Int J Mater Form (2008) 1:89–99DOI 10.1007/s12289-008-0377-5

ORIGINAL RESEARCH

Numerical simulation of spin coating processes involvingfunctionalised Carbon nanotube suspensions

E. Cueto · A. W. K. Ma · F. Chinesta · M. R. Mackley

Received: 7 April 2008 / Accepted: 9 May 2008 / Published online: 18 June 2008© Springer/ESAFORM 2008

Abstract This paper reports the numerical simulationof spin coating for functionalised Carbon Nanotube(CNT) suspensions. Spin coating is a process commonlyused to deposit uniform thin films onto flat substratesby means of high rotation velocity and centrifugalforce. The functionalised CNTs modelled in this studywere chemically treated in a way such that aggrega-tion was prevented through electrostatic repulsion be-tween CNTs. The functionalised CNTs in the semi-dilute suspensions can be modelled as rigid fibres withtheir orientation dictated by the flow of the solvent.The evolution of CNT orientations was simulated usinga pre-averaged kinetic theory with an appropriate

Work partially supported by the Spanish Ministry ofEducation and Science through grant numberCICYT-DPI2005-08727-C02-01.

E. Cueto (B)Group of Structural Mechanics and Material Modelling,Aragón Institute of Engineering Research, I3A,Universidad de Zaragoza, Edificio Betancourt.María de Luna, 7 50012 Zaragoza, Spaine-mail: ecueto@unizar.esURL: http://gemm.unizar.es

A. W. K. Ma · M. R. MackleyDepartment of Chemical Engineering,University of Cambridge, New Museums Site,Pembroke Street, Cambridge, CB2 3RA, UK

M. R. Mackleye-mail: mrm5@cam.ac.ukURL: http://www.cheng.cam.ac.uk/research/groups/polymer/

F. ChinestaLaboratoire de Mecanique des Systemes et des Procedes,UMR 8106, CNRS-ENSAM-ESEM, Paris, Francee-mail: francisco.chinesta@paris.ensam.frURL: http://www.paris.ensam.fr/lmsp/

rotary diffusion coefficient accounting for randomisingevents. A Natural Element strategy with an updatedLagrangian framework was implemented to solve thefree-surface problem involving large domain deforma-tion and to avoid numerical problems associated withFinite Element modelling. The model reported hereincouples micro-scale CNT orientation with the macro-scopic suspension kinematics and it offers importantinsights in relation to the final properties of spin-coatedCNT films as well as the processing behaviour of CNTsuspensions.

Keywords Carbon nanotubes · Spin coating ·Numerical simulation · Meshless

Introduction

Carbon nanotubes have generated a tremendous in-terest in the scientific community following a journalpublication by Iijima (1991). Single-walled carbon nan-otubes are one-atom thick sheets of graphite (calledgraphene) rolled up into seamless cylinders with diame-ter in the order of a nanometer. They posses impressivemechanical properties:

– Young’s modulus: 1 to 5 TPa– Tensile Strength: 13-53 GPa– Elastic strain up to 5%.– Density 2160 kg/m3.

Given these properties, carbon nanotubes (CNTs)have been extensively researched as possible reinforce-ment (and conductive elements) for high-performancenanocomposites. In addition, nanotube based transis-tors have been made that operate at room temperature

90 Int J Mater Form (2008) 1:89–99

and that are capable of digital switching using a singleelectron (Postma et al. 2001).

In order to design manufacturing processes moreeffectively, it is important to understand and modelthe behaviour of CNT suspensions and the orientationof CNTs. Although several studies have focused onproducing polymer nanotube composites, many prac-tical challenges remain before their potential can befully realized. Dispersing the nanotubes individuallyand uniformly into the matrix seems to be fundamentalin producing composites with reproducible and optimalproperties.

This work is aimed at establishing a method forthe numerical simulation of CNT suspension form-ing processes. We focus our attention on spin coatingprocesses, although we believe that the technique isgeneral enough to be applied to other forming pro-cesses as well. Spin coating processes are used to applyvery thin, more or less uniform, films onto flat sub-strates. In short, a drop of solution is placed in the cen-ter of the plate, which is then rotated at high speed. Thecentrifugal force makes the solution to spread on theflat substrate until the desired thickness is achieved onthe film. The solvent is sometimes volatile in this kindof forming processes, an effect that is not considered inthis work.

From the simulation point of view, spin coatingprocesses involving CNTs suspensions present somevery important challenges. First of all, tracking theevolution of the free surface of the liquid, that under-goes very large deformations, remains a problem fortraditional, Eulerian or Arbitrary Lagrangian EulerianFinite Element techniques (Donea 1983), althoughsome early references are available on the literature(Emslie et al. 1958). To this end, Lagrangian frame-works, in which the nodes of the simulation move withthe material velocity, seem to ba an appropriate choicefor the simulation. The family of meshless methods(Belytschko et al. 1994; Liu et al. 1995; Sukumar et al.1998) seem to be an appealing choice, since they donot rely on a mesh (and consequently mesh distortiondoes not affect the accuracy of the results). In this workwe employ a particular meshless method known asthe Natural Element method and the method was prov-en to be capable of solving numerical problems thatinvolve large distortion (Martinez et al. 2004; Gonzálezet al. 2007).

The other remaining challenge is the constitutivemodelling of CNT suspensions. In a first attempt, func-tionalized CNTs are considered as rigid fibres with verylarge aspect ratio (between 103-104, possibly up to 106,but typically in the order of hundreds after disper-sion), whose orientation is governed by the flow of the

solvent, which is Newtonian in this study. A quadraticclosure relation is then applied to the preaveragedkinetic theory model. At the macroscopic scale, theflow kinematics of the suspension is governed by themomentum and mass conservation equations, whichare further coupled with the micro-scale orientation ofthe CNTs. The CNT orientation and the suspensionkinematics are updated at each time step as discussedin Section Discrete model.

Some experimental measurements have been per-formed on the rheology of the CNTs suspensions, in or-der to characterize its behaviour. The CNTs modelledin this paper were chemically treated such that theydid not aggregate and no mesostructures were observedoptically.

The paper includes a brief description of the NaturalElement Method as well as the main assumptions in-volved the numerical modelling of spin coating. Somenumerical examples are given to illustrate the potentialof the proposed process simulation technique.

Spin coating processes

As mentioned before, spin coating processes consist ofmaking a drop of liquid spread on a surface by means ofcentrifugal acceleration. This acceleration is achievedusing a disk that rotates at a high angular velocity, ω,see Fig. 1.

The difficulties in the numerical simulation of spincoating processes arise from (1) the tremendous defor-mation of the domain (in spin coating, the boundarycan deform from a liquid droplet to thin films with athickness of 30 − 50 μm) and (2) the multiscale natureof the problem.

In this work, the rotating plate is assumed to beperfectly horizontal, so that there is no radial com-ponent of gravity. Following Emslie et al. (1958), weassume that Coriolis forces are negligible, comparedto centrifugal forces. To model the geometry of thedrop, axial symmetry around the rotation axis is alsoassumed. A system of reference that rotates with thedisc was used in describing the equations of motion.

ωFig. 1 Schematic representation of spin coating processes

Int J Mater Form (2008) 1:89–99 91

Numerical modelling of the spin coating process

CNT suspensions are modelled as short fibre sus-pensions described by the following equations (seeMartínez et al. 2003, and references therein)

– The balance of momentum equations, where weonly consider the centrifugal forces as in the treat-ment of Emslie et al. (1958)

Divσ = −ρω × (ω × r) = f r (1)

where σ is the stress tensor, ρ represents the den-sity, × denotes the tensor product, and ω the rota-tion speed of the spin coater as schematically drawnin Fig. 1.

– The incompressibility condition

Divv = 0 (2)

where v represents the velocity field.– The constitutive equation, with a quadratic closure

relation for the fourth order orientation tensor andother simplifying assumptions, results

σ = −pI + 2η{

D + NpTr(aD)a}

(3)

where p denotes the pressure, I is the unit tensor,η is the equivalent suspension viscosity, D is thestrain rate tensor, Np a scalar parameter that de-pends on both the tube concentration and its aspectratio, and a is the second order orientation tensordefined by

a =∮

ρ ⊗ ρ�(ρ)dρ (4)

where ρ is the unit vector defining the CNT axisdirection, ⊗ denotes the dyadic product, and �(ρ)

is the orientation distribution function, that shouldsatisfy the normality condition∮

�(ρ)dρ = 1 (5)

If �(ρ) = δ(ρ − ρ), with δ() the Dirac’s function, allthe orientation probability is concentrated in the di-rection defined by ρ, and the corresponding orien-tation tensor is a = ρ ⊗ ρ. In this case the quadraticclosure relation becomes exact. However, when thetubes are not perfectly aligned in a certain direc-tion the quadratic closure is no more exact and itconstitutes an approximation that is widely used inthe context of short fibre suspensions, see Martínezet al. (2003) and references therein.From a physical point of view, we can consider thatthe eigenvalues of the second order orientation ten-sor (a) represent the probability of finding the CNTin the direction of the corresponding eigenvectors.

– With a quadratic closure relation the orientationequation is expressed as

dadt

= �a − a� + k (Da + aD − 2Tr(aD)a)

−6Dr

(a − I

3

)(6)

D and � are the symmetric and skew-symmetriccomponents of Gradv, k is a constant that de-pends on the nanotube aspect ratio r (fibre lengthto fibre diameter ratio): k = (r2 − 1)/(r2 + 1) withk ≈ 1 given the high aspect ratio of the CNTs.Dr is a rotary diffusion coefficient accounting forrandomising events such as Brownian motion andfibre-fibre interaction in the suspension.

The flow model is defined in the volume occupiedby the fluid at time t, � f (t). On its boundary, � f (t) ≡∂� f (t) either the velocity or the traction is imposed:

v (x ∈ �1) = vg (7)

or

σ n (x ∈ �2) = Fg (8)

with �1 ∪ �2 = � f (t), �1 ∩ �2 = ∅, and where n(x) is theunit outwards vector, defined on the boundary at thepoint x.

In terms of the initial condition, the CNTs are as-sumed to be isotropically orientated (i.e., a = I/3).This model is solved on the domain defined by thenodes as they evolve. More details can be found atMartínez et al. (2003) and references therein.

In this modelling, the following form of the orien-tation vector is used for describing CNTs suspendedwithin the solvent:

ρ =⎛

⎝rθ

z

⎞

⎠ (9)

with θ ∈ [0, 2π [.The velocity field is expressed in cylindrical coordi-

nates, after the imposition of axial symmetry:

v =⎛

⎝vr

ω · rvz

⎞

⎠ (10)

where ω, as mentioned before, represents the angularvelocity of the spin coater. If, following Emslie et al.(1958), we assume a system of reference rotating withthe disk, we arrive at

v =⎛

⎝vr

0vz

⎞

⎠ (11)

92 Int J Mater Form (2008) 1:89–99

In this case, tensors D and � come from an axisym-metric flow, so their expression in cylindrical coordi-nates, as mentioned before, gives:

D=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂vr

∂r1

2

(1

r∂vr

∂θ+ ∂vθ

∂r− vθ

r

)1

2

(∂vr

∂z+ ∂vz

∂r

)

1

2

(1

r∂vr

∂θ+ ∂vθ

∂r− vθ

r

)1

r∂vθ

∂θ+ vr

r1

2

(1

r∂vz

∂θ+ ∂vθ

∂z

)

1

2

(∂vr

∂z+ ∂vz

∂r

)1

2

(1

r∂vz

∂θ+ ∂vθ

∂z

)∂vz

∂z

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (12)

and, provided that the flow is axially symmetric (vθ = 0,vr = vr(r, z), vz = vz(r, z)), and by combining the aboveequation with Eq. 11 D reduces to

D =

⎛

⎜⎜⎜⎜⎜⎝

∂vr

∂r0

1

2

(∂vr

∂z+ ∂vz

∂r

)

0vr

r0

1

2

(∂vr

∂z+ ∂vz

∂r

)0

∂vz

∂z

⎞

⎟⎟⎟⎟⎟⎠

. (13)

Similarly, the vorticity tensor takes the followingform:

� =

⎛

⎜⎜⎜⎜⎝

0 01

2

(∂vr

∂z− ∂vz

∂r

)

0 0 0

−1

2

(∂vr

∂z− ∂vz

∂r

)0 0

⎞

⎟⎟⎟⎟⎠

(14)

An isotropic orientation is defined in this model bythe uniform distribution

ψ(ρ) = 1

4π(15)

giving rise to an orientation tensor, for isotropicdistributions,

a = 1

3I (16)

These conditions are imposed at the initial time steponly.

Rheology of functionalized CNTs suspensions

A Fokker-Planck (FP) based orientation model wasused to describe the steady shear rheological responsesof the chemically treated CNT suspensions. Figure 2ashows the model fitting of surface-treated CNT sus-pensions with three different weight concentrations(0.05%, 0.2% and 0.33%). Taking the 0.3% CNT sus-pension as an example, the evolution of the apparent

viscosity ηa in the orientation model is controlled bythe values of Dr and Np and the best constant-Dr

fit was obtained for Dr = 0.005s−1 and Np = 7. InFig. 2b, different values of Dr were used to illus-trate the sensitivity of the model on the fitting pa-rameter Dr. In general, there is a good agreement

Fig. 2 a Orientation model fitting of 0.05%, 0.2% and 0.33%surface-treated CNTs suspended in epoxy resin. Np and Dr arethe fitting parameters. b Sensitivity of the model to Dr given theexperimental data of 0.33% treated CNT suspension and Np = 7

Int J Mater Form (2008) 1:89–99 93

between ηa predicted by the model and the exper-imental data, but for high γR, the predicted ηa isslightly higher than the experimental value. For in-stance, at shear rate γR = 60s−1, ηa of the 0.3% sus-pension was found experimentally to be 10.5Pa · s, butthe orientation model predicted a viscosity of 11.3Pa · s(with a constant Dr = 0.005s−1) and this gives an errorof about 8%. Although there is an error of a fewpercents in predicting high-shear-rate data, the use ofa constant Dr should be sufficient to provide a rea-sonably good estimation of ηa for general engineeringproblems, and this also saves the effort in determiningthe exact relationship between Dr and γR. However,in cases where a more accurate description of high-shear viscosity is needed, the dependence of Dr onshould be carefully evaluated (see for example, Folgarand Tucker 1984; Larson 1999). Moreover, our recentmodelling work suggested that the FP based orien-tation model cannot satisfactorily describe the shear-thinning characteristics for untreated CNT suspensionswhere optically resolvable CNT aggregates are present(see for example Rahatekar et al. 2006). For untreatedCNT suspensions, a new aggregation/orientation modelshould be used and in that model, diffusion coefficientdepends not only on the orientation of CNTs, but alsoon the entanglement state of CNTs. Detailed modelformulation and fittings for both treated and untreatedCNT suspensions can be found in the article entitled“The Rheological Modelling of CNT Suspensions” inthis special issue.

Basics of the natural element method

As mentioned in the introductory section, we have cho-sen to employ a meshless method, the Natural ElementMethod, to perform the simulation, mainly because itallows for a Lagrangian description of the flow kine-matics, convenient for solving free-surface flow prob-lems with large surface deformation. The basics of themethod are described next.

Natural neighbour interpolation

The vast majority of meshless methods are based onthe use of scattered data approximation techniques toconstruct the approximating spaces of the Galerkinmethod. These techniques must have, of course, lowsensitivity to mesh distortion, as opposed to FiniteElement methods. Among these techniques, theNatural Element Method employs any instance ofNatural Neighbour interpolation (Sibson 1981; Hiyoshiand Sugihara 1999) to construct trial and test functions.Prior to the introduction of these interpolation tech-niques, it is necessary to define some basic concepts.

The model will be constructed upon a cloud of pointswith no connectivity established a priori among them.We will call this cloud of points N = {n1, n2, . . . , nM} ⊂R

d, and there is a unique decomposition of the spaceinto regions such that each point within these regions iscloser to the node with which the region is associatedthan to any other in the cloud. This kind of spacedecomposition is called a Voronoi diagram of the cloudof points and each Voronoi cell is formally defined as(see Fig. 3):

TI = {x ∈ R

d : d(x, xI) < d(x, xJ) ∀ J �= I}, (17)

where d(·, ·) is the Euclidean distance function.The dual structure of the Voronoi diagram is the

Delaunay triangulation, obtained by connecting nodesthat share a common (d − 1)-dimensional facet. Whilethe Voronoi structure is unique, the Delaunay triangu-lation is not, there exist some degenerate cases in whichthere are two or more possible Delaunay triangula-tions (consider, for example, the case of triangulating asquare in 2D, as depicted in Fig. 3 (right)). Another wayto define the Delaunay triangulation of a set of nodesis by invoking the empty circumcircle property, whichmeans that no node of the cloud lies within the circlecovering a Delaunay triangle. Two nodes sharing afacet of their Voronoi cell are called natural neighboursand hence the name of the technique.

Fig. 3 Delaunaytriangulation and Voronoidiagram of a cloud of points

94 Int J Mater Form (2008) 1:89–99

In order to define the natural neighbour co-ordinatesit is necessary to introduce some additional concepts.The second-order Voronoi diagram of the cloud isdefined as

TI J = {x ∈ R

d : d(x, xI) < d(x, xJ) < d(x, xK)

∀ J �= I �= K}. (18)

The most common natural neighbour interpolationmethod, is the Sibson interpolant (Sibson 1980, 1981).Consider the introduction of the point x in the cloud ofnodes. Due to this introduction, the Voronoi diagramwill be altered, affecting the Voronoi cells of the nat-ural neighbours of x. Sibson (1980) defined the naturalneighbour coordinates of a point x with respect to oneof its neighbours I as the ratio of the cell TI that istransferred to Tx when adding x to the initial cloudof points to the total volume of Tx. In other words, ifκ(x) and κI(x) are the Lebesgue measures of Tx andTxI respectively, the natural neighbour coordinates ofx with respect to the node I is defined as

φI(x) = κI(x)

κ(x). (19)

In Fig. 4 the shape function associated to node 1 maybe expressed as

φ1(x) = Aabf e

Aabcd. (20)

It is straightforward to prove that natural element (NE)shape functions (see Fig. 5) form a partition of unity

x1

2

3

4

5

6

7

a

b c

de

f

Fig. 4 Definition of the natural neighbour coordinates of apoint x

Y

Z

X

Fig. 5 Typical function φ(x)

(Babuška and Melenk 1997), as well as some otherproperties like positivity (i.e., 0 ≤ φI(x) ≤ 1 ∀I, ∀x)and strict interpolation:

φI(xJ) = δI J. (21)

Sibson interpolants have some remarkable proper-ties that help to construct the trial and test functionalspaces of the Galerkin method (see Sukumar et al.1998; Hiyoshi and Sugihara 1999).

Besides properties like continuity and smoothness(everywhere except at the nodes for Sibson inter-polants), Sibson interpolants posses linear complete-ness (i.e., exact reproduction of a linear field) and itis noteworthy that they are able to exactly interpo-late prescribed essential (Dirichlet) boundary condi-tions, a condition usually not fulfilled by other meshlessmethods.

The α-shapes-based NE method

The identification of the free surface in an updatedLagrangian flow simulation deserves some comments.Using meshless methods, in which models are con-structed by a set of nodes only, boundary trackingcan be performed by employing different strategies.In particular, we have employed shape constructors toperform this task. Shape constructors are geometricalentities that transform finite point sets into a multiplyconnected shape in general. In particular, we employα-shapes (Edelsbrunner and Mücke 1994). α-shapesdefine a one-parameter family of shapes Sα (being α theparameter), ranging from the “coarsest” to the “finest”level of detail. α can be seen, precisely, as a measure ofthis level of detail.

Int J Mater Form (2008) 1:89–99 95

8 8.5 9 9.5 10 10.5 11 11.5 12

6.4

6.6

6.8

7

(a)

8 8.5 9 9.5 10 10.5 11 11.5 12

6.4

6.6

6.8

7

(b)

8 8.5 9 9.5 10 10.5 11 11.5 12

6.4

6.6

6.8

7

(c)

8 8.5 9 9.5 10 10.5 11 11.5 12

6.4

6.6

6.8

7

(d)

8 8.5 9 9.5 10 10.5 11 11.5 12

6.4

6.6

6.8

7

(e)

Fig. 6 Evolution of the family of α-shapes of a cloud of pointsrepresenting drop impacting on a flat surface. Shapes S0.03(a), S0.05 (b), S0.5 (c), S2.0 (d), S100.0 ∼ S∞ (e) are depicted

Details about the formal definition of the family ofα-shapes can be found in Edelsbrunner and Mücke(1994). In brief, the use of α-shapes to define the bound-ary of the domain relies in the choice of the level ofdetail needed to represent the domain, which is alwaysan analyst’s decision. It is obvious that the minimumnodal spacing parameter, say h, should be chosen so asto reproduce at least that level of detail α.

α-shapes provide a means to eliminate from thetriangulation those triangles or tetrahedra whose size isbigger than the specified level of detail. This criterion is

very simple: just perform a filtration to eliminate thosetriangles (tetrahedra) whose circum-radius is biggerthan the level of detail, α.

In Fig. 6 an example of the previously presentedtheory is presented. It represents some instances of thefinite set of shapes for a cloud in a intermediate stepof the simulation of drop impacting onto a flat surface.The experimental solution of this problem, togetherwith some snapshots of its numerical simulation, areshown in Figs. 7 and 8.

Note that the key question in using α-shapes is not tofind the precise value of α for a given configuration ofthe nodal cloud. Instead, we must set the problems interms of what level of detail we are interested in takinginto account for a particular geometry.

The use of α-shapes, however, has another relevantinfluence in the NE Method. As demonstrated in Cuetoet al. (2000), the construction of natural neighbourinterpolation on an α-shape of the domain alters thedistance measure. Natural neighbour interpolation isperformed on the basis of Voronoi diagrams, which em-ploy euclidean distance measure in their most generalform. This leads to some lack of interpolation alongnon-convex boundaries. This interpolation is recoveredif we construct the natural neighbour interpolants overan α-shape of the domain.

Thus, the use of α-shapes in the construction ofupdated Lagrangian simulations of fluid flow providesan appealing way to track the boundary of domainwhile ensuring appropriate interpolation of essen-tial boundary conditions, that can be imposed di-rectly in the discrete system of equations, as in theFinite Element Method.

Discrete model

A mixed C0 − C−1 NE interpolation has been appliedto discretize the strong form of the problem introducedbefore. More details on the application of mixed NE in-terpolations in incompressible and nearly incompress-ible media can be found in Cueto et al. (2001).

In the formulation here presented, a C0 interpolationscheme – smooth everywhere except at the nodes –has been chosen in the velocity field approximation,

Fig. 7 Sequence of the drop deformation under very low ambient pressure. Photos courtesy of Lei Xu and Sidney Nagel

96 Int J Mater Form (2008) 1:89–99

8 10 126.5

7

7.5

8

8.5

8 10 126.5

7

7.5

8

8.5

8 10 126.5

7

7.5

8

8.5

8 10 126.5

7

7.5

8

8.5

8 10 126.5

7

7.5

8

8.5

Fig. 8 Evolution of the free surface in the drop impact

whereas a discontinuous C−1 interpolation has beenused in the pressure approximation:

vh(x) =n∑

I=1

φI(x) v I (22)

ph(x) =n∑

I=1

ψI(x) pI =n∑

I=1

1

npI (23)

where v I and pI represent the nodal velocities andpressures, respectively, and n is the number of naturalneighbours of the point x under consideration.

This kind of approximation does not verify theLBB condition. However, it has been shown that itsbehaviour is very similar to that of the bilinear velocity-constant pressure finite element (see González et al.2004a). No spurious modes or locking have beenobserved in all the simulations computed in this paper.

Thus, with the fluid domain � f (t) extracted at time tfrom the cloud of nodes by using the α-shape technique,as described in the previous section, and the velocityand pressure NE interpolation defined by Eqs. 22 and23, we can proceed to a standard discretisation of themixed variational formulation of the flow equations∫

� f (t)σ : D∗d� =

∫

� f (t)f rv

∗d� (24)

∫

� f (t)Divv p∗d� = 0 (25)

with

σ = −pI + 2η{

D + NpTr(aD)a}

(26)

where a null traction is assumed on the flow front and aprescribed velocity is enforced on the other part of thefluid domain boundary. Essential boundary conditionsinvolve setting the velocity of the nodes in contact withthe rotating plate to zero.

The orientation equation is solved at each time incre-ment. With the flow kinematics known at time t, vt(x),the position of nodes can be updated at the same timethat the fibres orientation evolution, given by Eq. 6 iscomputed by using the method of characteristics. Thesimplest explicit updating consists in writing

x t+�tI = x t

I + vtI�t, ∀I (27)

and

at+�tI =at

I +{�t

I atI − at

I�tI + kDt

I atI + kat

I DtI

− 2kTr(at

I DtI

)at

I −6Dr

(at

I − I3

)}�t, ∀I

(28)

where DtI and �t

I are the symmetric and skew-symmetric components of the gradient of velocitytensor, at time t in the node xI , respectively.

Finally, in the kinematics resolution stage, the fibreorientation described by the second order orientationtensor a is assumed to be known at the nodes at presenttime step, at

I . The value of a at the integration points

Int J Mater Form (2008) 1:89–99 97

Fig. 9 Snapshots of thevelocity field at 1st (a), 10th

(b), 20th (c) and 30th (d) timesteps. �t = 0.01s

Y

1 2 3 4 5 60

1

2

3

4

5 Vx

1413121110987654321

(a)

Y

00 1 2 3 4 5 60

1

2

3

4

5 Vx

1413121110987654321

(b)

X

Y

0 1 2 3 4 5 60

1

2

3

4

5 Vx

1413121110987654321

(c)X

X X

Y

0 21 3 4 5 60

1

2

3

4

5 Vx

1413121110987654321

(d)

used to evaluate Eqs. 24 and 25 is computed by usingthe NE interpolation

at(x) =I=n∑

I=1

φI(x) atI (29)

The only difficulty in applying Eq. 28 to updatethe orientation is related to the non-derivability of theNE shape functions at their definition nodes. Thus,we can evaluate the velocity gradient tensor from theexpression

vt,k(x) =

I=n∑

I=1

φI,k(x) vtI (30)

where the subscripts k denotes the spatial derivativewith respect to the k−coordinate. The velocity gradientcan be computed everywhere except at the nodes, sinceφI,k(xI) is not defined. A possible solution for this reliesin the use of Stabilised Conforming Nodal integrationtechniques (Chen et al. 2001; Gonzalez et al. 2004b) orto use some kind of projection from integration points

and then averaging (Zienkiewicz and Zhu 1992; Cuetoet al. 2003).

Numerical results

The previously presented model was applied to thesimulation of spinning a drop of CNT suspension, withthe assumption that the initial shape is Gaussian. Themodel was composed of 977 nodes under axisymmetricassumptions for the flow (but not for the orientationfield, which is three-dimensional). Non-slip boundaryconditions were assumed at the horizontal plane,whereas symmetry boundary conditions are assumed atthe axis of symmetry. Some snapshots for the simulatedevolution of the velocity field for different time stepsare shown in Fig. 9.

The results on the orientation field for the CNTs areshown in Figs. 10 and 11. It can be noticed how CNTstend to adopt an horizontal orientation in the r − zplane, due to the gradient of velocities in this plane,very close to pure shear due to the non-slip boundary

98 Int J Mater Form (2008) 1:89–99

0 1 2 3 4 5 60

0.5

1

(a)

0 1 2 3 4 5 60

0.5

1

(b)

0 1 2 3 4 5 60

0.5

1

(c)

0 1 2 3 4 5 60

0.5

1

(d)

Fig. 10 Orientation field for the CNT suspension at someintermediate steps of the simulation (r − z orientation field). 1st

(a), 10th (b), 20th (c) and 30th (d) time steps

conditions. On the θ − z plane, however, and by virtueof the diverging velocity field, CNTs tend to align to theθ direction, although the ellipses are not as distorted asthose on the r − z plane and the degree of alignment isless significant.

Conclusions

A model has been presented for the numerical simula-tion of spin coating of CNTs suspensions. The modelis composed of a meshless (NE) strategy for the flowthat allows for a proper description of the free-surfaceflow and a micro-macro description of the nanotubescale that allows for a suitable description of CNTorientation field.

The use of a meshless strategy in an updatedLagrangian framework, combined with an α-shape ap-proach, to describe the large motion of the free surfaceseems to be an appealing choice, due to its simplicity.In this way since nodes move with material velocity,the orientation field can be stored at the nodes andbe updated easily by a simple algorithm based on themethod of characteristics.

0 1 2 3 4 5 60

0.5

1

(a)

0 1 2 3 4 5 60

0.5

1

(b)

0 1 2 3 4 5 60

0.5

1

(c)

0 1 2 3 4 5 60

0.5

1

(d)

Fig. 11 Orientation field for the CNT suspension at some inter-mediate steps of the simulation (r − θ plane). 1st (a), 10th (b), 20th

(c) and 30th (d) time steps

Other possibilities, such as the kinetic description de-rived from the FP equation or its equivalent stochastic(Itô) counterpart could also be applied in this sameframework. This remains an important part of ongoingresearch and the findings will be reported in a futurepaper.

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