Modeling nanomaterials

8/12/2019 Modeling nanomaterials

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559

16Modeling at the Nano Level:Application to Physical Processes

Serge Lefeuvre

Eurl Creawave, Toulouse, France

Olga Gomonova

Siberian State Aerospace University, Krasnoyarsk, Russia

16.1 Introduction

The processing of heterogeneous materials was, from the beginning of human activity, the

fruit of experiments transmitted as hand-turns or empiric expressions. Nowadays, experi-ments remain compulsory but are integrated into more accurate descriptions such as finiteelements description. The partial differential equations, solved in orthogonal spaces, arenow solved in more complicated geometries thanks to finite element method (FEM), butthe constants characteristic of heterogeneous materials keep usually a touch of empiri-cism. For instance, the use of polynomial approximations is still largely spread.

The modeling at the nano level is an attempt to achieve a more precise description ofthe blend, even if it remains impossible to describe the exact geometry of each nano grain.Apart from preserving the grain proportion of each component, the grain to grain descrip-tion opens the way to the description of the surface activities. This point is crucial since

more the volume is small, more its surface is active.For a long time, the nano-level approach was understood as a pileup of spheres as shownin Figure 16.1.

CONTENTS

16.1 Introduction ........................................................................................................................ 55916.2 Filtration: An Ideal and a Fictitious Soil ......................................................................... 560

16.2.1 Porosity of Fictitious Soil ...................................................................................... 56116.3 Meshing of Geometry Objects: Examples in 2D............................................................ 567

16.3.1 Examples of Objects............................................................................................... 56716.4 3D Modeling: Examples .................................................................................................... 57516.5 3D and Capillaries ............................................................................................................. 580

16.6 Fluid Flow in the Capillaries ............................................................................................ 58016.7 Microwave Heating ........................................................................................................... 58116.8 Conclusion .......................................................................................................................... 582References .....................................................................................................................................583


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560 Computational Finite Element Methods in Nanotechnology

One of the best ways of modeling heterogeneous materials respecting granulometry isto work at the level of nano grains. Mostly, during the investigation of materials and theirproperties, researchers are interested in calculating the main characteristics of these mate-rials, such as porosity, permeability, and permittivity [1,7].

Working at the nano level implies working on large numbers of objects. One way todraw these objects is to start from meshes already used in FEM techniques. Meshes are

very attractive because they are automatically produced and because they completely fillthe domain. At last, because the mesher reduces the dispersion of the meshes to get a fairdescription, the use of nano objects deduced from these meshes leads to a monodispersecompound of nanoparticles.

In this chapter, it is shown how to transform meshes into objects and how to modifythem just using matrix analysis. The size and the number of the objects taken into consid-eration are restricted by the memory number and the computational time of the computerand also by the need for readable figures. In most examples, 200300 objects are takeninto account depending on the number of PDEs to be solved after a general re-meshing.

The chapter is divided into two main parts: 2D and 3D modeling. Both of them begin

with raw objects, straight transposition of meshes, and present modifications, namely,homothetic reduction to get capillaries, grain joins, and so on.In dealing with large numbers of objects, a pioneer was Leibenzon who worked on the

porosity of soils to understand, among others, the process of filtration. Some principal well-known facts on the theory of granulated materials rest on his famous works, e.g.; on Ref. [6].

16.2 Filtration: An Ideal and a Fictitious Soil

Particles of natural fluids in natural soil [6] (such as water, gas, oil) move through the poresof the soil; that is, these particles are transferred through the finest channels which areformed by the not closely contacted grains of the soil. Such kind of fluid motion in the soilis calledfiltration [2,3].

FIGURE 16.1Model of a granular material.


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561Modeling at the Nano Level

Viscosity of fluid is very important because of utterly small cross section of pores andbecause of slow velocities of fluids inside the pores. As a rule, the motion inside the pores isaccepted to be laminar, but it can be like vortical transfer because of curving canals and modi-

fication of their cross section. Seeing that soil particles are the granules of awkward shapesand different sizes, it is possible to find solution of equations which describe the motion of theviscous fluid in such kind of medium. That is why some simplified models were constructed.

There are two kinds of soils: an ideal and a fictitious. In case of ideal soil, all pores areconsidered as cylindrical, and axes of these cylinders are parallel to each other. In case offictitious soil, all its granules are supposed to be spheres of the same diameter.

16.2.1 Porosity of Fictitious Soil

Assume that we have some natural soil of volume V1. All the granules of soil occupyvolume V2of the volume V1. Hence, a volume of pores in the V1equals V3= V1 V2.

A value

m

V

V

V V

V

V

V= =

= 3

1

1 2

1

2

1

1

is calledporosity. It is obvious that 0 < m< 1.Slichter determined a value of porosity of fictitious soil by means of simple geometrical

way [6,8,9]. The value of porosity obviously depends on configurations of the spheres (whichrepresent grains of the soil). As all the spheres have got the same size, the distance betweenthe centers of two of the closest spheres equals to diameter of the sphere. Therefore, centersof each eight contacting spheres are situated in summits of a parallelepiped, all planes of

which are rhombuses (Figure 16.2).This rhombohedron is a basic model of the fictitious soil in Slichters method. Studying

of the geometrical properties of this model can give a possibility to calculate the valueof porosity m. Different dispositions of the soils spheres have two limiting states. One ofthese states corresponds to the closest contact of spheres, another one implies not so closecontact. But in both the situations, the spheres are contiguous.

It is evident that angle of the rhombohedron planes is included into interval [60; 90](Figure 16.3). For every angle of the rhombohedron, there is another angle which is addedup to 180. That is why eight pieces of full spheres which are cut from eight concernedspheres form one whole sphere.

P

S

N M

O L

Q

FIGURE 16.2Model of a fictitious soil: rhombohedron. (After Leibenzon, L.S., Motion of Natural Fluids in Porous Medium,Technical and Theoretical Literature Publishing, Moscow, Leningrad, 1947; Slichter, C.S., Theoretical investiga-tion of the motion of ground water, U.S. Geological Survey 19th Annual Report, Part II, 1899; Slichter, C.S., Themotions of underground waters, U.S. Geological Survey Water Supply Paper 67, 1902.)


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In Figure 16.4, there are diagonal sections SPLMand NOQRof the rhombohedron.Let us obtain a value of the angle of the parallelogram SPLM, for example. For this, we

will circumscribe a full sphere from the vertex Oof the rhombohedron. The radius of thesphere equals d(Figure 16.5).

The diagonal section, jointly with the faces OADand OAB, crosses this full sphere inarcs which form spherical right-angled triangleABCwith right angle BCA. PerpendicularBEdropped on diagonal OCis the altitude hof the rhombohedron. From the triangleABCfollows

cos = cos cos .AB BC AC

But AB AC BC = = =

, , ,2

PP

O OL L

Q Q

FIGURE 16.3Limited dispositions of spheres.

P

RS NM

OL Q

FIGURE 16.4Diagonal cross sections of the rhombohedron.

B

d

D

C

d

h

OA

E

/2

FIGURE 16.5Intersections of the full sphere.


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Hence,

cos cos

cos

.

=

2

(16.1)

From (16.1), we get

sin cos

cos cos

cos

sin sin

cos

sin

= =

=

=1 2

2

2

2

422

2 2

2

2 2

2

2ccos sin

cos

sin

coscos .

2 2

2

2

2 2

2

2

2

4 2 1

=

=

As42

1 2 1 1 1 22cos ( cos ) cos ,

= + = + that

sin tan cos .

= +

22 1

From this expression, we obtain finally

sin

sin cos

cos

cos sin

coscos .

= + =

++

22 2

22

2 11

2 12

(16.2)

Further, from the right-angled triangle BEO, we can find

h d= sin . (16.3)

As area of the base of the rhombohedron is d2sin , then volume of the rhombohedronequals

V hd12

= sin .

With respect to formulas (16.2) and (16.3), the last expression becomes

V

d1

3 21

1=

+

+

sin cos

cos.

(16.4)

V2is a sum of all eight pieces of full spheres located inside the rhombohedron; and as itwas stated earlier, V2equals to the volume of one whole sphere:

V

d2

3

6=

. (16.5)


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The porosity m, with respect to formulas (16.4) and (16.5), becomes

m

V

V

d

d= =

+

+1 1 6

1

12

1

3

3 2

( cos )

sin cos.

Substitute sin2= (1 cos )(1 + cos ) into the last expression and obtain the fundamentalSlichters formula:

m =

+1

6 1 1 2

( cos ) cos. (16.6)

It follows from the Slichters formula that porosity of fictitious soil, which consists ofspherical particles, does not depend on diameters of these particles; it depends only on

their disposition and value of the angle .The limit values of the angle are 60 and 90; therefore, an interval of theoretical

porosity, taking into account formula (16.6), is

0.259 0.476. m

As one can see from Figure 16.2, the area of a free space among the full spheres in a planewhich contains centers of these spheres, equals S:

S S S= 1 2 ,

whereS1is an area of a rhombusS2is a sum of areas of the circles parts inside this rhombus

It is easy to see that all four parts of the circles inside the rhombus form one whole circlewith area

S

d2

2

4=

.

The area of rhombus S1

S d12

= sin .

Hence,

S d=

sin .

4

2

Slichter introduced the following relation

n

S

S

S

S= =

1

2

1

1


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and called it free space. This value describes the area of a fluid which goes through thenarrowest place of a pore channel. Taking into consideration expressions for S1and S2, thevalue of nequals

n = 1

4

sin. (16.7)

and it includes into the interval

0.0931 0.2146. n

From the formula (16.7), one can see that the fictitious soil value of n does not depend ondiameters of the granules.

These results were obtained by Slichter. We decided to improve his results, because wewant to take into consideration not only the closest spheres contact but also the differentsizes of spheres. That is why we can inscribe spheres of a smaller diameter into the freespace among the granules. We consider the following model.

If we take into consideration not only the closest disposition of soil granules but alsodifferent sizes of these granules, it will be able to inscribe new spheres of a smaller radiusinto free space among the granules. So, we can get two variants of dislocation of the grains(Figure 16.6a and b).

Let us obtain values offree space(n) and porosity (m) for the first variant of dispositions ofthe grains (Figure 16.6a). Let Rbe the radius of the grain of bigger size and rbe the radiusof a smaller grain. Based on uncomplicated mathematical calculations, one can obtain that

r R= ( ) ,2 1

area of the square OPQLis

S R= 4 2

,

and value of the free space among the grains is

S R R r12 2 2

4= .

P

P

O OL L

Q

Q

(a) (b)

FIGURE 16.6Improved model.


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Further, volume of a cube, vertexes of which are located in the centers P, Q, L, Oof wholespheres, equals

V R=8 3

,

and value of a free space among them is

V R R r1

3 38

4

3

4

3=

3 .

Finally, one can obtain values offree spaceand porosity, respectively:

n S

S

R R R

R

m V

V

R R R

= =

=

= =

3

1 2 2 2 2

2

1

3 3

4 2 14

1 2 22

8 4

3

4

32

( ) ,

( =

1

81

5 2 6

6

3

3

).

R

Applying the same way of reasoning to the second case of dislocation of the grains (Figure16.6b), one can obtain the following meanings of correspondent values:

r R

S R

S S R r

V R

V V R r

=

=

=

=

=

2

2

2 3

31

2 3

2

4 3

4

3

8

3

12

3

13

;

;

;

,

33.

And values offree spaceand porosity in this case are as follows:

n

S

S

R R R

R= =

=

12 2 2

2

2

2 3 2 2 3

31

2 31

17 8 3

6 3

,

m

V

V

R R R

R= =

= 13 3 3

3

3

4 3

4

3

8

3

2 3

3 1

4 31

52 3 81

27 3

.


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16.3 Meshing of Geometry Objects: Examples in 2D

The aforementioned analytical approach is very powerful but remains limited because itis unable to describe any size of pores, up to no pores at all.Working on the model represented in Figure 16.1, one can confront some difficulties.

One of them arises during changing the value of porosity: It is possible but complicatedto get the needed value using only the spheres and circles, even varying radii. And in thiscase, the real value of opened porosity seems to be very approximate. Another problemfollows from the previous onetime of calculating and simulation. Even working with 2Dmodel (which is in fact an intersection of the sample), it takes too much time to resolve thegiven task. Therefore, taking into consideration all these conditions, authors decided toconstruct and work with another model of grain material.

To construct the corresponding model, the authors dealt with any automatic meshing,

used in FEM processes, which provides the user with all the necessary mathematicaltools.

Among these tools are matrix of coordinates of nodes and matrix of meshes. Using thesematrices, it is possible to process all the needed geometrical transformations: reconstruc-tion of the grains elements, homothetic transformations, displacement, etc., to match allthe elements contained in a heterogeneous material which are already known by usingelectronic microscopy [5].

The meshes used in FEM open the way to a new type of description, since they letabsolutely no vacuum. The counterpart is that the meshes have an angular geometryas, for instance, triangles or tetrahedrons. To counterbalance this inconvenience, it is

possible to include, before meshing, objects of given shape such as circles or spheres orany handmade form built using Bezier method. Bezier method is very useful in shapingbecause it traces smooth shapes with a minimum number of variables and so facilitateslater computations.

The geometric objects are produced by meshing: Each mesh is transformed into an objectand then adjusted as needed using geometric transformation. This is done using Comsol-MATLABwhich exchanges their data quite easily: Comsol provides a mesherand a solverand exportsdata matrices to MATLAB which shapes and draws objects which, in turn, willbe imported by Comsol.

16.3.1 Examples of ObjectsThis part of the paragraph gives different examples in order to show the flexibility of themethod.

As it has already been said, the information about mesh and geometric objects isimported by Comsol Multiphysics into the main matrices: Pt, the matrix of nodes (i.e.,the coordinates of each node), and Tr, matrix of meshes (which contains the list of all thenodes belonging to each mesh). For convenience, these matrices are contracted into onematrix TrPwhich contains, instead of the number of the nodes, the set of the coordinates,classified as x,y,andz. This matrix also has the interest to ignore the doubles, triples, etc.,sometimes introduced by the mesher for its own convenience.

The matrix TrPcontains all the necessary information to reshape and draw the objects.The objects can be modified as needed. For instance, it is possible to reduce them byhomothetic transformation in order to let appear capillaries or to join two neighbors or to


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divide them. The order of objects, the lines of TrP, is given by the meshing. It is possibleto modify it as needed, for instance, to begin by the objects closed to a side or closed to apoint.

All these operations are made in MATLAB and then imported by Comsol; all the nanoobjects are remeshed again for computing.

Example 16.1 Triangles in a square

Comsol draws a square width: 0.001 m, corner: (0,0). The chosen meshes give the sizeof the triangles. For instance, the ultrafine meshing produces 25,127 meshes, that is,1582. This amount of meshes would give as many triangles with a mean side of somenanometers if the initial square would have a side around 0.1 m. The final meshing ofthis arrangement will be at least 300,000 meshes which can be solved on a Station. Here,in order to obtain simple figures, the coarse method is preferred. Then, it is necessaryto run an application, for instance, electricity. Comsol exports data in a txt file to get amatrix of coordinates:

Pt = [0.0 2.0E-40.0 1.0E-46.4200256E-5 1.6731408E-40.0 0.06.4200256E-5 6.731408E-5

7.461472E-4 9.097E-48.342704E-4 9.3145535E-46.922944E-4 8.1939995E-4

7.804176E-4 8.4115536E-48.685408E-4 8.629107E-4];

and a matrix of the numbers of elements (triangular):

Tr = [2 3 51 3 22 5 43 6 58 9 11

399 402 401404 405 407403 405 404404 407 406405 408 407];

Comsol Multiphysics also exports the values of the variable at each node, but this infor-mation is not needed at this point. Then, the following short file produces a matrix TrP

[line,col] = size(Tr);TrP = zeros(line,6);for tt = 1:linept1 = Tr(tt,1);

pt2 = Tr(tt,2);pt3 = Tr(tt,3);TrP(tt,:) = [Pt(pt1,1) Pt(pt2,1) Pt(pt3,1) Pt(pt1,2) Pt(pt2,2) Pt(pt3,2)];endTrP;


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which is

TrP = [0 6.4e-005 6.4e-005 1.0e-004 1.7e-004 6.7e-005

0 6.4e-005 0 2.0e-004 1.7e-004 1.0e-0040 6.4e-005 0 1.0e-004 6.7e-005 0

6.4e-005 1.3e-004 6.4e-005 1.7e-004 1.3e-004 6.7e-0051.0e-004 6.4e-005 1.6e-004 0 6.7e-005 6.7e-005];

Each line of the matrix TrPgives the summits of triangle; the coordinates are ordered:the xs first and then theys.

Then, it is just to add, in a loop [1:line], the expression

fprintf(1,strcat(g_,num2str(tt), = line2([,x1s,,,x2s,,,x3s,],[,y1s,,,y2s,,,y3s,]);n) )

where x1sand others are the string form of the abscissas of the first node to get on thescreen

g_1 = line2([0,6.42e-005,6.42e-005],[0.0001,0.00016731,6.7314e-005]);

which will be imported by Comsol as a triangle to give Figure 16.7 (consists of 272triangles).

This geometry may support a lot of physical expressions. For instance, it could bea capacitance with a given repartition of permittivity on each triangle. Since the sizeof the sample is very small compared to the wavelength, only the following electricalequation is considered, that is, electric equation for a capacitance:

+ =

( ) , i V

r0 0

FIGURE 16.7Triangles reconstructed from the mesh information.


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The boundary conditions are the following: ground, electric potential, electric insula-tion, and distributed impedance

n J = +( ) . i Vr0

which gives the possibility of surface properties different from volume ones.The mean permittivity is obtained after integration of the normal current Itover a

boundary. For instance, if the upper side is at the potential V, the downer at the ground,and the left and right are isolated, Itis

I jC V j V t m= = 0

withj 2= 1. In 2D, Itand Chave the meaning of a density in the z direction, and the exact

length of the side disappears in the case of a square.If the triangles have a pure permittivity and if there is a continuity between them,

the value of Itwill be a pure imaginary number (e.g., 0.005596j [A/m]). But if the insideboundaries between the triangles are supposed to be lossy for any reason, as Comsolgives the possibility under the label Distributed impedance, then the behavior of thesquare is changed.

Figure 16.8 shows the current lines (plain lines) and the electric field (arrows). Theelectric field points the influence of the conductive layer. The frequency measurementgives an insight on the influence of the conductive boundaries, that is,

f e I iy= = +=1 9 49 868574 0 0945990 001[ ] . . [ ].( . )Hz A m/

FIGURE 16.8The lines of current are plain; the arrows stand for the electric field.


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Note that the following conditions were chosen in this example:for the triangles, r= 10;ground fory= 0;

V y= 1 for = 0.001;

= 1e5 [S/m], = 10 for = 0, and = 0.001;r x x

for internal boundaries, = 1e5 [S/m], r= 10.The disturbances are due to the conductivity of the side of the triangles compared to

the permittivity of their surface.

Example 16.2 Capillaries in a square

Capillaries are introduced by reducing the size of the triangles. There are many possibleways to get this result. Among them, the homothetic reduction with the mass center asbase point is the most simple because it is fully automated. Starting from TrPand givingrhas the reduction factor, the new nodes are immediately obtained. But this result is tobe corrected so that the bases of the triangles on the side of the square form a straightline. Moreover, it is necessary to add an outside square to close the domain and get thefollowing result.

To reconstruct triangles according to the meshing and to transform them into inde-pendent objects, each number of nodes correlates with its coordinate. After that, it ispossible to apply any transformation, for instance, to change sizes of this triangles-objects applying homothety relative to the center of gravity of each triangle.

For this purpose, the mass center of each triangle (point PtG) using the well-knownformula was calculated, and every point PtGwas considered as a center of homothety. Bychanging a value of the homothety coefficient rh,we obtain needed sizes of the triangles:

% ptG - center of gravity of triangleptG = ([Pt(pt1,1)+Pt(pt2,1)+Pt(pt3,1),Pt(pt1,2)+Pt(pt2,2)+Pt(pt3,2)])/3;% coefficient of homothetyrh = 0.85;xx = [Pt(pt1,1) Pt(pt2,1) Pt(pt3,1)];yy = [Pt(pt1,2) Pt(pt2,2) Pt(pt3,2)];% homothetyxx = ptG(1,1)+rh*(xx-ptG(1,1) );

yy = ptG(1,2)+rh*(yy-ptG(1,2) );geomplot(poly1([xx], [yy]) );hold(on)

As a result, the following model was constructed (Figure 16.9). In this model, the tri-angles represent the nanoparticles separated by pores. Obviously, the mathematicaltreatment can provide any desired grading of the particles size.

As noted, one can adjust a value of a free space among the triangles by changing thecoefficient of homothety rh. This way, the needed value of porosity of the material canbe obtained. Here, reduction ration rh= 0.8 (that corresponds to an approximate valueof porosity 30%).

It is interesting to operate this drawing with NavierStokes equation because the tri-

angles are not active in the flow transfer or just active through the friction on their sides(no slip) or leaking wall. With the first physical conditions this drawing produces the veloc-ity field shown on Figure 16.10. The arrangement of the Figure 16.9 produces the velocityfield shown on the Figure 16.10 on which the lighter areas represent the higher velocities.


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Figure 16.10shows the amplitude of the fluid velocity and the arrows its direction.

Comsol Multiphysics provides the velocity in each output; this result immediately givesthe mean value of the permeability function, among others, of the size of the capillaries.The conditions which were chosen for this example are as follows:

Incompressible NavierStokes equation:

( ) ( ( ( ) )),

;

u u u u

u

T = +

=

0

The triangles are inactive.Capillaries: fluid density = 1000 [kg/m3], dynamic viscosity = 0.01 [Pas];

Boundaries: x= 0.001, P= 0.715 [Pa]; x= 0, P = 0;y= 0 andy =0.001, wall;The computed value of the integral velocity field is 1.223114e-10 [m 2/s].

Example 16.3 Microwave sintering

The same geometry can be used to simulate the sintering of a mixture of large grains(triangles) coated with smaller ones supposed to fill the precedent capillaries [4,5]. Themodeling uses two equations: one is for the electric capacitance and the other is for theheating. The source term of the heat equation is only in the capillaries (the very smallnano grains have a good ability to catch the electrical energy and transform it intothermal energy while the larger grains do not have this ability). The large grains heatby conduction, and since the source is in the very heart, the heating is very fast. The

surface of the square is not well heated; usually an infrared heating has to be added toget a fair homogeneity.

Figure 16.11 shows the temperature field with a lower temperature inside than ontheir boundaries. This is a typical requirement for sintering applications.

FIGURE 16.9Capilleries in triangles.


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FIGURE 16.10Velocity field.

FIGURE 16.11Temperature field.


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model. Figure 16.13 shows the fluid velocity, amplitude (shadow), and direction arrows.

As expected, the bean is an obstacle with a shadow effect in the back and a badly irri-gated area in the front.Figure 16.14is a lens effect on the upper left corner of the previous sample.Obtained value of integral velocity field is 6.821822e-11 [m2/s]. As compared with the

preceding case, the fluid flow is normally decreased by the bean.

Example 16.5 Introduction of corks in the geometry

Starting from the capillary geometry, it is possible to add, by hand, new objects in theirjunctions. These objects could simulate corks, for instance in wet clay, which stop theflow. When they are heated, they dry and shrink, opening the way to the flow in thecapillaries (Figure 16.15). For instance, they catch energy and their gas permeability

increases with their temperature. At the beginning, the capillaries are full of gas whichescapes when the permeability of the corks increases.

Gas is confined in capillaries closed by corks. Electrical energy opens the doors.Figure 16.16shows the gas pressure at a given time.

16.4 3D Modeling: Examples

The principle of 3D modeling is very similar to 2D:

To start with a cube, 1 mm side with a lower left summit at (0; 0; 0)

To choose a coarse meshing and run any application

To export the data in txt file

FIGURE 16.13Velocity field, the lighter areas represent higher velocities.


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The exported matrices look like

% Matrix of coordinatesPt = [

0.0 0.0 0.0

0.0 2.5E-4 2.5E-4

104

104

9.5

8.5

7.5

0.5 1 1.5 2 2.5 3

8

9

FIGURE 16.14Enlargement of the upper left corner of Figure 16.13.

FIGURE 16.15Distribution of gas pressure, the higher pressures are in the centre of the square and the lower on the sides.


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0.0 0.0 5.0E-4

0.0 5.0E-4 5.0E-4

0.0 2.5E-4 7.5E-4

0.0 5.0E-4 5.0E-4

7.5E-4 5.0E-4 2.5E-4

7.5E-4 7.5E-4 5.0E-45.0E-4 5.0E-4 5.0E-4

0.0010 5.0E-4 5.0E-4];

and

% Matrix of elements (tetrahedrons)

Tr = [

2 5 8 33 2 7 8

3 7 9 8

8 3 5 97 3 1 2

238 233 235 239

237 233 231 232

238 235 232 234239 236 233 235

240 239 237 238];

As for the 2D case, it is convenient to form the matrix TrPwhich gathers, in each line, allthe coordinates classified as xs,ys, andzs. This matrix has 12 columns. Then, it is enoughto add, in a loop on TrPlines, the following sentence

FIGURE 16.16Corks in nodes of capillary net.


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fprintf(1,strcat(g_,num2str(tt), = tetrahedron3([,xxstr,;,yystr,;,

zzstr,]);n) )

to get on the screen

g_1 = tetrahedron3([0 0 0.00025 0;0.00025 0.00025 0.00025 0;0.000250.00075 0.0005 0.0005]);

xxstris the string corresponding to the four xxs and similarly foryys andzzs.

The set of all the g_i imported into Comsol produces a cube full of tetrahedrons(Figure 16.17).

Figures 16.18and 16.19 give the result of the experimentation of this drawing seen as acapacitance: The ground is at the lower side (z= 0), the potential at the higher (z= 0.001),and the lateral sides are said to be isolated. The tetrahedrons have the same permittiv-ity and all the internal boundaries have an electrical conductivity . The behavior of the

capacitance is highly dependent on the ratio between and . In the first figure,

;it is the opposite in the second one. The current lines are those of a pure capacitance inFigure 16.18, that is perfectly straight when they are highly disturbed in Figure 16.19.

The equipotential surface, the grey voile on the Figure 16.19, also show the influenceof the conductive boundaries. This kind of result helps to understand the influence ofsurface defects, for instance, in a sintering: The measurement of the output current versusfrequency gives an insight into the internal boundaries.

To obtain these results, the following conditions were chosen:For tetrahedrons r= 10;On the boundaries: V= 1 forz= 0.001; ground forz= 0;

xy

z

FIGURE 16.17Cube full of tetrahedrons.


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For lateral faces of the cube: isolationInside boundaries: = 1000, r= 10.

The only difference between the two results is the frequency: 1e-6 Hz for the first, 1e9 Hzfor the second. The use of non-continuous boundaries is lighter to be solved since it avoidsthe use of capillaries to simulate the grain joins.

At last, if the tetrahedrons have different physical values and if the repartition is known,

this type of model gives a mean value of the cube behavior. When the repartition is notknown, it is necessary to look to statistics, that is, to make different tries and choose.

FIGURE 16.18Current flows straight lines in the capacitance if the tetrahedrons have no surface conductivity.

xy

z

FIGURE 16.19

Deformed current lines and equipotential surface (grey voile) in the capacitance if the tetrahedrons have a highsurface conductivity.


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16.7 Microwave Heating

Figures 16.22 and 16.23 show the result of the microwave heating of the capillaries full ofliquid water, nanoparticles, or any kind of susceptor. The electrical energy transformedinto heat energy is, by the same time, diffused to the tetrahedrons. The result is deeply

xyz

FIGURE 16.21Velocity field.

FIGURE 16.22Temperature of the tetrahedrons surface.


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dependant on the ratio between the input and the output power into and from the cap-illaries. Moreover, the density of the capillary network is also an important factor ofhomogeneity.

16.8 Conclusion

Automatic meshing and its associated nano objects is a convenient way to produce, in2D and 3D as well, full spaces without any forgotten vacuum. As a consequence, it letsthe boundaries play their specific behavior. At last, it is compatible with all the geometri-cal shapes, circles, spheres, and so on but also shapes extracted from experimentation as

shown with the bean.Finally, automatic meshing can be seen as a way to fill with nanoparticles spaces let

empty between objects previously implanted.Two different applications were described throughout the chapter:

1. Computation of mean values of physical constants of heterogeneous material, forinstance, thermal and electric conductivity, permittivity, etc., with or without thegrain joins influence

2. Evaluation of capillary flows leading to an inside knowledge of permeability, let-ting out of the computation the nanoparticles which act only by their surfaces, but

keeping them

On these geometries, it is possible to use all the classical FEM used for drying, sintering,microwave heating [4], and so on.

FIGURE 16.23Net heating in light and lines of thermal currents.


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As a consequence, when a problem is solved in a small domain, it is possible to enlargethe solution to larger domains, just by matrices association. Comsol Multiphysics andMATLAB softwares were used (run on an HP Z800, 16 proc, 64 Gbits).

References

1. Akulich P.V. and Grinchik N.N., Modeling of heat and mass transfer in capillary-porousmaterials,Journal of Engineering Physics and Thermophysics, 71, 2, 1998, 225233.

2. Barenblatt G.I., Entov V.M., and Ryzhik V.M., Fluid Transfer in Natural Seams, Nedra, Moscow,Russia, 1984 [in Russian].

3. Gubkin I.M., Study of Oil, Nauka, Moscow, Russia, 1975 [in Russian].

4. Lefeuvre S. and Gomonova O., Microwave heating at the grain level, Proceedings of COMSOLConference 2010, Paris, France, November 1719, 2010.

5. Lefeuvre S., Federova E., Gomonova O., and Tao J., Microwave sintering of micro- and nano-sized alumina powder, Advances in Modeling of Microwave Sintering: 12th Seminar ComputerModeling in Microwave Engineering and Applications, Grenoble, France, March 89, 2010,pp. 4650.

6. Leibenzon L.S.,Motion of Natural Fluids in Porous Medium, Technical and Theoretical LiteraturePublishing, Moscow, Leningrad, 1947 [in Russian].

7. Selyakov V.I. and Kadet V.V., Percolation Models of Transfer Processes in Porous Media, Nedra,Moscow, Russia, 1995.

8. Slichter C.S., Theoretical investigation of the motion of ground water, U.S. Geological Survey

19th Annual Report, Part II, 1899. 9. Slichter C.S., The motions of underground waters, U.S. Geological Survey Water Supply Paper67, 1902.


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Modeling nanomaterials