UDK 669.71'21:519.68	ISSN 1580-2949
Original scientific article/Izvirni znanstveni članek	MTAEC9, 45(4)335(2011)
CFD ANALYSIS OF EXOTHERMIC REACTIONS IN Al-Au NANO MULTI-LAYERED FOILS
CFD-ANALIZA EKSOTERMNIH REAKCIJ V VEČPLASTNIH NANOFOLIJAH Al-Au
Karlo T. Raic'i, Rebeka Rudolf2 3, Primož Ternik4, Zoran Žunic5, Vojkan Lazic6, Dragoslav Stamenkovic6, Tatjana TanaskoviC7, Ivan Anžel2
1University of Belgrade, Faculty of Technology and Metallurgy, Karnegijeva 4, Belgrade, Serbia 2University of Maribor, Faculty of Mechanical Engineering, Maribor, Slovenia 3Zlatarna Celje, d. d., Kersnikova ul.19, 3000 Celje, Slovenia 4Private Researcher, Bresterniška ulica 163, 2354 Bresternica, Slovenia 5AVL-AST, Trg Leona Štuklja 5, 2000 Maribor, Slovenia 6University of Belgrade, School of Dentistry, Clinic for Prosthodontics, Belgrade, Serbia 7College of Vocational Studies-Belgrade Polytechnics, Belgrade, Serbia karlo@tmf.bg.ac.rs
Prejem rokopisa - received: 2011-02-18; sprejem za objavo - accepted for publication: 2011-03-07
This work presents the possibility of numerical modelling using Computational Fluid Dynamics (CFD) in the field of nano-foils. The governing equations were solved using a Finite Volume Methodology (FVM). The computational domain was discretized using a uniform Cartesian grid with the appropriate mesh size along the x and y directions employing the corresponding number of grid points. The field variables were discretized at the cell centres and the spatial, as well as the time, derivatives were approximated using the second-order accurate numerical scheme. The time-evolution of the temperature and concentration fields, as well as the atomic diffusion coefficient, will be presented for the appropriate Al-Au nano-foil geometry and boundary conditions.
Key words: Au-Al nano-foils, finite volume method, temperature and concentration transfer
V delu je predstavljena možnost numeričnega modeliranja z uporabo računske dinamike fluidov (CFDC) v primeru nanofolij. Ustrezne enačbe so bile rešene z uporabo metodologije končnih prostornin (FVM). Področje izračunov je bilo opredeljeno z uporabo enakomerne kartezijeve mreže s primerno velikostjo vzdolž smeri x in y z ustreznim številom mrežnih točk. Spremenljivke polj so bile opredeljene za središče celic in prostorske ter časovne derivative ter aproksimirane z uporabo numerične sheme z drugim redom velikosti natančnosti. Časovni razvoj temperaturnih in koncentracijskih polj in koeficient atomske difuzije sta predstavljena za ustrezno geometrijo Al-Au-nanofolij in mejne pogoje.
Ključne besede: Al-Au nanofolije, metoda končnih elementov, prenos temperature in koncentracije
1 INTRODUCTION	process, near-net-form fabrication, and rapid material
synthesis.
Combustion synthesis, also called reaction synthesis,	The ignition of combustion synthesis can be subdivi-
is a process in which two or more materials with large	ded into two different modes. The first mode, important
exothermic heats of mixing, known as reactants, are	for present research, is termed self-propagating high-
combined together. When the reactants are heated	temperature synthesis, or SHS. In this method, the
sufficiently, they begin to spontaneously intermix on the	compact is heated locally using an external heat source.
atomic scale, releasing heat in the process. A large body The local heating initiates the reaction locally, releasing
of literature exists on this process, and it has been heat that drives the reaction forward. The reaction moves reviewed extensively.1
across the compact in a self-propagating manner, driven
In general, combustion synthesis has been accomp-
by its own heat. The other method of ignition is termed
lished using mixed reactant powders that are pressed into
thermal explosion, or simultaneous ignition. a pellet of a certain green density. The combustion is	,	, " .....
initiated by an external energy source to heat the com-	However, powder compacts have several limitations
pact. Once combustion begins, the energy released by when used for combustion synthesis. The density of the
intermixing contributes to an increase of the local tempe-	final product is often limited by the green density of the
rature, resulting in faster intermixing and combustion. A	powder compact. If the combustion synthesis reaction is
large amount of materials are reported to have been	solid state, the particles can only inter diffuse with each
synthesized with this general method, including carbides,	other where they are in physical contact. The size of the
borides, silicides, nitrides, sulfides, hydrides, inter-	powder particles is often large compared to the charac-
metallics, and complex composites. Combustion syn-	teristic diffusion distance for a given system, making it
thesis has many advantages, such as the ability to create	difficult to achieve full intermixing. Particles of highly
high-melting-temperature materials in a low-temperature	reactive components will often have a passivating outer
coating that acts as a barrier to the diffusion with other reactants. Moreover, voids between the powder particles limit thermal diffusion through the compact, reducing the ability of the reaction to be driven by its own heat.
With modern thin-film deposition techniques, fully dense multilayer materials with similar exothermic reactions can be fabricated (Figure 1). Such reactive multilayer foils consist of hundreds or thousands of nanometer-scale alternating layers of two or more materials, known as reactants, which can mix exother-mically. When heat is applied locally to the foil, it undergoes SHS in a fashion similar to powder compacts. Structures of reactive multilayer foils offer several potential improvements over powder compacts.
The individual layers in a reactive foil are usually on the scale of tens of nanometers, which significantly decreases the diffusion distances involved in interatomic diffusion between the reactants. The thickness of the individual reactant layers can be controlled during fabrication, allowing significant control over the properties of the foil. The individual layers are in intimate contact with each other, which increases both thermal and atomic diffusion in the system while also eliminating voids in the system. Moreover, having thin layers of reactants that are in intimate contact with each other makes the SHS reaction propagate faster in most reactive foils than in a powder compact with the same components.
In an A-B multilayer system (here Al-Au), when one end of the foil is ignited with a small thermal pulse (such as a spark) at room temperature, local atoms will diffuse normal to the layers with A-A and B-B bonds being exchanged for A-B bonds, as shown in Figure 1.
Figure 1: Schematic representation of a self-propagating reaction in a multilayer foil that is propagating from left to right. The unreacted foil consists of alternating layers of elements A and B with an intermixed region between the layers.
Slika 1: Shemati~na slika samonapredujo~e reakcije z leve na desno v ve~plastni foliji. Nezreagirana folija je iz izmeni~nih plasti elementov A in B s podro~jem me{anja med plastmi.
In this process, the heat is released and conducted parallel to the layers. If the atomic diffusion and energy release are sufficiently fast, then the reactions are self-propagating. These reactions can travel as fast as 25 m/s and can reach temperatures above 1500 °C.
Numerous studies of interface reactions have been performed in different multilayer systems. A common feature of these multilayers is the apparently large free energy, typically several tens of kJ/mol, available for the phase formation. This leads us to conclude that all possible product phases should be able to form from the very beginning of the reaction.
The aim of the present work is to support the development of novel reactive Au-Al nano-multilayered material that enables the rapid bonding of similar and dissimilar materials by using a Computational Fluid Dynamics (CFD) analysis. The advantage of numerical modelling is that, once the model is set up and established, a wide range of scenarios can be investigated with relatively little effort, and complex two-dimen-sional2 as well as three-dimensional3 problems may be solved using numerical models. The potential use for the reactive Al-Au nano-foils is as a controllable, localised heat source for joining applications4,5. To optimize the performance of the Al-Au foils it is necessary to have a clear understanding of the physical processes that dictate the reaction temperature, the rate of heat generation and the velocity at which the reaction propagates along a foil.
2 NUMERICAL MODELLING
The formulation of the mathematical model used to describe the self-propagating reactions in multi-layered Al-Au nano-foil is based on the following assumptions: (1) the effects of phase changes in the foil (e.g., melting of the reactants and/or products) are neglected; (2) the atomic diffusion is represented by a single binary diffusion coefficient D; (3) the physical properties of the foil (e.g., density p, thermal conductivity X and specific heat Cp) are assumed to be dependent on the composition. Last but not least, the width of the foil (z) is large in comparison to the thickness (y) and small in comparison to the length (x). As a result, it is possible to treat the thermal and atomic diffusion in the foil as a two-dimensional problem.
2.1 Governing equations
Under the assumptions above the atomic mixing is described using a time-dependent, conserved scalar field C(x,y,t), defined such that C = 1 for pure Au and C = 0 for pure Al. The evolution of C is governed by:
dC dt
d dC
dx
D
Cj ^ dxj J
(1)
The atomic diffusivity is assumed to be independent of the composition and to follow the Arrhenius dependence on the temperature, according to:
E
D=D0 exp
RT
(2)
where D0 is the Arrhenius pre-exponent, E is the activation energy and R is the universal gas constant. The values and E = 25.20 kJ/mol as used in the present study are taken from the work of Fouracre6.
The time-evolution of the concentration field is coupled with the temperature equation:
dT dt
d
dx
X dT
j ^^^p J
Q(C)
PC
p
(3)
where the physical properties (thermal conductivity X, density p and specific heat Cp) in any control volume of the domain are given by the corresponding fraction of the phase (Au and Al) in that volume. They are mathematically obtained as:
X = X AuC + X Al(1-C) p = PauC + PAI(1-C ) Cp = cp C + cp (1-C)
p pAu	pAr ^
(4)
The rate at which the heat is generated (dQ(C)/dt) is assumed to be proportional to the rate at which the composition of the foil changes. In the simplest case, the linear relationship between the composition and the energy released can be assumed7. However, a parabolic function may be closer to the real situation.8 9
Q(C) =-pCp(Tf0 -T0)C'-
(5)
and this was used in the present study.
Tf0 and T0 are the adiabatic temperature of the reaction and the initial temperature of the foil.
2.2 Geometry, boundary and initial conditions
The laminar and time-dependent thermal and species transport in a two-dimensional nano-foil was considered as depicted in Figure 2.
At the inlet we prescribed the constant temperature T = 1000 K, while the convective heat flux condition (for natural convection along the vertical plate) was used at
Figure 2: Schematic of multi-layered Au-Al nano-foil. L = 2000 nm, ^Au = ^AI = 100 nm
Slika 2: Shema ve~plastne Au-Al nanofolije. L = 2000 nm, öAu = öai = 100 nm
Figure 3: Boundary and initial conditions Slika 3: Za~etni in mejni pogoji
the outlet of the nano-foil. The initial conditions were as follows:
•	Temperature T = 298 K in the whole nano-foil.
•	Concentration C = 1 for Au and C = 0 for Al, see Figure 3.
For the nano-foil, the following physical properties were used for Au:
•	Density p = 19320 kg/m3
•	Specific heat Cp = 130 J/kgK
•	Thermal conductivity X = 318 W/mK
and Al:
•	Density p = 2700 kg/m3
•	Specific heat Cp = 910 J/kgK
•	Thermal conductivity X = 237 W/mK
2.3 Numerical procedure
The governing equations were solved using with the Ansys CFX numerical code which employs a standard finite-volume methodology10 with all the variables defined at the centre of control volumes populating the physical domain being considered. Each equation is integrated over the control volume to obtain a discrete equation which connects the variable at the centre of the volume with its neighbours.
The computational domain was discretized using a uniform Cartesian grid of mesh size Ax = Ay along the x and y directions, respectively. The corresponding number of grid points is denoted by Nx = 800 and Ny = 80. For a spatial discretization of diffusive terms, the second-order accurate scheme based on central-differences was used, while temporal derivatives were approximated with a second-order accurate backward differences for a fixed time step At = 10-9 s.
3 RESULTS AND DISCUSSION
The time-evolution of both the temperature and concentration field is presented in Figure 4. It can be seen that that the temperature (i.e., thermal) transport is essentially unidirectional (mostly taken in the axial direction) and occurs parallel to the layering.
In contrast to other studies7,8,9,11 where the atomic
diffusion was modelled as a one-dimensional pheno-
Figure 4: Time evolution of the temperature and concentration field Slika 4: Časovni razvoj temperaturnega in koncentracijskega polja
menon (occurring normal to the layering) the present study shows that the atomic diffusion is actually a two-dimensional phenomenon (see Figure 4). This is due to the fact that the diffusion coefficient D, as defined by Eq. (2), varies with the temperature (as a matter of fact it increases as the temperature increases) in the axial direction. Of course, if the diffusion coefficient is kept constant, the concentration diffusion would occur only in the streamwise direction (due to the concentration gradient).
Finally, in the present study the concentration diffusion is a much faster process due to the higher value of the diffusion coefficient D in comparison to the value of the thermal diffusion coefficient a.
4 CONCLUSIONS
In the present study, the time-dependent temperature and concentration diffusion in the multi-layered Al-Au nano-foil was studied numerically using mathematical modelling and a finite-volume-based CFD code. The atomic diffusion coefficient D was taken to be independent of the composition and to follow the Arrhenius dependence on the temperature while the physical properties (and therefore the thermal diffusion) in any control volume of the domain were taken to be dependent on the corresponding fraction of the phase (Au and Al) in that volume.
The present numerical results show that the temperature diffusion is essentially unidirectional (in the axial direction) while the atomic diffusion occurs in both (i.e., axial and streamwise) directions due to the variable diffusion coefficient and the concentration gradient. The latter is an important fact (influencing the thickness of the intermixed region between layers as well as the heat
release in the process) which has been ignored in most of the existing analytical as well as computational models.
5	REFERENCES
1	A. J. Swiston, T. C. Hufnagel, T. P. Weihs, Joining bulk metallic glass using reactive multilayer foils. Scripta Materialia, 48 (2003), 1575-1580
2	P. Ternik, New contributions on laminar flow of inelastic non-Newtonian fluid in the two-dimensional symmetric expansion: Creeping and slowly moving flow conditions. Journal of Non-Newtonian Fluid Mechanics, 165 (2010), 1400-1411
3	P. Ternik, J. Marn, Numerical study of blood flow in stenotic artery. Applied Rheology, 19 (2009), 13060
4K. T. Raic, R. Rudolf, B. Kosec, I. Anzel, V. Lazic, A. Todorovic, Nanofoils for soldering and brazing in dental joining practice and jewellery manufacturing. Mater. Tehnol., 43 (2009), 3-9
5K. T. Raic, R. Rudolf, A. Todorovic, D. Stamenkovic, I. Anzel, Liquid metal-ceramic interfaces in dental practice and jewellery manufacturing. Mater. Tehnol., 44 (2010) 2, 59-66
6	R. A. Fouracre, Electron microscope observations of chemical diffusion in the Al/Au system. Thin Solid Films, 135 (1986), 189-201
7	A. J. Gavens, D. Van Heerden, A. B. Mann, M. E. Reiss, T. P. Weihs, Effect of intermixing on self-propagating exothermic reactions in Al/Ni nanolaminate foils. Journal of Applied Physics, 87 (2000), 1255-1263
8	A. B. Mann, A. J. Gavens, M. E. Reiss, D. Van Heerden, G. Bao, T. P. Weihs, Modeling and characterizing the propagation velocity of exothermic reactions in multilayer foils. Journal of Applied Physics, 82 (1997), 1178-1188
9S. Jayaraman, A. B. Mann, M. Reiss, T. P. Weihs, O. M. Knio, Numerical Study of the Effect of Heat Losses on Self- Propagating Reactions in Multilayer Foils. Combustion and Flame, 124 (2001), 178-194
10	P. Ternik, Planar sudden symmetric expansion flows and bifurcation phenomena of purely viscous shear-thinning fluids. Journal of Non-Newtonian Fluid Mechanics, 157 (2009), 15-25
11	E. Bensoin, S. Cerutti, O. M. Knio, T. P. Weihs, Effect of reactant and product melting on self-propagating reactions in multilayer foils. Journal of Applied Physics, 92 (2002), 5474-5481