International audienceSuspensions involving nanoparticules - in particular nano bers and nanotubes - are in much use in the development of functional materials. Thus in order to optimize the usage of these materials and their fabrication, it is essential to have a thorough knowledge of the microstructure and its evolution. In this work, the objective is to develop a two-scale kinetic theory description of concentrated suspensions including the modelling of nanotube aggregates and their evolution

Abisset-Chavanne, Emmanuelle

Ausias, Gilles

Chinesta, Francisco

Ferec, Julien

Mezher, Rabih

HAL-Univ-Nantes

On the modelling of the aggregates’ elasticity in aconcentrated suspension of CNTsEmmanuelle Abisset-Chavanne, Rabih Mezher, Julien Ferec, Gilles Ausias,Francisco ChinestaTo cite this version:Emmanuelle Abisset-Chavanne, Rabih Mezher, Julien Ferec, Gilles Ausias, Francisco Chinesta.On the modelling of the aggregates’ elasticity in a concentrated suspension of CNTs. 2ndECCOMAS Young Investigators Conference (YIC 2013), Sep 2013, Bordeaux, France. <hal-00855860>HAL Id: hal-00855860https://hal.archives-ouvertes.fr/hal-00855860Submitted on 30 Aug 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.YIC2013Second ECCOMAS Young Investigators Conference2{6 September 2013, Bordeaux, FranceOn the modelling of the aggregates' elasticity in a concentratedsuspension of CNTsE. Abisset-Chavanne a;, R. Mezher a, J. Ferec b, G. Ausias b, F. Chinesta aa GeM-Ecole Centrale Nantes1, rue de la Noe44000 Nantes, Franceb LIMATB-Universit Europenne de Bretagnerue de St. Maud56325 Lorient, Franceemmanuelle.abisset-chavanne@ec-nantes.frAbstract. Suspensions involving nanoparticules - in particular nanobers and nanotubes - are in muchuse in the development of functional materials. Thus in order to optimize the usage of these materialsand their fabrication, it is essential to have a thorough knowledge of the microstructure and its evolution.In this work, the objective is to develop a two-scale kinetic theory description of concentrated suspensionsincluding the modelling of nanotube aggregates and their evolution.Keywords: Kinetic theory; Carbon nanotubes; Concentrated suspension.1 INTRODUCTIONIn order to improve the mechanical and thermal performances of a uid, a solution is to introduce micro/nanoparticles that change its behaviour and properties. A challenge is then to be able to predict themacro properties of the modied uid that depend on the particles type, their concentration and the owinduced microstructure evolution. In this work, the focus is made on the modelling of high concentratedsuspensions.Indeed, depending on the concentration, the suspensions involving rods present dierent morphologies.When the concentration is dilute enough, the microstructure can be described by tracking a populationof rods that move with the suspending uid and orient depending on the velocity gradient according tothe Jeery's equation [1]. In that case the motion and orientation of each bre is assumed decoupled fromthe others. When the concentration increases, bre-bre interactions occur leading to the formation ofclusters of entangled particles. The objective here is to derive a model that describes this microstructureand its evolution.To do so, the model used is dened at the mesoscale, between (ne) micro and (fast) macro descriptions,and developed in the framework of kinetic theory [2]. The approach used to build it can be summed up inthree points: The microstructure is dened at the microscale, via the introduction of dierent conformationalcoordinates (particle orientation, aspect ratio, disaggregation ratio...). Starting from this description, a distribution function  { and the associated conformation ten-sor a( ) { is dened at the mesoscale, representing the fraction of particles exhibiting a certainconformation present in the suspension at a certain position and a certain time. The conservation equation of  and a( ) are then written in order to describe the evolution of themicrostructure during the suspension ow.2 E. Abisset-Chavanne and al. j Young Investigators Conference 2013This approach allows then to address the system at the macroscale while keeping the ne physics througha number of conformational coordinates introduced to describe the microstructure and its time evolution.In this work, the principle of the microstructure modelling is recalled. A focus is then made on theintroduction of the relaxation in the conformation through an apporpriate term in the equation governingits time evolution.2 TWO-SCALES MODELLING OF CONCENTRATED SUSPENSIONSAt the microscale, each nanotube is modelled as a rigid segment joining two opposite beads. Its orientationis dened by the unit vector p and all the forces applied on it are assumed acting on both beads.When the concentration of the suspension is high enough, clusters composed of entangled rods are observedexperimentally. In two dimensions, these clusters can be sketched as entangled aggregates of N rods.In this structure, a rod is submitted to two kinds of forces: the hydrodynamic one due to the surroundinguid, and one due to the rods entanglement. The latter is assumed scaling with the dierence between therigid motion velocity (the one that the rod would have if the cluster would be rigid) and the real one.Finally the moment equilibrium of the cluster leads to the following rotary velocity for a rod involved ina cluster:_p =11 + _pJ +1 + _pR (1)where _pJ stands for the Jeery's rotary velocity dened by:_pJ = rv  p  (pT  rv  p)  p (2)and _pR stands for the rigid cluster rotary velocity dened by:_pR =W  p = (+ a D D  a)  p (3) is the \aggregation ratio" and represents the competition between the eect of the hydrodynamic forceand the one due to the rod entanglement. Indeed, if  << 1, the hydrodynamic forces eect is larger thanthe entanglement one, the cluster tends to disaggregate in the ow. A rod of the cluster tends then tobehave as if it was alone (Jeery's velocity). On the contrary, if  >> 1, the entanglement eect is muchlarger than the one due to hydrodynamic forces, the cluster is completely aggregated and behaves as arigid one.Starting from this microscopic description, we can derive a mesoscopic description of the clusters charac-terised by the distribution function  (x; t;p), representing the rod distribution constituting the cluster.The second order moment time derivative of the associated conformation tensor a( ), which controls themicrostructure evolution, writes:_a =11 + _aJ +1 + _aR (4)where_aJ = rv  a+ a  (rv)T   2a(4) : rv and _aR =W  a+ a WT (5)This model allows to represent the kinematic of the aggregates during the suspension ow and gives goodresults when compared to direct mechanical simulations at the microscopic scale. However, in order to beable to compare it to real rheological experiments, we have to introduce the relaxation due to the clusters'elasticity, responsible to the noticed suspension elasticity.3 RELAXATION: INTRODUCTION OF THE AGGREGATES' ELASTIC-ITYThe experimental observations show that, during a simple shearing ow, the aggregates not only rotatewith the ow but also undergo a stretching, and come back to an equilibrium conformation once the owE. Abisset-Chavanne and al. j Young Investigators Conference 2013 3stops. In order to take into account this \elasticity" of the aggregate, we introduce an extra term in theevolution equation that tends to bring the cluster back to an equilibrium conformation ar:_a =11 + _aJ +1 + _aR  K (a  ar) (6)The equilibrium conformation is dened starting from the history of the cluster deformation. Indeed, onecan easily understand that if the aggregate undergoes a short-time deformation, it comes back to its initialconformation. However, if the deformation state is prolonged, the cluster experiences internal recongura-tions and does not come back to its initial conformation but to one close to the deformed state. To modelthis behaviour, we introduce a stable conformation including fading memory function.The objective then, is to compare this model to experimental tests in order to validate it and to begin thereexion about its identication [3].4 CONCLUSIONSIn this work, a model at the mesoscopic scale has been developed to model the aggregates observed in thecase of concentrated CNTs suspension. This model is built using the kinetic theory that allows to includethe microstructure characteristics at the mesoscale without a signicant loss of information.In order to take into account the elasticity of such structures, a relaxation term has been added includingfading memory functions to mimic the internal reorganisation occurring during a prolonged deformed state.REFERENCES[1] Jeery, G.B.. The motion of ellipsoidal particles immersed in viscous uid. Proc. R. Soc. London A102:161-179, 1922.[2] Chinesta, F.. From Single-Scale to Two-Scales Kinetic Theory Descriptions of Rods Suspensions. Archives inComputational Methods in Engineering, 2012.[3] Ma, A., Mackley, M., Chinesta, F.. The microstructure and rheology of carbon nanotube suspensions Inter-national Journal of Material Forming 1:75-81, 2008.

On the modelling of the aggregates' elasticity in a concentrated suspension of CNTs

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On the modelling of the aggregates' elasticity in a concentrated suspension of CNTs

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