On the multiscale description of electrical conducting suspensions involving perfectly dispersed rods
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Abstract
Nanocomposites allow for a significant enhancement of functional properties, in particular electrical conduction. In order to optimize materials and parts, predictive models are required to evaluate particle distribution and orientation. Both are key parameters in order to evaluate percolation and the resulting electrical networks. Many forming processes involve flowing suspensions for which the final particle orientation could be controlled by means of the flow and the electric field. In view of the multiscale character of the problem, detailed descriptions are defined at the microscopic scale and then coarsened to be applied efficiently in process simulation at the macroscopic scale. The first part of this work revisits the different modeling approaches throughout the different description scales. Then, modeling of particle contacts is addressed as they determine the final functional properties, in particular electrical conduction. Different descriptors of rod contacts are proposed and analyzed. Numerical results are discussed, in particular to evaluate the impact of closure approximations needed to derive a macroscopic description.
Keywords
Nanocomposites Electrical properties Multiscale modeling FokkerPlanck equation Interaction tensorBackground
Nanocomposites composed of carbon nanotubes (CNTs) in a polymer matrix exhibit a significant enhancement of electrical conductivity, mechanical and thermal properties [1, 2]. Due to the large length to diameter aspect ratios (from 100 to 10,000), they create conducting networks at low volume fractions [3].
In many forming processes (injection, extrusion, among many others), however, the CNT flowinduced orientation can alter dramatically the effective properties [4]. Moreover, the flow can induce aggregation and disaggregation mechanisms that also affect the final properties of the processed part [5]. It is well known that extrusion [6] and injection processes [7] can in some cases cause a conductingtoinsulating transition.
An important goal is to develop robust processes that maximize both electrical conductivity and mechanical properties, which asks for a suitable compromise in terms of flowinduced microstructure. There are many works focusing on the effects of shear rate [8], network structure [9, 10, 11], extensional flow [12], CNT orientation [13] and the resulting properties [14].
An important recent observation is that the flowinduced concentration of CNTs is not uniform [15, 16, 17, 18, 19]. Indeed, CNTs have the tendency to aggregate (in what follows we do not make distintion between agglomeration and aggregation, we use the last term because it was the one considered in our former works). As a result, all properties and mechanisms must be reformulated in the context of suspensions and networks involving clusters composed of CNTs, instead of considering a population of perfectly distributed isolated CNTs [5].
In our recent studies [20, 21, 22], we have proposed kinetic theory models to predict the kinematics and rheology of either rigid or deformable aggregates of CNTs in a Newtonian fluid matrix. In addition to providing a simple description of a rich microstructure, the proposed models are able to point out collective effects that are observed experimentally [23]. In these studies, we considered hydrodynamic effects only. Electrical mechanisms are discussed in the present paper.
In terms of modeling and simulation, two different approaches are usually adopted to take account of electrical effects. The simplest one considers a given network and proceeds to evaluate its direct current (DC) electrical properties. To that purpose, three main steps are followed: (1) generation of the composite’s microstructure, (2) creation of an equivalent resistance network corresponding to this microstructure, and (3) calculation of this network in a continuous or discrete manner [24].
The above approach allows for a detailed analysis of the electrical properties for a given microstructure, which usually remains “frozen”.
The second approach consists in predicting the network itself, and then carrying out the electrical analysis. In this case, it is usual to proceed at the mesoscopic scale using methods of Dissipative Particle Dynamics (DPD) for describing packed assemblies of oriented fibers suspended in a viscous medium [25]. Computer simulations are performed in order to explore how the aspect ratio and degree of fiber alignment affect the critical volume fraction percolation threshold required to achieve electrical conductivity. The fiber network impedance is assessed using Monte Carlo simulations after establishing the structural arrangement with DPD. Thus, these simulations allow one to predict the microstructure (CNT dispersion and aggregation), and, even though most such simulations do not consider the flow coupling, there are no major difficulties to include it as well. Computational micromechanics approaches for modeling effective conductivity in CNTnanocomposites in general were addressed in [26, 27].
The main limitations of these two common approaches are that (1) they concern a computational domain that is only representative of a small region of the whole process and part, and (2) they analyze a particular configuration, which implies many individual solutions in order to perform a valuable statistical treatment of the results.
In the present work, we propose an alternative approach to evaluating electrical properties in flowing suspensions of perfectly dispersed CNTs in a Newtonian fluid (the dispersion is ensured by assuming an appropriate functionalization). Starting from a microscopic description, we derive both mesoscopic and macroscopic descriptions. The main advantage of mesoscopic models is their ability to address systems of macroscopic size, while keeping track of the detailed physics through a number of conformational coordinates for describing the microstructure and its time evolution. At the mesoscopic scale, the microstructure is defined by means of the orientation distribution function that depends on physical space, time and CNT orientation. The moments of this distribution constitute a coarser description often used in macroscopic modeling, at the cost of compulsory closure approximations whose impact on the results is either ignored or unknown. Finally, the modeling of particle contacts is addressed as they determine the final functional properties, in particular electrical conduction. Different descriptors of rod contacts will be proposed and analyzed.
Orientation induced by the electric and flow fields
In this section, we first give the equation governing the orientation of a rod immersed in a Newtonian fluid of viscosity \(\eta \) in presence of an electric field \(\varvec{\epsilon }(\mathbf {x},t)\) and a velocity field \(\mathbf {v}(\mathbf {x},t)\). Then, the proposed model will be coarsened for describing a population of rods within the framework of kinetic theory. Finally, a macroscopic model will be derived.
Microscopic description
We consider a suspending medium consisting of a Newtonian fluid in which are suspended N rigid slender rods (e.g. CNTs) of length 2L. As a first approximation, the fiber presence and orientation are considered not to affect the flow kinematics defined by the velocity field \(\mathbf {v}(\mathbf {x},t)\), with \(\mathbf {x} \in \varOmega \in \mathbb {R}^3\).
The microstructure is described at the microscopic scale by the unit vector defining the orientation of each rod, i.e. \(\mathbf {p}_i\), \(i=1,\ldots , N\). In absence of electric field, one fiber can be defined by \(\mathbf {p}\) or \(\mathbf {p}\), which implies a symmetry property for the orientation distribution function. When considering the electric field induced charges, however, that symmetry is broken and the orientation is defined univocally. We assume that \(\mathbf {p}\) points from the negativelycharge bead to the positive one (Fig. 1).
If the suspension is dilute enough, rodrod interactions can be neglected and a micromechanical model can then be derived by considering a single generic rod whose orientation is defined by the unit vector \(\mathbf {p}\).
Equation (3) determines the suspension rheology whereas Eq. (5) governs the microstructure evolution.
Mesoscopic description
Because the rod population is very large, the description that we just proposed, despite of its conceptual simplicity, fails to address the situations usually encountered in practice. For this reason, coarser descriptions are preferred. The first plausible coarser description applies a zoomout, wherein the rod individuality is lost in favour of a probability distribution function [30, 31, 32, 33].
FokkerPlanck equation
Parametric solutions of the FokkerPlanck equation
The FokkerPlanck equation (9) is defined in a multidimensional space involving the physical space \(\mathbf {x}\), the time t and the conformational coordinates associated to the rod orientation \(\mathbf {p}\), with \(\dot{\mathbf {p}} \) given by Eq. (8).
One of the most appealing features of this technique is its ability of solving multidimensional models. Thus, with the PGD, the parameters of a physical model can be considered as extracoordinates, such that by solving only once the resulting multidimensional model one has access to the general parametric solution that can be used to evaluate the impact of each parameter on the solution (that is, for performing sensitivity analyses) or to perform fast inverse identification [44, 45, 46].
In 3D physical space, this formulation involves 18 dimensions. With the PGD, the computational complexity is associated with the solution of some 2D or 3D problems related to the calculation of the functions \(F_i^x(\mathbf {x})\), \(F^p_i(\mathbf {p})\) and \(F_i^E({\tilde{\mathbf {E}}})\), and a series of 1D problems for calculating the remaining functions involved in Eq. (12). The generic PGD solution procedure, that is, the constructor of the unknown lowdimensional functions involved in Eq. (12), is described in detail in the book [44].
Macroscopic description
FokkerPlanck based descriptions are rarely considered in industrial applications precisely because the computational complexity that high dimensionality induced by the use of conformation coordinates implies. For this reason, mesoscopic models are commonly coarsened one step further to obtain macroscopic models defined in standard physical domains, involving only space and time coordinates.
Evolution equations for the orientation moments
On closure relations
When considering both microstructure descriptors, \(\mathbf {a}^{(1)}\) and \(\mathbf {a}^{(2)}\), closure relations are needed for approximating the third and fourthorder moments, \(\mathbf {a}^{(3)}\) and \(\mathbf {a}^{(4)}\) respectively.
There is a vast choice of possible closure relations of the fourthorder tensor \(\mathbf {a}^{(4)}\) usually encountered in standard suspension models [48, 49, 50].
When there is no difference between \(\mathbf {p}\) and \(\mathbf {p}\) to describe rod orientation, the distribution function is symmetric and odd moments vanish. In the present case, the symmetry is broken and odd moments do not vanish, thus requiring appropriate closures.

The cubic term \(\mathbf {a}^{(1)} \otimes \mathbf {a}^{(1)} \otimes \mathbf {a}^{(1)}\) fulfills the symmetry relations (25), whereas \(\mathbf {a}^{(1)} \otimes \mathbf {a}^{(2)}\) and \(\mathbf {a}^{(2)} \otimes \mathbf {a}^{(1)}\) do not satisfy in general conditions (25);

Quadratic terms obtained from tensor products of \(\mathbf {a}^{(2)}\) and constant vectors or quadratic terms coming from the tensor product of \(\mathbf {a}^{(1)}\) twice and constant vectors do not verify in the general case the symmetry conditions;

Linear terms obtained from tensor products of \(\mathbf {a}^{(1)}\) and constant matrix do not verify in the general case the symmetry conditions;

If we define vectors \(\mathbf {I}_1^T=(1,0)\) and \(\mathbf {I}_2^T=(0,1)\), then tensors \(\mathbf {I}_1 \otimes \mathbf {I}_1 \otimes \mathbf {I}_1\) and \(\mathbf {I}_2 \otimes \mathbf {I}_2 \otimes \mathbf {I}_2\) verify the above mentioned symmetry conditions.
Description of the rod network
We have seen how to model and predict the microstructure induced by the electric field. We addressed in [28] the following questions: (1) how to compute the electric field \(\mathbf {E}(\mathbf {x},t)\), (2) how to evaluate the induced conductivity properties, and finally (3) how to determine preferential electrical paths in the computational domain of interest \(\varOmega \).
In what follows, the electric field is assumed known from the solution of the Laplace equation in the domain occupied by the suspension (see [28] for details). We focus here on the multiscale description of rod contacts.
Mesoscopic description
In order to quantify the rod network, we introduce the number density of rod contacts \(\mathcal {C}(\mathbf {x},t,\mathbf {p})\) for a rod with orientation \(\mathbf {p}\), depending on the two main microstructure descriptors: (1) the CNT concentration \(\phi (\mathbf {x},t)\), and (2) the orientation distribution \(\varPsi (\mathbf {x},t,\mathbf {p})\).
The microstructure description based on the use of the density of contacts \(\mathcal {C}(\mathbf {p})\) is very close to that obtained following the rationale proposed by Toll [51, 52] (in continuity with the works of Doi and Edwards [53] and Ranganathan and Advani [54]). In our approach, the effect related to the finite diameter of the rods is neglected, but its inclusion is straightforward.
The density of contacts \(\mathcal {C}(\mathbf {p})\) is appropriate because it constitutes a rich description of the microstructure. Its calculation, however, requires knowledge of the distribution function \(\varPsi (\mathbf {p})\) that is only available when operating at the mesoscale by solving the FokkerPlanck equation.
Towards a fullymacroscopic description
The question arising immediately concerns the existence of an appropriate fullymacroscopic description.
This route allows us to finally establish a link between our descriptor \(\mathcal {L}\) (32) and the standard orientation tensors, which allows for a fullymacroscopic description.
Now, from the knowledge of \(\mathcal {L}\), we can obtain the directional properties along direction \(\mathbf {p}\) by calculating \(( \mathbf {p}^T \cdot \mathcal {L} \cdot \mathbf {p} ) \ \mathbf {p}\), or equivalently \((\mathcal {L} : ( \mathbf {p} \otimes \mathbf {p}) ) \ \mathbf {p}\).
Numerical results
Evaluating the closure relations
We now evaluate and validate the different closure relations introduced above.
In all cases, the FokkerPlanck equation (9) is solved for the orientation distribution \(\varPsi (\mathbf {p},t)\) by means of the PGD. The initial orientation distribution is assumed isotropic. From \(\varPsi (\mathbf {p},t)\), the first and secondorder moments \(\mathbf {a}^{(1),FP}(t)\) and \(\mathbf {a}^{(2),FP}(t)\) are calculated, respectively from Eqs. (13) and (14). This procedure does not involve any closure relation. The solutions \(\mathbf {a}^{(1),FP}(t)\) and \(\mathbf {a}^{(2),FP}(t)\) can thus be considered as reference solutions.
The same moments are now computed via Eqs. (40) and (41) using closure approximations. When integrating Eq. (40) to obtain \(\mathbf {a}^{(1)}(t)\), we take the secondorder moment from the solution of Eq. (41) and the thirdorder moment is approximated by using the closure (57). When integrating Eq. (41) to obtain \(\mathbf {a}^{(2)}(t)\), we use the closure (58) for the thirdorder moment and the standard hybrid closure (42) for the fourthorder moment.

Shear flow: \(G_{xx}=0\), \(G_{xy}=1\) and \(G_{yy}=0\) with \(D_r=0.1\) and \(D_r=0.01\).

Extensional flow: \(G_{xx}=1\), \(G_{xy}=0\) and \(G_{yy}=1\) with \(D_r=0.1\) and \(D_r=0.01\).
In Figs. 7 and 8, the approximate closure solutions \(\mathbf {a}^{(1)}\) and \(\mathbf {a}^{(2)}\) are compared with the reference solutions \(\mathbf {a}^{(1),FP}\) and \(\mathbf {a}^{(2),FP}\) obtained from the direct solution of the FokkerPlanck equation. Agreement is in general quite satisfactory, but the development of even more precise closures will require a considerable amount of work.
It is well known that closure relations can induce artifacts, as for example the disappearance of hysteric behaviour in reversal flows. To check the model response in such conditions, we apply a simple shear flow defined by \(\mathbf {v}^T = (\dot{\gamma }y,0,0) \) for \(t \in [0,T=5s];\) the flow is reversed for \(t \in [T,2T]\) by changing the sign of \(\dot{\gamma }\), and it is again reversed in subsequent time intervals. In other words, the shear flow is reversed at times \(t=n T\), \(n=1,2,\ldots \)
Parametric solution of the FokkerPlanck equation
When controlling the applied flow and electric fields, one could require many solutions of the model in order to reach optimal conditions with respect to an output of interest. In these circumstances, the parametric solution of the model would be extremely valuable. As the PGD parametric solution of the FokkerPlanck equation is performed offline and only once, it is indeed well worth the effort [44], despite of the necessity of solving a highdimensional problem, involving the physical coordinates (space and time), the conformational coordinates (orientation) and a number of extracoordinates (model parameters) as described in "Parametric solutions of the FokkerPlanck equation".
In what follows, we consider a homogeneous simple shear flow \(\mathbf {v}^T=(\dot{\gamma }y,0,0)\), \(\dot{\gamma }=1\). Its homogeneity avoids the dependence of the orientation distribution function \(\varPsi \) on the space coordinates \(\mathbf {x}\). We assume planar orientation, with a single conformational coordinate to describe it, i.e. the angle \(\theta \) with respect the \(x\)coordinate axis. An electric field of intensity E is applied along the \(y\)coordinate axis, i.e. \(\mathbf {E}= E \mathbf {i}_y\) (with \(\mathbf {i}_y\) the unit vector defining the \(y\)coordinate axis direction). In this section, \(\mathbf {E}\) refers to the effective electric field denoted by \(\tilde{\mathbf E}\) in "Parametric solutions of the FokkerPlanck equation".
For more details on the practical implementation of the PGD for solving parametric PDE’s, see [44, 45, 46].
Evaluating interaction descriptors
Finally, we check whether the different interaction descriptors discussed in "Description of the rod network" perform as expected.
Conclusions
In this work, we have revisited the microstructural kinematics of electrically conductive rods immersed in a flowing suspension to which an electric field is applied. The evolution equation for the microscopic rod orientation was derived by considering standard balances of forces and moments. These kinematics were then introduced into a mesoscopic description based on the FokkerPlanck equation, whose parametric solution was formulated within the separated representation framework at the heart of the method of Proper Generalized Decomposition. The mesoscopic description was coarsened one step further in order to derive a fully macroscopic description based on the use of the first or the first two moments of the orientation distribution. Both approaches require the use of appropriate closure relations. In the present work, we proposed consistent closures for the thirdorder orientation tensor, which despite being non optimal produce results that are in reasonable agreement with reference solutions obtained directly from the FokkerPlanck description.
On the other hand, prediction of the electrical properties of the suspension as a whole requires the quantification of the rod interaction. We quantified the directional conductivity by calculating the directional density of contacts that depends on the orientation distribution and the local rod concentration. This approach, already considered in our former work [28], requires however the mesoscopic calculation of the orientation distribution. In this work, we proposed an alternative fullymacroscopic route. For that purpose, we defined an interaction tensor \(\mathcal {L}\) which by construction is equivalent to the interaction tensor proposed by Ferec in [55]. As the latter can be approximated with reasonable accuracy from the orientation tensors of even order, a fully macroscopic approach is thus finally obtained.
Notes
Authors’ contributions
MP carried out most of the numerical simulations and contributed to most of the works. EACh extended standard multiscale rood suspensions in presence of electrical fields; AB developed models related to electrical conduction; AA applied the PGD for the solution of the parametric FokkerPlanck equation and FCh and RK supervised the works and elaborated the manuscript. All authors read and approved the final manuscript.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
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