Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles
In the present work, a detailed thermodynamic consideration for the magnetic free energy of the composite material consisting of the ferromagnetic powder particles embedded into a polymer matrix is given. We estimate their magnetostatic interaction energy and its dependence on the microscopic distri...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
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| Cite this: | Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles / A.A. Likhachev, Yu.N. Koval // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 9. — С. 1221-1230. — Бібліогр.: 13 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1518622025-02-09T17:07:28Z Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles Магнитомеханические эффекты в упругих полимерных композитах с ферромагнитными порошковыми частицами Магнетомеханічні ефекти у пружніх полімерних композитах з феромагнетними порошковими частинками Likhachev, A.A. Koval, Yu.N. Электронные структура и свойства In the present work, a detailed thermodynamic consideration for the magnetic free energy of the composite material consisting of the ferromagnetic powder particles embedded into a polymer matrix is given. We estimate their magnetostatic interaction energy and its dependence on the microscopic distribution of the magnetization and the magnetic field in the composite material. We also define the hydrostatic component of the mechanical force developed in a composite and the volume change effect caused by the magnetostatic interactions in such composites. В данной работе даётся детальный термодинамический анализ магнитной свободной энергии композитного материала, который состоит из ферромагнитных порошковых частиц, внедрённых в упругую полимерную матрицу. Мы даём оценку их энергии магнитного взаимодействия и её зависимости от микроскопического распределения намагниченности и магнитного поля внутри композитного материала. Мы также определяем гидростатическую компоненту механических напряжений, которые развиваются в композите, и эффект изменения объёма, обусловленный магнитостатическими взаимодействиями порошковых частиц в таких композитах. В даній роботі дається детальна термодинамічна аналіза магнетної вільної енергії композитного матеріялу, що складається з феромагнетних порошкових частинок, втілених у пружню полімерну матрицю. Ми даємо оцінку їхньої енергії магнетної взаємодії та її залежність від мікроскопічного розподілу намагнетованости та магнетного поля всередині композитного матеріялу. Ми також визначаємо гідростатичну компоненту механічних напружень, що розвиваються у композиті, та ефект зміни об’єму, зумовлений магнетостатичною взаємодією порошкових частинок у таких композитах. 2018 Article Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles / A.A. Likhachev, Yu.N. Koval // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 9. — С. 1221-1230. — Бібліогр.: 13 назв. — англ. 1024-1809 PACS: 61.50.Ks, 62.20.fg, 64.70.K-, 75.80.+q, 81.40.Jj, 82.35.Np, 83.60.Np DOI: 10.15407/mfint.40.09.1221 https://nasplib.isofts.kiev.ua/handle/123456789/151862 en Металлофизика и новейшие технологии application/pdf Інститут металофізики ім. Г.В. Курдюмова НАН України |
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Электронные структура и свойства Электронные структура и свойства |
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Электронные структура и свойства Электронные структура и свойства Likhachev, A.A. Koval, Yu.N. Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles Металлофизика и новейшие технологии |
| description |
In the present work, a detailed thermodynamic consideration for the magnetic free energy of the composite material consisting of the ferromagnetic powder particles embedded into a polymer matrix is given. We estimate their magnetostatic interaction energy and its dependence on the microscopic distribution of the magnetization and the magnetic field in the composite material. We also define the hydrostatic component of the mechanical force developed in a composite and the volume change effect caused by the magnetostatic interactions in such composites. |
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Article |
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Likhachev, A.A. Koval, Yu.N. |
| author_facet |
Likhachev, A.A. Koval, Yu.N. |
| author_sort |
Likhachev, A.A. |
| title |
Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles |
| title_short |
Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles |
| title_full |
Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles |
| title_fullStr |
Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles |
| title_full_unstemmed |
Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles |
| title_sort |
magnetomechanical effects in the elastic polymer composites containing ferromagnetic powder particles |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
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2018 |
| topic_facet |
Электронные структура и свойства |
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https://nasplib.isofts.kiev.ua/handle/123456789/151862 |
| citation_txt |
Magnetomechanical Effects in the Elastic Polymer Composites Containing Ferromagnetic Powder Particles / A.A. Likhachev, Yu.N. Koval // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 9. — С. 1221-1230. — Бібліогр.: 13 назв. — англ. |
| series |
Металлофизика и новейшие технологии |
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2025-11-28T09:43:58Z |
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2025-11-28T09:43:58Z |
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1850026802829852672 |
| fulltext |
PACS numbers: 61.50.Ks, 62.20.fg, 64.70.K-, 75.80.+q, 81.40.Jj, 82.35.Np, 83.60.Np
Magnetomechanical Effects in the Elastic Polymer Composites
Containing Ferromagnetic Powder Particles
A. A. Likhachev and Yu. N. Koval
G. V. Kurdyumov Institute for Metal Physics, N.A.S. of Ukraine,
36 Academician Vernadsky Blvd.,
UA-03142 Kyiv, Ukraine
In the present work, a detailed thermodynamic consideration for the magnet-
ic free energy of the composite material consisting of the ferromagnetic pow-
der particles embedded into a polymer matrix is given. We estimate their
magnetostatic interaction energy and its dependence on the microscopic dis-
tribution of the magnetization and the magnetic field in the composite mate-
rial. We also define the hydrostatic component of the mechanical force devel-
oped in a composite and the volume change effect caused by the magnetostat-
ic interactions in such composites.
Key words: magnetomechanical effect, magnetostriction, elastic polymer,
ferromagnetic powder, composite material.
В даній роботі дається детальна термодинамічна аналіза магнетної вільної
енергії композитного матеріялу, що складається з феромагнетних порош-
кових частинок, втілених у пружню полімерну матрицю. Ми даємо оцін-
ку їхньої енергії магнетної взаємодії та її залежність від мікроскопічного
розподілу намагнетованости та магнетного поля всередині композитного
матеріялу. Ми також визначаємо гідростатичну компоненту механічних
напружень, що розвиваються у композиті, та ефект зміни об’єму, зумов-
лений магнетостатичною взаємодією порошкових частинок у таких ком-
позитах.
Ключові слова: магнетомеханічний ефект, магнетострикція, пружні по-
лімери, феромагнетні порошки, композитні матеріяли.
Corresponding author: Alexander Anatolievich Likhachev
E-mail: alexl@imp.kiev.ua
Citation: A. A. Likhachev and Yu. N. Koval, Magnetomechanical Effects in the Elastic
Polymer Composites Containing Ferromagnetic Powder Particles, Metallofiz.
Noveishie Tekhnol., 40, No. 9: 1221–1230 (2018), DOI: 10.15407/mfint.40.09.1221.
Ìåòàëëîôèç. íîâåéøèå òåõíîë. / Metallofiz. Noveishie Tekhnol.
2018, т. 40, № 9, сс. 1221–1230 / DOI: 10.15407/mfint.40.09.1221
Îттиски доступнû непосредственно от издателя
Ôотокопирование разрешено только
в соответствии с лицензией
2018 ÈМÔ (Èнститут металлофизики
им. Ã. В. Êурдюмова ÍÀÍ Óкраинû)
Íапечатано в Óкраине.
1221
mailto:gfirst@imp.kiev.ua
https://doi.org/10.15407/mfint.40.09.1221
https://doi.org/10.15407/mfint.40.09.1221
1222 A. A. LIKHACHEV and Yu. N. KOVAL
В данной работе даётся детальнûй термодинамический анализ магнитной
свободной энергии композитного материала, которûй состоит из ферро-
магнитнûх порошковûх частиц, внедрённûх в упругую полимерную мат-
рицу. Мû даём оценку их энергии магнитного взаимодействия и её зави-
симости от микроскопического распределения намагниченности и маг-
нитного поля внутри композитного материала. Мû также определяем
гидростатическую компоненту механических напряжений, которûе раз-
виваются в композите, и эффект изменения объёма, обусловленнûй маг-
нитостатическими взаимодействиями порошковûх частиц в таких ком-
позитах.
Ключевые слова: магнитомеханический эффект, магнитострикция,
упругие полимерû, ферромагнитнûе порошки, композитнûе материалû.
(Received June 17, 2018)
1. INTRODUCTION
In the last years, the study of heterogeneous materials consisting of
magnetic micro- and nanoparticles imbedded into a nonmagnetic ma-
trix has been increasing due to its importance for understanding mi-
cromagnetic interactions in these systems as well as for possible appli-
cations [1, 2]. When the magnetic particles are magnetized and the ma-
trix material is elastic, an elastic magnet is obtained. This material
may exhibit elastomagnetic effects [3, 4], and it can be used for sensors
and actuators [5, 6]. In order to have optimum performances for the
mentioned applications, the material must have a high content of mag-
netic particles.
During these years, several different materials were tested: magne-
toelastic composite with the filling particles made of magnetostrictive,
hard or soft ferromagnetic material [7, 8]. Some possible applications
in the airplane and car industries as actuators or antifriction compo-
nents [9], heat-shrinkable elastic ferromagnets with the variable mag-
netic and conductive properties [10] were discussed. Those papers were
dealing with a similar kind of materials and focusing on the theoretical
and experimental correlation of the material elasticity with its mag-
netic behaviour.
In particular, they analysed a composite of particles uniformly dis-
persed inside the matrix material: a) the particles having an asymmet-
ric shape, preferably with a main anisotropy axis; b) the particles,
which were soft ferromagnetic or small permanent magnets; c) the
composites having an elastic behaviour, due to the matrix properties,
up to a relative deformation. In these conditions, a strong coupling
acts between magnetization axis and the main shape anisotropy axis of
the particles. Therefore, a change of the magnetizing field along an
axis different from the easy magnetization one gives a rotation of the
MAGNETOMECHANICAL EFFECTS IN THE ELASTIC POLYMER COMPOSITES 1223
particles due to the mechanical torque, in order to align the magnetic
moments with the applied field [4]. The macroscopic effect of these lo-
cal rotations can be a deformation of the whole material. The inverse
effect [11] consists in the change of magnetization axis due to a defor-
mation of the elastic material. As an example, an elongation of the
elastomagnetic material produces a rotation of each particle and a con-
sequent rotation of its magnetic moment because it is strongly coupled
with the particles geometry. This also gives a variation of the magneti-
zation component along the elongation axis. In conclusion, the inverse
elastomagnetic effect can be used to have a strain sensor detecting de-
formation by means of the induced magnetization changes at constant
temperature and magnetizing field [3, 12, 13].
A general target of the present work is to produce a detailed ther-
modynamic consideration for the magnetic free energy of the compo-
site material consisting of the ferromagnetic powder particles embed-
ded into a polymer matrix. We estimate their magnetostatic interac-
tion energy and its dependence on the microscopic distribution of the
magnetization and the magnetic field in the composite. We also define
the hydrostatic component of the mechanical force and the volume
change effect caused by the magnetostatic interactions in such compo-
sites.
2. MAGNETIC FREE ENERGY, MAGNETIC FORCES, AND STRAIN
EFFECT IN POLYMER COMPOSITES CONTAINING
FERROMAGNETIC PARTICLES
Generally, the magnetic forces can be produced in any ferromagnetic
material when it magnetizes. Everything depends on the fact, if the
material is deformable and if the magnetic free energy per unit volume
Fmag(h, ε) is dependent not only on the external magnetic field h, but
also on the strain of the material ε. Here, we have defined the magnetic
free energy to be zero at h = 0. So, it represents only a magnetic part of
the total free energy of the material F
mag(h, ε) = F(0, ε) + F
mag(h, ε). In
such a case, both the macroscopic magnetization of the material m(h, ε)
and the field-induced magnetic forces σmag(h, ε) can be represented on
the basis of the general thermodynamic relationships as follows:
mag mag mag mag( , ) ( , ) , ( , ) ( , ) .F F
∂ ∂ = − = − ∂ ∂ h
m h h h h
h ε
ε ε σ ε ε
ε
(1)
In absence of the magnetic field, any deformable material can be
strained elastically or unelastically applying the external mechanical
load σ. In that case, the strain response of the material can be repre-
sented by its zero-field stress–strain relationship σ = σ0(ε). In general
case, when both the mechanical forces and the magnetic ones are ap-
1224 A. A. LIKHACHEV and Yu. N. KOVAL
plied, they work altogether simultaneously, and a corresponding
strain effect can be found from the following force balance equation:
mag 0( , ) ( ).+ =hσ σ σ σ ε (2)
So, in two partial cases σ = 0 and h = 0, we obtain very similar relation-
ships:
= =mag 0 0( , ) ( ) and ( ),hσ ε σ ε σ σ ε (3)
where the first one defines implicitly the magnetic-field-induced
strain (MFIS) effect in a particular ferromagnetic material, which is
generally dependent both on its magnetic properties and also on the
mechanical behaviour of the material represented by its zero-field
strain–stress relationship σ = σ0(ε). For instance, in the ordinary mag-
netostrictive materials, it is given by a well-known linear and com-
pletely reversible Hook’s law. In other recently discovered large MFIS
systems like ferromagnetic shape-memory NiMnGa alloy, the straining
mechanism is not elastic and based on the field-induced twinning oc-
curring in these systems.
3. HYDROSTATIC MAGNETIC FORCES AND VOLUME CHANGE
IN POLYMER COMPOSITES CONTAINING FERROMAGNETIC
PARTICLES
Here, we will apply the previously discussed idea to understand what
the hydrostatic magnetic forces can be developed in systems consisting
of the multiple ferromagnetic particles imbedded into some elastically
soft polymer matrix and what the volume changes can be expected in
similar materials.
For that, we introduce the total magnetic free energy F of such a
composite system of a volume V and containing totally the volume Vm
of all ferromagnetic particles randomly distributed in the polymer ma-
trix. We also assume that the saturation magnetization Ms of the bulk
ferromagnetic material per its unit volume is known and that the satu-
ration magnetization of each particular is the same.
In that case, F(V, Vm, h) can be represented in the following general
form:
( , , ) , ( , ),m
m m m
V
F V V h VF h VF z h
V
= =
(4)
where Fm(z, h) is the magnetic free energy per unit volume of the com-
posite, h is the external magnetic field, and z = Vm/V is the volume
fraction of the ferromagnetic material in the composite. This gives us
MAGNETOMECHANICAL EFFECTS IN THE ELASTIC POLYMER COMPOSITES 1225
the possibility to define the hydrostatic pressure developed in a mag-
netic field as follows:
( , ) , ( , ) ( , ) .m
m m m m
hh
V
P z h VF h F z h z F z h
V V z
∂ ∂ = = − ∂ ∂
(5)
We can also define the magnetization per unit volume of the composite
material accordingly:
( )( , ) , ( , ) .m
m m z
V
V
M z h F h F z h
h V h
∂ ∂ = − = − ∂ ∂
(6)
4. MAGNETOSTATIC ENERGY IN POLYMER COMPOSITES
Generally, the total magnetic free energy of a composite material F
containing N ferromagnetic particles magnetized to its full saturation
consists of the Zeeman’s and magnetostatic (demagnetizing) energy
contributions. It can be represented as follows:
1
1
.
2
N
P P P P
P
F v
=
= − +
∑ hm h m (7)
Here, vP are the particle volumes, mP = Mseh—their local magnetiza-
tions all fully magnetized parallel to the external magnetic field h, and
hP is the local demagnetizing field averaged over the particle volume.
The local demagnetizing field is produced by the surface magnetic
charges induced both at the external surface of the composite material
and at the particle interface, where normal magnetization components
have jumps. Therefore, it can be written in the following manner:
4 4 ( ),P P P= − π − π −h Dm D m m (8)
where mP = Mseh, m is the macroscopic magnetization of the fully satu-
rated composite material:
1 1
1
, .
N N
P P s h m P
P P
v zM V v
V = =
= = =∑ ∑m m e (9)
Here, D, DP are the demagnetizing matrices representing both the
composite material and the ferromagnetic particle dependent only on
their shapes. So, finally, for the magnetic free energy per unit volume
of the composite material, we obtain:
2 21
( , ) 4 ( ( ) )( ) .
2m P P s sF z h D z D D z M zM h= π + − − (10)
1226 A. A. LIKHACHEV and Yu. N. KOVAL
Here, D, PD are the components of the demagnetizing matrices par-
allel to the external magnetic field applied. It is important that PD is
defined as the average demagnetizing factor of the ferromagnetic par-
ticle system:
1 1
1
, .
N N
P P P m P
P Pm
D v D V v
V = =
= =∑ ∑ (11)
In particular, it means that, if all the particles are randomly oriented,
then their average demagnetizing factor must be equal PD = 1/3 even
if all of them are not really the spherical ones. In other words, the fer-
romagnetic material distribution in the polymer matrix is statistically
isotropic in this case.
Using Eqs. (5) and (10), one can also obtain the magnetic pressure:
2 21
( , ) 4 ( ) .
2 P sP z h D D z M= − π − (12)
The corresponding calculation results are represented in Fig. 1.
As follows from these results, the maximal magnetic force, which is
achieved in the completely saturated composite samples with the ran-
domly oriented particles, is strongly dependent on the volume fraction
of the ferromagnetic material and the demagnetizing factor of the
composite samples. It may have different signs as the demagnetizing
Fig. 1. Dependence of the magnetic pressure developed in a fully saturated
composite containing the randomly oriented ferromagnetic particles at the
different demagnetizing factors D of the composite samples.
MAGNETOMECHANICAL EFFECTS IN THE ELASTIC POLYMER COMPOSITES 1227
factor changes within the range 0 < D < 1. In case D = 0 (the sample is a
long cylinder parallel to the magnetic field), the magnetic force will
produce a compression effect. In the other case D = 1 (the sample is a
thin plate aligned perpendicular to the magnetic field), the magnetic
force will cause an extension. A general characteristic scale of the
magnetic forces is defined by the material constant P0 = 2π(Ms)
2
pro-
portional to the saturation magnetization squared.
5. STRAIN EFFECT PRODUCED BY MAGNETIC FORCES
As follows from Eq. (3), the relative volume change must be propor-
tional to the magnetic force developed in a composite material and re-
verse proportional to its effective bulk elastic modulus C(z).
1
vol ( , ) ( ) ( , ), ( ) ( ( )) .
V
z h K z P z h K z C z
V
−δ
= ε = = (13)
Here, K(z) is obviously a bulk elastic compressibility of the composite
material. As follows from the mechanical testing experiments shown
in Fig. 2, the elastic moduli of the polymer composites containing the
rigid powder particles are strongly dependent on its volume fraction.
These results indicate that the bulk elastic modulus is sharply goes
up as the volume fraction of rigid particles increases so as the com-
pressibility decreases approximately linearly. Therefore, for some
practical reasons, we can use a following simple linear model for the
Fig. 2. Fraction dependence of bulk elastic modulus and elastic compressibil-
ity.
1228 A. A. LIKHACHEV and Yu. N. KOVAL
elastic compressibility:
0( ) (1 ).K z K z= − (14)
It means that only the compressibility of polymer matrix K0 gives a
contribution into K(z) proportional to its volume fraction (1 − z). So,
finally, we obtain the field induced volume effect:
2 2
0( , ) 2 ( ) ( ) (1 ).vol s Pz h K M D D z zε = − π − − (15)
It is strongly dependent on both the sample and ferromagnetic parti-
cle shapes represented by the difference between their demagnetizing
factors. It is also strongly dependent on the volume fraction of ferro-
magnetic material. The general scale of the field-induced effects is de-
fined by the material constant proportional to the elastic compressibility
of the polymer and squared saturation magnetization of bulk ferromag-
netic material. The fraction dependent factor has a maximum at the vol-
ume fraction at about z = 2/3. These final results are shown in Fig. 3.
6. CONCLUSIONS
The magnetic free energy of the composite material consists of the
magnetostatic energy of separate ferromagnetic powder particles,
their magnetostatic interaction energy, and the Zeeman’s energy. It is
strongly dependent on the volume fraction of magnetic powder and the
Fig. 3. Elastomagnetic volume change effect and its dependence on the volume
fraction of ferromagnetic powder and demagnetizing factor of the composite
sample.
MAGNETOMECHANICAL EFFECTS IN THE ELASTIC POLYMER COMPOSITES 1229
magnetic field applied.
Both the macroscopic magnetization and the hydrostatic magnetic
forces can be obtained from the magnetic free energy as functions of
the volume fraction and the magnetic field applied by using the general
differential thermodynamic relationships.
The micromagnetic model of the polymer composite consisting of
the ferromagnetic powders and elastically soft polymer matrix devel-
oped here has given the expression for the magnetostatic energy and
the magnetic forces as functions of the volume fraction, magnetic field
and the macroscopic demagnetizing factors of the composite samples.
It has given a possibility to estimate the maximal field induced strain
effect and its dependence on the volume fraction of the ferromagnetic
material.
The experimental study of the elastic moduli of the polymer compo-
sites containing the different volume fraction of powder particles has
shown its strong dependence on the volume fraction of powder. One
can use these results to estimate the fraction dependence of the field
induced strain effect, its dependence on the volume fraction of the fer-
romagnetic material, and its maximal value.
Finally, we have concluded that the field induced volume effect is
strongly dependent on the sample and ferromagnetic-particle shapes
represented by the difference between their demagnetizing factors. It
is also strongly dependent on the volume fraction of ferromagnetic ma-
terial. The general scale of the field-induced effects is defined by the
material constant proportional to the elastic compressibility of the
polymer and squared saturation magnetization of bulk ferromagnetic
material. The fraction dependent factor has a maximum at the volume
fraction at about z = 2/3.
This work has been supported by Science Technology Centre of
Ukraine (Project No. 5522).
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|