Towards a kinetic theory description of electricalconduction in perfectly dispersed CNT
nanocomposites
M. Perez⇤, E. Abisset-Chavanne⇤, A. Barasinski⇤,A. Ammar⇤⇤, F. Chinesta⇤ & R. Keunings⇤⇤⇤
⇤ Ecole Centrale de Nantes, France⇤⇤ ENSAM, Angers, France & UMSSDT, ENSIT Tunis, Tunisie
⇤⇤⇤ Université catholique de Louvain, Belgium
February 24, 2015
2
Contents
0.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110.2. Orientation induced by the electric field . . . . . . . . . . . . . . . . . . 12
0.2.1. Microscopic description . . . . . . . . . . . . . . . . . . . . . . . . 120.2.2. Mesoscopic description . . . . . . . . . . . . . . . . . . . . . . . . 150.2.3. Macroscopic description . . . . . . . . . . . . . . . . . . . . . . . . 16
0.3. Introducing randomizing mechanisms . . . . . . . . . . . . . . . . . . . 200.4. Proper Generalized Decomposition and parametric solutions . . . . . . 220.5. Electrical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
0.5.1. Evaluating the electric field . . . . . . . . . . . . . . . . . . . . . . 240.5.2. Evaluating the electrical properties . . . . . . . . . . . . . . . . . . 250.5.3. Determining electrical paths . . . . . . . . . . . . . . . . . . . . . 26
0.6. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280.6.1. Fluid at rest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
0.6.1.1. Microscopic rod kinematics with Dr = 0 and constant elec-tric field E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
0.6.1.2. Kinetic theory model with Dr 6= 0 and constant electricfield E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
0.6.1.3. Macroscopic modeling with constant electric field E . . . . 320.6.1.4. Solution of the electrostatic problem for obtaining E(x) . . 320.6.1.5. Calculation of the steady-state density of contacts C(x,', ✓) 340.6.1.6. Construction of the electrical network . . . . . . . . . . . . . 35
0.6.2. Introducing the fluid flow . . . . . . . . . . . . . . . . . . . . . . . 35
9
0.1. Introduction
Nanocomposites composed of carbon nanotubes – CNTs – in a polymer matrixexhibit a significant enhancement of electrical conductivity, mechanical and thermalproperties [COL06] [XU06]. Due to the large length to diameter aspect ratios (from100 to 10.000), they create conducting networks at low volume fractions [OUN03].
In many forming processes (injection, extrusion, etc.), however, the CTNs’ flow-induced orientation can alter dramatically the effective properties [MA09]. Moreover,the flow can induce aggregation and disaggregation mechanisms that also affect the fi-nal properties of the processed part [MA08b]. It is well known that extrusion [HWA10]and injection processes [VIL08] can in some cases cause a conducting-to-insulatingtransition.
An important goal is to develop robust processes that maximize both electricalconductivity and mechanical properties, which asks for a suitable compromise interms of flow-induced microstructure. There are many works focusing on the effectsof shear rate [OBR07], network structure [EKE11] [BAU10] [KAS07], extensionalflow [HAG06], CNT orientation [ABB10] and the resulting properties [ALI08].
An important recent observation is that the flow-induced concentration of CNTsis not uniform [ALI12] [BAU09] [KHA04] [SCH97] [MA08]. Indeed, CNTs havethe tendency to aggregate. As a result, all properties and mechanisms must be re-formulated in the context of suspensions and networks involving clusters composedof CNTs, instead of considering a population of perfectly distributed isolated CNTs[MA08b].
In our recent studies [CHI13] [ABI13] [ABI14], we have proposed kinetic theorymodels to predict the kinematics and rheology of either rigid or deformable aggregatesof CNTs in a Newtonian fluid matrix. In addition to providing a simple description ofa rich microstructure, the proposed models are able to point out collective effects thatare observed experimentally [PET00]. In these studies, we considered hydrodynamiceffects only. Electrical mechanisms are discussed in the present chapter.
In terms of modeling and simulation, two different approaches are usually adoptedto take account of electrical effects. The simplest one considers a given network andproceeds to evaluate its direct current (DC) electrical properties. To that purpose, threemain steps are followed: (i) generation of the composite’s microstructure, (ii) creationof an equivalent resistance network corresponding to this microstructure, and (iii) cal-culation of this network in a continuous or discrete manner [DAL06].
The above approach allows for a detailed analysis of the electrical properties for agiven microstructure, which usually remains "frozen".
10
Contents 11
The second approach consists in predicting the network itself, and then carryingout the electrical analysis. In this case, it is usual to proceed at the mesoscopic scaleusing methods of Dissipative Particle Dynamics – DPD – for describing packed as-semblies of oriented fibers suspended in a viscous medium [RAH05]. Computer sim-ulations are performed in order to explore how the aspect ratio and degree of fiberalignment affect the critical volume fraction percolation threshold required to achieveelectrical conductivity. The fiber network impedance is assessed using Monte Carlosimulations after establishing the structural arrangement with DPD. Thus, these sim-ulations allow one to predict the microstructure (CNT dispersion, aggregation, etc.),and, even though most such simulations do not consider the flow coupling, there areno major difficulties to include it as well.
The main limitations of these two common approaches are that (i) they concerna computational domain that is only representative of a small region of the wholeprocess and part, and (ii) they analyze a particular configuration, which implies manyindividual solutions in order to perform a valuable statistical treatment of the results.
In the present work, we propose an alternative approach to evaluating electricalproperties in flowing suspensions of perfectly dispersed CNTs in a Newtonian fluid.Starting from a microscopic description, we derive both mesoscopic and macroscopicdescriptions. The main advantage of mesoscopic models is their ability to addresssystems of macroscopic size, while keeping track of the detailed physics through anumber of conformational coordinates for describing the microstructure and its timeevolution. At the mesoscopic scale, the microstructure is defined by means of theorientation distribution function that depends on physical space, time and CNT ori-entation. The moments of this distribution constitute a coarser description often usedin macroscopic modeling, at the cost of compulsory closure approximations whoseimpact on the results is either ignored or unknown.
0.2. Orientation induced by the electric field
In this section, we first give the equation governing the orientation of a rod im-mersed in a Newtonian fluid of viscosity ⌘ in presence of an electric field ✏(x, t) anda velocity field v(x, t). Then, the proposed model will be coarsened for describinga population of rods within the framework of kinetic theory. Finally, a macroscopicmodel will be derived.
0.2.1. Microscopic description
We consider a suspending medium consisting of a Newtonian fluid in which aresuspended N rigid slender rods (e.g. CNTs) of length 2L. As a first approximation, thefiber presence and orientation are considered not to affect the flow kinematics definedby the velocity field v(x, t), with x 2 ⌦ 2 R3.
12 Electrical conduction in perfectly dispersed CNT nanocomposites
The microstructure is described at the microscopic scale by the unit vector definingthe orientation of each rod, i.e. pi, i = 1, · · · , N . In absence of electric field, one fibercan be defined by p or �p, which implies a symmetry property for the orientationdistribution function. When considering the electric field induced charges, however,that symmetry is broken and the orientation is defined univocally. We assume that ppoints from the negatively-charge bead to the positive one (Fig. 1).
If the suspension is dilute enough, rod-rod interactions can be neglected and amicro-mechanical model can then be derived by considering a single generic rodwhose orientation is defined by the unit vector p.
In what follows, the following notation is used (assuming Einstein summation con-vention):
– if a and b are first-order tensors, then the single contraction "·" reads (a · b) =aj bj ;
– if a and b are first-order tensors, then the dyadic product "⌦" reads (a⌦b)jk =aj bk;
– if a and b are respectively second and first-order tensors, then the single con-traction "·" reads (a · b)j = ajk bk;
– if a and b are respectively third and first-order tensors, then the single contrac-tion "·" reads (a · b)jk = ajkm bm;
– if a and b are second-order tensors, then the single contraction "·" reads (a ·b)jk = ajm bmk;
– if a and b are second-order tensors, then the double contraction ":" reads (a :b) = ajk bkj ;
– if a and b are respectively third and second-order tensors, then the double con-traction ":" reads (a : b)j = ajkm bmk;
– if a and b are respectively fourth and second-order tensors, then the double con-traction ":" reads (a : b)jk = ajkml blm.
We thus consider the system illustrated in Fig. 1, that consists of a rod immersedinto a fluid flow, with two beads at the extremities that have respectively an electri-cal charge +q and �q induced by the electric field. Hydrodynamic forces are alsoassumed to act on both beads. Thus, the resultant force acting on bead pL reads
F(pL) = E+ ⇠L (rv · p� ˙p) , [1]
where ⇠ is the friction coefficient and E ⌘ q✏. It has been assumed that the velocitygradient is constant at the scale of the rod (first-gradient theory).
Contents 13
Figure 1. Hydrodynamic and electrostatic forces applied on a rod immersed in a Newtonianfluid
Obviously, if F is applied on bead pL, then for the opposite bead �pL the resul-tant force reads
F(�pL) = �E� ⇠L (rv · p� ˙p) . [2]
Thus, neglecting inertia, the linear momentum balance is ensured since F(�pL) =�F(pL). As the resulting torque must also vanish, the only possibility is that the forceF acts along p, that is F = �p, with � 2 R. Thus, we can write
�p = E+ ⇠L (rv · p� ˙p) . [3]
Premultiplying Eq. [3] by p and taking into account that p · p = 1 and consequentlyp · ˙p = 0, we obtain:
� = E · p+ ⇠L�p
T ·rv · p�, [4]
implying
F = (E · p)p+ ⇠L�p
T ·rv · p�p, [5]
14 Electrical conduction in perfectly dispersed CNT nanocomposites
and
˙
p =
1
⇠L(E� (E · p)p) +
�rv · p�
�p
T ·rv · p�p
�. [6]
This can be rewritten as:
˙
p =
1
⇠L(I� p⌦ p) ·E+
�rv · p�
�p
T ·rv · p�p
�=
˙
p
E+
˙
p
J , [7]
where ˙pE refers to the rotary velocity due to the electrostatic forces and ˙pJ representsthe hydrodynamic contribution that coincides with the Jeffery solution for infinite as-pect ratio ellipsoids [JEF22].
0.2.2. Mesoscopic description
Because the rod population is very large, the description that we just proposed, de-spite of its conceptual simplicity, fails to address the situations usually encountered inpractice. For this reason, coarser descriptions are preferred. The first plausible coarserdescription applies a zoom-out, wherein the rod individuality is lost in favour of aprobability distribution function [BIR87] [DOI87] [PET99] [KEU04].
In the case of rods, one can describe the microstructure at a certain point x andtime t from the orientation distribution function (x, t,p) which gives the fraction ofrods that at position x and time t are oriented in direction p. Obviously, the function satisfies the normality condition:
Z
S (x, t,p) dp = 1, 8x, 8t, [8]
where S is the surface of the unit ball that defines all possible rod orientations.
The balance equation that ensures conservation of probability conservation reads:
@
@t+
@
@x(
˙
x ) +
@
@p(
˙
p ) = 0. [9]
Contents 15
For inertialess rods in first-gradient flows, we have ˙x = v(x, t) and the rod rotaryvelocity is given by Eq. [7]:
˙
p =
1
⇠L(I� p⌦ p) ·E+
�rv · p�
�p
T ·rv · p�p
�. [10]
Equation [9], combined with Eq. [10], is known as a Fokker-Planck equation. It is asuitable compromise between the macroscopic scale that defines the overall process,and a finer microscopic description of the electric field induced orientation of eachindividual rod. The price to pay is the increase of the model dimensionality, sincethe orientation distribution is defined in a high-dimensional domain, i.e. (x, t,p) 2⌦⇥ I ⇥ S .
0.2.3. Macroscopic description
Fokker-Planck based descriptions are rarely considered in industrial applicationsprecisely because of the curse of dimensionality that the introduction of conformationcoordinates (the rod orientation in the case considered here) implies. Indeed, standardmesh-based discretization techniques, such as finite differences, finite elements or fi-nite volumes, fail when addressing models defined in high-dimensional spaces. Forthis reason, mesoscopic models are commonly coarsened one step further to obtainmacroscopic models defined in standard physical domains, involving only space andtime coordinates.
In this section, we illustrate the transition from the mesoscopic to the macroscopicscale. At the macroscopic scale, the orientation distribution function is substituted byits moments for describing the microstructure, as proposed in [ADV87].
We consider the first to fourth orientation moments a(1), a(2), a(3) and a(4) re-spectively, defined as:
a
(1)=
Z
Sp dp, [11]
a
(2)=
Z
Sp⌦ p dp, [12]
a
(3)=
Z
Sp⌦ p⌦ p dp, [13]
16 Electrical conduction in perfectly dispersed CNT nanocomposites
and
a
(4)=
Z
Sp⌦ p⌦ p⌦ p dp. [14]
Remark 1. In view of the non-symmetry of , the odd moments of the distributionfunction do not vanish.
The time derivative of a(1) reads
˙
a
(1)=
Z
S˙
p dp =
Z
S
�˙
p
E+
˙
p
J� dp =
1
⇠L
Z
S(I� p⌦ p) ·E dp+
Z
S
�rv · p�
�p
T ·rv · p�p
� dp =
1
⇠L
⇣E� a(2) ·E
⌘+rv · a(1) � a(3) : rv. [15]
In the previous expression, a closure issue appears because of a(2) and a(3). A closurerelation is thus needed in order to express the second and third-order moments as afunction of the first moment.
Should we try to avoid a closure relation by considering the time derivative of a(2),we obtain
˙
a
(2)=
Z
S(
˙
p⌦ p+ p⌦ ˙p) dp =
Z
S
��˙
p
E+
˙
p
J�⌦ p+ p⌦
�˙
p
E+
˙
p
J�� dp =
Z
S
�˙
p
E ⌦ p+ p⌦ ˙pE� dp+
Z
S
�˙
p
J ⌦ p+ p⌦ ˙pJ� dp = ˙aE + ˙aJ , [16]
Contents 17
where
˙
a
E=
Z
S
�˙
p
E ⌦ p+ p⌦ ˙pE� dp =
1
⇠L
Z
S(((I� p⌦ p) ·E)⌦ p+ p⌦ ((I� p⌦ p) ·E)) dp =
1
⇠L
⇣E⌦ a(1) + a(1) ⌦E� 2a(3) ·E
⌘, [17]
and
˙
a
J=
Z
S
�˙
p
J ⌦ p+ p⌦ ˙pJ� dp =
Z
S
�rv · p�
�p
T ·rv · p�p
�⌦ p dp+
Z
Sp⌦
�rv · p�
�p
T ·rv · p�p
� dp =
rv · a(2) + a(2) · (rv)T � 2a(4) : rv. [18]
We notice that the issue persists: a closure relation is needed in order to express thethird and fourth-order moments a(3) and a(4) as a function of the lower-order mo-ments.
In what follows, we consider only the first route, i.e. we considers a(1) as the singlemicrostructural descriptor. The simplest closure relation that we could consider is thequadratic one:
a
(2),qcr ⇡ a(1) ⌦ a(1)
tr�a
(1) ⌦ a(1)� , [19]
18 Electrical conduction in perfectly dispersed CNT nanocomposites
where tr (•) denotes the trace of (•), and that only applies if tr�a
(1) ⌦ a(1)�6= 0.
The quadratic closure relation [19] becomes exact when rods fully align in a certaindirection. For representing isotropic orientation distributions, we add a new term toEq. [19] that results in the hybrid closure relation:
a
(2),hyb ⇡ �I+ �a(2),qcr. [20]
If we consider
� =1
d
✓1�
⇣a
(1) · a(1)⌘2◆
, [21]
with d = 2, 3 in 2D and 3D respectively, and
� =⇣a
(1) · a(1)⌘2
, [22]
we obtain as shown later a reasonable agreement with results computed from the prob-ability distribution function (i.e. without any closure relation). It is easy to prove thatthe above hybrid closure approximation is exact in the case of isotropic distributions aswell as when rods fully align along a certain direction. Many other possible and moreaccurate closure relations could be elaborated, but in the present work we only con-sider the hybrid closure [20]. Finally, for the third-order moment, we use the closureapproximation
a
(3),hyb ⇡ a(2),hyb ⌦ a(1). [23]
Thus, we obtain the following closed-form macroscopic evolution equation
˙
a
(1)=
1
⇠L
⇣I� a(2),hyb
⌘·E+rv · a(1) � a(3),hyb : rv, [24]
which is written, via Eqs [19-23], in terms of only the first-order moment a(1) of theorientation distribution function.
Contents 19
0.3. Introducing randomizing mechanisms
The main issue of the model presented above is that it tends to predict full align-ment of rods, that is
p(t ! 1) = ˆp, [25]
and
a
(1)(t ! 1) = ˆp, [26]
where ˆp is a particular orientation that results from the competition between the elec-tric field and the hydrodynamic drag.
In actual flowing systems, full alignment is prevented by rod-rod interactions.Moreover, in the case of dilute suspensions of CNTs, Brownian forces have a random-izing effect. These effects can be accurately described collectively with a diffusionterm in the Fokker-Planck equation:
@
@t+
@
@x(v ) +
@
@p(
˙
p ) =
@
@p
✓Dr
@
@p
◆, [27]
where the diffusion coefficient Dr quantifies the randomizing nature of rod-rod inter-actions and Brownian forces. In semi-concentrated suspensions, the diffusion mecha-nisms scale with the shear rate [FOL84].
The rotary velocity is given again by
˙
p =
1
⇠L(I� p⌦ p) ·E+
�rv · p�
�p
T ·rv · p�p
�. [28]
By moving the right-hand side of Eq. [27] to the left one, we get
@
@t+
@
@x(v ) +
@
@p
✓˙
p �Dr@
@p
◆= 0, [29]
20 Electrical conduction in perfectly dispersed CNT nanocomposites
which yields the Fokker-Planck equation [27] in the equivalent form
@
@t+
@
@x(v ) +
@
@p
⇣˙
˜
p
⌘= 0. [30]
Here, the equivalent rotary velocity ˙˜p is given by:
˙
˜
p =
1
⇠L(I� p⌦ p) ·E+
�rv · p�
�p
T ·rv · p�p
��Dr
@ @p
, [31]
which is the microscopic description of the rod rotary velocity in presence of isotropicdiffusion effects.
Now coming back to the macroscopic scale and following the rationale consideredpreviously, we find that the evolution of the first moment of the orientation distributionfunction is governed by
˙
a
(1)=
1
⇠L
⇣E� a(2) ·E
⌘+rv · a(1) � a(3) : rv �Dr a(1). [32]
In the absence of electric field, E = 0 and of flow, v(x, t) = 0, we obtain a steady-state isotropic distribution, i.e. a(1)(t ! 1;E = 0;v = 0) = 0.
The evolution equation of the second moment of the orientation distribution func-tion reads
˙
a
(2)=
1
⇠L
⇣E⌦ a(1) + a(1) ⌦E� 2a(3) ·E
⌘+
rv · a(2) + a(2) · (rv)T � 2 · a(4) : rv � 2dDr✓a
(2) � Id
◆, [33]
where d = 2 in 2D and d = 3 in 3D. The fourth-order tensor a(4) requires the use ofappropriate closures [ADV90] [DUP99] [KRO08].
Contents 21
0.4. Proper Generalized Decomposition and parametric solutions
The Fokker-Planck equation [27] is defined in a multidimensional space involvingthe physical space x, the time t and the conformational coordinates associated to therod orientation p,
@
@t+
@
@x(v ) +
@
@p(
˙
p ) =
@
@p
✓Dr
@
@p
◆, [34]
with
˙
p =
1
⇠L(I� p⌦ p) ·E+
�rv · p�
�p
T ·rv · p�p
�. [35]
We proposed a few years ago [AMM06] [AMM07] a discretization techniquebased on the use of separated representations in order to ensure that the complex-ity scales linearly with the model dimensionality. This new approach is now knownas Proper Generalized Decomposition – PGD –. This technique consists in express-ing the unknown field as a finite sum of functional products, i.e. expressing a genericmultidimensional function u(x1, · · · , xd) as
u(x1, · · · , xd) ⇡i=NX
i=1
F 1i (x1) · · ·F di (xd). [36]
Here, both the one-dimensional functions F ji and the appropriate number of terms Nare unknown a priori. The interested reader can refer to [MOK07] [PRU09] [AMM10][AMM10b] [MOK10] [CHI10] [CHI11] [CHI11b] [CHI13b] [CHI14] and the refer-ences therein for a deep analysis of this technique and its applications in computationalrheology.
One of the most appealing features of this technique is its ability of solving mul-tidimensional models. Thus, with the PGD, the parameters of a physical model canbe considered as extra-coordinates, such that by solving only once the resulting mul-tidimensional model one has access to the general parametric solution that can beused to evaluate the impact of each parameter on the solution (that is, for performingsensitivity analyses) or to perform fast inverse identification.
In the case of the Fokker-Planck model [34-35] presented above, which only in-volves as coordinates the space x, the time t and the conformational coordinates p,
22 Electrical conduction in perfectly dispersed CNT nanocomposites
its solution depend parametrically on the following parameters: (i) the diffusion co-efficient Dr, (ii) the product of the rod size and the friction coefficient L⇠, (iii) theelectric field E and (iv) the components Gij of the velocity gradient rv, according to
rv =
0
@G11 G12 G13G21 G22 G23G31 G32 G33
1
A . [37]
Remark 2. In the absence of flow, Eq. [34] only depends on space through the possiblespatial dependence of the electric field E, i.e. E(x). That is, the Fokker-Planck modeldefines a local problem in physical space x. Thus, two routes could be envisaged:(i) solving model [34-35] at each position x (possibly at the nodes xi of a grid Gassociated to the domain ⌦) where the electric field is known E(xi) for calculating (t,p;x), and (ii) globalizing the model by considering x as a real coordinate insteadof a simple parameter as was proposed in [CHI10]. In what follows, we consider thesecond alternative.
Remark 3. When the electric field depends on space, only the parameters of an appro-priate approximation of it can be considered as model parameters. Thus, if we assumethe approximation of the electric field given by E(x) ⇡
PMi=1 �i ·Gi(x), the approx-
imation coefficients �i become the model parameters.
Within the PGD framework, and assuming a uniform electric field E, the paramet-ric orientation distribution (x, t,p, Dr, L⇠,E,rv) is written in a separated formfor circumventing the curse of dimensionality associated with standard mesh-baseddiscretization techniques. The simplest separated representation reads
(x, t,p, Dr, L,E,rv) ⇡
i=NX
i=1
F xi (x) · F ti (t) · Fpi (p) · F
Di (Dr) · FLi (L⇠) · FEi (E) · G11i (G11) · G12i (G12)·
G13i (G13)·G21i (G21)·G22i (G22)·G23i (G23)·G31i (G31)·G32i (G32)·G33i (G33)[38]
In 3D physical space, this formulation involves 20 dimensions. With the PGD, thecomputational complexity is associated with the solution of some 2D or 3D problemsrelated to the calculation of the functions F xi (x), F
pi (p) and F
Ei (E) and a series of
1D problems for calculating the remaining functions involved in Eq. [38]. The genericPGD solution procedure, that is, the constructor of the unknown low-dimensionalfunctions involved in Eq. [38], is described in detail in the book [CHI14].
Contents 23
0.5. Electrical properties
We have seen how to model and predict the microstructure induced by the electricfield. We now address the following questions: (i) how to compute the electric fieldE(x, t), (ii) how to evaluate the induced conductivity properties, and finally (iii) howto determine preferential electrical paths in the computational domain of interest ⌦.
0.5.1. Evaluating the electric field
Although the electric field is generally affected by the microstructure, that is, bythe presence and orientation of the CNTs, one can consider as a first approximationthat it remains unperturbed. In any case, the coupling of both is quite standard andrelies on the use of appropriate homogenization techniques.
For an isotropic and homogeneous medium, the electrostatic potential V(x, t) isgoverned by the Laplace equation
r2V(x, t) = 0, [39]
with x 2 ⌦. The time dependence of the potential V is due to the possible timedependence of the boundary conditions. Boundary conditions are prescribed on thedomain boundary � ⌘ @⌦, that is composed of two parts �N and �D such that �N [�D = � and �N \ �D = ;. Along �N , it is assumed that the normal derivative of thepotential V vanishes:
rV(x 2 �N , t) · n = 0, [40]
where n denotes the outward unit normal to �N . Along �D, the potential is prescribed:
V(x 2 �D, t) = Vg(x 2 �D, t). [41]
Once the electric potential V(x, t) is known, the electric field E(x, t) is computed asE(x, t) = �rV(x, t).
24 Electrical conduction in perfectly dispersed CNT nanocomposites
0.5.2. Evaluating the electrical properties
In order to quantify the electrical properties, we introduce the density of rod con-tacts C(x, t,p) for a rod with orientation p, depending on the two main microstructuredescriptors: (i) the CNT concentration �(x, t), assumed constant without loss of gen-erality, i.e. �(x, t) = � (a finer description requires defining and solving the associatedadvection-diffusion equation), and (ii) the orientation distribution (x, t,p).
When calculating the density of contacts, there is not difference between p and�p. Thus, we use the symmetrized orientation distribution function S(x, t,p) de-fined as follows:
S(x, t,p) =
(x, t,p) + (x, t,�p)2
. [42]
We consider the 3D case that allows for a more realistic visualization of the pro-cedure proposed for calculating the density of contacts, defined as C(x, t,p). For thesake of clarity, we omit all dependences on space x and time t.
The fraction of rods oriented in direction p is � S(p). The center of gravity ofall of these is located at position P . All rods having their center of gravity at positionQ inside the sphere of radius 2L centered at point P could interact with the former.
In fact, because of electronic tunneling effect, the contact between the fibers is notnecessarily needed. There is a minimum distance, �, from which the electrical con-ductivity occurs. The interaction between rods aligned in directions p and p0 whosecenters of gravity are closer than 2L can be evaluated by a simple geometrical con-struction as shown in Fig.2.
The derivation of the expression of the density of contacts C(p) when its depen-dence on the space and time coordinates is omitted reads
C(p) = �I
B
Z
S�(r,p,p0) S(p0) dp0 dr, [43]
where B is the ball of radius 2L centered at point P and where �(r,p,p0) is a functionequal either to 1 if the distance between the fibers is lower than �, or to zero otherwise.
Remark 4. Due to the small size of the considered rods (CNTs), we can assume thatthe orientation distribution and concentration � at positions P and Q coincide.
Contents 25
Figure 2. Evaluating the density of rod contacts
Remark 5. It can be expected that the conductivity at position x and time t alongthe direction p scales with the number of contacts � C(x, t,p) S(x, t,p). Thus theconductivity becomes local and directional.
Remark 6. One can expect that percolation along direction p occurs locally (at po-sition x and time t) when the number of contacts � C(x, t,p) S(x, t,p) is higherthan a threshold value T . Lower values imply an infinite local directional electricalresistivity. Thus, percolation could be considered local and directional.
0.5.3. Determining electrical paths
We have just determined a local and directional behaviour for the electrical con-ductivity. In order to evaluate macroscopic quantities, however, we must operate at themacroscopic scale related to the domain of interest ⌦.
26 Electrical conduction in perfectly dispersed CNT nanocomposites
Figure 3. Cell related to node (i, j, k)
We consider a 3D rectangular domain ⌦ = ⌦x ⇥ ⌦y ⇥ ⌦z . The domain ⌦ isequipped with a grid F resulting from the Cartesian product of the regular one-dimen-sional grids associated to the three domains ⌦x, ⌦y and ⌦z , respectively. Thus, a nodeof the 3D grid with coordinates (xi, yj , zk), can be characterized by three indices(i, j, k).
Even though within the kinetic theory approach the current could have any di-rection p 2 S , we consider only 26 directions (inspired by Lattice-Boltzmann tech-niques) as depicted in Fig. 3. Once we know the local directional resistivity at node(i, j, k) scaling with � C(x, t,p) S(x, t,p), we can compute the electrical resistancesbetween node (i, j, k) and each one of its neighbouring nodes.
At this stage, an electric circuit associated to the grid F becomes perfectly defined.For a potential difference applied between two regions of the domain boundary, thecurrent through each segment of the circuit can be calculated by using Kirchhoff’sfirst law that states that the total current entering a junction (a node) must equal thetotal current leaving it. Now, as each current in a segment depends on both the voltagedifference between the two nodes defining it and on its resistance according to Ohm’slaw, we can obtain a linear system that allows us to compute the voltage at each nodeand then the current circulating in each segment. In this manner, it is very easy toidentify preferential electrical paths, as well as to evaluate the current reconfigurationby removing in a deterministic or stochastic manner some segments of the network.Here, model order reduction allows for fast and accurate simulations of a variety ofelectrical grid scenarios.
Contents 27
Figure 4. Orientation of a fiber
0.6. Numerical results
0.6.1. Fluid at rest
First, we validate the different conceptual bricks previously introduced by applyingan electric field only.
0.6.1.1. Microscopic rod kinematics with Dr = 0 and constant electric field E
Starting from a given orientation of the fiber, assuming ⇠ = 1 and a constantelectric field with components E = (1, 1, 1), the time evolution of the orientation pis computed. For this purpose, p is expressed as pT = (sin ✓ cos', sin ✓ sin', cos ✓),as shown in Fig. 4.
The rotary velocity reads
˙
p =
1
⇠L(I� p⌦ p) ·E. [44]
Equation [44] is integrated from different initial orientations p(t = 0). Varioussolutions p(t;p(t = 0) = p0) are depicted in Figs. 5, 6 and 7. It can be noticed thatin all the cases, the rods align in the direction in which the electric field applies, asexpected from Eq. [44].
28 Electrical conduction in perfectly dispersed CNT nanocomposites
−1−0.500.51
−1−0.5
00.5
1
−1
−0.5
0
0.5
1
XY
Z
Figure 5. Orientation evolution p(t;'0 = 0�; ✓0 = 0�)
−1−0.5
00.5
1
−1−0.5
00.5
1
−1
−0.5
0
0.5
1
XY
Z
Figure 6. Orientation evolution p(t;'0 = 0�; ✓0 = 90�)
0.6.1.2. Kinetic theory model with Dr 6= 0 and constant electric field E
We consider the Fokker-Planck equation
@
@t+
@
@p(
˙
p ) =
@
@p
✓Dr
@
@p
◆, [45]
Contents 29
−1−0.5
00.5
1 −1−0.50
0.51
−1
−0.5
0
0.5
1
YX
Z
Figure 7. Orientation evolution p(t;'0 = 180�; ✓0 = 135�)
and the rotary velocity given again by:
˙
p =
1
⇠L(I� p⌦ p) ·E. [46]
There are two options for discretizing the surface of the unit sphere: either a Carte-sian or a spherical coordinate system. With spherical coordinates, the equation is de-fined in the domain (', ✓) 2 S ⌘ [0, 2⇡) ⇥ [0,⇡]. This essentially simplifies theapplication of discretization techniques, such as finite differences and finite elements,and allows for a computation with higher accuracy. The main difficulty associatedwith the use of spherical coordinates is the singularity at angles ✓ = 0 and ✓ = ⇡. Inthe present work, we consider a procedure similar to the one considered in [MA08].
We consider the Fokker-Planck equation [45] for different values of the diffusioncoefficient, with the rotary velocity given by Eq. [46] and an isotropic initial orien-tation distribution (', ✓, t = 0) = 14⇡ . Figure 8 shows the orientation evolutionfor two different values of the diffusion coefficients. It can be noticed that the steadystate orientation distribution is more localized for the lowest diffusion coefficient. Asexpected, the orientation distribution becomes asymmetric and localized around thedirection along which the electric field applies.
30 Electrical conduction in perfectly dispersed CNT nanocomposites
t = 0s
−10
1
−1
0
1
−1
0
1
XY
Z
−10
1
−1
0
1
−1
0
1
XY
Z
0
0.5
1
1.5
t = 1s
−10
1
−1
0
1
−1
0
1
XY
Z
−10
1
−1
0
1
−1
0
1
XY
Z
t = 2s
−10
1
−1
0
1
−1
0
1
XY
Z
−10
1
−1
0
1
−1
0
1
XY
Z
t = 3s
−10
1
−1
0
1
−1
0
1
XY
Z
−10
1
−1
0
1
−1
0
1
XY
Z
t = 20s
Figure 8. Evolution of the orientation distribution function defined on the surface of the unitsphere (left) Dr = 0.2 and (right) Dr = 0.5
Contents 31
0.6.1.3. Macroscopic modeling with constant electric field E
Taking into account Brownian effects, we have seen that the time derivative of thefirst orientation moment a(1) is given by
˙
a
(1)=
1
⇠L
⇣E� a(2) ·E
⌘�Dr a(1), [47]
where the hybrid closure relation is adopted
a
(2),hyb ⇡ �I+ � a(1) ⌦ a(1)
tr�a
(1) ⌦ a(1)� , [48]
with
� = 0.33
✓1�
⇣a
(1) · a(1)⌘2◆
, [49]
and
� =⇣a
(1) · a(1)⌘2
. [50]
Figure 9 shows the time evolution of the components of a(1)(t) from an initialorientation state characterized by a(1)
T
(t = 0) = ( 14⇡ ,14⇡ ,
14⇡ ), for three different
values of the diffusion coefficient Dr and a constant electric field E = (1, 0, 0.5).Since the electric field acts along a given direction and rods tend to orient in thatdirection, it is expected that the components of a(1)(t) – a(1)1 (t), a
(1)2 (t) and a
(1)3 (t) –
approach 1, 0 and 0.5, respectively, when t ! 1 as Brownian diffusion gets smaller.This is confirmed in the results shown in Fig. 9.
0.6.1.4. Solution of the electrostatic problem for obtaining E(x)
We consider the rectangular 3D domain ⌦ = ⌦x ⇥⌦y ⇥⌦z = (0,W )⇥ (0, L)⇥(0, H) depicted in Fig. 10 (W = 2, L = 1 and H = 1), wherein the Laplace problemdefined in Eq. [39] is solved.
A potential V(0, 0 < y < L, H2 z < H) = 0 is prescribed on the upper part ofthe yz-plane at x = 0, and the potential V(W, 0 < y < L, 0 < z H2 ) = 1 on the
32 Electrical conduction in perfectly dispersed CNT nanocomposites
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Orie
ntat
ion
tens
or c
ompo
nent
s
time (s) (Dr=0)
a(1)1a(1)2a(1)3
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Orie
ntat
ion
tens
or c
ompo
nent
s
time (s) (Dr=0.2)
a(1)1a(1)2a(1)3
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Orie
ntat
ion
tens
or c
ompo
nent
s
time (s) (Dr=0.5)
a(1)1a(1)2a(1)3
Figure 9. Orientation evolution a(1)(t) for Dr = 0, Dr = 0.2 and Dr = 0.5
Contents 33
0 0.51 1.5
20
0.5
10
0.5
1
Z
XY
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 10. Potential field V(x)
lower part of the yz-plane at x = W . On the remaining part of the domain boundary,a vanishing flux in prescribed.
We selected 11⇥ 6⇥ 6 nodes for depicting the solutions. The computed potentialV(x) is shown in Fig. 10. Figure 11 depicts at those nodal positions the electric fieldE(x) obtained from
E(x) = �rV(x). [51]
For the sake of simplicity and a better visualization, Fig. 12 shows only one plane ofthe grid F , in particular the xz-plane at y = 0.
0.6.1.5. Calculation of the steady-state density of contacts C(x,', ✓)Once we have computed the electric field E(x) at each node of the grid F , the
steady-state orientation distribution function (x,', ✓) can be calculated as previ-ously described. Then, that distribution is symmetrized to obtain S(x,', ✓) at eachnodal position x according to
S(x,', ✓) =
(x,', ✓) + (x,�',�✓)2
. [52]
34 Electrical conduction in perfectly dispersed CNT nanocomposites
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.51
0
0.2
0.4
0.6
0.8
1
YX
Z
Figure 11. Electric field E(x)
Now, we calculate at each node of the grid F the density of contacts C(x,', ✓).Figure 13 depicts the density of contacts C(x,', ✓) for a value of the diffusion coeffi-cient accounting for randomizing effects, Dr = 0.5, when applying the electric fieldobtained in the previous step.
For the sake of simplicity, Figs. 14 and 15 show only one plane of the grid F , inparticular the xz-plane at y = 0, for the values of the diffusion coefficient Dr = 0.2and Dr = 0.5.
0.6.1.6. Construction of the electrical network
After solving the linear system according to Kirchoff’s first law, the current circu-lating in each segment of the grid F was obtained and it is depicted in Fig. 3. Figure16 shows currents leaving nodes. Finally, Fig. 17 depicts currents leaving the nodelocated at (x, y, z) = (1.6, 1, 0.4).
0.6.2. Introducing the fluid flow
In this section, we combine electric field and flow.
Contents 35
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0
0
0.2
0.4
0.6
0.8
1
YX
Z
Figure 12. Electric field E(x) in xz-plane at y = 0
0.6.2.1. Microscopic rod kinematics with Dr = 0 and constant electric field E
Starting from a given orientation of a fiber immersed in a simple shear flow char-acterized by an homogeneous velocity field vT = (�̇y, 0, 0) with �̇ = 1, assum-ing ⇠ = 1 and a constant electric field with components E = (1, 1, 1), we com-pute the time evolution of the orientation p. For this purpose, p is expressed asp
T= (sin ✓ cos', sin ✓ sin', cos ✓).
The rotary velocity reads
˙
p =
1
⇠L(I� p⌦ p) ·E+rv · p�
�p
T ·rv · p�p. [53]
Equation [53] is then integrated from different initial orientations p(t = 0). Varioussolutions p(t;p(t = 0) = p0) are depicted in Figs. 18, 19 and 20. It can be no-ticed that rod alignment results from a compromise between the applied electrical andvelocity fields, as expected from Eq. [53].
36 Electrical conduction in perfectly dispersed CNT nanocomposites
Figure 13. Contact density C(x,', ✓) for Dr = 0.5
0.6.2.2. Kinetic theory model with Dr 6= 0 and constant electric field EWe consider the Fokker-Planck equation
@
@t+
@
@p(
˙
p ) =
@
@p
✓Dr
@
@p
◆, [54]
and the rotary velocity given again by:
˙
p =
1
⇠L(I� p⌦ p) ·E+
�rv · p�
�p
T ·rv · p�p
�. [55]
We solve the Fokker-Planck equation for different values of the diffusion coefficient,with the rotary velocity given by Eq. [55] and an isotropic initial orientation distribu-tion (', ✓, t = 0) = 14⇡ .
Figure 21 depicts the orientation evolution for two different diffusion coefficients.It can be noticed that the steady-state orientation distribution is more localized for thelowest diffusion coefficient.
Contents 37
Figure 14. Contact density C(x,', ✓) in xz-plane for Dr = 0.2
0.6.2.3. Macroscopic modeling with constant electric field E
Taking into account Brownian effects, the first orientation moment a(1) evolvesaccording to
˙
a
(1)=
1
⇠L
⇣E� a(2) ·E
⌘+rv · a(1) � a(3) : rv �Dr a(1), [56]
where again the hybrid closure relation is considered.
Figure 22 shows the time evolution of the components of a(1)(t) from an initialorientation characterized by a(1)
T
(t = 0) = ( 14⇡ ,14⇡ ,
14⇡ ), for three different values
of the diffusion coefficient Dr, a constant electric field E = (1, 0, 0.5) and, as before,an homogeneous velocity field vT = (�̇y, 0, 0) with �̇ = 1. Since the electric fieldacts in a direction different from that of the velocity vector, it is expected that a(1)(t)is the result of a compromise between both effects, as Fig. 22 confirms.
0.6.2.4. Calculation of the steady-state contact density C(x,', ✓)Considering the same electric field as previously in absence of flow, we calculate
at each node of the grid F the density of contacts C(x,', ✓) from the steady-state
38 Electrical conduction in perfectly dispersed CNT nanocomposites
Figure 15. Contact density C(x,', ✓) in xz-plane for Dr = 0.5
0 0.2 0.4 0.60.8 1 1.2 1.4
1.6 1.8 20
0.5
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y
Z
10
20
30
40
50
60
Figure 16. Current distribution
Contents 39
1.561.57
1.581.59
1.6
0.970.98
0.991
0.37
0.38
0.39
0.4
0.41
0.42
0.43
0.44
XY
Z
10
20
30
40
50
60
Figure 17. Currents leaving a particular node
−0.50
0.51
−1−0.5
00.5
1
−1
−0.5
0
0.5
1
XY
Z
Figure 18. Orientation evolution p(t;'0 = 0�; ✓0 = 0�)
40 Electrical conduction in perfectly dispersed CNT nanocomposites
−1−0.5
00.5
1
−1−0.5
00.5
1
−1
−0.5
0
0.5
1
XY
Z
Figure 19. Orientation evolution p(t;'0 = 0�; ✓0 = 90�)
−1−0.500.51
−1−0.5
00.5
1−1
−0.5
0
0.5
1
YX
Z
Figure 20. Orientation evolution p(t;'0 = 180�; ✓0 = 135�)
symmetrized orientation distribution function S(x,', ✓). Figure 23 shows C(x,', ✓)for a value Dr = 0.5 of the diffusion coefficient accounting for randomizing effects.
For the sake of simplicity, Figs. 24 and 25 represent only one plane of the grid F ,in particular the xz-plane at y = 0, for the values of the diffusion coefficient Dr = 0.2and Dr = 0.5.
Contents 41
t = 0s
−10
1
−1
0
1
−1
0
1
XY
Z
−10
1
−1
0
1
−1
0
1
XY
Z
0
0.5
1
1.5
t = 1s
−10
1
−1
0
1
−1
0
1
XY
Z
−10
1
−1
0
1
−1
0
1
XY
Z
t = 2s
−10
1
−1
0
1
−1
0
1
XY
Z
−10
1
−1
0
1
−1
0
1
XY
Z
t = 3s
−10
1
−1
0
1
−1
0
1
XY
Z
−10
1
−1
0
1
−1
0
1
XY
Z
t = 20s
Figure 21. Evolution of the orientation distribution function defined on the unit sphere surfacefor (left) Dr = 0.2 and (right) Dr = 0.5
42 Electrical conduction in perfectly dispersed CNT nanocomposites
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Orie
ntat
ion
tens
or c
ompo
nent
s
time (s) (Dr=0)
a(1)1a(1)2a(1)3
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Orie
ntat
ion
tens
or c
ompo
nent
s
time (s) (Dr=0.2)
a(1)1a(1)2a(1)3
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Orie
ntat
ion
tens
or c
ompo
nent
s
time (s) (Dr=0.5)
a(1)1a(1)2a(1)3
Figure 22. Orientation evolution a(1)(t) for Dr = 0, Dr = 0.2 and Dr = 0.5
Contents 43
Figure 23. Contact density C(x,', ✓) for Dr = 0.5
0.6.2.5. Construction of the electrical network
After solving the linear system according to Kirchhoff’s first law, we have obtainedthe current circulating in each segment of the grid F . Figure 26 plots currents leavingeach node.
0.7. Conclusions
In this work, we have revisited the multi-scale description of perfectly dispersedsuspensions involving CNTs when combining hydrodynamic and electrostatic forces.
First, we have generalized Jeffery’s equation by accounting for electrically-inducedorientation mechanisms. Then, coarser models were derived, in particular a meso-scopic model involving the probability distribution function governed by a Fokker-Planck equation, and a macroscopic model involving different moments of that distri-bution function.
Knowledge of the local orientation distribution allowed for the calculation of adirectional density of CNT contacts, yielding a local directional electrical resistance.
44 Electrical conduction in perfectly dispersed CNT nanocomposites
Figure 24. Contact density C(x,', ✓) in xz-plane for Dr = 0.2
The latter has a double interest, as it allows for defining (i) a sort of directional perco-lation, and (ii) an electrical network from which the preferential current paths can beidentified statistically.
Obviously, the possibility of carrying out this type of analysis at the process andpart scales opens numerous possibilities. For example, one can think of the control ofthe microstructure by tuning the electric field acting on the flowing system in order toreach an optimal orientation in the molded part.
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Figure 25. Contact density C(x,', ✓) in xz-plane for Dr = 0.5
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46 Electrical conduction in perfectly dispersed CNT nanocomposites
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.51
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Z
Y
10
20
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Figure 26. Current distribution
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