Mathematical Modeling of Nanomaterials - COMSOL.com

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Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt
Mathematical Modeling of Nanomaterials
1
D. J. Strauss P
P, C. TrenadoP
2
1
2
PInstitute of New Materials, Saarbrücken, Tstrauss@inm-gmbh.deT, P
PInstitute of New Materials,
Saarbrücken, Ttrenado@inm-gmbh.deT,
1 Introduction
Mathematical modeling at the Institute of New Materials has played a crucial role in
supporting the manufacturing and design of new technologies of nanomaterials, whose
applications range from transportation, electronics and optics engineering to environmental
sciences. In this paper, we focus our attention to two mathematical models together with their
corresponding FEMLAB simulations: The first part of the paper is about computational
modeling of fire-retardant nanocomposite gels and the second one deals with Magnetic
nanoparticles for in vivo applications. We provide specific background and numerical results
for each model to finally state conclusions to our modeling.
1.1 Computational Modeling of Fire-Retardant Nanocomposite Gels
1.1.1 Background
Fire retardant intumescent materials start to bubble in response to fire and swell into bust
foam. They provide protective coatings through absorption of heat by endothermic reactions
and through insulating properties of the final structure. Recently, aqueous intumescent
nanocomposites based on nanoscaled SiO2 particles have been introduced by the Institute
for New Materials and may lead to a significant improvement of general fire protective
systems. See Fig. 1 for an application of this gel.
A reliable mathematical model representing the heat flow characteristics of these materials
does not exist up to now, but it is indispensable for an optimization of fire retardant systems
containing this composite. Moreover, such a model might provide a deeper understanding of
the mechanisms determining the excellent thermal insulating properties of the composite and
may thus help to derive optimization objectives. In this study, we present the first macroscale
heat flow model of this nanocomposite. We evaluate our model by using experimental data.
This model might be used for optimizing macroscale parameters of fire retardant systems
containing this composite such as the layer thickness, especially, in multilayer arrangements,
and for diverse boundary conditions. Furthermore, chemical manufacture objectives for the
optimal bubble growth and outgassing can be derived from this model. From a modeling
point of view, this model is also a prerequisite for a more complex heterogeneous multiscale
modeling framework in which the matrix changes are obtained from an embedded micronanoscale
model. Such a multiscale approach allows for the simulation of materials in
arbitrary settings and with extreme and unusual boundary conditions, which in turn impacts
enormously in the experimental cost and material component optimization.
In this respect, an evolutionary programming approach can be combined with heterogeneous
multiscale models to obtain application specific parameters for the model by making use of
unconstraint programming of high dimensional hypersurfaces. This notion has been applied
for the optimization of the intumescent retardant material which has been developed at the
INM.

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt
Figure 1. The intumescent nanocomposite gel for the realization of retardant windows.
1.1.2 Model description
Our model represents the phase change from a virgin material (the gel) to a porous material
(the final structure with bubble inclusions and after the carbonization of the organic
compounds) by a moving mushy front, mainly driven by outgassing. Here we do not consider
a (hard) Heavyside phase front, usually used in problems of the Stefan type, which is more
natural from a physical point of view. For this, a generalized form of the transient heat
equation with a drain term representing the absorption of heat by endothermic reactions is
employed.
An additional mass conservation equation is coupled to this heat equation using a
temperature dependent empirical function for the mass loss due to outgassing.
This mass conservation equation is also used to derive the change of the porosity
(normalized inverse of the density) of the material and the effective dynamic thermal
conductivity (according to the Maxwell model).
The dynamic porosity is related to the averaged bubble growth and growth limits are
introduced by taking the stability of the bubbles into account. The stability of bubbles is
strongly related to the insulating properties of the porous phase and the final glass matrix
after the mushy transformation front has passed. The equations are solved in a domain Ω
using the finite element method.
To the derive the unknown empirical parameters as well as to validate the model, a number
of experimental measurements were performed in a specially designed high temperature
oven using (mechanically) stiff gel plates with a varying proportion of a hardening agent,
which determines the growth and stability of bubbles, and different thicknesses of the plates.
Genetic programming was applied to fit the model to the experimental data as well as the
boundary conditions, in addition the general model conduction +convection + radiation was
used to determine the free boundary.
1.1.3 Results
Our model provided a high correlation to the oven measurements using 1D and 2D numerical
simulations. In particular, our model allowed for a representation of the temperature
gradients in space over time and exhibited a good qualitative correlation in the phase change
behavior, which was analyzed in a different set of experiments such as thermo-gravimetric
analysis. The influence of the hardening agent was also reflected in the numerical
simulations. In Fig. 2 (left) the outgassing function depending on the temperature is shown
for particular sets of parameters. The temperature profile at the backside (none heating side)
of a stiff gel plate with a thickness of 20mm for different outgassing settings which are related
to the proportion of hardening agent in the plates is shown Fig. 2 (right). The plateaus in
curve reflect the active insulation properties due to outgassing of the aqueous nanocomposite.

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt
400
Q=1
Q=1.5
350
300
Q=1
Outgassing
Temperature
250
200
150
Q=1.5
100
50
0 100 200 300 400 500
Temperature (°C)
0
0 1000 2000 3000 4000 5000 6000 7000
Time (sec)
Figure 2. Left: the outgassing function over time; Right: the temperature at the insulation side for the
different outgassing settings
1.1.4 FEMLAB Modeling
The different finite element capabilities of FEMLAB were used to deal with the simulation of
heat transport in the intumescent material. In Fig. 3 we show an investigation of the spherical
bubble size to check effective thermal conductivity models of porous media, e.g., Maxwell or
Raleigh. Fig. 4 shows two similar composed constructions (steal, copper, and iron letters on
a marble block). The left construction has a layer of the INM intumescent nanocomposite
which provides thermal insulation while the construction on the right is in direct contact to the
heat source. The thermal insulation by the nanocomposite is clearly noticeable (heating time
approx. 30 min).
Figure 3: Bubble investigation
Figure 4: Two composed constructions
1.1.5 INM measurements used in fitting the numerical model
In Fig. 5 we show the volume of the sample in dependence on the temperature (“Härter” -
hardening agent). In Fig. 6 we show the thermal conductivity of the solid phase (the
destroyed and trampled down char) in dependence on the temperature.

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt
Figure 5 Figure 6
In Fig. 7 we show the density of the sample in dependence on the temperature and for
different proportions of the hardening agent. In Fig. 8 we show the loss of mass in the
sample in dependence on the temperature and for different proportions of the hardening
agent.
Figure 7 Figure 8
1.2 Magnetic nanoparticles for in vivo applications:
A Numerical modeling study
1.2.1 Background
In vivo applications of biocompatible magnetic nanoparticles in a carrier liquid controlled by
an external magnetic field from outside the body has recently been proposed for specific
drug delivery such as in locoregional cancer therapies or the occlusion aneurysms. They can
also be used as guided contrast agents in myocardial imaging after myocardial infarction.
However, the choice of the optimal clinical setting still remains a challenge for every of the
mentioned applications. A numerical heterogeneous multiscale model can be used for the
optimal a priori determination of the free parameters and might help to overcome this
problem. In this study, we are going to present an approach to the implementation of such a

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt
multiscale model. We applied a hybrid scheme which is based on Maxwell and Navier-
Stokes equations to paramagnetic liquids with particles of the size of 10nm (ferrofluids). For
a considered area of the body, the Maxwell equations for the static magnetic case were
solved. The vector potential was then coupled to a liquid flow problem described by the
Navier-Stokes equations by a volume force acting on the magnetic liquid. Time-dependent
boundary conditions were used to describe the systolic blood flow regime. The system of
coupled partial differential equations was solved by the finite element method on adaptive
meshes. For a representative geometry which can be extracted by medical imaging, our
hybrid model allowed the study of the hydrodynamics of the magnetic liquid. Moreover, our
model is open for an embedding of a nanoscale model which represents the particle
dynamics. It is concluded that the proposed model is a prerequisite for the optimal
computational choice of the free parameters of magnetic liquids and the external magnetic
field for in vivo applications.
1.2.2 Model equations
Maxwell‘s Equations for the Static Case:
∇ × H = J ∇ ⋅B = 0
Using a constitutive relation: B = µ ( H + M)
These equations can be combined to:
−1
∇× ( µ ∇× A − M)
= J
Where A denotes a magnetic vector potential, B is the Magnetic Flux density, M the
magnetization vector, J is the current density vector and H is the magnetic field vector.
The magnetization (gamma) of the ferrofluid is given by
⎛
⎜
⎛ −
γ =
α arctan⎜
βµ
⎝ ⎝
1
∂A ( x,
y)
⎞ ⎛ ∂ ⎞⎞
−1
A(
x,
y)
⎟,
α arctan⎜
βµ ⎟⎟
∂x
⎠
⎝ ∂y
⎠⎠
Incompressible Navier-Stokes Equation with Magnetic Force Term
∂u
ρ − ∇ ⋅
∂t
T
( ∇u
+ ( ∇u)
) + ρ(
u ⋅∇)
u + ∇p
= F
m
+ F
g
1.2.3 FEMLAB Modeling
Here we show two simulations, Fig. 9 is based on the macroscale model already described.
Fig. 10 shows a particle scale simulation, which is in development. It is noticeable that above
a time t, the magnetic nanoparticle carriers in the flow are disturbed in the vicinity of a
magnetic field.

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt
Figure 9. Macroscale model
Figure 10. Particle scale simulation
1.3 Conclusion
We have developed a new model for an aqueous intumescent nanocomposite material which
was recently introduced for fire retardation. The model provided a high correlation to
experimental data obtained in oven measurements. We also have developed a numerical
simulation for the behaviour of magnetic nanoparticles in a carrier liquid for in vivo
applications.
The implementations were possible by means of the finite element and different meshing
capabilities of FEMLAB, which combined with evolutionary algorithmic tools allowed to obtain
the scientific simulations used for our models.
References
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Magnetic nanoparticles for in Vivo applications: A Numerical modeling study
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