Farouk Belkadi

Nadège Troussier

Frederic Huet

Thierry Gidel

Eric Bonjour
email: ebonjour@ens2m.frr'

Benoît Eynard
email: benoit.eynard@utc.fr

PLM-based approach for collaborative design between OEM and suppliers: case study of aeronautic industry

Keywords: Suppliers Integration, PLM, UML, Innovative Organisation

To achieve different assembly operations on the aircraft structure, the aeronautic OEM needs to create and manage various fixture tools. To cope with these needs, the OEM begun to adopt the supplier integration into the tooling development process. This paper presents a conceptual PLM-based approach to support new business partnership of different suppliers. The new business partnership aims to improve the role of supplier in the different tasks of design, configuration and fabrication of the tooling. The use of the PLM concepts is proposed to enhance the collaboration between OEM and the equipment's suppliers. UML models are proposed to specify the structure of the PLM solution. These models describe the relation between the aircraft assembly project, and the tooling design process.

Introduction

The role of supplier in a successful assembly process of aircraft component is very important. Because of the specific aircraft structure, assembly department needs to constantly design new fixture tools used for new assembly operations. It obviously happens when the aeronautic OEM creates new aircraft model, and also when this OEM modifies the existing models to satisfy a particular customer requirement.

To deal with assembly tool costs and time to market optimization challenges, the collaboration between OEM and suppliers should rather go into strategic partnership, covering the whole tool's lifecycle. The purpose of our research is to develop a new business partnership that enables efficient collaboration between OEM and suppliers. This partnership would enhance the suppliers' role in the design process of assembly tools. The case study concerns the tooling design activities and manufacturing process in the aeronautic industry. The construction of this business partnership is obtained according the following perspectives:

• Definition of its mission and organization,

• Identification of new methodologies to optimize it's operating processes, • Realization of a collaborative IT framework to support its activities.

This paper focuses on the last perspective and describes a conceptual framework to specify an innovative PLM-based approach. The originality of our approach comes from the high abstraction level of the proposed models based on the situation concept. These concepts are valuable to describe different organization forms. It mainly provides specification of IT system that can gives innovation aided by enhancing the project organization in context of extended enterprise and by favouring interoperability between heterogeneous shared between OEM and supplier systems. (for instance, between SAP system to capture Aircraft information at the OEM level and DELMIA tool to identify the tooling behaviour in the assembly process, at the supplier level).

First, we present an overview of the context study and the interest of PLM approach to solve this problematic. Second, a literature review is presented concerning the use of PLM methodologies to support the OEM supplier partnership. Third, we develop our conceptual models of the IT structure. The specification of the PLM-based approach is defined according to a unified modelling that describes, at the same abstraction level, the product (assembly tools or equipment) data and process data. Fourth, the concept of project view is presented. Using the UML activity diagram, we detailed some functionalities of the future collaborative system to manage the equipment design project.

Context and aims of the study

Traditionally, in the aeronautic industry, the tooling supplier is a basic manufacturer of the assembly tools. The design and manufacture processes of these tools are considered as a sequential one. First the design department delivers the engineering documents of the different aircraft parts; the production engineering department specifies and designs the detailed assembly processes and needed tools to carry out the assembly operations. Then, the production engineering department sends the detailed specifications to the supplier for tools manufacturing.

Figure 1 shows this configuration. On the one hand, three departments are engaged in the global process of assembly tools purchasing: production service specifies the assembly needs, the equipment's R&D designs the tooling structure and the purchase service negotiates and sends the order to supplier. On the other hand, several suppliers located in different geographical locations are contracted with to produce the various parts of the tool. After the completion of the tool, it is sent directly to the production shop for use. During the manufacturing process of the assembly tool, some modifications may occur on the initial configuration of aircraft components. These modifications imply changes on the specification of the assembly process and thus of the assembly tool. The whole cycle of the assembly tool ordering is then repeated to cope with the new specifications.

This approach proved its limits in the current context. The supplier is not integrated in the first stages of the tools specification and likewise, the OEM has not access to the manufacture process of the tool. Thus, much iteration is occurred before obtaining the final tool definition fulfilling the requirements of production engineering department. Several problems have been observed during the preliminary study: Important time and costs of the assembly tools manufacturing (and consequently for the assembly process of the aircraft parts, delivery date not respected); difficulty to manage the assembly tool range by the OEM (no-use of standards, bad maintenance…); the OEM has to manage several product data interfaces with various partners.

In the future configuration, an innovative PLM-based approach is proposed to support a new business partnership approach. PLM is used for the seamlessly integration of all the information specified throughout all phases of the equipment's life cycle to everyone in the new organization (OEM and a new global supplier network) at every managerial and technical level. Figure 2 shows the proposed configuration of the new business partnership. In this configuration, design tasks', configuration and fabrication of the assembly tool are performed collaboratively with the new global supplier network. Suppliers are already informed by new modifications of the assembly operations and design themselves the new tool.

This description shows important evolutions in the configuration of the development process that can be summarized by considering the shift from a linear and sequential process to a much more "interactionnist" one [START_REF] Kline | An overview of innovation[END_REF]. This reconfiguration should lead to significant improvement in cost and time saving in association with a greater innovative potential [START_REF] Nishiguchi | Suppliers' innovation: undestated aspects of japanese industrial sourcing[END_REF], [START_REF] Kim | Performance effect of partnership between manufacturers and suppliers for new product development: the supplier's standpoint[END_REF].

But, what is at stake in this study goes beyond the development process and the impact or the evolution has to be considered at the (inter)organizational level. We can both consider the renewal of the expected shared competences and new governance modalities for these new relationships [START_REF] Foss | Theories of the firm : contractual and competence perspective[END_REF], [START_REF] Mahmoud-Jouni | Les coopérations interentreprises dans les projets de développement[END_REF]. First, in the traditional development process, the supplier's competences were exclusively manufacturing ones. In the new process, the expected incomes evolve towards innovation capacity, sub-system integration, proactive behavior during the process… This leads to consider the new role of the suppliers, not only as an efficient manufacturer, but more as a service supplier, collaborating in the definition and conception stages [START_REF] Calvi | Le rôle des services achats dans le développement des produits nouveaux : une approche organisationnelle[END_REF]. Thus, knowledge transfer and learning capacities are at the core of these new activities for these suppliers [START_REF] Nooteboom | Learning and innovation in organizations and economies[END_REF]. So, as we can see, this evolution will have to be encompassed in a wider evolution of the competences that is expected from the suppliers.

Second, to promote greater innovative potential, interactions between the different partners will have to be carefully managed, because of the change in the nature of their transactions. We can at least anticipate three significant modifications in their relationships. The advantage of the new process is a more important distribution of risk between the partners, previously only assumed by the OEM. In the new context, risk is distributed between all the involved actors. This collaborative organization implies that the partners reveal some of their competences, to combine and fertilize them. Thus, the management of core competences and the equilibrium between the individual interests of each actor and the collective objectives has to be questioned [START_REF] Hamel | Strategy as Stretch and Leverage[END_REF], [START_REF] Ouchi | Markets, bureaucracies[END_REF]. And, to promote innovation, upstream monitoring and planning will necessarily have to be adapted, in order to facilitate the emergence of new opportunities, which were not anticipated at the beginning of the collaboration. This seems all the more important, that previous studies have shown that innovation through cooperation is linked to a sort of "plasticity" of the relationship, allowing to discover new opportunities and sources of learning [START_REF] Huet | Capacités d'innovation et coopération de PME : des effets auto-renforçants[END_REF].

These different preliminary elements shed light on the extended impact of this evolution in the development process, both in the vertical interactions (between OEM and suppliers) and in the horizontal ones (between suppliers). The innovation and collaboration objectives show that governance will have to rely on a new equilibrium between contractual prescriptions and trust based relationships [START_REF] Adler | Market hierarchy and trust, The knowledege Economy and the future of capitalism[END_REF]. Indeed, to promote innovation, contracts will necessarily remain uncomplete and could lead to strong inertia during the collaboration, while trust, both on competence and behavior, will bring more flexibility in front of novelty and knowledge mutualisation. This vertical and horizontal integration that necessitate risks and benefits sharing, implies developing common practices and methodologies. By sharing project management approach, problem solving methods or design methodology, the partners would in turn shared objectives and decision processes.

Even if this contribution is focused on the definition of a PLM platform, these different elements have to be mentioned. They necessarily won't be neutral for the definition and appropriation of this new support by the different partners.

PLM is used for the seamlessly integration of all the information specified throughout all phases of the equipment's life cycle to everyone in the new organization (OEM and a new global supplier network) at every managerial and technical level. The following section presents a literature review about PLM concept.

About Product Life Management

PLM is defined as a systematic concept for the integrated management of all product related information and processes through the entire lifecycle, from the initial idea to end-of-life [START_REF] Saaksvuori | Product Lifecycle Management[END_REF]. In [START_REF] Jun | Research issues on closed-loop PLM[END_REF], PLM is considered as a strategic business approach that applies a consistent set of business solution in support of the collaborative creation, management, dissemination, and use of product information across the extended enterprise.

Such as in the automotive industry, the aeronautic industry is seen to adopt the supplier integration into the development process. The new management culture considers necessary the PLM approach to get these goals [START_REF] Gomes | Applying a benchmarking method to organize the product lifecycle management for aeronautic suppliers[END_REF]. Tang [START_REF] Tang | Product lifecycle management for automotive development focusing on supplier integration[END_REF] present a literature review of PLM approaches used in automotive industry to improve collaboration between OEM and suppliers. The lifecycle currently support the OEM supplier partnership can be grouped in collaborative environment with three main phases [START_REF] Schilli | Collaborative life cycle management between suppliers and OEM[END_REF]:

• Designing the systems to be used in the OEM's product.

• Supply chain integration to produce and deliver the requested systems to OEM.

• Provide services for the components for both OEM and supplier systems.

The IT solution to support PLM results from the integration between enterprise resource planning (ERP), product data management (PDM) and other related systems, such as computer aided design (CAD) and costumer relationship management (CRM) [START_REF] Schuh | Process oriented framework to support PLM implementation[END_REF]. A critical aspect of PLM systems is their product information modeling architecture. In the literature, several representations of the product data are presented [START_REF] Terzi | Development of a metamodel to foster interoperability along the product lifecycle traceability[END_REF], [START_REF] Moka | Managing engineering knowledge: methodology for knowledge based engineering applications[END_REF]. The unified representation of the product knowledge can favor semantic interoperability of CAD/CAE/ tools at the conceptual level [START_REF] Szykmana | A foundation for interoperability in next-generation product development systems[END_REF]. UML is currently used to support product models [START_REF] Nishiguchi | Suppliers' innovation: undestated aspects of japanese industrial sourcing[END_REF]. STEP and XML are used to obtain interoperability at the implementation level [START_REF] Fenves | CPM2: A revised core product model for representing design information[END_REF]. Sudarsan [START_REF] Sudarsan | A product information modeling framework for product lifecycle management[END_REF] propose a product information-modeling framework that aimed at support PLM information needs and intended to capture product data, design rationale, assembly, tolerance information, the evolution of products and product families.

However, product model is not the unique element in a PLM structure. Nowak [START_REF] Nowak | Towards a design process model enabling the integration of product, process and organization[END_REF] present architecture of a collaborative aided design framework integrating Product, Process and Organization models for engineering performance improvement. Danesi [25] propose the P4LM methodology which allows the management of Projects, Products, Processes, and Proceeds in collaborative design and that aims to allow the integration of information coming from different partners which are involved in a PLM application. This framework allows a topdown approach by defining functions in an abstraction level according to four modules (Project, Product, Proceed and Process).

In aim to get best integration of suppliers in the automotive design and manufacturing processes, Trappey [START_REF] Trappey | Applying collaborative design and modularized assembly for automotive ODM supply chain integration[END_REF] develops and implements an information platform called advanced production quality planning (APQP) hub. The information hub mainly provides a collaborative environment that enhancing the visibility of the supply chain operations and contributes in collecting and delivering APQP documents among both the OEM and all supply chain. This information platform applies the concept of modularized assembly and consists of five major functions: Categorized part library, project based collaborative design, real-time information exchange, on-line confirmation of modularized products, and on-line negotiation and ordering. Each one of the obvious functions is implemented according to an interactive process.

Our work deals with the integration of Product, Process, and Organization dimensions of a design project. Several models are developed to support, at the conceptual level, this integration.

The Package model of the PLM approach

At the conceptual level, our approach is based on the concept of working situation proposed in [START_REF] Belkadi | Modelling Framework of a Traceability System to Improve Knowledge Sharing and Collaborative Design[END_REF]. According to this model, each process and activity in a collaborative design project is considered as an interactional entity that refers to links between various entities of the situation. These entities may bring together different physical elements such as human resources and material resources (product, drawings, documents, CAD tools, planning tools, etc). It may bring together, also, other interactional entities (activities, processes, communities).

The nature of the contribution made by each entity to the interactions is formalized in this approach using the concept of specific role that is a systemic extension of the role proposed in the organization theory [START_REF] Uschold | The Enterprise Ontology[END_REF]. Five kinds of specific roles are distinguished:

• The "actor" role concerns every entity who/which participates directly in the interaction and who/which is responsible for the end result. • The "customer" role brings together all the entities that are going to be receiving the end result of the interaction. • The "manager" role concerns every entity who/which regulates the functioning of an interaction. • The "support" role includes every entity who/which give help in an interaction.

• The "object" role concerns every entity on whom/which the interaction acts. Figure 3 describes the global package model structuring the PLM data. According to this model, the assembly tool (equipment) may be considered as a mediation artifact since it is simultaneously a support of the assembly aircraft project (that is performed in the OEM Assembly Workshop) and, the main object of the equipment project (that are realized by the new trade organization). Aircraft project plays the role of customer of the equipment project. The different needs of the equipment project are specified according to the different activities of the aircraft assembly process. Thus, Aircraft processes take also the role of customer in the equipment processes.

Figure 3 The package model structuring our PLM approach.

The processes packages (of aircraft and equipment) group various processes that organize the functioning of related projects. For example, design and fabrication processes are the principal processes of the equipment project, its play the role "actor". The process data concerns both the assembly process of aircraft parts and design process of the equipment. Saved information is used to recognize the activities evolution of each partner (new requirement of the OEM, new kind of assembly tools proposed by suppliers …).

The data model package organizes, according to various sub-models, the equipment's information that are produced and manipulated during different stages of the whole equipment lifecycle. The detailed data are stored in different documents represented by "documents package". For example, the structural model contains information about the physical composition of the equipment. The detailed structure is stored in CAO documents.

The project view

We consider the concept of project as a form of operational interaction including a set of processes in order to obtain specific goals. The project structure consists of a systemic decomposition into different sub-projects regarding to the product complexity.

Figure 4 The project view Figure 4 shows the Meta model of the project structure, the project is considered as an interactional entity according to the situation concept (cf. section 2). Each project contributes to one or several goals. The class "goals" make dependence between the project view, the process view and the task view.

The project Meta model presents the contribution of different elements similarly at the organizational level (organization of different human resources and communities) and at the operational level (organization of different processes). The contribution of all project elements is presented by an instantiation of: {the class entity, the class role (replaced by a specific subclass) and interactional entity (in this case project)}. For instance, the Aircraft project is associated to the Equipment project by mean of the "Customer" class.

The main idea is that both the aircraft design project and equipment design project are described under the same main project reference. When a manufacturing order of an assembly tool is submitted, a new sub project for this need is created. All aircraft sub-project that are concerned by this equipment are related to the above project in the global situation framework. For this use case, three specific roles where to be considered:

• The aircraft R&D takes the role of actor in the aircraft design process and the indirect "customer" in the equipment design process (send the original needs through the production department). We note this entity "Aircraft_R&D". • The production department takes the role of support in the aircraft design process; it performs different production and assembly operations. At the same time, it takes the role of direct "customer" in the equipment design process (define the assembly procedure and assembly tool functions). We note this entity "Aircraft_Prod". • The R&D service of the new business partnership takes the role actor of the equipment design process. We note this entity "Equipment_R&D". • The manufacturing service of the new business partnership takes the role support of the equipment design process. This entity is noted "Equipment_Prod". When a modification in the aircraft structure is occurred, the system informs the members of the business partnership and sends him the new requirement to consider in the specification of the related assembly tool. Figure 6 presents the interaction process for this case.

Conclusion

In this paper, we have proposed a modeling framework to support, at the conceptual level, a new PLM approach to improve information sharing in collaborative design, and then to enhance the integration of supplier in the design and manufacturing processes. The final goals of the project is to reduce costs and time to market of the assembly tools, and consequently thus of the aircraft product.

The new business partnership implies to establish new collaboration strategy between OEM and supplier. Other benefits can be obtained from this framework by monitoring the evolution of collective work and facilitating its coordination.

The developed framework deals with the integration of Product, Process and Organization dimensions of a design project, and, in future works, the corresponding extension of CAD/CAM and PDM existing tools. The proposed Product model gives the structure of the product data base. It uses a generic semantic that can favor, in our sense, the conceptual interoperability between different product data coming from different partners.

Although our work is developed initially to resolve a particular problem in a special firm of the aeronautic industry, the use of a modeling framework based on the generic concepts of entities and interactions in the working situation may gives more interests.

In this contribution, one specific dimension has been developed, related to the PLM platform, to support the shift from a sequential to an interactionnist development process. Face to the complexity of such a change, success will not only rely on this support dimension. This PLM platform will have to be considered in a more global system/organisation, to take into account the entanglement of technology, market and usage dimensions. At an operational level, this integrated approach will enhance the chances of success and at a more analytic level, it will allow to precise the conditions of application and transposition in other contexts.

Further research work will be performed to improve and validate these different issues. A prototype is under development and is being tested thanks to our industrial case study.

Figure 1 .

 1 Figure 1. The current OEM-Supplier partnership.

Figure 2

 2 Figure 2 Future configuration of OEM-Supplier partnership.

Figure 6

 6 Figure 6 Scenario of modifying requirement

Farouk Belkadi, Nadège Troussier, Frederic Huet et al.

Several modeling tools are used to describe process achievement (IDEF, GRAi, UML, etc.). In our approach, we used the Activity diagram of UML formalism (figure 5) to describe the interaction process during the creation of a new project. At the beginning, Aircraft R&D creates a new project and sends initial specifications to the production department. It defines the different operations of the assembly process and specifies the functions of the assembly tool to be realized. After, it searches in the furniture warehouse a tool which satisfies these functions. If no tool is founded, production department creates new equipment sub project and sends information to the supplier network (equipment R&D and manufacturing).

In fact, the real process is established in concurrent way. When, the equipment R&D starts the design process, manufacturing service is simultaneously schedules the manufacturing operations and researches the available technological solutions. Thanks to the collaborative system, the specification and manufacturing of the assembly tool is performed progressively and co-jointly by different partners according to the global scenario.


A Aïssa-El-Bey

K Abed-Meraim

Y Grenier
email: grenier@tsi.enst.fr

Blind audio source separation using sparsity based criterion for convolutive mixture case

In this paper, we are interested in the separation of audio sources from their instantaneous or convolutive mixtures. We propose a new separation method that exploits the sparsity of the audio signals via an p-norm based contrast function. A simple and efficient natural gradient technique is used for the optimization of the contrast function in an instantaneous mixture case. We extend this method to the convolutive mixture case, by exploiting the property of the Fourier transform. The resulting algorithm is shown to outperform existing techniques in terms of separation quality and computational cost.

Introduction

Blind Source Separation (BSS) is an approach to estimate and recover independent source signals using only the information within the mixtures observed at each channel. Many algorithms have been proposed to solve the standard blind source separation problem in which the mixtures are assumed to be instantaneous. A fundamental and necessary assumption of BSS is that the sources are statistically independent and thus are often separated using higher-order statistical information [START_REF] Cardoso | Blind signal separation : statistical principles[END_REF]. If extra information about the sources is available at hand, such as temporal coherency [2], source nonstationarity [3], or source cyclostationarity [4], then one can remain in the second-order statistical scenario, to achieve the BSS. In the case of non-stationary signals (including audio signals), certain solutions using time-frequency analysis of the observations exist [5]. Other solutions use the statistical independence of the sources assuming a local stationarity to solve the BSS problem [6]. This is a strong assumption that is not always verified [7]. To avoid this problem, we propose a new approach that handles the general linear instantaneous model (possibly noisy) by using the sparsity assumption of the sources in the time domain. Then, we extend this algorithm to the convolutive mixture case, by transforming the convolutive problem into instantaneous problem in the frequency domain, and separating the instantaneous mixtures in every frequency bin. The use of sparsity to handle this model, has arisen in several papers in the area of source separation [8,9]. We first present a sparsity contrast function for BSS. Then, in order to achieve BSS, we optimize the considered contrast function using an iterative algorithm based on the relative gradient technique.

In the following section, we discuss the data model that formulates our problem. Next, we detail the different steps of the proposed algorithm. In Section 4, some simulations are undertaken to validate our algorithm and to compare its performance to other existing BSS techniques.

Instantaneous mixture case

Data model

Assume that N audio signals impinge on an array of M ≥ N sensors. The measured array output is a weighted superposition of the signals, corrupted by additive noise, i.e.

x(t) = As(t) + w(t) t = 0, . . . , T -1 (1)
where

s(t) = [s 1 (t), • • • , s N (t)] T is the N × 1 sparse source vector, w(t) = [w 1 (t), • • • , w M (t)]
T is the M × 1 complex noise vector, A is the M × N full column rank mixing matrix (i.e., M ≥ N ), and the superscript T denotes the transpose operator. The purpose of blind source separation is to find a separating matrix, i.e. a N × M matrix B such that s(t) = Bx(t) is an estimate of the source signals.

Before proceeding, note that complete blind identification of separating matrix B (or equivalently, the mixing matrix A) is impossible in this context, because the exchange of a fixed scalar between the source signal and the corresponding column of A leaves the observations unaffected. Also note that the numbering of the signals is immaterial. It follows that the best that can be done is to determine B up to a permutation and scalar shifts of its columns, i.e., B is a separating matrix iff :

Bx(t) = P Λs(t) (2)
where P is a permutation matrix and Λ a non-singular diagonal matrix.

Sparsity-based BSS algorithm

Before starting, we propose to use 'an optional' whitening step which set the mixtures to the same energy level and reduces the number of parameters to be estimated. More precisely, the whitening step is applied to the signal mixtures before using our separation algorithm. The whitening is achieved by applying a N × M matrix W to the signal mixtures in such a way Cov(W x) = I in the noiseless case, where Cov(•) stands for the covariance operator. As shown in [2], W can be computed as the inverse square root of the noiseless covariance matrix of the signal mixtures (see [2] for more details). In the following, we apply our separation algorithm on the whitened data :

x w (t) = W x(t).
We propose an iterative algorithm for the separation of sparse audio signals, namely the ISBS for Iterative Sparse Blind Separation. It is well known that audio signals are characterized by their sparsity property in the time domain [8,9] which is measured by their p norm where 0 ≤ p < 2. More specifically, one can define the following sparsity based contrast function

G p (s) = N i=1 [J p (s i )] 1 p , (3)
where

J p (s i ) = 1 T T -1 t=0 |s i (t)| p . (4)
The algorithm finds a separating matrix B such as,

B = arg min B {G p (B)} , (5)
where

G p (B) G p (z) , (6)
and z(t) Bx w (t) represents the estimated sources. The approach we choose to solve (5) is inspired from [10]. It is a block technique based on the processing of T received samples and consists in searching iteratively the minimum of (5) in the form :

B (k+1) = (I + (k) )B (k) (7) z (k+1) (t) = (I + (k) )z (k) (t) (8)
where I denotes the identity matrix. At iteration k, a matrix (k) is determined from a local linearization of G p (B (k+1) x w ). It is an approximate Newton technique with the benefit that (k) can be very simply computed (no Hessian inversion) under the additional assumption that B (k) is close to a separating matrix. This procedure is illustrated in the following :

At the (k + 1) th iteration, the proposed criterion (4) can be developed as follows:

J p (z (k+1) i ) = 1 T T -1 t=0 z (k) i (t) + N j=1 (k) ij z (k) j (t) p = 1 T T -1 t=0 |z (k) i (t)| p 1 + N j=1 (k) ij z (k) j (t) z (k) i (t) p .
Under the assumption that B (k) is close to a separating matrix, we have

| (k) ij | 1
and thus, a first order approximation of J p (z

(k+1) i
) is given by :

J p (z (k+1) i ) ≈ 1 T T -1 t=0 |z (k) i (t)| p + p N j=1 e( (k) ij ) e |z (k) i (t)| p-1 e -φ (k) i (t) z (k) j (t) -m( (k) ij ) m |z (k) i (t)| p-1 e -φ (k) i (t) z (k) j (t) (9
) where e(x) and m(x) denote the real and imaginary parts of x and φ

(k) i (t) is the argument of the complex number z (k) i (t).
Using equation (9), equation (3) can be rewritten in more compact form as :

G p B (k+1) = G p B (k) + e T r (k) R (k)H D (k)H (10)
where (•) denotes the conjugate of (•), T r(•) is the matrix trace operator and the ij th entry of matrix R (k) is given by : R

(k) ij = 1 T T -1 t=0 |z (k) i (t)| p-1 e -φ (k) i (t) z (k) j (t) (11)
D (k) = diag R (k) 11 , . . . , R (k) N N 1 p -1 . (12)
Using a gradient technique, (k) can be chosen as :

(k) = -µD (k) R (k) (13)
where µ > 0 is the gradient step. Replacing (13) into (10) leads to,

G p B (k+1) = G p B (k) -µ D (k) R (k) 2 . ( 14
)
So µ controls the decrement of the criterion. Now, to avoid the algorithm's convergence to the trivial solution B = 0, one normalizes the outputs of the separating matrix to unit-power, i.e. ρ

(k+1) zi 1 T T -1 t=0 |z (k+1) i (t)| 2 = 1, ∀ i.
Using first order approximation, this normalization leads to :

(k) ii = 1 -ρ (k) zi 2ρ (k) zi . ( 15
)
After convergence of the algorithm, the separation matrix B = B (K) is applied to the whitened signal mixtures x w to obtain an estimation of the original source signals. K denotes here the number of iterations that can be either chosen a priori or given by a stopping criterion of the form B (k+1) -B (k) < δ where δ is a small threshold value.

Convolutive mixture case

Unfortunately, instantaneous mixing is very rarely encountered in real-world situations, where multipath propagation with large channel delay spread occurs, in which case convolutive mixtures are considered. In this case, the signal can be modeled by the following equation :

x(t) = L l=0 H(l)s(t -l) + w(t) (16)
where H(l) are M × N matrices for l ∈ [0, L] representing the impulse response coefficients of the channel and the polynomial matrix

H(z) = L l=0
H(l)z -l is assumed to be irreducible (i.e. H(z) is of full column rank for all z). If we apply a short time Fourier transform (STFT) to the observed data x(t), the model in (16) (in the noiseless case) becomes approximately

S x (t, f ) ≈ H(f )S s (t, f ) (17)
where S x (t, f ) is the mixture STFT vector, S s (t, f ) is the source STFT vector and H(f ) is the channel Fourier Transform matrix. It shows that, for each frequency bin, the convolutive mixtures reduce to simple instantaneous mixtures. Therefore we can apply our ISBS algorithm for each frequency and separate the signals. As a result, in each frequency bin, we obtain the STFT source estimate

S b s (t, f ) = B(f )S x (t, f ) . ( 18
)
It seems natural to reconstruct the separated signals by aligning these S b s (t, f ) obtained for each frequency bin and applying the inverse short time Fourier transform. For that we need first to solve the permutation and scaling ambiguities as shown next.

Removing the scaling end permutation ambiguities

In this stage, the output of the separation filter is processed with the permutation matrix Π(f ) and the scaling matrix C(f ).

G(f ) = Π(f )C(f )B(f ) . ( 19
)
The scaling matrix C(f ) is a N × N diagonal matrix found as in [11] by

C(f ) = diag[B(f ) # ].
For the permutation matrix Π(f ), we exploit the continuity property of the acoustic filter in the frequency domain [12]. To align the estimated sources at two successive frequency bins, we test of the closeness of G(f n )G(f n-1 ) # to a diagonal matrix. Indeed, by using the representation (19), one can find the permutation matrix by minimizing :

Π(f n ) = arg min f Π    i =j ΠC(f n )B(f n )G(f n-1 ) # 2 ij    . (20)
In our simulations, we have used an exhaustive search to solve (20). However, when the number of sources is large, the exhaustive search becomes prohibitive. In that case, one can estimate Π(f n ) as the matrix with ones at the ij th entry satisfying

|M(f n )| ij = max k |M(f n )| ik
and zeros elsewhere with

M(f n ) = C(f n )B(f n )G(f n-1 )
# . This solution has the advantage of simplicity but may lead to erroneous solution in difficult context. An alternative solution would be to decompose Π(f n ) as product of elementary permutations 1 Π (pq) . The latter is considered at a given iteration, only if it decrease criterion (20), if

|M(f n )| 2 pq + |M(f n )| 2 qp > |M(f n )| 2 pp + |M(f n )| 2 qq
Finally, we obtain :

Π(f n ) = nb of iterations 1≤p<q≤N Π (pq) , (21)
Π (pq) being either the identity matrix or the above permutation matrix Π (pq) depending on the binary decision rule define above. We stop the iterative process, when all matrices Π (pq) are equal to the identity. We have observed that one or, at most, two iterations are sufficient to get the desired permutation. Finally, we apply the updated separation matrix G(f ) to the frequency domain mixture :

S b s (t, f ) = G(f )S x (t, f ) . ( 22
)

Simulation results

We present here some numerical simulations to evaluate the performance of our algorithm. We consider an array of M = 2 sensors receiving two audio signals in the presence of stationary temporally white noise of covariance σ 2 I (σ 2 being the noise power). 10000 samples are used with a sampling frequency of 8Khz (this represents 1.25sec recording). In order to evaluate the performance in the instantaneous mixture case, the separation quality is measured using the Interference to Signal Ratio (ISR) criterion [2] defined as :

ISR def = p =q E |(BA) pq | 2 ρ q E (|(BA) pp | 2 ) ρ p (23)
where

ρ i = E(|s i (t)| 2
) is the i th source power evaluated here as 1-(a) represents the two original sources and their mixtures in the noiseless case. In Fig. 1-(b), we compare the performance of the proposed algorithm in instantaneous mixture case, to the Relative Newton algorithm developed by Zibulevsky et al. in [9] where the case of sparse sources is considered and to SOBI algorithm developed by Belouchrani et al. in [2]. We plot the residual interference between separated sources (ISR) versus the SNR. It is clearly shown that our algorithm (ISBS) performs better in terms of ISR especially for low SNRs as compared to the two other methods. In Fig. 2-(a), we represent the evolution of 1 Π (pq) is defined such as way that for a given vector y, e y = Π (pq) y iff e y(k) = y(k), for k / ∈ {p, q}, e y(p) = y(q) and e y(q) = y(p). the ISR as a function of the iteration number. A fast convergence rate is observed. In Fig. 2-(b), we compare, in the 2 × 2 convolutive mixture case the separation performance of our algorithm, Deville's algorithm in [13], Parra's algorithm in [14] and extended version of Zibulevsky's algorithm to the convolutive mixture case. The filter coefficients are chosen randomly and the channel order is L = 128. We use in this experiment the ISR criterion defined for the convolutive case in [14] that takes into account the fact the source estimates are obtained up to a scalar filter. We observe a significant performance gain in favor of the proposed method especially at low SNR values. Moreover, the complexity of the proposed algorithm is equal to 2N 2 T + O(N 2 ) flops per iteration whereas the complexity of the Relative Newton algorithm in [9] is 2N 4 + N 3 T + N 6 /6.

1 T T -1 t=0 |s i (t)| 2 . Fig.
0 0.2 0.4 0.6 0.8 1 1.2 -1 0 1 s1(t) 0 0.2 0.4 0.6 0.8 1 1.2 -1 0 1 s2(t) 0 0.2 0.4 0.6 0.8 1 1.2 -1 0 1 x1(t) 0 0.2 0.4 0.6 0.8 1 1.2 -1 0 1 Time (sec) x2(t)

Conclusion

This paper presents a blind source separation method for sparse sources in instantaneous mixture case and its extension to the convolutive mixture case. A sparse contrast function is introduced and an iterative algorithm based on gradient technique is proposed to minimize it and perform the BSS. Numerical simulation results have been given evidence the usefulness of the method. The proposed technique outperforms existing solutions in terms of separation quality and computational cost in both instantaneous and convolutive mixture cases.

Fig. 1 .

 1 Fig. 1. (a) Up the two original source signals and bottom the two signal mixtures. (b) Interference to Signal Ratio (ISR) versus SNR for 2 audio sources and 2 sensors in instantaneous mixture case.

Fig. 2 .

 2 Fig. 2. (a) ISR as a function of the iteration number for 2 audio sources and 2 sensors in instantaneous mixture case. (b) ISR versus SNR for 2 × 2 convolutive mixture case.


Giuseppe Boniello

Christophe Tribet

Emmanuelle Marie

Vincent Croquette

Dražen Zanchi

Rolling and aging in temperature-ramp soft adhesion

Keywords: colloids, temperature-responsive polymer coating, Brownian dynamics, hindered diffusion, adhesion

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Adhesion of colloids on a flat surface presents interesting aging dynamics if either of surfaces is elastic or soft [1]. Soft adhesion with aging dynamics can also be produced by coating surfaces with polymer whose ends can stick to the adjacent surface, as it has been recently reported [2]. During soft adhesion, ends of polymers protruding from engaged surfaces stochastically explore the opposite side, increasing the number of attached contacts, so that the soft contact domain evolves in time. Control of these very final events preceding the immobilisation are of interest for research on functional materials, lubrication, cell adhesion etc. [3][4][5][6]. In the present paper we focus on colloids coated by a controlled molar fraction f of T-responsive polymers that switch interactions between colloids and the surface from repulsive to attractive at T = T c ≈ 32 • C. We analyze the details of the colloid Brownian motion occurring just before their adsorption to the surface. Near-surface Brownian dynamics can be analyzed by direct video 2D or 3D tracking, either by real-time analysis of the diffraction pattern [7,8], by total internal reflection microscopy (TIRM) [9][10][11] or by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM) [12,13]. The mean-square displacement (MSD) reveals how the diffusion is hindered by near-surface effects [14][15][16][17]. In the 3D tracking the resident time distribution (RTD), corresponding to the z-histogram of the trajectory, has been used for studying the particle-surface interaction potential [11]. If the adsorption (or self-aggregation) is irreversible, as it is in our present case, the situation is clearly out-of-equilibrium process. In such case, the aging effects in the pre-adhesion phase were identified in constant shear flux by Kalasin and Santore [2] and for a particle held (and released) by optical tweezers in suspension above a flat surface, by Kumar, Sharma, Ghosh and Bhattacharya [1,18]. To bring the system out from equilibrium we use T-switchable attraction between colloids and the surface. Namely, T-responsiveness of the polymer Poly(N-isopropylacrylamide) (PNIPAM, T c = 32 ± 1 • C [19]), at varying surface molar fraction f , was already exploited for analysis of self-association kinetics of coated microparticles [20,21].

In the Experimental section of the paper we present methods and results. The theoretical section reports on a simple adsorption model allowing to understand all experimental findings. Several issues that our results and their interpretation could raise are pointed out in Discussion section. Final section contains concluding remarks.

II. EXPERIMENTAL SECTIONN

A. Materials and methods

Silica beads (0.96µm in diameter, Bangs Laboratories, SS03N) are dispersed in sodium hydroxide solution 1M by sonication for 15 min and dialysed against water (Slide-A-Lyzer, M W cutoff 3500 kDa, Thermo Scientific). The solution is diluted in Phosphate-buffered Saline (PBS) solution (0.15 M). Particle coating is obtained by mixing 22µL of beads in PBS with 100µL of polymer solution 10g.L -1 (f % PLL-g-PNIPAM, (100 -f )% PLLg-PEG, PLL: M w = 15 -30 kg/mol; PEG: M w = 20 kg/mol; PNIPAM: M w = 7 kg/mol) in PBS. The re- sulting suspension is incubated for 30 min at room temperature. Polymer excess is removed by 5 centrifugation cycles (1,800 g for 5 min) replacing the supernatant by deionized water.

Flat substrate: borosilicate glass coverslips are cleaned with ethanol and plunged in a 1M sodium hydroxide ultrasonic bath for 30 min. After rinsing with deionized water and drying, the experimental cell is prepared by superposing bi-adhesive tape and a mylar film. A 52 mm x 5 mm x 50µm channel is thus created between the borosilicate glass slide and the mylar film. The bottom of the channel is functionalized by coating the glass surface with PLL-g-PEG (same as the one used for coating particles), ensuring a steric repulsion for T < T c . This is achieved by injecting polymer solution (1g.L -1 in PBS) in the cell and left incubated for 30 min at room temperature. The cell is then rinsed with deionized water and dried by compressed air.

T-ramp experiment: the particles suspension is injected in the cell at 26 • C, the temperature is increased at 10 • C/min up to 38 • C and kept constant until the end of the acquisition. Attraction between beads and the flat plate is thereby triggered by crossing the critical temperature T c = 32 ± 1 • C. The advantage of the method is two-fold. First, it allows to switch from repulsive to attractive regime without requiring a fine temperature control around the transition temperature. Secondly, we get rid of additional effects, such as thermal inertia or uncertainty on PNIPAM transition. 3D beads motion is observed by slightly defocus microscopy in parallel illumination decorating bead image with interference rings observed with a CMOS camera. Particles are tracked in real time using a PicoTwist apparatus and PicouEye software [7] allowing sub-pixel resolution (∼ 5 nm) at 50 frames/s, see Figure 1a.

To measure the fraction of adsorbed particles over time, large number of particles (typically 40-50 per run) are followed by CCD camera at 10 frames/s, as long as all visible beads are immobilized. The recorded movies are converted to binary masks (1: particle, 0: background). Cumulative distributions of binding times, i.e. the fraction of adsorbed particles, are extracted from the correlation of each frame with the last one:

P a (t) = x,y I(t)I(t f ) x,y I(t f ) (1)
where x,y stands for the sum of all the pixel values on the same resulting image. At a given coverage ratio f the in-plane average diffusion D eff (f ) is obtained by fitting the mean-square displacement (MSD) extracted from in-plane tracking over the final time interval [t 0 -∆t, t 0 ] before stopping at t = t 0 . In our experiment ∆t = 16 s was chosen: long enough to contain several up-down excursions, and short enough to give meaningful results even for rapidly adsorbing particles. The MSD for i-th particle on stage is extracted from x i (t) and y i (t) tracks using the relation

MSD x,i ≡ [x i (t + t ) -x i (t )] 2 t = 2D eff,i t , (2)
and equivalently for M SD y . The average ... t runs over t , lying within the observed time interval ∆t. For all values of f MSDs are linear in time over several seconds, indicating that the movement is diffusive. The diffusion constant is obtained by fitting the equation 2 for each of typically N ∼ 10 particles per run and taking the average:

D eff = D eff,i i .
Finally, for each 3D track the RTD over z direction is extracted by constructing histogram of the z track record within typically 6-7 µm above the flat surface, and averaged over N particles from the same run.

B. Results

The fraction of immobilised particles as function of time is shown on Figure 1b. Samples with higher PNI-PAM coverage f adsorb faster. The characteristic sigmoidal shape of the adsorption kinetics indicates that adsorption rate increases gradually, corroborating the aging nature of the pre-adhesion dynamics of surface engaged particles.

If particles are in contact with the surface and rolling (or crawling) most of the time, their in-plane average diffusion D eff is reduced, as it was observed in experiments with finely thermostated DNA-coated microbeads [17]. Our systems shows similar behaviour. Figure 2a shows D eff (f )/D 0 , where D 0 = k B T /6πµR = 0.67 µm 2 / s is the diffusion constant for 1 µm beads in water far from wall at 38 • C. As we increase f from 2% to 100%, D eff /D 0 increases from ∼0.2 to ∼0.65. Naively, one would expect the contrary, that is, stickier are the particles, and slower is their motion. As we will show in next section, the puzzle is solved by assuming that the surface engaged beads are much slower than the untethered ones, and that the time that beads spend in contact with the surface decreases as f increases. In fact, beads with high coverage stick rapidly after engagement, while these with low f spend most of their time rolling, which lowers their diffusion coefficient considerably.

The normalized RTD over z is shown on Fig. 2b, as extracted from the z-record histogram. It shows that the beads with high f spend most of their pre-arrest time away from surface, while beads with lower coverage f can roll un tumble without sticking, implying that in average they spend more time closer to the surface.

III. THEORY

According to the theory of Mani, Gopinath and Mahadavan (MGM) [22] a typical time scale for soft contact aging is given by

τ * ∼ a l 1 × τ visc. × 1 2 Kl 2 1 /k B T , (3)
where τ visc. = 3µ/(nKl 1 ) is the viscous settling time, a is the microparticle radius, K is the spring constant of a single polymer involved in sticking, l 1 is its rest length, µ is the viscosity and n is the number of available sticky polymer tethers per unit surface. Thus, τ * is the result of interplay between visco-eleastic time and the attachment efficiency measured by Kl 2 1 /(2k B T ) factor. The resulting τ * = µa/nk B T is independent of the elastic constant of the tethers. On the contrary, the settling vertical distance during the draining does depend on spring constant. However, the latter does not intervene in our present model. For our case the most important is the inverse proportionality between τ * and n. Notice that the binding energy per sticking tether does not enter in the MGM theory since the polymer-substrate sticking rate is determined by the first passage time. Moreover, according to [1,2,18], the pre-arrest dynamics of soft adhesion proceeds step-wise, i.e. the 3D approach is well separated from soft contact (2D) rolling phase, indicating that the untethered and the rolling colloids can be seen as separated populations.

We construct the rate equations for three fractions: 1) the untethered fraction represented by the probability P b (t), 2) the fraction of rolling colloids of age τ , whose probability to be found between the ages τ and τ + dτ is P r (t, τ ) dτ , and 3) the population of colloids in arrest, with the probability

P a (t) = 1 -P b (t) - t 0 P r (t, τ )dτ . (4)
The time zero is chosen to coincide with the onset of the attractive interactions. Consequently, rolling particles cannot be older than t. To describe properly the evolution of the rolling population, we introduce following physical quantities: the sedimentation rate κ, the re- desperion rate function a(τ ), and the irreversible stopping rate function b(τ ). κ corresponds to the fraction of particles within the observation window absorbed per unit time by a totally absorbing sink at z = 0. It can be determined from the stationary solution of Fokker-Planck equation with z-dependent viscous drag due to near-wall hydrodynamic corrections [23] in gravitational and van der Waals potential. The observation window in our case corresponds to the layer of 6-7 µm above the surface. For our choice of parameters we calculated the theoretical value κ = 0.17 s -1 . a(τ ) is the re-dispersion rate of rolling colloids of age τ back into bulk, and b(τ ) is the irreversible stopping rate of rolling colloids of age τ .

high$f" α ∼$f" low$f" α ∼$f" 1/2 " 0 5 10
The rate equation for P r (t, τ ) is determined by rates κ, a(τ ) and b(τ ) as follows:

dP r (t, τ ) dt = - ∂P r (t, τ ) ∂τ + δ(τ )κP b (t) + -a(τ )P r (t, τ ) -b(τ )P r (t, τ ) , (5)
where δ(τ ) is Dirac function, ensuring that any newly sunk particle is a "just born" rolling one (τ = 0). The first term on the RHS reproduces simple uniform time evolution. To illustrate, suppose that we have some initial distribution P r (t = 0, τ ) = F (τ ), in the absence of any source or sink terms (κ = 0 and a = b = 0 respectively) at some later time t the solution is F (τ -t), that is, the time evolution is trivial shifting by -t. Nontrivial (and interesting) modulation in the time-evolution is brought in by sources and sinks. The rate equation for untethered colloids is

Ṗb = -κP b + t 0 a(τ )P r (t, τ ) dτ , (6)
which, together with Eqs. 4 and 5, determines completely the evolution of the system.

Since in our experiment the particles are unresolved over τ , starting from eq.5 a simplified equation is derived, in which all rolling particles are represented by P r (t) ≡ t 0 P r (t, τ )dτ .

(7)

We will suppose that the re-dispersion rate a(τ ) and the arrest rate b(τ ) depend on τ over characteristic aging time scale τ * , allowing us to write:

a(τ ) = k off Φ(τ /τ * ) , b(τ ) = kχ(τ /τ * ) , ( 8
)
where Φ is a monotonically decreasing and χ monotonically increasing function limited between 0 and 1. We introduce the effective aging functions ρ and g as follows:

t 0 Φ(τ /τ * )P r (t, τ )dτ = ρ(t/τ * )P r (t) (9)
and t 0 χ(τ /τ * )P r (t, τ )dτ = g(t/τ * )P r (t) .

These relations are purely formal and do not allow to obtain functions ρ and g from the original aging functions a and b. In this regard the present theory is merely a phenomenology because the aging functions are not given explicitly in terms of the parameters of the model. However, the relations 9 and 10 show that it is always possible to recast the system of rate equations for P r (t, τ ), to the system for the total number of rolling beads P r (t), and that the effective aging functions ρ and g vary at the same aging time scale τ * as the original functions a and b. By integrating over τ the equation 5 and using the definition 7 we obtain the rate equations

Ṗb = -κP b + k off ρ(t/τ * )P r Ṗr = κP b -[k off ρ(t/τ * ) + k g(t/τ * )] P r , (11)
while the third population of colloids is in arrest,

P a (t) = 1 -P b (t) -P r (t)
.

According to [2,18,22] the "age" can be associated with the number of stuck point contacts within the engaged soft domain. We chose ρ(x) = e -x and g(x) = 1 -ρ(x) because we want the re-dispersion rate to decrease and the arrest rate to increase upon aging. Initial conditions of the systems are P r (0) = P a (0) = 0 and P b (0) = 1. The central issue of this work is to find out how the parameters of the model, Eqs. 11, depend on f . Following MGM, we suppose that the aging time τ * scales as n -1 , but with extended meaning of n as the effective surface density of sticky tethers visited by the contact domain during rolling, Fig. 3b. Sticky tethers for us are the dangling polymers that are able to reach the opposite bare surface by overcoming the steric shield. This is possible in the immediate vicinity of the spots containing collapsed PNIPAM. The effective number of available sticky tethers is therefore proportional to the number of PNIPAM spots visited by the contact domain. For low f , we expect that average width of the contact domain is wider than typical distance between PNIPAM spots. The number of tethers involved is proportional to the linear density of PNIPAM patches along the rolling path, i.e. n ∼ f 1/2 , while for higher f , n becomes proportional to f , since the inter-patch distance becomes smaller than the width of the searching trail. Notice that the present argumentation does not prejudice about details of PNIPAM disposition over the surface, as far as PLL-g-PNIPAM is disposed in a discrete number of spots. In particular, the PLL-g-PNIPAM molecules can be grouped in patches with some size distribution. In order to confirm that present argumentation makes sense, we assumed that all parameters of Eq.11 depend on f over a single, monotonically increasing, scaling function α(f ), proportional to the number of available tethers within the searching area. Accordingly, for the aging time we pose τ * = τ * 0 /α(f ). Since the stopping rate constant k is supposed to increase with α(f ), we put k = k 0 α(f ). Detachment rate constant should decrease with increasing number of sticking points: we use k off = k off0 /α(f ). Calculated P a (t) using Eqs. 11 are fitted to adsorption kinetics by adjusting solely the value of α for each f , see figure 1b. The absolute scale for α being arbitrary, we choose α(f = 2%) = 1. The fitting parameters are k 0 = k off0 = 0.05 s -1 and τ * 0 = 350 s for all curves. Figure 3a) shows the best fitting scaling function α(f ). It is consistent with ∼ √ f tendency for low f and is steeper for high f , which confirms our picture based on interplay between the discreteness of PNIPAM spots and the finite width of contact domain.

We want now to reproduce the measured in-plane diffusion D eff , Fig. 2a). Since the process is non-stationary and irreversible, we must know in what moment (t 1 ) and for how long (∆t) the tracks are recorded, so that the average fraction of time that beads spend in "b" and in "r" state evolves in time. We assume that the most probable time t 1 is equal t max (f ), the time of maximal adsorption rate, corresponding to the maximum slope of kinetics, Fig. 1b. The tracking duration was ∆t = 16 s, a compromise regarding the number of particle excursions between "b" and "r" and the lifetime of the moving particle starting from t = 0. D eff is calculated as the time average of instantaneous diffusion d inst (t) over ∆t centred at t max , which mimics the way in which experimental D eff is extracted from the tracking. Formally, it writes

D eff ≈ 1 ∆t tmax+∆t/2 tmax-∆t/2 d inst (t) dt , (12)
where the instantaneous diffusion constant is Values of α(f ) used here are the ones that fit the adsorption kinetics, figure 1b). Bold line: most probable time intervals of particle tracking. The effective diffusion constant Deff is obtained as average over the bold portions, see eq. 12.

d inst (t) = [D b P b (t) + D r P r (t
Eqs. 11. It is important not to confound the quantity d inst (t) with the instantaneous diffusion "constant" of a single particle, that can be extracted from the tracking and is a fast fluctuating quantity. In fact, d inst (t) is interpreted as the ensemble average of diffusion constant at the time t. [START_REF]The equation 12 ignores the temperature variation within the observed T-ramp segment. Namely, the temperature dependence enters via D0(T ) = kBT /6πµ(T )R and µ(T ) decreases from 0.77 to 0[END_REF] Calculated time evolution of d inst (t) for a range of coverage values f is shown in dashed lines on figure 4. The portions drawn in bold correspond to the most probable time intervals of tracking recording. According to eq. 12, the resulting effective in plane diffusion constant D eff is the average over these intervals. Notice that for the very highest values of f the acquisition interval spread even over times before the PNIPAM collapse transition at t = 0.

The resulting calculated profile of D eff (f )/D 0 , shown on Figure 2a) is fitted to experimental data by adjusting D b and D r , while we used the same dependence α(f ) (Fig. 3a) that fits the adsorption kinetics, Fig. 1b. Best fitting values are D b = 0.73D 0 and D r = 0.15D 0 .

In order to fit the residence time distribution (RTD), figure fig2prl, to our theory, we associate the RTD to the probability distribution function (PDF), supposed to have the form

W (z) = Pb W b (z) + (1 -Pb )W r (z) , ( 13
)
where Pb is the average P b (t) over the observation time of ∆t centred at t = t max :

Pb = 1 ∆t tmax+∆t/2 tmax-∆t/2 P b (t) P b (t) + P r (t) dt . (14)
It is in fact the average fraction of time that particle spends away from the surface during the recording time interval. We take the equilibrium barometric law for PDF of detached particles

W b (z) ∼ e -mgz/k B T , ( 15
)
where m is the buoyant mass, and we assume that rolling particles have a phenomenological distribution

W r (z) ∼ e -arz/k b T , (16)
a r being the apparent weight of rolling particles, a r mg. Calculated distributions W (z) are shown in Figure 2b), together with Pb (f ) in inset. We see that in the asymptotic part of RTD, corresponding to the untethered particles, is fairly well reproduced by our model, i.e. the untethered population decreases as predicted, with decreasing f .

IV. DISCUSSION

In the light of rate equations-based theory, Eq. 11, we understand why the sample with the highest f have also the highest D eff . At high f a large majority of moving particles are in the suspension far from the surface during tracking, because the rolling regime is very short (i.e. the stopping is faster than the free sedimentation k κ, so that the beads get arrested as soon as they touch the surface), see inset of Fig. 2b. For low f this is not the case any more, the beads spend most of their time engaged in rolling motion, which is much slower. In that sense, the most interesting regime for us is the one of moderately low f , since the rolling and aging are the signatures of soft adhesion. For f 5%, the theory predicts a rapid increase in D eff (f )/D 0 with decreasing f , as expected, because particles without PNIPAM never stick nor roll on the surface. This regime is controlled by stopping rates constant k, which scales as √ f for small f . The points measured at f = 5% and 2% confirm this tendency.

Another instructive point to discuss concerns the value of D b = 0.73D 0 that fits experimental data. One expects it to correspond to the equilibrium near-wall hindered diffusion of untethered particles. The corresponding stationary PDF taking only gravitational potential is W b (z), given by eq. 15, which indeed fits the RTD, see figure 2. The diffusion constant at distance z for parallel (in-plane) motion is D 0 /φ (z), where φ (z) is the hinderance factor due to hydrodynamic interactions of spherical particle moving near a flat surface. Analytic form of φ (z) is a standard result, reported in literature [START_REF] O'neill | [END_REF][26][START_REF] Russel | Colloidal Dispersions[END_REF].

Taking average over z we get

D b = D eff | f =0 = D 0 W b (z) φ (z) dz = 0.72D 0 , (17)
which is fairly close to the value that fits experiment. Interestingly, taking into account also the Van der Waals interaction we get D b = 0.44D 0 , which is too low. This indicates that untethered colloids live in gravitational potential only, since the steric shield prevents them from approaching the surface and feeling the VdW forces.

One could wonder if our whole interpretation in terms of rolling and aging is over-complicated, and that experimental findings could be understood by effects of concentration depletion in the vicinity of partially absorbing sink at z = 0, and casting only two populations, the untethered and the arrested one. Indeed, it sounds plausible that higher adsorption implies lower concentration near the surfaces, and consequently some shift of the overall population upwards, where diffusion hinderance is less effective. Consequence is that the average D increases with increasing f , just as we want. For that reason, we calculated D , which under present assumptions equals D b , given by eq. 17. Calculations of W b (z) for partially absorbing wall imply finite flux solution of the Fokker-Planck equation [15]. We find that D eff increases linearly with f , as shown in Figure 2a). Cases with gravitational potential alone and including also Van der Waals part are both in disagreement with the experimental points, which shows that model that ignores the possibility of rolling cannot explain D eff (f ). For a similar system, the increase of D upon raising temperature above T c has been reported in [START_REF] Tu | [END_REF] in equilibrium conditions, where the phenomenon was attributed to electrostatic effects. This interpretation cannot be applied to our case in which, for high f , the particles spend most of their time away from the surface, at distances much higher than the Debye length.

Finally, let's discuss the assumption made in our model that the f dependence is contained simply in the parameter α(f ) with the meaning of the effective surface density of available sticking tethers during the soft sticking. This assumption assumes that individual "stick" and "release" events between one PEG end and the adjacent surface are independent on PNIPAM coverage f . In another words, when collapsing at T c , PNIPAM patches simply allow a finite number of PEGs to reach the adjacent surface, without affecting the sticking event of each PEG with the surface, which includes the diffusive search of the surface by the PEG end. A priori, this is a reasonable assumption because within the PLL-g-PEG coating the surface density of PEG chains is well below the over-crowded coverage [6], which would affect the sticking/release dynamics per PEG chain at low f . The agreement between the model and the experimental findings confirms that the present interpretation makes sense.

V. CONCLUSION

In conclusion, the soft adsorption kinetics depends on f over a single function α(f ), which is a measure of the number of discrete sticky patches within the soft contact area during Brownian rolling, extending the theory of Mani, Gopinath and Mahadavan [22]. At low/high f the effective inter-patch distance is larger/smaller than the contact diameter. From the point of view of PNIPAM spots (Figure 3b) the trail of the rolling contact domain crosses over from 1D to 2D, implying crossover of α(f ) from ∼ √ f to ∼ f . Two most remarkable effects are (i) a characteristic sigmoidal profile of the adsorption kinetics due to aging, and (ii) a decrease of the in-plane diffusion constant in pre-arrest Brownian dynamics with decreasing f due to reduction of pre-adhesion rolling time.

FIG. 1

 1 FIG. 1. a) Typical 3D tracking record. Bead was captured irreversibly at t = 15.2 s. Inset: schematic visualisation of temperature ramp experiment. b) Fraction of adsorbed particles as a function of time in a T-ramp of 10 • C/min between 26 • C and 38 • C, for a range of PNIPAM ratios f . Solid lines are calculated best fitting adsorption profiles Pa(t).

FIG. 2

 2 FIG. 2. a) Experimental points: effective in-plane diffusion Deff(f ) obtained by fitting the MSDs immediately before stopping. Bold line: result of our theory. Dotted lines: estimation of the near-wall diffusion hindrance by hydrodynamic effects near partially absorbing wall in gravitational potential. Dashed line: same estimation, including Van der Waals potential. b) Residence time distribution (RTD) extracted from tracking records z(t) record, compared to calculated PDF W (z). Inset: Pb (f ), the calculated average time fraction that particle spend away from the surface during the recording time interval.

FIG. 3

 3 FIG. 3. a) Dot symbols: scaling parameter α(f ) fitting the experiments. Low f and high f regimes are visible. b) Schematic interpretation of the two regimes. Shaded trail is the bead surface portion visited by the contact domain during Brownian rolling. At low f the number of tethers involved is proportional to the linear density of PNIPAM patches along the rolling path, while at high f it crosses over to the surface density.

  FIG. 4. Dashed line: d ef f (t)/D0 within the present model based on aging rolling particles, for a range of coverages f .

		1.0 d inst (t)/D 0
		2%
		5%	0.8
		20%	0.6	100 % 90 %
		33%
		40%	0.4			33 %		50 %
		50%						2 %
		90%	0.2			5 %
	-20	100%	0	20	40	60	80	100	t/s

)]/[P b (t) + P r (t)], D b and D r are effective diffusion constants for untethered and rolling particles respectively while P b (t) and P r (t) are calculated by

ACKNOWLEDGMENTS

We thank Maurizio Nobili for critical reading of the manuscript and Ken Sekimoto for stimulating and helpful discussions. This work was supported by ANR DAP-PlePur 13-BS08-0001-01 and program "investissement d'avenir" ANR-11-LABX-0011-01.


Didier Florence Cagnard

Naturally Occurring Asbestos in France: Geological Mapping, Mineral Characterization and Regulatory Developments

Naturally Occurring Asbestos in France: Geological Mapping, Mineral Characterization and Regulatory Developments

Cagnard, Florence, French Geological Survey, f.cagnard@brgm.fr Lahondere, Didier, French Geological Survey, d.lahondere@brgm.fr In France, the asbestos banning is subject to a national decree (n° 96-1133), published in 1996. The regulatory texts and standards adopted to control this banning concern in particular asbestos-bearing manufactured products, but remain difficult to apply to asbestos-bearing natural materials (ie. rocks, soils).

Considering problems related to such asbestos-bearing natural materials, the Ministry of Ecological and Solidary Transition has mandated the French Geological Survey to locate the impacted areas. Mappings were priority carried out in geological domains where NOA was predictable (French Alps, Corsica). These studies integrated field expertise, sampling and laboratory analyses, in order to characterize the potential of geological units to contain NOA. Furthermore, some expertises were carried out on geological formations exploited in France to produce aggregates. These studies concerned the quarries exploiting massive basic or ultrabasic rocks, likely to contain NOA, and quarries exploiting alluvium likely to contain asbestos-bearing rock pebbles.

These studies highlight the difficulty of establishing robust diagnoses for natural materials. Indeed, distinction between cleavage fragments resulting from the fragmentation of non-asbestos particles and proper asbestos fibers is particularly problematic for laboratories. Thus, a recent study of the National Agency for Health Safety, Food, Environment and Work (2015) recommends to apply the asbestos regulation for elongated mineral particles (L/D > 3:1, L > 5 μm, D < 3 μm) with chemical composition corresponding to one of the five regulated amphibole species, irrespective of their mode of crystallization (asbestiform or non-asbestiform).

The upcoming regulatory changes are part of a decree published in 2017, including the prior identification of asbestos in natural soils or rocks likely to be impacted by the execution of work. Specific protocols will be defined for sampling, analysis and characterization of natural materials that may contain asbestos.