Anisotropic thermal conductivity in epoxy-bonded magnetocaloric compositesBruno Weise, Kai Sellschopp, Marius Bierdel, Alexander Funk, Manfred Bobeth, Maria Krautz, and Anja Waske Citation: Journal of Applied Physics 120, 125103 (2016); doi: 10.1063/1.4962972 View online: http://dx.doi.org/10.1063/1.4962972 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/120/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Improvement of magnetic hysteresis loss, corrosion resistance and compressive strength through spark plasmasintering magnetocaloric LaFe11.65Si1.35/Cu core-shell powders AIP Advances 6, 055321 (2016); 10.1063/1.4952757 Enhanced thermal conductivity in off-stoichiometric La-(Fe,Co)-Si magnetocaloric alloys Appl. Phys. Lett. 107, 152403 (2015); 10.1063/1.4933261 Enhanced magnetostrictive effect in epoxy-bonded TbxDy0.9−xNd0.1(Fe0.8Co0.2)1.93 pseudo 1–3 particulatecomposites J. Appl. Phys. 117, 17A914 (2015); 10.1063/1.4916507 Consequences of the magnetocaloric effect on magnetometry measurements J. Appl. Phys. 108, 043923 (2010); 10.1063/1.3466977 La ( Fe , Co , Si ) 13 bulk alloys and ribbons with high temperature magnetocaloric effect J. Appl. Phys. 107, 09A953 (2010); 10.1063/1.3335892

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Anisotropic thermal conductivity in epoxy-bonded magnetocaloriccomposites

Bruno Weise,1,a) Kai Sellschopp,1,a) Marius Bierdel,1 Alexander Funk,1,a) Manfred Bobeth,2

Maria Krautz,1 and Anja Waske1,b)

1Institute for Complex Materials, IFW Dresden, P.O. Box 270116, D-01171 Dresden, Germany2Institute for Materials Science and Max Bergmann Center of Biomaterials, TU Dresden,01062 Dresden, Germany

(Received 6 July 2016; accepted 5 September 2016; published online 28 September 2016)

Thermal management is one of the crucial issues in the development of magnetocaloric refrigeration

technology for application. In order to ensure optimal exploitation of the materials “primary”

properties, such as entropy change and temperature lift, thermal properties (and other “secondary”

properties) play an important role. In magnetocaloric composites, which show an increased cycling

stability in comparison to their bulk counterparts, thermal properties are strongly determined by the

geometric arrangement of the corresponding components. In the first part of this paper, the inner

structure of a polymer-bonded La(Fe, Co, Si)13-composite was studied by X-ray computed tomogra-

phy. Based on this 3D data, a numerical study along all three spatial directions revealed anisotropic

thermal conductivity of the composite: Due to the preparation process, the long-axis of the magneto-

caloric particles is aligned along the xy plane which is why the in-plane thermal conductivity is larger

than the thermal conductivity along the z-axis. Further, the study is expanded to a second aspect

devoted to the influence of particle distribution and alignment within the polymer matrix. Based on

an equivalent ellipsoids model to describe the inner structure of the composite, numerical simulation

of the thermal conductivity in different particle arrangements and orientation distributions were per-

formed. This paper evaluates the possibilities of microstructural design for inducing and adjusting

anisotropic thermal conductivity in magnetocaloric composites. Published by AIP Publishing.[http://dx.doi.org/10.1063/1.4962972]

I. INTRODUCTION

Magnetocaloric refrigeration is believed to be a competi-

tive alternative for conventional vapour compression technol-

ogy and therefore has drawn large interest from both scientific

(a comprehensive overview can be found in Ref. 1) and indus-

trial research (see recent press-releases2,3). Instead of a gas

which is used conventionally, a solid body exhibiting a

magnetocaloric effect serves as refrigerant material for

magnetocaloric refrigeration. The material is magnetized and

demagnetized during a refrigeration cycle and thereby expels

or absorbs heat to or from the surrounding, respectively. In

the search for suitable materials, the alloys based on La(Fe,

Si)13 and Fe2P-type compounds are considered to be most

promising for room-temperature application.4 It is crucial that

these materials can withstand up to several millions of magne-

tization/demagnetization cycles, which is one of the reasons

why composites with increased mechanical stability have

been developed.5–8 Since the refrigerant material is a solid,

the generated heat has to be transferred out of the solid body

to a surrounding heat transfer fluid in order to create a hot and

cold end of the regenerator bed within a certain operating fre-

quency. This frequency scales with the cooling power of the

device; therefore, a large thermal conductivity from the solid

body to the surrounding cooling liquid is required.9 On the

contrary, lateral heat transfer within the refrigerant material is

disadvantageous since the operating thermal span and, there-

fore, the cooling power of the device are reduced. From this

technical viewpoint, the demand of an anisotropic heat con-

ductivity of the magnetocaloric material arises. Up to now,

experimental values only for isotropic thermal transport prop-

erties have been reported,10–13 and less attention has been

paid towards anisotropy. In this paper, the 3D microstructure

of an epoxy-bonded composite plate is determined by X-ray

computed tomography (XCT) in order to access potentially

preferred particle orientation. The inner structure then is

linked to the thermal properties, the thermal conductivity in

this case, which is experimentally and numerically determined

along each plate edge (x-, y-, z-direction). Based on that,

hypotheses to increase the anisotropy of the thermal conduc-

tivity are assessed by modeling different particle arrange-

ments by means of an equivalent ellipsoids model.

II. 3D MICROSTRUCTURE OF COMPOSITES

A. Preparation and characterization

Pulko et al.6 investigated a series of polymer-bonded mag-

netocaloric composites in order to assess their competitiveness

in comparison to conventionally sintered LaFe13�x�y CoxSiyplates. From these series, one sample has been chosen in order

to study its inner constitution in detail by X-ray computed

a)Also at Institute for Materials Science, TU Dresden, D-01069 Dresden,

Germany.b)Also at Institute for Materials Science, TU Dresden, D-01069 Dresden,

Germany; URL: http://www.ifw-dresden.de/magnetocalorics; a.waske@ifw-

dresden.de.

0021-8979/2016/120(12)/125103/6/$30.00 Published by AIP Publishing.120, 125103-1

JOURNAL OF APPLIED PHYSICS 120, 125103 (2016)

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tomography (XCT).14 The sample consists of LaFe13�x�y

CoxSiy-powder with a mean diameter d < 130 lm intermixed

with 45 vol. % Amerlock Sealer epoxy and is considered as

sample A in the publication of Pulko et al.6 More information

about the magnetocaloric properties of the composite can be

found in this paper. X-ray computed tomography was per-

formed with a GE Phoenix Nanotom M. The sample volume of

ð2� 1:5� 0:3Þmm3 was irradiated with X-rays from a tung-

sten target (Diamond window), using an acceleration voltage

U ¼ 130 kV and current I ¼ 100 lA. During a 360� rotation, a

set of 1900 2D absorption images was obtained with an expo-

sure time of t¼ 0.75 s and a Voxelsize Vx ¼ 2 lm. Phoenix

datosjx2 software was used for volume reconstruction and FEI

Avizo 9.0 software for advanced image processing.

B. Image processing and modelling

From XCT, a 3D reconstruction of the composite was

obtained (Figure 1(a)). Each voxel of the reconstructed vol-

ume has a defined 8-bit grey value. From the corresponding

histogram, the phases of the composite can be separated by

thresholding. To isolate the magnetocaloric particles from

the surrounding matrix, first an unsharp masking algorithm

has been applied in order to sharpen the contrast between the

particles and the surrounding matrix. Second, a median filter

was used to decrease the noise in the data-set. Finally, the

grey value distribution was reduced to a binarized distribu-

tion. This step is shown in Fig. 1(b), where the matrix and

air surrounding the particles is black and the particles are

illustrated in red. As a next step, particles are separated by

means of the watershed algorithm,15 which removes remain-

ing voxel-bridges between adjacent particles. Now, the iso-

lated particles can be labeled individually (Fig. 1(c)), and

further analysis such as preferred orientation can be per-

formed as will be explained as follows.

The orientation distribution in this work is volume

weighted, i.e., each value contributes to the distribution pro-

portional to the volume of the particle V. The volume can be

obtained from the 3D dataset by simply counting all voxels

belonging to the particle

V ¼ð

V

1 d ~V ) V � VVoxel

Xparticle

1: (1)

Other integral quantities like the center of mass

~rcom ¼1

V

ðV

~r d ~V ; (2)

and the inertia tensor referenced to the center of mass

I ¼ð

V

ðð~r �~rcomÞ2 � E� ð~r �~rcomÞ � ð~r �~rcomÞÞ d ~V ; (3)

can also be calculated for each particle from the 3D dataset

using the substitution of the volume integral in Eq. (1). In

these equations,~r denotes the position vector, E is the iden-

tity matrix—a 3� 3 diagonal matrix where all the diagonal

elements equal 1—and the density of the particle is assumed

to be constant and set to 1 for the sake of simplicity. The

eigenvectors of the inertia tensor are regarded as the princi-

pal axes of the particle, which are used to describe the orien-

tation of the particle. The eigenvector with the lowest

eigenvalue, i.e., with the lowest principal moment of inertia,

corresponds to the direction with the largest dimension of

the particle.

C. Model of equivalent ellipsoids

In order to assess the influence of the particle orientation

on the thermal properties of the composites, the complex

geometry of the particles is translated into an equivalent

ellipsoid model. Based on the XCT-data of the composite,

the volume, the center of mass, and the inertia tensor of each

individual particle are determined. Then, each particle is

replaced by an ellipsoid with the same center of mass and

inertia tensor. This method for the simplification of particle

shape is well known in literature (see, e.g., Refs. 16–18). It

can be seen in Figure 2 that the particle orientation is con-

served by this means. Nevertheless, it has to be mentioned

that the volume of the ellipsoid is slightly bigger than that

of the initial particle. When replaced by an equivalent ellip-

soid, the orientation of the particle can be manipulated easily

with the help of a self-written MATLABVR

program in further

simulations. As a consequence, all further results on thermal

properties are based on real XCT-data sets, i.e., the orienta-

tion distribution of the equivalent ellipsoids represents the

FIG. 1. Tomographic slice of the composite including all grey values (a), binarization step to prepare the separation of individual particles (b), and separated

particles by watershed algorithm marked with individual colour code (c). On the right side, the reconstructed 3D volume with individual particles marked in

all tomographic slices is shown. The sample volume was ð2� 1:5� 0:3Þmm3.

FIG. 2. Illustration of equivalent ellipsoids model. The equivalent ellipsoid

representing the particle is defined by three half-axes corresponding to the

axes of inertia of the real particle.

125103-2 Weise et al. J. Appl. Phys. 120, 125103 (2016)

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arrangement of the irregularly shaped particles in the

composite.

D. Analysis of 3D microstructure

Similar to a pole figure for texture depiction in materials,

the orientation distribution of the half-axes of the equivalent

ellipsoids in the composite is shown in Figure 3. h describes

the angle between each half-axis of the particle and the global

z-axis of the composite (cf. Fig. 1). The range of h is restricted

to the interval [0�; 90�] because of the twofold symmetry of

the ellipsoids.

Obviously, the long and the middle half-axes are oriented

perpendicular to the z-axis (out-of-plane direction) of the

composite. Due to that, intuitively, the short half-axis of

the ellipsoids would be oriented parallel to the z-direction.

However, a random distribution of the short-axis is observed

which can be attributed to the statistical deviation of the orien-

tation angles of the ellipsoids within the composite. In other

words, during the preparation of the epoxy-bonded compo-

sites as described in Ref. 6, elongated irregularly shaped

La(Fe, Co, Si)13 particles align along the global xy plane (in-

plane). Note that from Fig. 3(a) mean orientation angle of the

long-axis of 70:6� was determined.

It is obvious that geometrical anisotropy will affect the

thermal properties of the composites. The influence of the

orientation texture on the thermal conductivity of the compo-

sites will be addressed in Sec. III.

III. THERMAL CONDUCTIVITY OF COMPOSITES

A. Theoretical bounds

The overall thermal conductivity in a multicomponent

system can be described in analogy to Ohm’s law:19 The lim-

iting cases are either a series or parallel arrangement of the

components, i.e., the upper and lower bound of the overall

thermal conductivity in the present composite can be deter-

mined from

kmax;parallel ¼ fkparticles þ ð1� f Þkmatrix; (4)

and

kmin;series ¼kparticleskmatrix

1� fð Þkparticles þ fkmatrix; (5)

with f¼ 58 vol. % (from XCT) as the volume fraction

of the particles in the composite and thermal conductivi-

ties kparticles ¼ 8:93 Wðm�1 K�1Þ (Ref. 6) and kmatrix

¼ 0:24 Wðm�1 K�1Þ,20 respectively. From this approach,

the thermal conductivities determined by computational

methods in Sec. III B have to be in the interval

0.55 W m�1 K�1 <k < 5.28 W m�1 K�1.

B. Measurements and finite element method (FEM)simulation

According to Equation (6), the thermal conductivity

k can be directly derived from the thermal diffusivity a,

specific heat capacity cp, and density q

k ¼ acpq: (6)

To access, the thermal conductivity of the composite measure-

ments of thermal diffusivity was performed with a NETZSCH

LFA 457 MicroFlash device in zero-field using the Cape-

Lehman model.21 Due to geometrical restrictions of the device,

only the out-of-plane thermal diffusivity, az, was accessible.

Specific heat measurements were performed in a quantum

design physical property measurement system in zero-field.

According to Pulko et al.,6 the density of the composite was

4.7 g cm�3. Figure 4 shows the measured data of azðTÞ(dashed), cpðTÞ (dashed-dotted), and the resulting curve (solid)

of the thermal conductivity kzðTÞ. At T¼ 300 K, the measured

thermal conductivity is kz ¼ 2:41 W m�1 K�1.

The binarized reconstructed volume of the XCT-scans

served for the simulation of the thermal conductivity of the

FIG. 3. Orientation distribution of the half-axes of equivalent ellipsoids.

FIG. 4. Temperature dependent thermal diffusivity az and heat capacity

cp of the composite in zero-field (right axes). The thermal conductivity

kz (left axis) was calculated from these measurements according to

Equation (6).

125103-3 Weise et al. J. Appl. Phys. 120, 125103 (2016)

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composite in all three directions (x-, y-, z-axis). For FEM simu-

lation, the binarized histogram as described in Section II B

was used to generate a voxel-based mesh comprising

65 847 936 elements. Following that, the thermal conductivity

of the two different phases (kparticles ¼ 8:93 W m�1 K�1,

kmatrix ¼ 0:24 W m�1 K�1) was allocated to the corresponding

phases within the mesh. Table I summarizes the results of the

numerical simulation of the thermal conductivity along the x-,

y-, and z-axis of the composites and the in-plane and isotropic

value of k that can be directly derived from kðx;y;zÞ. First, it is

to note that the experimentally and numerically determined

out-of-plane values of heat conductivity (kz) correspond very

well. Additionally, simulated kz is significantly lower than kx

and ky. Evidently, the anisotropic alignment of the La(Fe, Co,

Si)13 particles along the xy plane (as shown in Fig. 3) leads to

anisotropic thermal conductivity of the composite.

To summarize, in the presented composite, the thermal

conductivity is larger in the xy plane than along the z-axis.

This is opposite to what is desired for practical use: In a

regenerator plate-bed, thermal conductivity should be large

along the z-axis, in order to transfer heat out of the plate

body and low in xy plane to reduce parasitic heat exchange

within the refrigerant body.

It is therefore consistent to study the influence of the

degree of orientation on the thermal conductivity both in-

plane (xy plane) and out-of-plane (along z-axis) to assess the

limits of thermal conductivity that can be achieved by parti-

cle alignment across a regenerator composite-plate.

C. Influence of particle orientation on thermalconductivity

In this section, the influence of particle orientation on

thermal conductivity is studied using the aforementioned

equivalent ellipsoid model. For this purpose, the model first

is created with the self-written MATLAB program. Then, a

mesh for FEM simulation is generated using the þFE mod-

ule of the ScanIP software environment (Simpleware Ltd.,

Exeter, UK). It has to be mentioned that meshing did not

lead to satisfying results for all particle sizes in the sample.

Meshing errors occurred for particle sizes below d < 10 lm.

Due to this fact, this fraction was neglected during the

meshing process. Although the fraction of coarse particles

increases slightly with this method, the total volume fraction

of the particles (f ¼ 58 vol:%) remained approximately

constant because of the small volume of the excluded

particles. The FEM simulation for the determination of

thermal conductivity is finally performed with the software

COMSOL MultiphysicsVR

. In preliminary simulations, the

influence of mesh density was studied so that for the follow-

ing simulations the mesh density could be chosen in a way

that it does not affect the results. Furthermore, it has to be

mentioned that the absolute values of thermal conductivity

in the ellipsoid model are higher than in the original dataset

since the volumes of the equivalent ellipsoids are slightly

bigger than those of the original particles.

1. Thermal conductivity after changing the preferredorientation from in-plane to out-of-plane

It was shown in Sec. II that the long and middle half-

axis of the particles in the as-prepared composite is aligned

along the xy plane. In this section, the influence on thermal

conductivity of the particle orientation along the z-axis

will be studied by numerical simulation. As input data, the

geometric information of the equivalent ellipsoids of the as-

prepared composite is taken. The long half-axis of the equiv-

alent ellipsoids, representing the magnetocaloric particles,

is oriented in 15� steps from 0� (xy plane ¼̂ in-plane) to 90�

(z-axis ¼̂ out-of-plane).

Depending on this orientation angle, / values for the in-

plane and out-of-plane thermal conductivity can be simu-

lated as shown in Figure 5. An increasing orientation of the

long half-axis along the z-axis of the composite gives rise to

a relative enhancement of kz of about 21% to a maximum

value of 5.16 W m�1 K�1 at / � 67�, which approximately

corresponds to the mean orientation angle of the longest

half-axis in the initial distribution. Note that the change from

xy plane as preferred orientation to / � 90� along the z-axis

gives about 16% of enhancement. The in-plane thermal con-

ductivity on the other hand decreases at the same time, since

the geometrical anisotropy is increased due to the change

of the preferred particle orientation from the xy plane to the

z-axis. The orientation dependent in-plane and out-of-plane

thermal conductivity can be described as a sinusoidal

TABLE I. Anisotropic thermal conductivities determined by laser flash

method and FEM simulation. If the same temperature gradient is applied in

all directions, kin�plane is the mean value of the thermal conductivity along

the x- and y-axis. The isotropic thermal conductivity can be derived analo-

gously by the mean value of the thermal conductivity along all three axes.

Thermal conductivity (W m�1 K�1) Measured Simulated

kx … 2.67

ky … 2.65

kin�plane … 2.66

kz 2.41 2.47

kiso … 2.60 FIG. 5. Simulated thermal conductivity as function of orientation angle / of

the equivalent ellipsoids.

125103-4 Weise et al. J. Appl. Phys. 120, 125103 (2016)

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function of the orientation angle, i.e., due to symmetry rea-

sons values of kz and kin–plane decrease/increase again above

/ � 67� where some ellipsoids are already turned out of

z-axis again (insets Fig. 5). Due to overlapping of ellipsoids

and finite sample size, a re-orientation of the ellipsoids

results in a change of the particle-to-matrix ratio. The par-

ticles volume fraction decreases from 70% at / ¼ 0� to 60%

at / ¼ 90�. This causes a decrease of the mean (isotropic)

thermal conductivity superposing the influence of the re-

orientation of the ellipsoids (grey values in Fig. 5). In real

composites with oriented particles, equal volume fractions

can be created for all rotation angles. Therefore, it can be

assumed that the isotropic thermal conductivity will not

depend on the rotation angle in real composites. Building on

this assumption, we corrected our data to compensate the

influence of decreasing volume fraction with increasing /(blue and orange values in Fig. 5). Admittedly, the quantita-

tive values of heat conductivity are slightly overestimated in

all directions because of the slightly larger volume of the

inertia equivalent ellipsoids compared to the initial particles.

Nevertheless, the tendency of the evolution of k by reorienta-

tion is well represented.

The initial values of kz and kin–plane reflect the particle

arrangement of the as-prepared composite, where a larger

heat conductivity in the xy plane is present.

2. Thermal conductivity for different orientationdistribution functions

The above mentioned simulation is based on the ori-

entation distribution predefined by the as-prepared com-

posite (Fig. 3). A manipulation of this distribution will

have an impact on the thermal conductivity. In the follow-

ing, different orientation distributions differing in their

standard deviation r will be examined. As a condition,

the preferred orientation of the particles is already along

the z-axis. For illustration, particle arrangements with

broad (high standard deviation) and narrow (low standard

deviation) orientation distributions are schematically

shown in Fig. 6(a).

In order to evaluate the influence of r on kz as initial

scenario, a particle arrangement corresponding to reor-

iented particles along the z-axis with orientation angle

/ ¼ 90� (cf. Fig. 5) was chosen. Here, rinit ¼ 25:5� corre-

sponds to the orientation distribution determined by XCT

from the as-prepared sample (as discussed in Fig. 3). As

can be seen in Figure 6, the thermal conductivity kz

increases with decreasing standard deviation of the particle

orientation distribution r. In other words, a narrow distri-

bution leads to enhanced thermal conductivity although

kz saturates by approaching r ¼ 0�. As a result, already

a rotation of all equivalent ellipsoids along the z-axis of

/ � 10� is enough to give a larger heat conductivity. It is

to expect that kz reaches values close to kiso for increasing

r > 25:5�. However, the increase in kz is only about 5.6%.

In comparison to the effect of particle reorientation from

the xy plane along the z-axis, the increase of kz by narrow-

ing the orientation distribution is rather low.

IV. CONCLUSION

The 3D microstructure of a polymer-bonded magneto-

caloric composite (cf. Ref. 6) was studied by X-ray com-

puted tomography, and its thermal conductivity along the x-,

y-, and z-axis was determined by numerical simulation

(FEM). The simulated result of kz was experimentally con-

firmed by measuring out-of-plane thermal conductivity with

the laser flash method.

Based on the XCT-data, a texture along the global xyplane of the composite has been revealed. During the com-

paction process, plate- and rod-shaped particles align along

their long and middle half-axes of their equivalent ellipsoids.

This texture leads to an increase of the in-plane thermal con-

ductivity (kx, ky), whereas kz is decreased.

Besides the characterization of an as-prepared compos-

ite, the second aspect of the paper was devoted to the possi-

bilities of enhancing the anisotropic thermal conductivity as

it is crucial for regenerator designs. Numerical simulation of

different particle arrangements and particle orientation distri-

butions has been evaluated. To summarize, a reorientation of

particles along the axis where thermal transport is preferred

gives a considerable (16% in the present composite) increase

of the thermal conductivity in comparison to the natural

FIG. 6. Illustration of the orientation of equivalent ellipsoids in the compos-

ite with large and low standard deviation r of the distribution of the long

half-axis (red arrow) along the z-axis (a). Out-of-plane thermal conductivity

kz as function of r (b).

125103-5 Weise et al. J. Appl. Phys. 120, 125103 (2016)

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plane-anisotropy which is present in composites prepared by

conventional cold-pressing. Although smaller, a further

increase of the thermal conductivity can be achieved by a

narrow particle distribution function. Considering these

influences, this paper gives hints for an optimized prepara-

tion (e.g., curing in a magnetic field, optimized particle

shapes, etc.) of composite plates for regenerator beds.

ACKNOWLEDGMENTS

The authors thank Robert M€uller for supporting the

finite element calculations and the Center for Information

Services and High Performance Computing (ZIH) at TU

Dresden for computational resources. B.W. thanks the Korea

Institute of Industrial Technology (KITECH) for financial

support. A.W. would like to gratefully acknowledge funding

from the DFG SPP “Ferroic Cooling” under Grant No. WA

3294/3-2.

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