aluminum hybrid foams

Author’s Accepted Manuscript Experiments, Modeling and Simulation of the Magnetic Behavior of Inhomogeneously coated Nickel/Aluminium Hybrid Foams A. Jung, D. Klis, F. Goldschmidt www.elsevier.com/locate/jmmm

PII: DOI: Reference:

S0304-8853(14)01019-1 http://dx.doi.org/10.1016/j.jmmm.2014.10.106 MAGMA59551

To appear in: Journal of Magnetism and Magnetic Materials Received date: 6 July 2014 Revised date: 8 September 2014 Accepted date: 2 October 2014 Cite this article as: A. Jung, D. Klis and F. Goldschmidt, Experiments, Modeling and Simulation of the Magnetic Behavior of Inhomogeneously coated Nickel/Aluminium Hybrid Foams, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2014.10.106 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Experiments, Modeling and Simulation of the Magnetic Behavior of Inhomogeneously coated Nickel/Aluminium Hybrid Foams A. Junga,∗, D. Klisb,∗∗, F. Goldschmidta,∗∗ a

Universit¨ at des Saarlandes, Institute of Applied Mechanics, Campus A4 2, 66123 Saarbr¨ ucken, Germany

b Universit¨ at

des Saarlandes, Laboratory for Electromagnetic Theory, Campus C6 3, 66123 Saarbr¨ ucken, Germany

Abstract Open-cell metal foams are used as lightweight construction elements, energy absorbers or as support for catalytic coatings. Coating of open-cell metal foams is not only used for catalytic applications, but it it leads also to tremendous increases in stiffness and energy absorption capacity. A non-line of sight coating technique for complex 3D structures is electrodeposition. Unfortunately, due to the 3D porosity and the related problems in mass transport limitation during the deposition, it is not possible to produce homogeneously coated foams. In the present contribution, we present a semi-non-destructive technique applicable to determine the coating thickness distribution of magnetic coatings by measuring the remanent magnetic field of coated foams. In order to have a closer look at the mass transport mechanism, a numerical model was developed to predict the field scans for different coating thickness distributions in the foams. For long deposition times the deposition reaches a steady state whereas a Helmholtz equation is sufficient to predict the coating thickness distribution.The applied current density could be identified as the main influencing parameter. Based on the developed model, it is possible to improve the electrodeposition process and hence the homogeneity in the ∗

Principal corresponding author, phone +49-681 302 2169, Email address: [email protected] (A. Jung) ∗∗ Co-author Email addresses: [email protected] (D. Klis), [email protected] (F. Goldschmidt) Preprint submitted to Journal of Magnetism and Magnetic Materials

October 3, 2014

coating thickness of coated metal foams. This leads to enhanced mechanical properties of the hybrid foams and contributes to better and resource-efficient energy absorbers and lightweight materials. Keywords: hybrid metal foams, nickel coating, electrodeposition, magnetic flux density, FEM modeling 1. Introduction Metal foams are bio-inspired materials mimicking the microstructure of natural load-bearing structures like wood and bones. Based on this cellular microstructure and the related deformation mechanism, metal foams are used as energy absorbers. Due to their high specific stiffness-to-weight ratio, a further field of application is as a lightweight material [1, 2, 3]. The widest field of application belongs to aluminium foams, which are of significant interest in aerospace, automotive and mechanical engineering [2, 4]. Open-cell metal foams consist of a 3D interconnected network of gas-filled pores and solid struts. Based on the high inner surface of open-cell metal foams, they are used as heat exchangers and suppport for catalytic coatings in chemical engineering, as electrodes in lithium-ion batteries or in fuell cells [2, 4, 5]. Also the mechanical properties of open-cell foams can be enhanced by thin but stiff nanocrystalline coatings. For example, a coating of 150 µm nickel on a 10 ppi (pores per inch) aluminium foam for example improves the stiffness by a factor of 3.5 and leads to a tenfold increase of the energy absorption capacity [6, 7, 8, 9]. According to the open porous structure, open-cell foams can be infiltrated by fluids. Coating of metal foams is mainly of great interest for the functionalization of the large inner surface area. The usage as filter and catalyst support leads to the development of different coating technologies to improve the corrosion resistance or to apply catalysts. Such coating techniques are dip-coating, high velocity oxygen flame spraying, vapour deposition techniques or electrodeposition. Electrodeposition and vapour deposition are the only techniques applicable for the production of nanocrystalline coatings, but vapour deposition is more expensive. In electrodeposition, the foam will be used as cathode in the electroplating process for the electrochemical deposition of catalysts [5] or the reinforcement of lightweight aluminium foams improving the mechanical properties. Currently, this new research area is hardly investigated [7, 10, 11]. In 2008, Boonyongmaneerat et al. [10] tried 2

to reinforce aluminium foams with a thickness of 4 mm with a stiff, nanocrystalline coating of a nickel-tungsten alloy by electrodeposition. They outlined positive effects of the coating on the stiffness, strength and energy absorption capacity, but according to the mass increase due to the coating, there was no positive effect on the specific properties. One year later, Bouwhuis et al. [11] dealed with the electrodeposition of nickel on about 13 mm thick aluminium foams. They confirmed the findings of Boonyongmaneerat et al. for nickel. The general problem is the inhomogeneous coating over the cross section of the foam. The coating thickness in the foam center is about 10% of the coating thickness on the boundary regions of the foam. In 2010, Jung et al. [7] developed a new cage-like anode for the electrodeposition of quite homogeneously coated Ni/Al hybrid foams, for which the coating thickness in the foam center could be increased up to 80% in comparison to the outer foam regions. As already shown by Euler [12, 13, 14], the electrochemical deposition process and hence the coating thickness distribution in porous electrodes is strongly coupled with the mass transport limitation and electromagnetic shielding based on the Faraday effect of the foam microstructure. First efforts for a deeper understanding of the mass transport limitation during the electrodeposition of open-cell metal foams were made by Jung et al. [8, 15]. They used the magnetic properties of the nickel coating to semi-non-destructively determine the coating thickness distribution of Ni/Al hybrid metal foams based on field scans of the magnetic flux density and developed a mass transport limitation model for direct current and pulsed electrodeposition based on these scans. In this work, we study the mass transport limitation during the electrodeposition of metal foams using magnetic field scans and numerical simulations. The results from the field scans will be correlated to the simulations, whereas the nickel distribution can be calculated by inverse methods from the field scans using a Helmholtz distribution as stationary diffusion-reaction equation to describe the nickel distribution in the foam samples. Hence, the results of this work contribute to a better understanding of the complex deposition process in open-cell metal foams and can help to improve the coating process. The findings are not only related to nickel coating and open-cell metal foams but can be also adopted to other coating materials and materials with an open-cellular microstructure.

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2. Methods and Experiments 2.1. Electrodeposition on Metal Foams Electrodeposition is a common non-line of sight technique for the preparation of metals and alloys in form of thin coatings up to millimeter thick selfsupporting structures. In this work, cubic aluminium alloy foams (AlSi7 Mg0.3 , Celltec Materials, Dresden, Germany) with an edge length of 50 mm and a pore size of 10 ppi were coated by direct current plating (DC) and pulsed electrodeposition (PED) with nanocrystalline nickel with a theoretical overall coating thickness of 20 µm and a crystallite size of 50±8 nm. A commercial nickel sulfamate electrolyte (Enthone GmbH, Langenfeld, Germany) with a nickel content of 110 g/L nickel was used at a pH of 3.8 and a temperature of 40 ◦ C. To obtain a qualitatively good coating on aluminium, the pretreatment steps of pickling and electroless plating preventing the aluminium from dissolution in the acid nickel electrolyte have been performed according to the known literature [8]. To guarantee a quite homogeneous coating, a cagelike anode filled with nickel pellets (Ampere GmbH, Dietzenbach, Germany), as described by Jung et al. [6, 7, 16], was used. 2.2. Flux Density Distribution The determination of the coating thickness distribution on coated foam samples is the basis for the improvement of the coating homogeneity by changing the plating setup and parameters. The first attemps to get information on the coating thickness distribution in the foams was done by Bouwhuis et al. [11] by cutting coated Ni/Al hybrid foams into slices and observing the inhomogeneity in the coating distribution by means of scanning electron microscopy (SEM). With this method, it is possible to distinguish between nickel and aluminium according their different atomic numbers and electronic structures. Based on the higher backscattering in heavier elements, nickel appears lighter than aluminium. This method is only applicable for small samples that can be introduced into a SEM microscope. For larger samples, some coated struts must be extracted from different foam positions and the coating thickness must be determined for each strut [11]. SEM provides only visual information, which has to be translated into relevant data using grey scale analysis. An advantage is the applicability to all coating materials. For magnetic coatings like nickel, Jung et al. [15, 8] developed a semi-nondestructive method to visualize the coating distribution by scanning the mag4

netic flux density distribution of the remanent magnetic field of a homogeneously magnetized foam. This is in analogy to the magnetic field trapping now routinely performed on bulk high-TC superconductors to determine inhomogeneities in the flux density distribution [17, 18, 19]. Christides at al. [20] applied magnetic flux density scanning also to bulk permanent magnets. To perform the field scans, the coated Ni/Al hybrid foam cubes are cut into 5 mm thick rectangular slices. The thickness of the slices corresponds to the size of one single pore. Each slice is introduced into a Helmholtz coil with a homogeneous magnetic field of 256 mT orientated in normal direction to the cutting direction to guarantee a well-defined initial magnetic state for each foam slice. As the nickel was previously unmagnetized, the remanent field follows the initial magnetization curve. The coating distribution of each foam slice is determined by measuring the remanent magnetic flux density distribution Bx3 (x1 , x2 ). Therefore, the surface area of the foam slices was scanned with a commercial Hall probe (Arepoc, Bratislava, Slovakia, magnetic resolution 0.1×10−1 T). Performing an automatized scan of the surface area, the Hall probe was mounted in a fixed x3 -direction 1.5 mm above the sample on a x−y −z-stage. The x1 - and x2 - step size was 2 mm, respectively. The measuring setup and the measuring planes on the foam slices are shown in Fig. 1. Hall y 1m

MP1 MP2 MP3 MP4 MP5 MP6

bb

5m

aa

x z

m m

Figure 1: Foam slices and measuring planes MP 1 to MP 6 (a) and measuring setup for the flux density scans (b).

In order to consider the magnetic scatter field, the measuring area was three times the dimensions of the foam slice in the x1 - and x2 -direction. Hence, each foam slice is placed in the center of a rectangular area of 150× 150 mm2 . Under the assumption that the spatial nickel distribution is reflected in the distribution of the measured magnetic field, the coating thickness distribution can be calculated based on the magnetic field. This will be done in section 3. Additional information about this method used on coated 5

metal foams can be found in previous work [6, 8, 15]. 3. Theory and Modeling 3.1. Mass Transport Limitations during Electrodeposition Inhomogeneities in the coating of such complex 3D structures as metal foams by electrodeposition arise not only from the mass transport limitations but also from the electromagnetic shielding due to the Faraday effect caused by the foam’s microstructure. In contrast to the inhomogeneities originating from the mass transport limitations, inhomogeneities caused by the Faraday effect cannot be eliminated without extensive changes to the microstructure. Hence, in the following, we focus on the mass transport limitation effects during DC plating of metal foams. In DC electrodeposition of planar bulk electrodes, there is an evolution of two concentration zones in front of the electrode (see Fig. 2 (a)). The first zone far away from the electrode belongs to the bulk electrolyte with the constant concentration of metal ions c∞ . The second zone directly evolves in front of the electrode due to the electrochemical conversion and leads to the depletion of the electrolyte from c∞ to the surface concentration cs on the electrode surface. For a given current density, the conversion of metal ions at the electrode surface and the diffusion of metal ions from the bulk electrolyte to the depletion zone in the vicinity of the electrode are in an equilibrium. The concentration profile in this stationary, so-called Helmholtz diffusion zone is characterized by Fick’s law of diffusion with a linear gradient of the metal ion concentration from c∞ to cs . The thickness of this diffusion zone can be reduced by further improving mass transport using convection. For 3D porous electrodes, the concentration profile is more complex. The described concentration profile for planar electrodes must be expanded into the microstructure of the foam electrode, and there is not only a two dimensional but also a three dimensional plating process. During the electrodeposition, each strut of the metal foam acts as a planar electrode in the above-mentioned way. The plating mechanism is outlined in Fig. 2 (b). An open porous microstructure is plunged into an electrolyte. In the case that no current is applied, there is a homogeneous distribution of metal ions in the bulk elektrolyte as well as in the foam pore volume [1]. Applying a direct current leads to the spontaneous deposition of all metal ions in the foam itself and in the vicinity of the outer foam surface [2]. To continue the electrochemical coating process, the metal ions have to diffuse from the bulk electrolyte to 6

the foam center [3]. The diffusion process is superimposed by the consumption of metal ions due to the electrochemical deposition at each strut from the outer geometric foam surface up to the center of the foam. Jung et al. [8] developed a model for the description of the mass transport limitations and the concentration profil in cellular structures. This mass transport limitation based on a diffusion reaction and an electrochemical conversion is the reason for the inhomogeneity in the coating thickness distribution over the cross section of the foam. AA

CC center Center deltaN

Outer surface Foam

Elec

Elec.

A1

J1

Conc

J2 Cinf

Cinf J3

A2

distance 0

Cinf

A3

BB A3

Current

A1 A2

A4 0

Cinf

A4 time

Figure 2: Mass transport limitation in front of a planar electrode (a), current transient for DC plating (b) and mass transport limitation in a foam (c): [1] before deposition, foam plunged into the electrolyte, [2] after switch on of the current, [3] during deposition, [4] after deposition, foam removed from the electrolyte.

3.2. Theory From Maxwell’s equations for the stationary case, we know curl H = j, div B = 0. 7

(1) (2)

with the magnetic field strength H, the magnetic flux density B, and the total electric current density j [21]. The field strength and the flux density are linked by the constitutive law B = µ0 µr H + M .

(3)

Therein, µ0 and µr denote the vacuum permeability and the relative magnetic permeability. The magnetization M is defined as the concentration of magnetic dipoles per volume P m . (4) M = lim ∆V →0 ∆V After a magnetization process, ferromagnetic materials like nickel show a non-vanishing magnetization. For magnetic fields that are strong enough, this so-called remanent magnetization is independent of the magnetization history [22]. In our case, the 256 mT are not high enough to saturate the nickel, but the material was previously unmagnetized so that M Ni has a unique value defined by the initial magnetization curve. During the field measurement process described in section 2.2, there is no macroscopic current flowing in the foam so that curl H = 0.

(5)

This equation is always fulfilled in a simply connected domain if the magnetic field strength is represented as the gradient of a magnetic scalar potential H = grad Ψ.

(6)

Now, the divergence-free condition of the flux density assumes the form of Poisson’s equation div (µ0 µr grad Ψ + M ) = 0.

(7)

This equation is also known as the magnetic scalar potential formulation [21] and can easily be solved numerically, e. g. with the finite element method. For the purpose of our simulation, the microstructure of the foam can be regarded as infinitely small, so that for the macroscopic magnetization P m ∆VNi = M Ni cNi , (8) M macro = lim ∆V →0 ∆VNi ∆V with the volumetric concentration of Nickel cNi . That way, the inhomogeneities in the coating thickness of the foam can be modeled by a spatially varying M macro . 8

3.3. Modeling and Numerical Investigations of the Magnetic Field Based on the assumption that the flow of nickel ions in the electrolyte is only driven by a gradient in concentration ϕ(x, t) a diffusion equation has to be used to describe the time-dependent spatial distribution of nickel in the electrolyte. Furthermore taking into account the effect that the nickel deposition on the foam reduces its concentration in the fluid, a reactiondiffusion equation [23, 24, 25] of the form ∂ϕ = div ( D grad ϕ) + f (x, t, ϕ) ∂t

(9)

is chosen. In this context, the diffusion coefficient D is assumed to be independent of space, time, and concentration. As the deposition rate is proportional to the concentration of nickel in the electrolyte, a reaction equation f (x, t, ϕ) = a ϕ with the growth rate a for the deposition is used as a first approach. This relationship allows to calculate the time-dependent deposition rate (Fig. 3). In this case, the deposition rate becomes constant for larger time scales. This steady state can be descibed by a Helmholtz equahelmholtz stationary time 1.0y

decomprate

0.98y

0.96y

0.94y

092y 0x

0.2x

0.4x

0.6x

0.8x

1.0x

space

Figure 3: Spatial distribution across a domain Ω1 [0, 1] with dirichlet boundary conditions ∂ΩD = 1 for reaction-diffusion, a stationary state, and a Helmholtz equation.

tion. Considering the long deposition time (see section 4), it is reasonable to 9

neglect the transient effects. Consequently, the normalized spatial distribution of the thickness of nickel ψ(x) on the aluminium can be described by a Helmholtz equation as well a . (10) div (grad ψ) − λ ψ = 0 with λ=− D Depending on the parameters of the electrodeposition (time, current, convection, etc. ), the values of λ vary. This approach gives a spatial distribution of nickel (Fig. 4) at a certain moment. This distribution is the starting point for helm 0a 09 08 07 06 05 04 03 02 01 00 c500s

c20

c50

c100

c500

Figure 4: Spatial distribution of nickel depending on the parameter λ R the simulation of the magnetic behavior of the foam. Comsol Multiphysics was used to apply the magnetic properties in the domain. For the simulation, solid plates as well as plates with holes that embody the foam were used. In our case, the foam has a porosity of 77%. According to equation (8), the spatial distribution of the magnetization is proportional to the distribution of the thickness of the nickel layer. So the magnetization of the domain is applied according to M = (0, 0, ψ(x))T . (11)

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To be able to conduct an analysis and to compare the simulated and measured data, the magnetic field in the air above the plane is calculated. 4. Results and Discussion The relative magnetic flux density of three foams with different plating parameters was measured. Therefore, for each foam slice, the magnetic flux density was normalized to the maximal magnetic flux density of the first slice (MP 1). The foams ware plated with DC plating at a current density of j = 1 mA/cm2 and j = 8 mA/cm2 , respectively. The third foam was plated using pulsed electrodeposition with a duty cycle of 50% and a frequency f of 20 Hz at an averaged current density jm = 2 mA/cm2 . Fig. 5 shows the magnetic flux density distribution for each foam slice of the foam plated with DC 8 mA/cm2 in the range of the measuring area from -50 to 50 mm in the x1 and x2 -direction with the foam slice in the center. The measuring plane MP 1 corresponds to the outer foam surface and thus to a homogeneously coated zone. The relative magnetic flux density distribution has the shape of an elliptical paraboloid. The nickel coating thickness in the foams decreases from the outer surface to the foam center. This results in a dip in the center of the parabolic magnetic flux density distribution and a reduction of the maximum relative magnetic flux density. Fig. 6 presents the maximal relative magnetic flux density for each sample as a function of the position in the foam. The magnetic flux density decreases from the outer surface (MP 1) to the foam center (MP 6). Lower current densities for the plating result in a higher relative magnetic flux density in the foam center and hence in a more homogeneous coating distribution. The best homogeneity is reached by the pulsed electrodeposited foam though this can be explained as follows: A duty cycle of 50% in combination with a frequency of 20 Hz means that for 25 ms a current pulse of 4 mA/cm2 was applied, followed by current-off time of 25 ms. Hence, there was an average current density of jm = 2 mA/cm2 . Due to the current-off time, the metal ions are able to diffuse to the center of the foams without being deposited. This undisturbed diffusion leads to a higher concentration of metal ions in the center of the foam for the next current pulse despite of the higher average current density. In order to make clear that a consideration of the real foam microstructure in the simulation is not necessary, Fig. 7 compares the spatial relative magnetic flux density distribution (from -50 to 50 mm of the measuring plane with the 11

10 09 08 07 06 05 04 03 02 01 00 50 x0

10 09 08 07 06 05 04 03 02 01 00 50x0

10 09 08 07 06 05 04 03 02 01 00 x0

x 00

x 00

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00y 50

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10 09 08 07 06 05 04 03 02 01 00 x0

09

m50

m50

m50

10

x 00

00y 50

m50

x 00

00y 50

x0

m50

Figure 5: Distribution of the relative magnetic flux density for the measuring planes 1 to 6 for the foams coated with 8 mA/cm2 DC.

foams in the center) of the artificial foam structure and the solid cube in Fig. 4 with the parameter λ = 50 for the measuring planes MP 1, MP 3, and MP 5. Both spatial distributions correspond very well quantitatively to each other and qualitatively to the measured spatial relative magnetic flux density distribution in Fig. 5. The simulation of the magnetic fields by means of consideration of the foam microstructure needs a much finer mesh than the simulation of the solid plates. Hence, the simulation is computationally very expensive when taking into account the microstructure. Based on the good correlation between the field scan simulations of the artificial foam plates and the solid plates outlined in Fig. 7, the further computational experiments are done on solid cubes. For a better comparison of the effect of the spatial distribution of nickel on the relative magnetic flux density distribution, the simulated magnetic fields for the solid cubes in Fig. 4 are outlined as contour plots in Fig. 8. With the increase of the parameter λ, the concentration gradient of nickel from the outer foam surface to its center increases. This is similar to the effect of increasing the current density. After increasing λ, the maximal relative magnetic flux density shows a strong decrease from the outer surface (MP 1);

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a1 09

simulation

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c500 c100 c50 c20

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relflux

06 05 04

messure PED24mA

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DC1mA DC8mA

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01 00 25

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distance Figure 6: Relative magnetic flux density distribution as a function of the position in the foam; measurements and simulation. a 10 09 08 07 06 05 04 03 02 01 00 x0

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x 00

00 y 50 m50

Figure 7: Comparison of the simulated magnetic fields for an artificial foam cube (a) and a solid cube (b) with the spatial nickel distribution for λ = 50 outlined in Fig. 4.

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the dip into the parabolic-shaped flux density distribution is much more pronounced as well. The maximal relative magnetic flux density for each value of λ is also shown in Fig. 6. In a parameter identification for the foam coated with 1 mA/cm2 , λ = 20 was identified. The decrease in the maximal magnetic flux density for a defined position while increasing λ is not linear. As seen for λ = 500, even for the large concentration gradient and gradient in the magnetization from MP 1 to MP 2, there is a non-zero saturation remanent flux density and hence a stationary concentration of metal ions in the foam. For larger current densities, there is a significant deviation between the measurements and the simulation. This is an effect of the side reaction during the electrodeposition. The fact that there is a good correlation between the measurements and the simulation for low current densities outlines that the afore-mentioned assumption of a linear reaction rate with f = aϕ fits the real situation very well. For larger current densities such as 8 mA/cm2 , an increasing amount of current is used for the decomposition of the water in the electrolyte and leads to the formation of hydrogen. Hence, there is a reduced deposition rate for nickel. If the current is applied, all metal ions will be deposited in the foam structure. The concentration in the center is very low. Due to the reduced nickel reduction rate as a fact of the hydrogen evolution and based on the evolving hydrogen bubbles, there is a higher nickel concentration and magnetic flux density in the outer region of the foam than was expected by the simulation. The low magnetic field in the foam center at higher current densities is also based on the hydrogen evolution. In contrast to the outer foam regions, the hydrogen bubbles are trapped by the microstructure though and block the deposition by reducing the active foam surface. This explains the pronounced decrease of the magnetic flux density and hence of the nickel coating thickness for MP 3 to MP 6. 5. Conclusions The strengthening of metal foams by means of the electrodeposition of nanocrystalline coatings such as nickel is an important method to improve the mechanical properties of metal foams as it leads to tremendous increases in stiffness and energy absorption capacity. The deposition process strongly depends on the mass transport limitations, mainly associated with the limited diffusion and superposition with the electrochemical deposition leading to an inhomogeneous spatial distribution of the coating thickness in the foam. Field scans of the remanent magnetic flux density were used to semi-non14

l1

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s1

s2

s3

s4

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50

y

s6

0

m50 m50

0

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0.5

1.0

Figure 8: Simulation of the relative magnetic flux density distribution as function of the position in the foam for different parameters λ. For each contour plot, only the sector from -50 to 50 mm in the x1 - and x2 -direction of the measuring plane are represented with the foam placed in the origin.

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destructively visualize the coating thickness distribution in the foams. In order to achieve a deeper understanding of the deposition process and the results of the field scans, numerical simulations based on a multiphysical approach were performed. To describe the superposition of the diffusion of the metal ions and the deposition, a reaction-diffusion equation was used to describe the inhomogeneous spatial coating thickness distribution. It could be demonstrated that, based on the long deposition time, it is sufficient to model these inhomogeneities using a Helmholtz equation describing a stationary coating process. The remanent magnetic fields of samples with an inhomogeneous spatial nickel distribution were modeled using magnetostatics. The foam microstructure is infinitely small in comparison to the resolution of the Hall probe, hence, the assumption that the macroscopic magnetization only depends on the magnetization of nickel and the nickel concentration in the foam, was made. As a consequence, inhomogeneities in the coating thickness were modeled by a spatial varying macroscopic magnetization. The simulated fields correspond very well with the experimental results for foams coated under low current densities. The experimental results as well as the numerical results show increasing coating homogeneities for lower current densities and deposition rates, respectively. For larger current densities, there are some deviations between the simulation and the experiment arising from side reactions as hydrogen evolution. The elimination of this deviation and the investigation of the exact effect of pulsed electrodeposition will be the topic of further advanced studies. The present contribution demonstrates that, for future work, it is possible to determine the spatial coating thickness distribution of 3D porous electrodes in electrodeposition based on inverse calculations of the simulated magnetic fields and the measured field scan. In future work, to improve the computational results, the effect of side reactions such as the hydrogen evolution should be included by introducing a dependency of the growth rate a on the current density and the caused hydrogen evolution, respectively. The presented work is a milestone for the better understanding and elimination of mass transport limitations in porous electrodes and could be used for the targeted enhancement of the mechanical properties of hybrid metal foams and hence for the development of lightweight construction materials and energy absorber with superior properties. Due to the fact that this method is not limited to the electrodeposition process and to metal foams, this experimental and numerical method is also ˙ the battery and fuel of great interest for other fields of application, e. gfor cell industry, in which porous electrodes are used. In this fields the current 16

distribution and the spatial distribution of electrodeposited catalysts are of interest. The presented method can be used to improve the homogeneity of catalytic coatings and hence to improve the efficiency of fuel cells or catalysed production processes in chemical industry. Acknowledgement The authors thank Dr. M. R. Koblischka for his contributions in many fruitfull discussions and R. Keller for his experimental support. References [1] M. Ashby, The mechanical properties of cellular solids, Metall Trans A 14 (9) (1983) 1755–1769. [2] J. Banhart, Eigenschaften und anwendungsgebiete offenporiger metallischer werkstoffe, Mat.-wiss. u. Werkstofftech. 31 (2000) 501–504. [3] J. Banhart, J. Baumeister, A. Melzer, W. Seeliger, M. Weber, Aluminiumschaum-Leichtbaustrukturen f¨ ur den Fahrzeugbau, ATZ Automobiltechnische Zeitschrift 100 (1998) 66–70. [4] J. Banhart, Manufacture, characterisation and application of cellular metals and metal foams, Prog. Mater. Sci. 46 (6) (2001) 559–632. [5] W. Yang, S. Yang, W. Sun, G. Sun, Q. Xin, Nanostructured palladiumsilver coated nickel foam cathode for magnesium-hydrogen peroxide fuel cells, Electrochim. Acta 52 (1) (2006) 9 – 14. doi:DOI: 10.1016/j.electacta.2006.03.066. [6] A. Jung, H. Natter, R. Hempelmann, S. Diebels, M. R. Koblischka, U. Hartmann, E. Lach, Electrodeposition of Nanocrystalline Metals on Open Cell Metal Foams: Improved Mechanical Properties, ECS Transactions 25 (41) (2010) 165–172. [7] A. Jung, H. Natter, S. Diebels, E. Lach, R. Hempelmann, Nanonickel coated aluminum foam for enhanced impact energy absorption, Advanced Engineering Materials 13 (1-2) (2011) 23–28.

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