms of interactions are emerging such as tactile, mixed, tangible or spatial user interfaces, promoting ease of learning and use. To contribute to the democratization of 3D Virtual Environments (3DVE) and their use by persons who are not experts in 3D and occasional users, simultaneously considering Computer Graphics and Human Computer Interaction design considerations is required. In this position paper, we first provide an overview of a new analytical framework for the design of advanced interaction techniques for 3D Virtual Environment. It consists in identifying links that support the interaction and connect user's tasks to be performed in a 3DVE with the targeted scene graph. We relate our work to existing modeling approaches and discuss about our expectations with regards to the engineering of advanced interaction technique.

INTRODUCTION

With the evolution of technologies, computing capabilities and rendering techniques, the use of 3D Virtual Environments (3DVE) is becoming popular. 3DVE are no longer restricted to industrial uses and they are now available to the mass-market in various situations: for leisure in video games, to explore a city in Google Earth or in public displays [START_REF] Tan | Physically large displays improve performance on spatial tasks[END_REF], to design house furniture [START_REF] Lee | Tangible authoring of 3D virtual scenes in dynamic augmented reality environment[END_REF] or to explore cultural heritage sites in a museum [START_REF] Christou | A versatile large-scale multimodal VR system for cultural heritage visualization[END_REF]. However, in these mass market contexts, the user's attention must be focused on the content of the message and not distracted by any difficulties caused by the use of a complex or inappropriate interaction technique. This is especially true in a museum where the maximization of the knowledge transfer is the primary goal of an interactive 3D experience. Common devices, such as keyboard and mouse [START_REF] Pecchioli | ISEE: Information access through the navigation of a 3D interactive environment[END_REF] or joystick [START_REF] Wischgoll | An explorational exhibit of a pig's heart[END_REF] are therefore widely used in museums. To increase the immersion of the user, solutions combining multiple screens or cave-like devices [START_REF] Christou | A versatile large-scale multimodal VR system for cultural heritage visualization[END_REF] also exist. However, these solutions are cumbersome and expensive.

Meanwhile, the Human-Computer Interaction (HCI) research domain is rapidly evolving and growing in complexity with new advanced forms of interaction such as mobile [START_REF] Lee | 3D interaction using mobile device on 3D environments with large screen[END_REF], ambient computing [START_REF] Ishii | ambientROOM: integrating ambient media with architectural space[END_REF], spatial interfaces [START_REF] Laviola | 3D spatial interaction[END_REF] and tangible user interface [START_REF] Williams | TZee: exploiting the lighting properties of multi-touch tabletops for tangible 3d interactions[END_REF]. A common feature to these advanced forms of interaction is the attempt to involve and combine the use of multiple objects and entities taken in the physical and digital environments: interactive solutions are smoothly integrated in the user's activity and have been proved to be easier to apprehend by newcomers [START_REF] Shaer | Tangible User Interfaces: Past, Present, and Future Directions[END_REF]. Successful uses of such advanced interaction have been recently demonstrated in mass-market applications involving 3DVE for museums [START_REF] Hornecker | Interactions around a contextually embedded system[END_REF] [START_REF] Huang | Tangible photorealistic virtual museum[END_REF].

It thus appears that 3D interactive applications are more and more widespread, from professional context to public spaces and from expert users to very occasional users. In addition, advanced forms of interaction techniques offer new potentials such as being based on personal belongings (devices or artefacts), integrated in the physical environment, easy to apprehend. But developing 3D interactive applications on one hand and advanced interaction techniques on the other hand are two preoccupations that are mostly considered through separated approaches, leading to compartmentalized progresses. There is therefore a need for understanding and supporting the engineering of advanced interaction techniques for exploring and taking advantage of 3DVE.

In this position paper, we first provide an overview on a new analytical framework for helping and guiding the design of advanced interaction techniques for 3DVE. We motivate and illustrate the choice of its grounding elements and then discuss a number of existing modeling approaches potentially useful to complement or refine this framework. We finally discuss our expectations from the workshop, with regard to the proposed framework and more widely with regards to the engineering of advanced interaction technique.

OVERVIEW OF A NEW ANALYTICAL FRAMEWORK

We first present the two pillars of our modeling approach, tasks analysis and interactive scene graph. We then introduce the notion of links between these two pillars.

Tasks tree

The first pillar of the analytical framework is the result of a task analysis: a task tree. Task analysis (Figure 1 -left) consists in a decomposition of user's goal into tasks and sub-tasks which must be achieved to reach the user's goal. Task analysis is widely accepted in the HCI community as a starting point to the design of interactive techniques because it thus provides an algorithm of the user's activity, the logic and the dynamic of tasks accomplishment. But a task analysis does not express how the task is concretely performed with the system: no information related to the interaction technique is provided. Task analysis is particularly useful to understand and structure the user's activity, define functional specifications, identify data requirements, etc. [START_REF] Annett | Hierarchical Task Analysis[END_REF].

Different formalisms exist to represent the result of a task analysis. We choose to rely on the familiar Hierarchical Task Analysis (HTA) formalism [START_REF] Annett | Hierarchical Task Analysis[END_REF]. With HTA, the decomposition of the task is presented as a hierarchical tree of tasks and sub-tasks, enriched with some attributes (iteration, optional, parallel, etc.). Choosing a light approach ensures that non-experts are able to use it as a support for designing advanced multimodal interaction for 3DVE.

When used to describe tasks with an interactive 3DVE, most of the identified tasks are specific to the application domain and the leaves of the tasks tree are close to Bowman's tasks [START_REF] Bowman | D User Interfaces: Theory and Practice[END_REF] (navigation, selection, manipulation, system control). But none of the 3D elements impacted by the tasks can be specified. This is where the interactive scene graph comes to action.

Interactive scene graph

The second pillar of the analytical framework is the interactive scene graph which provides a structured description of the 3DVE to be used. This scene graph is therefore specific to each 3DVE. A 3DVE generally consists of 3D objects (meshes, widgets or basic geometrical elements such as cone, cube, cylinder, etc.), lights and virtual cameras. At a finer grain 3D objects are described as a set of vertices (geometry), faces and edges (topology). Manipulations of 3D objects (translations, rotations or scaling) must therefore take into consideration the underlying topology and geometry. To assist this process, the concept of scene graph [START_REF] Strauss | An object-oriented 3D graphics toolkit[END_REF] has been developed to organize the 3D elements and provide for developers a structure for the assembly of a 3D scene.

Scene graph is a widely accepted method used in the Computer Graphic (CG) community to describe the essential components of a 3DVE. Scene graphs are also relevant to our context because we need to understand and take into account the structure of the 3D scene to design the interaction with it. But, we are not interested in elements related to the implementation of the scene graph by the 3D API in charge of the rendering. We are also not interested in the way the scene graph may impact the use of 3D engines for solving issues like texture management or collision.

However, with the scene graph description, only geometric and topologic aspects are expressed. It is not clearly identified which parts of which components of the 3D scene are likely to be impacted by user's interaction. To this end, we propose to define the "interactive scene graph" (Figure 1 -right). Its aim is to highlight and characterize handled and not-handled objects, i.e. objects impacted or not by one of the user's tasks identified during the tasks analysis.

The definition of the "interactive scene graph" is based on the most relevant features used to support 3DVE user's interaction in 3D engines like Unity 3D [START_REF]Unity 3D -Game Engine[END_REF], Irrlicht [START_REF]Irrlicht Engine -A free open source 3D engine[END_REF]. It is also derived from the standard description language X3D [START_REF]Web3D Consortium : X3D for Developers[END_REF]. X3D supports the description of animated 3D scenes: behaviors among 3D nodes are expressed in script nodes or simple links among 3D nodes. The role of the "interactive scene graph" goes beyond the description of animations in the 3DVE: it emphasizes which are the elements and attributes of the 3D scene with which external elements may interact. Based on these existing approaches, we distinguish two types of handled objects: components and renderers.

Components are 3D objects composing the 3DVE (mesh, geometrical element, widgets). User's interaction may impact Components through the modification of two classes of attributes: state and manipulation. State attributes refer to the color, the texture or the visibility (display or not in the 3DVE) of the object. Manipulation attributes are more complex. Three levels of manipulation attributes coexist: the object, its faces and its points. First, one may modify position, orientation or scale of an object as a whole. Second, an object is made of a set of faces: depending on the object structure, modifying a face can be limited in terms of degrees of freedom in orientations and scales. Third and finest level, a face of an object is made of a set of points: at that level one can only acts on the 3D position of each point. Faces and points levels are thus useful to refine and characterize the deformation of a 3D object.

Renderers are objects such as lights or camera, taking part in the rendering of the 3D scene. User's interaction may impact Renderers through the modification of two classes of attributes: state and manipulation. States include attribute such as enabled/disabled and color. Manipulation attributes correspond to position and orientation, with orientation depicting the definition of the point of view of the camera. The next step consists in identifying existing links between elements of the 3DVE and user's sub-tasks of the task tree that are affecting them.

Linking user's task with elements of the 3D scene

It consists in identifying for a set of sub-tasks of the task tree, the attribute(s) of the interactive scene graph affected by the realization of each of these specific sub-tasks (Figure 1 -middle). As a result pairs of user's sub-task and attributes of the 3DVE are clearly highlighted. This set of links provides a complete view on the user's interaction that will be performed in the 3DVE to perform the user's activities required to reach his/her goal. Each link depicts the use of one interaction modality [START_REF] Nigay | A design space for multimodal systems: concurrent processing and data fusion[END_REF]: on the one hand each link may involve a different interaction modality, on the other hand every links is using the same and unique interaction modality. In addition, we anticipate that the operators (sequence, alternative, etc.) present in the task tree at higher-levels will influence the design of the links.

Highlighting the pairs of user's sub-task and attributes of the interactive scene graph therefore constitutes the description of the overall user's interaction with the 3DVE. Furthermore, it establishes a link between user's activities (task analysis) and the 3DVE content and behavior (the interactive scene graph). Therefore, this description constitutes a support to reason about the design of the overall system and takes advantage of HCI and CG specificities. It overcomes the description of one specific task or one specific technique out of the context of use and manipulation of the 3DVE. In the next section, we illustrate the overview of our analytical framework with an example of the literature.

ILLUSTRATIVE EXAMPLE

Portico [START_REF] Avrahami | Portico: tangible interaction on and around a tablet[END_REF] is an interactive system for enabling tangible interaction on and around tablet computers. We focus our illustration on one of their application example named penalty shootout. Let us describe the interactive setting with the use of our analytical framework. The aim of the user is to shoot a goal (Figure 2 -Atask T0). For this, the user has to watch the goal and goalie (T1), place the ball (T2) and shoot the ball (T3). The system does not support multiple balls thereby there is a unique ball available for the user. Now, the interactive scene graph (Figure 2 -B) is composed by a non-handled camera (renderer), a handled soccer ball (component) and some non-handled objects like the goal, the goalie and the soccer field. Finally, the designer has adopted a tangible interaction to establish the link between the task tree and the scene graph (Figure 2 -C). Concretely, the soccer ball is a physical object manipulated by the user and thus, the sole task (L1) impacting the interactive scene graph is the task T3. The link L1 of this example connects the task T3 to the manipulation attribute position of the 3D ball in the scene graph. The task T2 does not impact the virtual ball position because the virtual ball position change only when the physical ball touch the tablet.

From this description, we can identify that the interactive situation supports the manipulation task in a 3DVE through the link L1 and an implicit selection task through the manipulation of the physical ball. The remaining Bowman's 3D tasks (navigation, system control) are not supported by this interactive technique. The description of the link L1 also highlights a direct connection between a physical object and a virtual 3D element (the ball): physical behavior and representation are directly mapped to the behavior of the corresponding digital object. Although it is a rather simple example, it shows that the framework can help in visualizing when the user's focus has to be on the 3D scene with regards to the user's task realization. If several links are present it may also help identify inappropriate sequence of interaction, such as switching modalities while focusing on the same 3D parts.

To better structure this kind of reasoning, designing the link L1 is subject to a set of design aspects that we extracted from the literature and summarize in the following section.

DESIGN ASPECTS OF THE INTERACTION WITH 3D

The HCI and CG communities have already been working on the design and implementation of advanced interaction techniques with 3D. Different points of view have been adopted thus revealing multiple design aspects. We summarize these considerations along the three parts of our analytical framework: user's interaction, 3D system and links between them.

Regarding the user's interaction, relevant considerations include the specification of the users expertise with the manipulation of the application as defined in Rasmussen work [START_REF] Rasmussen | Information Processing and Human-Machine Interaction: An Approach to Cognitive Engineering[END_REF]. The definition of the application type (AR, VR, desktop application) is also considered as crucial in the 3-DIC model [START_REF] Figueroa | A Conceptual Model and Specification Language for Mixed Reality Interface Components[END_REF]. Given the use of advanced interaction technique, it is also required to specify which parts of the physical world are involved. It includes a description of the objects used and their constraints [START_REF] Ullmer | Token+constraint systems for tangible interaction with digital information[END_REF]. It also requires to specify how information are transferred from the user to the 3D system and vice-versa, as partly addressed in the ASUR [START_REF] Dubois | ASUR++: A Design Notation for Mobile Mixed Systems[END_REF] and MIM [START_REF] Coutrix | Mixed Reality: A model of Mixed Interaction[END_REF] models. It goes even to more precise description of the gestures and characterization of movements that will be performed by the users [START_REF] Manches | Physical manipulation: evaluating the potential for tangible designs[END_REF] as well as perceptual properties of interest such as visual, tactile and auditory properties.

Regarding the 3D system, as already mentioned X3D [START_REF]Web3D Consortium : X3D for Developers[END_REF] is the standard to describe the 3D nodes, structure and internal animation or behavior. Most of works however focus on the description of virtual reality interaction techniques such as ray casting and are focusing on their behavior or implementation. Among them the reusable library of 3D interaction technique [START_REF] Figueroa | A reusable library of 3D interaction techniques[END_REF], the Petri Net model [START_REF] Ying | An approach to Petri net based formal modeling of user interactions from X3D content[END_REF], ontology model [START_REF] Latoschik | A user interface framework for multimodal VR interactions[END_REF], the 3-DIC model [START_REF] Figueroa | A Conceptual Model and Specification Language for Mixed Reality Interface Components[END_REF] or IFFI [START_REF] Ray | Towards a system for reusable 3D interaction techniques[END_REF] and Viargo [START_REF] Valkov | Viargo -A generic virtual reality interaction library[END_REF] library are complementary alternatives.

Finally, in the literature, the link between our two pillars is often limited to the analysis of the required input device. Simple taxonomies offer an overview of the possibilities such as the Mackinlay taxonomy [START_REF] Mackinlay | A Semantic Analysis of the Design Space of Input Devices[END_REF]. More elaborated models like RVDT [START_REF] Ajaj | RVDT: a design space for multiple input devices, multipleviews and multiple display surfaces combination[END_REF] or InTml [START_REF] Figueroa | A reusable library of 3D interaction techniques[END_REF] deal with a particular aspect of input device, e.g. the type of data (float, integer, boolean) and the number of sensed DOF.

DISCUSSION

Obviously, developing advanced interaction techniques for 3DVE requires to confront multiple design considerations and to pay attention to both communities' preoccupations.

To do so, offering a structured and refined set of design attributes that reconcile these multiple aspects will lead to a better understanding of the links between a user's goal and attribute of a 3D scene. For example metrics might be extracted to efficiently compare techniques; properties might be defined to clearly express how design choices in the user's part impact design choices in the 3D system parts and conversely.

We believe that providing such a structured approach to describe the links between task and scene graph is a fruitful way to help reason about the design and implementation of advanced interaction techniques for 3DVE. The resulting model or notation will constitute a pseudo-formal description language of interaction techniques for 3DVE.

From such description, a semi-automatic implementation of the described advanced interaction for 3DVE could then be built in a platform for rapid development of multimodal interaction such as the Dynamo framework [START_REF] Avouac | Autonomic management of multimodal interaction: DynaMo in action[END_REF].

CONCLUSION

Applying advanced forms of interaction to 3D applications is required to contribute to a more effective use of 3D interactive environment. As multiple types of user and context are potentially targeted a user centered approach to the design of interactive 3D application is particularly expected.

In this paper we proposed a way to narrow two communities by involving well established design resource of each domain as the two pillars of a dedicated approach. We then identified a set of existing design and implementation supports for bridging the gap between these two pillars.

During the workshop, we hope to find the opportunity to further illustrate the use of this framework on different prototypes we have implemented in our lab. We then expect a fruitful discussion with the other participants of the workshop to identify additional existing design approaches relevant to this context of interaction with 3DVE, or relevant metrics, properties or considerations. In particular, we are interested to discuss what could be the ways to tightly anchor 3D specificities in the design of interaction technique. We are also interested in refining the links between our two pillars with relevant approaches. Finally, we hope to hear about similar approaches in different contexts, i.e. a context in which advanced HCI and another domain are involved and in which engineering supports of the two communities have been brought together. From such situation we expect to hear about lessons learnt, benefits and limits of such approaches.

Figure 1 :

 1 Figure 1: Overview of our analytical framework

Figure 2 :

 2 Figure 2: Task tree (A), scene graph (B) and link (C) of the Penalty Shootout interaction technique with Portico [4]


Jules Valente

Frédéric Gruy

Patrice Nortier

Emmanuelle Allain

Evidence of structural reorganization during aggregation of silica nanoparticles

Keywords: light scattering, aggregates, in situ turbidimetry, restructuring

Silica nanoparticles have been produced by neutralization of sodium silicate. The obtained suspension was unstable because of the high value of the ionic strength of the medium. Aggregation of the particles has been monitored thanks to a combination of in situ turbidity and dynamic light scattering (DLS) measurements. An optical model has been developed to extract both fractal dimension D f and primary particle radius r pp of the formed aggregates from these data. The obtained results clearly indicate a densification of the clusters as aggregation proceeds. A correlation between our experimental turbidity values and hydrodynamic radii was found. Comparison with calculated dimensionless numbers showed that a constant Df could not explain the observed trend. A fractal dimension dependent on the number of particles inside the aggregate N pp is thus suggested.

Introduction

Colloidal systems are currently used in many fields of application and the potential of such materials is still to be fully explored. As far as silica is concerned, fields of application include controlled abrasion, elastomer reinforcement, whiteness improvement, coating,… [START_REF]Colloidal silica : fundamentals and applications[END_REF] Very lately, colloidal suspensions of silica nanoparticles have been highlighted as potential tools for organ repair [START_REF] Meddahi-Pellé | Organ Repair, Hemostasis, and In Vivo Bonding of Medical Devices by Aqueous Solutions of Nanoparticles[END_REF]. When silica is produced from TEOS (tetraethylorthosilicate) using Stöber's method [START_REF] Stöber | Controlled Growth of Monodisperse Silica Spheres in Micron Size Range[END_REF], monodisperse isolated particles are obtained. However, when silica is industrially manufactured by neutralization of a silicate solution with acid (precipitated silica) or by hydrolysis of silicon tetrachloride SiCl 4 in vapour phase (pyrogenic silica), the resulting powder is made of strong aggregates. These aggregates pile to form large agglomerates. The presence of such structural organization is a prerequisite for certain of the previously mentioned applications. Typically, it has been shown that the hierarchical structures of precipitated silica and carbon black are very similar [START_REF] Schaefer | Multilevel structure of reinforcing silica and carbon[END_REF], which explains why the former has progressively replaced the latter as a reinforcing filler in rubbers over the years. However, no consensus exists regarding the morphological properties of the ideal filler [START_REF] Schaefer | Morphology of Highly Dispersing Precipitated Silica: Impact of Drying and Sonication[END_REF]. Investigations are thus still conducted to gain control over the morphology of the silica aggregates [START_REF] Schilde | A novel way to vary the structure of precipitated silica and calcium carbonate aggregates in a wide range by using grinding media during the precipitation process[END_REF][7] [START_REF] Wijnen | Diffusion-Limited and Reaction-Limited Aggregation of Aqueous Silicate Solutions[END_REF][9] [START_REF] Baldyga | Precipitation of amorphous colloidal silica from aqueous solutions-Aggregation problem[END_REF].

As with many colloids, the aggregates formed by coagulation of silica particles are known to be fractal [START_REF] Lin | Universality in colloid aggregation[END_REF]. This model of aggregate structure [START_REF] Meakin | Fractal aggregates[END_REF] is based on the existence of a parameter called the fractal dimension D f , which varies between 1 and 3 according to the density of the aggregate. Usual values of D f are associated with the mechanism of aggregation: 1.8 when aggregation is fast and limited by diffusion of the particles in the suspension and 2.1 when aggregation is slow and limited by the low collision efficiency of the particles. In the case of silica, fractal aggregates can sometimes undergo some restructuring. Aubert and Cannel [START_REF] Cannell | Restructuring of Colloidal Silica Aggregates[END_REF] could form aggregates with fractal dimensions of either 1.75 or 2.05 depending on the conditions of the experiment. The authors also observed that, given enough time, D f would naturally increase from 1.75 to 2.05 and that the speed of restructuring was a function of pH.

The origin of such restructuring is still unclear. A possible explanation would be that the aggregated particles are in a shallow secondary potential minimum [START_REF] Chinchalikar | Evolution of structure and interaction during aggregation of silica nanoparticles in aqueous electrolyte solution[END_REF][15] [START_REF] Johnsson | Combined Electrospray-Scanning Mobility Particle Sizer (ES-SMPS) and Time-Resolved Synchrotron Radiation-Small-Angle Xray Scattering (SR-SAXS) Investigation of Colloidal Silica Aggregation. Part II. Influence of Aggregation Initiator on Gel Stability[END_REF]. Aggregation would then be reversible: Martin et al. triggered aggregation of silica nanoparticles by addition of salts to their suspension and dispersion of the formed aggregates by dilution [START_REF] Martin | Fast Aggregation of Colloidal Silica[END_REF]. The reversibility of aggregation would allow the primary particles to diffuse inside the aggregates and form denser structures over time. Schlomach and Kind, out of their simulations [START_REF] Schlomach | Theoretical study of the reorganization of fractal aggregates by diffusion[END_REF], suggested an Arrhenius-type mechanism for intra-aggregate diffusion. Another theory is based on the nature of the bonds between silica particles: at first, two particles are linked by inter-particle hydroxo bridges between silanols groups [START_REF] Depasse | Coagulation of colloidal silica by alkaline cations Surface dehydration or interparticle bridging[END_REF]. These bonds are easily broken and flexible which gives mobility to the particles inside the aggregate. The hydroxo bridges later evolve into covalent bonds, thus making the aggregation irreversible.

Turbidity is one of the widest-spread techniques to monitor the aggregation of a suspension [START_REF] Gregory | Monitoring particle aggregation processes[END_REF]. It has, unsurprisingly, been applied to study aggregation of silica [START_REF] Trompette | Ion-specific effect on the gelation kinetics of concentrated colloidal silica suspensions[END_REF] [START_REF] Gruy | Formation of small silica aggregates by turbulent aggregation[END_REF]. In this paper, turbidity is used not only to monitor the aggregation of destabilized nanoparticles but also to get structural information about the formed aggregates. An optical model, based on scattering of fractal aggregates, has been built to extract both the fractal dimension and the primary particle size of the aggregates from the experimental turbidity spectra. Evolution of the calculated fractal dimension with time would evidence structural rearrangement of our aggregates. Section 2 of this paper deals with the method of preparation of our silica nanoparticles suspensions and with the characterization techniques employed during this work. The optical model and the associated fitting software we developed are introduced in section 3. In section 4, some of our experimental results are presented and analyzed thanks to our program. Finally, in section 5, the correlation observed between our experimental data is confronted to theoretical considerations.

Material and methods

Experimental set-up

Suspensions of silica nanoparticles have been prepared by neutralization of a SiO2 sodium silicate solution (3.55 SiO 2 :Na 2 O ratio) with 8 wt% sulfuric acid. These solutions have been prepared by dilution of commercially available solutions of 19 wt% SiO2 sodium silicate and 72 wt% sulfuric acid (Chimie Plus) with deionized water. The mixing of reactants has been performed in a 3 L stirred tank reactor. The temperature has been maintained to 85°C during the whole experiment.

The first suspension prepared according to this protocol presented, at the end of the neutralization, a final volume of 1.3 L and a volume fraction of solid Φ SiO2 of 7.5x10 -3 . The pH and the ionic strength in solution were respectively 7.5 and 0.23 M (The ionic strength of the solution has been calculated from the total concentrations of sodium and sulfate ions.). The second one presented a final volume of 1.6 L and a volume fraction of solid Φ SiO2 of 1.22x10 -2 . In this case, the pH and the ionic strength in solution were respectively 8.5 and 0.36 M. Once the mixing was over, aggregation of the particles was monitored by a combination of in situ turbidity and DLS measurements until the system gelled. The pH of the medium did not deviate from its original value during the time of the experiment.

Turbidity measurements

The turbidity measurements were carried out with a set-up comprising three elements: a light source, a turbidity probe and a commercial spectrometer (Fig. 1). A halogen-deuterium lamp (L7893, Hamamatsu) provides a polychromatic signal in the working range of 350-900nm, which is passed through an optical fibre (600 µm in diameter, T58-457, Edmund Optics) to the probe. The probe has been designed according to Crawley's recommendations [START_REF] Crawley | Size analysis of fine particle suspensions by spectral turbidimetry: potential and limits[END_REF]: two plano-convex spherical lenses (LPX-5.0-5.2-C, Melles-Griot) insure the parallelism of the light beam across the probe cavity. The length of this cavity sets the optical path length L opt , which is used in the determination of experimental turbidity. The transmitted signal is passed through a second optical fibre to the spectrometer (USB Red Tide USB650, Ocean Optics). Its wavelength detection range extends from 200 to 1100 nm, with an optical resolution of 2 nm. Turbidity τ is then calculated according to Eq. 1. I 0 corresponds to the intensity of the incident beam and I to the intensity of the transmitted beam. This calculation is carried out for every available wavelength λ. The reference spectrum I 0 (λ) has been recorded before addition of the sulfuric acid to the silicate and thus, before formation of any solid

Dynamic Light Scattering (DLS) measurements

Particle size distributions before the gelation point were determined by DLS measurements with a Zetasizer Nano ZS from Malvern Instruments. This appliance is equipped with a He-Ne laser (λ = 633 nm) and a backscatter detector at a fixed angle of 173°. 5 mL samples were regularly taken from the suspension. In order to keep them stable until their analysis, each sample was poured in 20 mL of deionized water at room temperature. Stabilization came from both dilution and cooling of the sample.

Besides, pH of the sample was immediately adjusted to 2, where aggregation of silica is the slowest according to literature [START_REF] Iler | The Chemistry of silica solubility, polymerization, colloid and surface properties, and biochemistry[END_REF], by addition of a small quantity of sulfuric acid.

In this work, only z-average radii are presented. These sizes are the most reliable data given by the Zetasizer Nano ZS as they are directly determined by cumulants analysis of the correlation function.

Z-average sizes are therefore intensity averages. Our samples are monomodal with reasonable polydispersity (the polydispersity index PDI determined by cumulants analysis is inferior to 0.3 for all samples) and thus, suited to the use of z-average sizes.

Powder characterization

After the gelation point, precipitated silica has been collected by centrifugation (5000 rpm, 40 minutes). The obtained paste was redispersed into deionized water and submitted to another 40 minutes of centrifugation in order to purify the sample of residual sodium and sulfate ions. The operation was repeated three times before the silica was dried overnight at 100°C. The samples were out-gassed at 160°C overnight before N 2 physisorption analysis (Micromeritics ASAP 2020). Specific surface areas, determined using the Brunauer-Emmet-Teller (BET) method, were respectively 329.4 m²/g and 282.3 m²/g for powders obtained after aggregation at pH 7.5 and 8.5.

The equivalent particle diameters are then 8.3 and 9.7 nm. Mesoporosity of the samples was analyzed using the Barrett-Joyner-Halenda (BJH) method. The resulting average pore widths are respectively 8.9 and 10.5 nm.

Optical model

Scattering theory

The purpose of working with turbidity at several wavelengths was to be able to extract morphological parameters from the experimental spectra. In the case of silica nanoparticles suspended in water, optical contrast is low: the ratio m of refractive indices for amorphous silica (n SiO2 =1.44) and water (n water =1.33) is close to 1. Particles are small compared to the wavelengths of the incident beam. As a result, the Rayleigh-Debye-Gans (RDG) approximation of the Mie theory can be used to calculate the optical properties of the growing aggregates. Turbidity of a suspension of aggregates with N pp primary particles of radius r pp is given by Eq. 2. In the frame of this work, the assumption will be made that all the aggregates in the suspension are identical.

sca pp SiO C r N * 4 3 * 3 pp 2 π φ τ = Eq. 2
C sca is the scattering cross section of an aggregate. In the frame of the RDG approximation, each primary particle inside the aggregate independently scatters the light. In a given direction, the waves scattered by all these particles superpose and generate interferences without multiple scattering. The scattering cross section of the aggregate (averaged over all the possible orientations) is determined by these interferences and obeys Eq. 3.

∫ = π θ θ θ 0 2 pp ) ( * ) ( d F S N C sca
Eq. 3 θ is the scattering angle, F(θ) is the shape factor, dependent on the primary particle morphology, and S(θ) is a structure factor accounting for the spatial arrangement of the particle inside the aggregates.

Expressions of both factors are given in Eq. 4 and in Eq. 5. In these equations, x is the product of the wavenumber k and the primary particle radius r pp , with k=(2π*n water )/λ.

( ) θ θ θ π θ sin * cos 1 * 2 sin 2 * ) 1 ( 9 4 * ) ( 2 2 4 2 2 pp +                       - = x f x m r F Eq. 4 with ) cos (sin 3 ) ( 3 x x x x x f - = for spherical primary particles dr r D R r k R r k N N N S N p ) ( * 2 sin * * * 2 2 sin * * * 2 sin * ) 1 ( ) ( 2 0 max max pp pp pp 2 pp ∫                     - + = θ θ θ Eq. 5
where r is a dimensionless distance, R max is the radius of the circumscribed sphere on the aggregate and D p (r) is a pair distribution density corresponding to the arrangement of the primary particles inside the aggregate. The pair distribution density is normalized. Its expression depends on the value of the dimensionless distance r. In the case of aggregates with few primary particles, the self-similarity of fractal aggregates is not strictly ensured. Only the relation between the radial number density ρ and the radial coordinate is respected:

∫ = 2 0 ) ( ' ) ( ' ) ( dr r D r D r D p p p • r < 1 ∫ ∫ - - + = r r p dx x f dx x f r D 1 0 2 1 1 1 ) ( ) ( ) ( ' • 1 ≤ r ≤ 2 dx x f r D r p ∫ - = 1 1 1 ) ( ) ( ' With ( )       - + - = + + 2 1 2 2 1 1 * 2 1 * * ) ( b b xr x r x r x f and ( ) ( )       - + - + + = + + + 2 1 2 2 2 1 2 2 1 2 * 2 * 2 * * ) ( b b b xr x r xr x r x r x f

Fitting program

As the aggregates studied in this work are fractal, their radius and number of primary particles are linked by Eq. 6. R is the radius of the aggregate and k f is the fractal prefactor. This factor is still the topic of many investigations and no unique theory exists concerning its value. Some authors express k f as an explicit function of D f [START_REF] Gmachowski | Calculation of the fractal dimension of aggregates[END_REF] whereas some others give it a specific value depending on the considered aggregation mechanism [START_REF] Brasil | Numerical characterization of the morphology of aggregated particles[END_REF]. Lately, it has even been suggested that k f and D f are not able to completely describe a fractal aggregate and that an extra morphological parameter must be used [START_REF] Heinson | A three parameter description of the structure of diffusion limited cluster fractal aggregates[END_REF].

f D f r R k N         = pp pp Eq. 6
The nature of the radius used in Eq. 6 also has a significant impact on k f . The result of a DLS analysis is a hydrodynamic radius R hyd . R hyd corresponds to the radius of the solid sphere with the same diffusion constant as the aggregate. Kaetzel et al. introduced a hydrodynamic dimension of translation D f h,t and a fractal prefactor of translation k f h,t [START_REF] Kaetzel | Dynamic light scattering for the characterization of polydisperse fractal systems: I. Simulation of the diffusional Behavior[END_REF], which both depend on the fractal dimension D f .

Their dependence on D f are given in Eq. 7 and Eq. 8. D f h,t and k f h,t are to be used in Eq. 6 when the considered R is R hyd.

, f f f t h f D D D k - - - = Eq. 8
However, in the calculation of C sca , the value of R max is needed. From the study of Ehrl et al. on simulated fractal aggregates, a relation between the fractal prefactor k f and D f could be inferred, when R max is the considered radius [START_REF] Ehrl | Generation and Geometrical Analysis of Dense Clusters with Variable Fractal Dimension[END_REF]. Eq. 9 is different from the equation [START_REF] Skvarla | Quantitative Interpretation of Anomalous Coagulation Behavior of Colloidal Silica Using a Swellable Polyelectrolyte Gel Model of Electrical Double Layer[END_REF] in their paper because the authors used a gyration radius, which is another possible descriptor for a fractal aggregate.

2 08 . 2 2 * * 46 . 4 f D f f f f D D D k         + = - Eq. 9
A MATLAB program has been designed to deduce both r pp and D f from experimental turbidity spectra. Direct calculation of these parameters from the experimental data would involve the inversion of Eq. 3, which is a complex task. Instead, the program is able to simulate turbidity spectra for given values of r pp and D f . When both are fixed, N pp can be inferred from the experimentally determined R hyd (Eq. 6). Calculation of C sca and τ is then straightforward as Φ SiO2 is known. The couple (r pp , D f ) is numerically adjusted so that the simulated and experimental spectra match at every wavelength. The optimized r pp and D f are the morphological parameters of the aggregate at the time the turbidity spectrum was acquired.

Results

As similar results were obtained with the two suspensions (see the discussion section), the whole approach is detailed only for the data obtained with aggregation at pH 7.5.

Monitoring of aggregation

The time t=0 min corresponds to the end of the mixing of the reactants. Initially, the turbidity value is very low: the silica nanoparticles in the suspension are small and do not scatter much. But over time, the turbidity steadily increases because of the aggregation of the particles (Fig. 2).The turbidity values are higher for shorter wavelengths, which is to be expected in the frame of the RDG approximation. The evolution of turbidity with time is pseudo-linear as was observed by Trompette and Meireles during their study realized with commercial silica suspensions [START_REF] Trompette | Ion-specific effect on the gelation kinetics of concentrated colloidal silica suspensions[END_REF]. In the final minutes of the experiment, the linearity is lost. As gelation is close, the optical properties of the suspension change greatly and may not be accurately described by our optical model. As a result, turbidity values after 120 minutes of experiment are not analyzed with our program.

Aggregation of the particles is also evidenced by the DLS measurements. Initially, the hydrodynamic radius is close to 10 nm. As the radius particle calculated from the specific surface area of the final powder is 4.15 nm, this result would suggest that the particles are already slightly aggregated. The evolution of the hydrodynamic radius with time is also linear. As silica particles are known to generate strong short-range repulsive forces [START_REF] Guleryuz | AFM measurements of forces between silica surfaces[END_REF][31], a reaction-limited aggregation regime could have been expected. In such case, an exponential evolution of the hydrodynamic radius should theoretically be observed [START_REF] Runkana | Reaction-limited aggregation in presence of short-range structural forces[END_REF]. A decrease in the measured hydrodynamic radius occurred between t=120 min and t=140 min. The formed aggregates have become too large to be properly characterized by DLS. This confirms that our fitting program should not be used with data acquired after 120 minutes of experiment.

Extraction of morphological parameters

Two fitting parameters

The fitting program was used to extract morphological parameters from the experimental spectra presented in Fig. 4. Upper and lower limits were set for both r pp and D f : the calculated values of r pp had to be between 3 and 10nm, calculated values of D f between 1 and 3.In order to save computational time, the analysis was carried out with spectra consisting of 32 experimental values at selected wavelengths in the range 350-900 nm. The results are presented in Table 1.

The relevance of the returned parameters is questionable. At long times (t ≥ 36 min), r pp increases from 3 nm to 10 nm. As no growth of the particles in the suspension is expected, r pp should not vary with time. On the other hand, at short times (0 < t < 36 min), the returned radii are fairly constant and close enough to the 4.15 nm expected from the BET analysis. Calculated fractal dimensions are rather low compared to the values of the literature. 1.8 is usually the lower limit of experimentally observed fractal dimensions [START_REF] Wu | Dependence of fractal dimension of DLCA clusters on size of primary particles[END_REF]. However, the concept of fractal aggregates has been developed for clusters of several thousands of primary particles. At short times, the characterized aggregates are small and may thus not be described accurately by the fractal theory. The evolution of the calculated N pp with time also evidences the lack of physical relevance of these first results. Extracting both r pp and D f from the experimental turbidity spectra appears as a too challenging task for our fitting program.

One fitting parameter

As no growth of the primary particles is expected during the time of the experiment, new series of simulations in which only D f is a fitting parameter and r pp is set to a constant value have been performed. This has been done for r pp =4.2 nm, the value derived from the BET analysis, and for r pp =5.0 nm, which is the average of the fitted radii between t=5 min and t=22 min. The results of these simulations are reported in Table 2.

The results obtained with both values of r pp follow the same trend: D f is fairly constant from t=5 min to t=22 min and steadily increases at longer times. D f is lower at short times for the greater r pp value but its final value is rather independent from the chosen r pp . In both cases, D f is around 2.2 which is the theoretical fractal dimension for reaction-limited aggregation. The evolution of D f with time evidences a restructuring of the aggregates. Besides, evolution of the calculated N pp with time is well approximated by an exponential function, which is compatible with reaction-limited aggregation (Fig. 5). The values of N pp are systematically higher for the smaller r pp .

Discussion

Turbidity and hydrodynamic radius were both found to follow a linear evolution over the aggregation time. Despite significant differences in the conditions of operation, a similar behavior was observed at pH 8.5 (Fig. 6). This suggests a certain universality of the restructuring evidenced thanks to our fitting program. However, direct comparison of the two experiments is difficult as the kinetics of aggregation is different in each case.

In order to mitigate these differences and make comparison easier, we calculated dimensionless turbidities τ adim : the experimental value was divided by the turbidity the suspension would have if all the primary particles were isolated, ie N pp =1 (Eq. 10). The resulting value is thus only dependent on the spatial arrangement of the primary particles inside the aggregate in the RDG framework. In the same fashion, the experimental hydrodynamic radius was divided by the radius of the primary particle, which was fixed at 5 nm. According to Eq. 6, this dimensionless radius R adim is a function of N pp and D f only. The resulting plot is presented in Fig. 7. A linear relation between the two kinds of dimensionless numbers is evidenced, with nearly identical slopes for both sets of data.

( )

( ) ( ) 1 * 1 * 4 3 * 3 XP adim 2 = = = = pp sca pp pp sca pp sca pp SiO N C N N C N C r π φ τ τ Eq. 10
In order to check whether such correlation is specific to our system or can be found in every suspension of aggregates with fractal properties, we calculated theoretical τ adim and R adim at different values of D f for N pp varying between 1 and 500. For these calculations, r pp was once again fixed at 5 nm. The results are plotted on Fig. 8. For constant D f , the evolution of τ adim significantly differs from what could be observed with our experimental data. In particular, calculated τ adim systematically levels off for large R adim values with the plateau being reached sooner for smaller D f . However, when using a fractal dimension dependent on N pp , the results significantly improved. A good linearity was found up to R adim =12. Simulated and experimental values do not superimpose perfectly but the evolutions are quite similar. The relation used (Eq. 11) is probably not the best that could be found but it reproduces reasonably well the evolution of fitted D f over time obtained for r pp =4.2 nm, using the calculated N pp in Table 2 (Fig. 9). The idea of a fractal dimension increasing with the number of primary particles inside the aggregate is in agreement with the theoretical considerations proposed by Gmachowski [START_REF] Gmachowski | Aggregate restructuring by internal aggregation[END_REF]. Besides, it enables us to prove our point: during our experiments, the aggregates underwent some restructuring, leading to larger D f over time.

Conclusion

In this paper, we have presented an optical model for scattering of silica aggregates with fractal properties. This model was implemented in a fitting program designed to extract the radius of primary particle r pp and the fractal dimension of aggregate D f from experimental turbidity spectra. Aggregation of two suspensions prepared by neutralization of sodium silicate with sulfuric acid was monitored by DLS and in situ turbidity measurements. The data issued from one of these experiments evidenced a densification of the aggregates with time. A correlation between our experimental turbidity values and hydrodynamic radii was found. By confrontation of these data with theoretical calculations, densification of the aggregates was once again evidenced. A fractal dimension dependent on the number of primary particles N pp has been suggested. There is room for improvement of the proposed expression. The kinetics of restructuring may possibly be determined and correlated with the physicochemical parameters of the suspension.

In situ SAXS measurements would also be a valuable asset, as it would enable us to confirm the evolution of D f we observed. The small number of primary particles in the aggregates of our suspensions may cause difficulties in the interpretation of SAXS experimental data. The classical method which relies on the use of the function q -Df (where q is the modulus of the scattering vector) may indeed not be applicable in our case. Cryo-TEM observations coupled with image analysis are also considered to validate our results.

Besides, the analytical approach described in this work could be employed with other systems for which the RDG approximation remains valid. It is typically the case of suspensions of submicronic polymer particles. These systems, as silica, have been widely used in investigations on aggregation.

Our method could thus prove to be a valuable tool to obtain time-resolved information about aggregating systems and potentially lead to a better understanding of aggregation in general.

List of tables

Table 1: Aggregate morphological parameters extracted from the experimental turbidity spectra at different times of the experiment (Upper and lower limits are respectively 3 nm and 10 nm for r pp , 1 and 3 for D f ) -pH 7.5 Table 2: Aggregate morphological parameters extracted from the experimental turbidity spectra at different times of the experiment when r pp is fixed and D f is a fitting parameter (Upper and lower limits are 1 and 3 for D f . The calculations are performed successively for r pp = 4.2 nm and r pp =5.0 nm) -pH 7.5

Highlights

• We monitored aggregation of silica particles thanks to turbidity and DLS analyses.

• We developed an optical model to quantatively interpret the turbidity spectra.

• Analysis of the data evidenced aggregate restructuring.

  α, a proportionality constant, and b equal to D f -3. Thus, the pair distribution density obeys the following equations:

τFig. 1 :Fig. 2 : 5 Fig. 3 : 5 Fig. 4 : 5 Fig. 5 : 5 Fig. 6 :Fig. 7 :Fig. 8 :Fig. 9 :

 1253545556789 Fig. 1: Experimental set-up for turbidity measurements Fig. 2: Evolution of experimental turbidity at several wavelenghts with time (the black dashed line is just a guide for the eye to appreciate the linearity of the evolution) -pH 7.5 Fig. 3: Evolution of hydrodynamic radius with time -pH 7.5 Fig. 4: Experimental turbidity spectra used in the fitting program -pH 7.5 Fig. 5: Evolution of the calculated N pp with time -pH 7.5 Fig. 6: Comparison of the evolution of turbidity and hydrodynamic radius with time in the two experiments Fig. 7: Dimensionless turbidity as a function of dimensionless radius -Experimental data Fig. 8: Dimensionless turbidity as a function of dimensionless radius -Simulations (the black dotted line is just a guide for the eye to appreciate the linearity of the evolution) Fig. 9: Comparison of the fitted D f with the values issued from D f (N pp )=1.6+0.6*tanh(0.003*N pp )

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

 1253545556789 Figure 1: Experimental set-up for turbidity measurements

Table 1 : Aggregate morphological parameters extracted from the experimental turbidity spectra at different times of the experiment (Upper and lower limits are respectively 3 nm and 10 nm for rpp, 1 and 3 for Df) -pH 7.5

 1

	Time (min)	0	5	10	15	22	36	51	66	91	106	121
	N pp (-)	18	8	18	21	21	298	538	53	45	65	108
	r pp (nm)	3.7	5.4	4.7	4.8	5.3	3	3	7,4	9,5	10	10
	D f (-)	2.7	1	1.4	1.3	1.2	2.1	2.2	1.5	1.4	1.5	1.7

Table 2 : Aggregate morphological parameters extracted from the experimental turbidity spectra at different times of the experiment when rpp is fixed and Df is a fitting parameter (Upper and lower limits are 1 and 3 for Df. The calculations are performed successively for rpp= 4.2 nm and rpp=5.0 nm) -pH 7.5

 2

Abbreviations