Behaviour of the Young’s modulus at the magnetocaloric transition in La(Fe,Co,Si)13

B. Kaeswurm1,2, A. Barcza3, M. Vögler1, P.T. Geiger1,2, M. Katter3, O. Gutfleisch1 and L.F. Cohen4

1. Technische Universität Darmstadt, 64287 Darmstadt, Germany.2. Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen Germany

3. Vacuumschmelze GmbH & Co. KG, 63450 Hanau, Germany4. Imperial College, London SW7 2AZ, United Kingdom

Keywords: Magnetocaloric, Ferromagnetic, Phase transition, La(Fe,Si)13, lattice softening, Young’s modulus,

Abstract— Magnetic solid state cooling applications require families of samples where the magnetic

transition is cascaded across the working range of the fridge. Although magnetic properties are

widely studied, information relating to the mechanical properties of such systems is less prevalent.

Here we study the mechanical properties of a series of magnetocaloric La(Co,Fe,Si)13 samples where

the Co content is varied to produce a range of transition temperatures. It was found that at room

temperature the flexural strength decreases and the Young’s modulus increases with increasing Co

content. Interestingly we find a significant reduction of Young‘s modulus at temperature around the

magnetic transition temperature. This reduction was less pronounced with increasing Co content.

We associate the softening with the magnetovolume coupling known to exist in these materials.

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1. Introduction

1.1. The magneto caloric effect and magnetic refrigeration

Magnetic refrigeration is an energy efficient solid state based cooling technology which may replace

conventional vapour cycle refrigeration in some applications in the future [1-6]. The technology is based on

large magnetocaloric effects (MCE) where in the most favourable materials, a coupled magnetic and

structural phase change leads to a thermal response upon application of a magnetic field. In many materials

exhibiting MCE, the transition may also be driven by pressure (barocaloric effect) [7-11]. The thermal

response of magnetic materials subjected to a field has been known since the 19 th century, for a review on

the discovery see A. Smith [12]. The effect is used in low temperature physics in the milli-Kelvin range

[13] and received renewed interest in 1997 with the discovery of a large magnetocaloric effect of 7.5 K in a

magnetic field change of 20 kOe at 276 K in Gd5(Si2Ge2) [14]. Since then a number of magnetocaloric

materials with transition near room temperature have been investigated such as Gd [15], Fe2P-based

materials [16], and the materials from La(FeSi)13 system [17-21]. The MCE is characterised in terms of the

adiabatic temperature change ∆ T ad and the isothermal entropy change∆ S for an applied magnetic field H .

1.2. Requirements for magnetic refrigeration

The challenges of magnetic refrigeration lie not only in the choice of a suitable material with appropriate

magnetocaloric properties ∆ S and ∆ T adbut also in the engineering of a device. Other physical properties

are desirable for application, such as high (anisotropic) thermal conductivity in order to provide efficient

heat transfer to the heat exchange fluid, mechanical integrity over many field and temperature cycles and

corrosion resistance. The intrinsic brittleness of many magnetocaloric materials also makes the process of

shaping the material into complex heat exchanger designs required for efficient heat transfer, demanding.

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1.3. Previous work on mechanical properties of magnetocaloric materials

Only a limited amount of work has been reported on the mechanical properties of magnetocaloric

materials. The mechanical properties of the magnetocaloric compound MnFe(P,Si,B) have been studied by

tracking properties such as electrical resistivity and micro hardness across the first order transition, and it

was found that B substitution improved the mechanical stability of this compound [22]. A few reports exist

in the literature on the chemical stability of La(Fe,Si)13 [23-28], but not many reports on mechanical

properties. Hot compacted La(Fe,Si)13 has been shown to display superior mechanical stability after

temperature cycling than induction melted samples [29]. While induction melted La(Fe,Si)13 shows a

reduction in MCE due to cracks when the material is subjected to multiple field cycles, the porous hot

compacted material maintains its mechanical stability. Investigation into the mechanical properties of

polymer bonded hydrogenated La(Fe,Si)13 showed the stress-strain relation to depend largely on the type of

epoxy used [30]. For reactively sintered samples such as those studied here, the mechanical properties were

shown to depend strongly on the heat treatment [31, 32].

1.4 The magnetocaloric material La(Fe,Si)13

The La(Fe,Si)13 material system is a promising candidate for magnetic refrigeration due to the large

magnetocaloric effect, nontoxicity and sufficient abundancy of the constituting elements [19, 33]. The

material system shows a conventional magnetocaloric effect [31, 32, 34] and inverse barocaloric effect

[35]. La(Fe,Si)13 undergoes a first order magnetic phase transition at around 200 K from ferromagnetic to

paramagnetic state upon heating [36]. The transition temperature TT can be adjusted by substitution or

addition of interstitial elements. This makes cascading of TT possible, showing the material useful for

applications. The crystal structure of the compound is that of cubic NaZn13 (Fm3c) with lattice parameters

between 11.46 Å and 11.56 Å [37, 38] in the ferromagnetic phase at low temperature, depending on the

stoichiometry. During transition from the the low temperature ferromagnetic (FM) to high temperature

paramagnetic (PM) state, the unit cell undergoes a volume reduction (magnetovolume effect) of 0.5 to 1.5

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% upon heating, also depending on the precise stoichiometry and heat treatment [31, 39, 40]. This causes

macroscopic strain in the heat exchangers consisting of polycrystalline bulk or flakes. Substitution of Co

for Fe in La(Fe,Si)13 increases the transition temperature. With increasing Co content the transition

becomes increasingly broader and the MCE and the volume change is reduced. For example for

LaFe11.74Co0.13Si1.13 a volume change of 1 % is observed, where as in contrast in LaFe10.71Co1.30Si1.00 the

volume change is reduced to 0.1 % [19].

In this work we study the Young’s modulus (E) and the flexural strength of La(Fe,Co,Si)13. The Young’s

modulus is a key parameter for mechanical engineering design, as well as being of special interest in the

temperature region around the phase transition, since a strong coupling between the magnetic and elastic

properties is expected. A change in elastic modulus with temperature (T) is expected in a magnetic

material. At increasing temperature the exchange energy decreases, reducing the magnetic alignment and

hence changing the magnetic contribution to the free energy. The second reported mechanical property is

the flexural strength acquired in a four-point bending setup. It is important to note that the flexural strength

is not a material property but a statistical sample parameter which is influenced by the fracture toughness of

the material and the size and geometry of defects such as pores, inhomogenities, surface defects and micro-

cracks. To the best of our knowledge this is the first report on the temperature dependent Young’s modulus

in the magnetocaloric material La(Fe,Co,Si)13.

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2. Material and Methods

2.1. Sample details and determination of magnetic properties

Tests were performed on four La(Fe,Co,Si)13 compounds of varying Co content, their compositions are

shown in table 1.

Table 1

Sample composition

A La1 Fe11.09 Co0.86 Si1.05

B La1 Fe10.81 Co1.18 Si1.01

C La1 Fe10.63 Co1.33 Si1.04

D La1 Fe10.54 Co1.43 Si1.03

At room temperature sample A is in paramagnetic and low unit cell volume state while the other three

samples are ferromagnetic and in their high unit cell volume state. The samples were prepared by reactive

sintering [31] and a subsequent thermal decomposition treatment [32, 41] to decompose the NaZn13 phase

into alpha-iron and La-rich phases. In this this state the samples were wire-cut into the geometry required

for mechanical testing. After wire cutting the samples were annealed at 1050°C to recombine the alpha-iron

and La-rich phases into the NaZn13 structure and reduce any residual stresses. Four- point flexural tests and

resonance frequency and damping analysis (RFDA) were performed on bars of 3 mm x 4 mm x 25 mm.

For nano-indentation testing and scanning electron microscopy (SEM), the samples were embedded into

epoxy resin and polished in order to obtain a smooth surface. While for the samples with the least Co

content (sample A) 90 % phase homogeneity is observed, those with the highest Co content show only 86%

of the 1:13 phase, with the remaining percentage being Fe rich phases, La rich phases and pores.

The magnetisation dependences were measured on samples of dimensions 1.5 mm x 2 mm x 3 mm using

a vibrating sample magnetometer. The magnetic field was applied along the long axis to minimise

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demagnetising effects. The transition temperature TT is defined as the inflection point of the temperature

dependent magnetisation curve and was determined by finding the extremum in the first derivative of the

magnetisation curve. The entropy change was calculated from magnetization isotherms measured at

varying temperatures M(H)T and application of the Maxwell relation [42]. Measurements of the adiabatic

temperature change of the samples were carried out in a bespoke device on samples of dimensions 3 mm x

4 mm x 6 mm, while a magnetic field is applied along the long axis to minimise demagnetising effects and

to be consistent with the magnetisation measurements. A thermocouple was attached to the sample, which

was kept under adiabatic conditions and its temperature was compared to that of the sample holder. For a

detailed description of the technique see the supplementary material in reference [10].

2.2. Determination of mechanical properties

2.2.1. Four-point flexural test

The flexural strength σf was examined using a four-point flexural test configuration according to ASTM

C1161 [43]. The tests were carried out with a uniaxial hydraulic press (MTS 810, MTS Systems GmbH,

Berlin, Germany). The samples were mounted between the inner and outer bearing cylinders. A preload of

20 N was applied, subsequently increasing the force with a rate of 20 N/s until fracture occurred. In order to

provide statistically significant information, n= 25 samples for each composition were tested. All tests were

carried out at room temperature. The flexural strength was calculated using equation (1).

σ f =3(S1−S2)

t 2bFmax (1)

Where S1 and S2 are the outer and inner bearing distances, t and b are the specimen thickness and width and

Fmax the maximum applied force when fracture occurred. The data was analysed using the Weibull

probability distribution, given in equation (2).

Pf ( σ f )=1−e−(σ f

σ 0)

w

(2)

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Where Pf ( σ f ) is the cumulative probability of failure, which is the probability that failure has occurred by at

stress σ f . With this strength distribution two parameters can be extracted from the data: (i) The Weibull

modulus w, which is an inverse measure of the distribution width and therefore an indicator for the

variation between the samples. (ii) The characteristic break strength σ0, which is the upper strength limit

where a sample will fracture with a probability of 63.2 %.

2.2.2. Resonance frequency and damping analysis

A resonance frequency and damping analysis (RFDA-HT1750, IMCE, Genk, Belgium) was carried out

in order to study the Young’s modulus E as a function of temperature. This method is a non-destructive

means of studying the elastic properties of a specimen. A bar shaped sample is mechanically exited by the

impulse of a small hammer every 30 seconds, and the acoustic signal is measured with a microphone and

analysed by fast Fourier transformation in order to obtain the resonance frequency fr. The Young’s modulus

is calculated according to ASTM E 1876-99 [44] by Eq. 3 and 4. In these equations t and b are the

specimen thickness and width as in equation 1, m is the mass and l the length of the sample. The sample is

mounted in the way that the tapping device hits the sample in the middle of the surface defined by the

length and width of the sample. Y is an empiric factor, containing the Poisson’s ratio ν, which is assumed to

be 0.275 as this lies in the middle of known values for ceramic magnets (0.27) and steel (0.28)[45-47]. The

experiment is performed in a furnace with a heating rate of 2 K/min.

E=0.9465(m f r2

b )( lt3 )Y

(3)

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Y=1+6.585 (1+0.0752 ν+0.8109 ν2 )( tl )

2

−[ 8.34 (1+0.2023 ν+2.173 ν2 )( tl )

4

1+6.338 (1+0.1408 ν+1.536 ν2 )( tl )

2 ](4)

2.2.3. Nanoindentation

Indentation testing was used to examine the resistance to permanent deformation. In addition,

indentation testing provides another measure of elastic modulus. Measurements were made with a

Nanomechanics Inc. nanoindenter having the capability to measure modulus and hardness under quasi-

static and dynamic conditions [48-50].

3. Results

3.1. Magnetocaloric properties

The magnetic properties of the set of samples are shown in Figure 1. Figure 1 a) shows the temperature

dependent magnetisation in a non-saturating field of 500 Oe. The cooling and heating branch are indicated

by blue and red arrows, respectively. The magnetic transition from ferromagnetism to paramagnetism at

varying TT is clearly visible for all four samples and shifts from 280 K under cooling for the sample with

lowest Co content (sample A) to 332 K for the sample with highest Co content (sample D). The field

dependence of TT upon heating is illustrated in Figure 1 b). For all four samples the transition shifts by 2 K

per 10 kOe. The entropy change shown in Figure 1 c) as calculated from measurements of isothermal

magnetisation curves up to 20 kOe shows the entropy to decrease with Co content from ΔS= -8.7 J kg-1K-1

at 282 K (sample A) to ΔS= -6.2 J kg-1K-1 at 334 K (sample D).

Figure 1 d) shows the temperature change under adiabatic conditions ΔTad in a field chance of 19 kOe.

Details of the magnetocaloric properties for all four samples including the full width half maxima values

(FWHM) of the ΔS and ΔTAD are shown in table 2.

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Figure 1: Magnetic properties of the four La(Fe,Co,Si)13 samples of different Co content studied in this

work: a) Temperature dependent magnetisation, b) field dependence of the transition, c) entropy change

and d) adiabatic temperature change (colour online).

Table 2: Magnetocaloric Properties

Sample TT ΔS FWHM of ΔS ΔTAD FWHM of ΔTAD

CoolingHapplied= 500 Oe

(0.05 T)

For a field change of 20 kOe (2T)

For a field change of 19 kOe (1.9T)

[K] [J kg-1K-1 ] [K] [K] [K]A 280.2 -8.7 18.6 3.3 20.9

B 312.7 -7.2 23.7 3.0 23.7

C 321.4 -6.6 26.2 2.6 26.2

D 331.8 -6.2 28.8 2.5 28.8

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3.2. Mechanical Properties

3.2.1. Flexural strength

The details of the flexural strength measurements are presented in table 3. A decrease of the characteristic

breaking strength is observed when comparing sample A (lowest Co content, TT= 280 K) to the samples

with a higher content of Co (B-D) σ0=157 MPa to 153 MPa. At the same time, the Weibull modulus

increases with increasing Co content. It would be usual to ascribe the decrease in flexural strength in terms

of inhomogeneities such as secondary phases and pores. We have used SEM images (not shown here) to

find direct evidence for this. We find that pore density is similar between samples and there does not appear

to be a strong variation in secondary phases between sample A and the three other samples. Hence it is not

possible to be conclusive on this point within the resolution of these measurements. Additionally it is to be

noted that at room temperature, where the measurements were performed, sample A is paramagnetic and in

its low unit cell volume state, while all other three compositions are ferromagnetic and in high unit cell

volume state. The data suggests that the flexural strength although similar across all samples, shows a

gradual decreasing value as Co is added. The Weibull modulus shows a systematic increase as the Co

content is increased, implying that there is a broader distribution of properties within the samples as the Co

doping is increased.

Table 3: Flexural Strength details

Sampl

e

Flexural

strength

Characteristic break

strength

Standard

deviation

Weibull

modulus w

Coefficient of

determination

MPa MPa MPa

A 119 - 190 173.3 17.76 10 0.98

B 124 - 182 157.4 15.83 11 0.89

C 133 - 170 154.7 10.13 17 0.92

D 121 - 162 152.8 10.45 16 0.97

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3.2.2. Young’s modulus

Using nanoindentation, a Young’s modulus of 130 ± 5 GPa and hardness of 8.5 ± 0.5 GPa was measured

at room temperature for all three compositions. Figure 2 shows Young’s modulus as a function of

temperature as determined by RFDA for sample D, the composition with highest Co content. At room

temperature an E of 105 GPa is measured. With increasing temperature, softening to 93 GPa at the

transition temperature is observed. The Young’s modulus shows a rapid recovery after the transition for

TT<T<TT+20 K. With continuous heating, the slope decreases in a strictly monotonously manner, reaching

maximum value E= 110 GPA at 530 K. At temperatures above 650 K, a linear behaviour is observed as

expected for a purely cubic crystal structure.

Figure 2: Young’s modulus E as a function of temperature T for sample D of composition La1 Fe10.54

Co1.43 Si1.03 (colour online).

Figure 3 a) shows a comparison Young’s modulus E as a function of temperature for samples B, C and D.

At room temperature a Young’s modulus of 105 GPa is observed for the sample with highest Co content

(red triangles, sample D), while a lower Young’s modulus of 100 GPa is observed in the sample with a low

Co content (blue squares, sample B). A softening of E is found in the vicinity of the transition temperature

in all compositions. Above the transition temperature the Young’s modulus rises again and at high

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temperature a higher E is observed for sample B compared to sample D. The inset shows that all samples

follow as similar overall temperature dependence, however in the vicinity of the transition a larger drop is

observed for lower Co content. This is in agreement with previous work showing a larger volume change

for samples with a lower Co content [19].

Figure 3: a) Young’s modulus E as a function of temperature b) first derivative of magnetisation versus

temperature under heating (colour online).

Figure 3 b) shows the first derivative of the temperature dependent magnetisation curve under heating,

shown in Figure 1 a). The data was smoothed using a second order polynomial. From this data it is clear

that the lattice softening as measured by RFDA in Figure 3 a) occurs over a much greater temperature

range than the magnetic transition. It is therefore the lattice that anticipates the magnetic transition.

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4. Discussion

The magnetocaloric properties shown in figure 1 are in agreement with previous studies of the material

system where the adiabatic temperature change is reduced and the width of the transition widened with

increasing Co content [19, 32]. It has been shown that with increasing Co content the transition can be

changed from first order to second order [51]. This explains the width of the ΔT and ΔS curve as well as the

hysteresis in the temperature dependent magnetisation measurement. The range of characteristic breaking

strength observed between 153 to 173 MPa is in agreement with previous results by Katter et al. [31] who

observed values of breaking strength between 130 and 190 MPa in the same material. This is also the same

order of magnitude of the Nd-Fe-B permanent magnets, which also show flexural strength between 130 to

300 MPa depending on various alloying elements and heat treatment [52]. The decrease in flexural strength

with samples of increasing Co content is attributed to a combination of influence of Co on the bonding

strength as well as inhomogeneties such as secondary phases and pores in this particular sample set.

A greater elastic modulus of E= 130 ± 5 GPa was measured during indentation. Within the resolution of

the technique no variation between the samples is observed. Using RFDA on the other hand a room

temperature value of 100 ± 2 GPa is observed for sample A and 105 ± 2 GPa for sample D. A reason for

this difference could be the fact that indentation testing was carried out on a local scale where porosity had

negligible influence on the Young’s modulus, while in RFDA the entire sample was measured. The change

in Young’s modulus at the phase transition is attributed to a change in vibrational properties. Local

variations in structure are coupled with acoustic phonon modes, causing small variations in strain which

can lead to large changes in the elastic properties. This has been observed in a variety of different materials

such as quartz [53, 54] and perovskite structures [55]. Lattice softening has been reported to occur over

temperature intervals over ~100 K to 300 K [55]. Analogues to the loss of magnetic properties at the Curie

temperature in magnetic materials, is the loss of ferroelectric properties in ferroelectric materials. A

reduction in Young’s modulus near the Curie temperature due to lattice softening has also been observed in

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ferroelectric materials [56, 57] where it is attributed to movement of ferroelectric domain walls [55, 56].

The linear decrease of the Young’s modulus at high temperatures was expected due to an increase of

thermal vibrations. Larger vibrations increase the mean atomic lattice constant and therefore weaken the

bonding strength between the atoms. The relation of the volume coefficient of thermal expansion of a solid

and its bulk modulus have been described by an equation of state [58] and various approximations[59-61].

The linear behaviour of the Young’s modulus at high temperatures was experimentally shown in

polycrystalline oxides by Wachtman et al. [62].

The shape memory alloy Ni2MnGa experiences a 6.56% c axis contraction on cooling through the

martensitic phase transition [31, 39, 40, 63]. For these materials a correlation has been shown between

phonon softening and the martensitic transition [64-66], as a necessary precursor to the change in crystal

structure that takes place. In La(Fe,Si)13 the material undergoes a ~1% volume contraction upon heating,

without structural change. This change has until recently been considered to be associated with lattice

stiffening. However, it has recently been demonstrated by Gruner et al. [67], that lattice softening indeed

occurs at the transition in La(FeSi)13. They showed that the temperature induced magnetic disorder causes

distinct modification in the vibrational density of states at the cubic to cubic FM to PM transition [67]. The

Young’s modulus measurements made here reinforce that description, and show a direct link between

lattice softening and the change of magnetisation with temperature.

Conclusion

The mechanical and magnetocaloric data presented here show systematic behaviour at the ferromagnetic

to paramagnetic phase transition in La(Fe,Co,Si)13 as a function of Co content. This is useful both to

understand the system’s fundamental properties and suitability of the material for future magnetic cooling

applications. By simple mechanical measurement, the determination of the Young’s modulus quite

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remarkably shows how the lattice anticipates the approaching magnetic transition, and softens in order that

the commensurate volume change is enabled.

Acknowledgements

The authors wish to thank Kurt E. Johanns for nanoindentation measurements as well as Markus Frericks

for experimental support. Samples were produced at VAC, measurements conducted at TU Darmstadt. The

research leading to these results has received funding from the European community's seventh framework

programme FP 7/2007-2013 under grant agreement n°310748 (DRREAM).

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