International Journal of Engineering Science · phase-transition fronts in deformable ferromagnets. As a particular case, namely in stationary conditions, the evolution equation is

International Journal of Engineering Science 47 (2009) 821–839

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International Journal of Engineering Science

journal homepage: www.elsevier .com/locate / i jengsci

A thermodynamic approach to ferromagnetism and phase transitions

M. Fabrizio a, C. Giorgi b, A. Morro c,*

a Department of Mathematics, Piazza di Porta S. Donato 5, 40126 Bologna, Italyb Department of Mathematics, Via Valotti 9, 25133 Brescia, Italyc University of Genova, DIBE, Via Opera Pia 11a, 16145 Genova, Italy

a r t i c l e i n f o

Article history:Received 30 December 2008Received in revised form 30 March 2009Accepted 13 May 2009Available online 31 May 2009

Communicated by K.R. Rajagopal

Keywords:Ferromagnetic–paramagnetic transitionSaturation effectContinuum thermodynamicsLogarithmic potential

0020-7225/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.ijengsci.2009.05.010

* Corresponding author. Tel.: +39 10 3532786; faE-mail address: [email protected] (A. Morro

a b s t r a c t

The paper provides a modelling of the magnetization curve and of the ferromagnetic–para-magnetic transition within a continuum thermodynamic setting. The general model of thenonlinear, time dependent behaviour of ferromagnetic materials is accomplished byregarding the magnetization vector as an internal variable, namely as a vector field whosetime evolution is a constitutive equation subject to the requirements of the second law ofthermodynamics. The exchange interaction of the magnetization is modelled through adependence of the free energy on the magnetization gradient. Consistent with the non-simple character of the material, the second law allows for a non-zero extra-entropy flux.A general three-dimensional scheme is elaborated which seems to be new in the literature.The three-dimensional setting is then established for stationary and homogeneous fieldsthus finding the collinearity and the corresponding form of the magnetic susceptibility.The whole evolution problem for the temperature and the magnetization is provided sothat temperature-induced transition processes are allowed. The model accounts also forthe dependence of the saturation magnetization on the temperature. Also for the sake ofcomparison with the existing literature, the evolution equations for the direction and theintensity of magnetization are derived. Known models, such as those of Landau–Lifshitzand Gilbert, are recovered as particular cases of saturated bodies. Next, the model is mademore specific so as to account in detail for the saturation, the residual or spontaneous mag-netization and the coercive field. First, the classical potential, which traces back to Ginz-burg, and the Weiss model are revisited. The corresponding lack of the saturation effector the description via implicit relations are emphasized. Hence, a new potential, with a log-arithmic dependence on the magnetization, is investigated which provides the residualmagnetization and the coercive field in an explicit way and satisfies expected propertiesof the residual magnetization as a function of the temperature.

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1. Introduction

Ferromagnetic materials exhibit a non-linear relation between the magnetic field H and the magnetization M. The non-linear relation provides the standard hysteretic phenomena. Nonlinearity and hysteresis occur below a characteristic tem-perature called the (magnetic) Curie temperature hc . Above the Curie temperature, the materials are paramagnetic in thatthe relation is linear with a coefficient, the magnetic susceptibility, which is inversely proportional to the differenceh� hc between the current temperature h and hc; such a dependence is the content of the Curie–Weiss law. The fact thatso different M�H curves are parameterized by the temperature allows us to cast the passage from one curve to anotherwithin the scheme of phase transitions.

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822 M. Fabrizio et al. / International Journal of Engineering Science 47 (2009) 821–839

There is a wide variety of approaches to the modelling of magnetization in ferromagnetic bodies. At the bottom, the mod-els deal with the time evolution of the magnetization. We mention that this problem has been addressed in the works ofLandau et al. [1,2] which model the time evolution of M in a magnetically-saturated body under the action of a time-depen-dent field H. Next, in the framework of micromagnetics and gyromagnetics, a similar phenomenological approach has beenset up by Gilbert [3,4] and then improved by Brown [5] who replaced the field H with an effective field Heff . By selecting anappropriate form of Heff , Mallinson [6] has derived a description of the switching effect in damped gyromagnetics. Recentpapers exhibit more involved models to account for the evolution of domain walls in ferromagnets (see, e.g., [7–9]). Suchmodels are well-motivated from micromagnetics but are essentially isothermal in character. Non-isothermal models areprovided by Maugin [10] about magnetoelasticity through internal variables and next by Maugin and Fomethe [11] to modelphase-transition fronts in deformable ferromagnets.

As a particular case, namely in stationary conditions, the evolution equation is expected to provide the M�H relation ofthe magnetization curve. Upon the assumption that M and H have a common fixed alignement, the evolution equation wasfirst set up in the form (see, e.g., [2,12])

_M ¼ a0l0H � a1ðh� hcÞM þ a2M3 þ a3DM:

This non-isothermal model accounts for the transition, at the Curie temperature hc , between the paramagnetic and the fer-romagnetic behaviours. Yet, the corresponding magnetization curve

a0l0H ¼ a1ðh� hcÞM � a2M3;

does not allow for saturation in the ferromagnetic regime (h < hc). In 1907, Weiss developed a theory of ferromagnetic do-mains structure known as mean field theory. In essence he arranged the Langevin potential in order to describe the non-iso-thermal paramagnetic–ferromagnetic transition and to account for the saturation phenomena [13]. Such a model ismotivated by statistical physics but is one-dimensional in character.

The purpose of this paper is threefold. The first fold is to model the nonlinear, time dependent behaviour of ferromagneticmaterials within a thermodynamic framework in three-dimensional setting. By analogy with [14], this is accomplished byregarding the magnetization M as an internal variable, namely as a phase field whose time evolution is given by a constitu-tive equation subject to the requirements of the second law of thermodynamics. The exchange interaction of the magneti-zation is modelled through a dependence of the free energy on the gradient of M. Consistent with the non-local character ofthe material, the second law allows for a non-zero extra entropy flux. A general form of the evolution equation is then de-rived and known models, such as those of Landau–Lifshitz, Gilbert and others, are recovered as particular cases of saturatedbodies. The second fold is to improve the model of the M�H relation in ferromagnetism so as to account in detail for thesaturation, the residual or spontaneous magnetization and the coercive field. This part is realized by starting with the clas-sical potential which traces back to Ginzburg and showing that a different behaviour (ferromagnetic–paramagnetic) occursaccording as h < hc or h > hc but the saturation effect does not follow. The Weiss model is then examined whence it followsthat transition and saturation are allowed though via implicit relations. A new singular potential, with a logarithmic depen-dence on the magnetization, is investigated which provides the residual magnetization and the coercive field in an explicitway and satisfies expected properties of the residual magnetization as a function of the temperature. The third fold is to pro-vide the whole set of evolution equations, for the temperature and the order parameter, in the three-dimensional frame-work. As a result, our model describes temperature-induced reversible transitions between the paramagnetic and theferromagnetic regimes. Hence, we can control the phase transition process by acting on the external heat source and the ap-plied magnetic field. Moreover, the scheme is set up in a general way so as to allow also for the dependence of the sponta-neous magnetization, relative to the saturation magnetization, on the temperature. These features seem to be new in theliterature.

The advantage of the present approach is the unified, thermodynamically-consistent scheme of constitutive equationsand evolution equations which in turn provide the magnetization curve. The pertinent equations prove to be characterizedby the free energy as a thermodynamic potential. The scheme is three-dimensional in character and, also for the sake of com-parison with the existing literature, the evolution equations for the direction and the intensity of M are derived. It is of inter-est that all of the schemes appeared in the literature (e.g., time-dependent or stationary, Lagrangian) are recovered asparticular cases. In particular, the evolution equation of the direction reduces to that of Landau and Lifshitz if the exchangeand anisotropic interactions are neglected. Also, in static conditions, Brown’s equation is obtained with a general form of theeffective magnetic field. Next, as an application, the appropriate potential for crystals of iron is determined through the datafor the residual magnetization versus h=hc. Moreover, a one-parameter free energy is shown to provide a satisfactory descrip-tion of both the ferromagnetic and the paramagnetic behaviour according as the temperature is below or above the Curietemperature.

2. Balance equations

An undeformable ferromagnetic material occupies the region X # R3. The electric field E, the magnetic induction B, theelectric displacement D and the magnetic field H satisfy Maxwell’s equations, in the space-time domain X� R,

M. Fabrizio et al. / International Journal of Engineering Science 47 (2009) 821–839 823

r� E ¼ � _B; r�H ¼ _Dþ J; ð2:1Þr � B ¼ 0; r � D ¼ q; ð2:2Þ

where J is the current density and q is the charge density. The superposed dot denotes the time derivative and r is the gra-dient operator. The balance of energy in electromagnetic materials is based on the view that E�H is the vector flux of energyof electromagnetic character. This view follows from Poynting’s theorem which merely shows that

�r � ðE�HÞ ¼ H � _Bþ E � _Dþ E � J ð2:3Þ

is a consequence of Maxwell’s equations.Since the body is undeformable, then the balance of energy is taken in the form (see [15])

_e ¼ �r � ðE�Hþ qÞ þ r;

where e is the internal energy density, q is the heat flux and r is the heat supply, namely energy per unit volume and unittime provided by external sources. By means of the identity (2.3) we have

_e ¼ H � _Bþ E � _Dþ J � E�r � qþ r: ð2:4Þ

The second law of thermodynamics is taken as the statement that the Clausius–Duhem inequality holds for any set offunctions which satisfy Maxwell’s equations (2.1) and (2.2) and the energy equation (2.4). Also because of possible nonlocaleffects, the entropy flux is likely to be different from q=h, h being the absolute temperature. Hence, letting g be the entropydensity and k the extra-entropy flux vector, we write the Clausius–Duhem inequality in the form

_g P �r � ðq=hÞ � r � kþ rh: ð2:5Þ

The extra-entropy flux k is required to satisfy the boundary condition
Z@X
k � n da ¼ 0 ð2:6Þ

for the whole body (see [16]). This allows (2.5) to provide the standard global statement of the second law,

ddt

ZXg dv P

ZX

rh

dv �Z@X

1h

q � n da:

By (2.4) and (2.5) we have

_e� h _g�H � _B� E � _D� J � Eþ 1h

q � rh� hr � k 6 0:

For later convenience we consider the free energy density

w ¼ e� hg:

Hence, the Clausius–Duhem inequality becomes

_wþ g _h�H � _B� E � _D� J � Eþ 1h

q � rh� hr � k 6 0: ð2:7Þ

Having in mind a model for ferromagnetism, we disregard polarization and let

D ¼ �0E; B ¼ l0ðHþMÞ: ð2:8Þ

This assumption is consistent with the fact that polarization does not contribute to magnetization in bodies at rest (see e.g.[12], p. 83, and [17]).

Upon substitution of (2.8) in (2.7) we find that the Clausius–Duhem inequality takes the form

_wþ g _h� l0H � _H� l0H � _M� �0E � _E� J � Eþ 1h

q � rh� hr � k 6 0: ð2:9Þ

Restrictions placed by the inequality (2.9) are now evaluated for a rather general set of constitutive equations.

3. Thermodynamic restrictions

Let w;g;q;k and _M be given by (constitutive) functions of the set of variables

C ¼ ðh;E;H;M;rh;rMÞ:

The vector quantities q, k and the time derivative _M are allowed to depend also on the higher-order gradients

rrh;rrM:


Maxwell’s equations read

l0ð _Hþ _MÞ ¼ �r� E; r�H ¼ �0_Eþ J; ð3:1Þ

r �M ¼ �r �H; �0r � E ¼ q: ð3:2Þ

Eq. (3.2) hold at any time t, as a consequence of (3.1), provided they hold at an initial time t0 and the continuity equation

r � Jþ _q ¼ 0

holds. Hence we can take the values of _H; _E as arbitrary whereas _M is provided by the pertinent constitutive equation,

_M ¼ MðCÞ:

The function M may be viewed as c times the d’Alembertian inertia couple density, c being the gyromagnetic ratio.The space dependence of E and H is required to be appropriate so that r� E;r�H satisfy Eq. (3.1) and r � _M;r � _E

satisfy

r � _Mþr � _H ¼ 0; �0r � _Eþr � J ¼ 0:

The chain rule allows us to write the inequality (2.9) in the form

ðwh þ gÞ _hþ wrhr _hþ ðwH � l0HÞ � _Hþ ðwM � l0HÞ � _Mþ ðwE � �0EÞ � _Eþ wrM � r _M� J � Eþ 1h

q � rh� hr � k 6 0;

ð3:3Þ

where the indices h;rh;H;M;E;rM denote partial derivatives. The arbitrariness of _h, r _h and _H, _E requires that

g ¼ �wh; wrh ¼ 0

and

wH ¼ l0H; wE ¼ �0E:

As a consequence,

w ¼ 12l0H2 þ 1

2�0E2 þWðh;M;rMÞ:

Upon some rearrangements, the inequality (3.3) becomes

ðWM � l0H�r �WrMÞ � _Mþr � ðWrM_M� hkÞ þ k � rh� J � Eþ 1

hq � rh 6 0: ð3:4Þ

A simple scheme arises by letting

hk ¼ WrM_M ð3:5Þ

and hence (2.6) requires that
Z@X
1h

WrM_M � nda ¼ 0: ð3:6Þ

Look now at the corresponding conditions which guarantee the validity of (3.4). By (3.5) we have

ðWM � l0H�r �WrMÞ � _Mþ k � rh ¼ ½hðWM �r � WrMÞ � l0H� � _M;

where

W ¼ Wh; _M ¼ MðCÞ:

Hence (3.4) reduces to

½hðWM �r � WrMÞ � l0H� � M� J � Eþ 1h

q � rh 6 0: ð3:7Þ

The inequality (3.7) holds if

J � E P 0; q � rh 6 0; ð3:8Þ

and

½hðWM �r � WrMÞ � l0H� � M 6 0: ð3:9Þ


The inequalities (3.8) are satisfied by Ohm’s and Fourier’s laws, namely

J ¼ rE; q ¼ �jrh

with positive-valued functions r and j of C. The inequality (3.9) is a restriction on the constitutive equation for the timederivative _M and hence on the time evolution of the magnetic polarization M. To save writing, we can express the inequality(3.9) as

N � M 6 0; ð3:10Þ

where

N :¼ dMW� l0H; dMW :¼ hðWM �r � WrMÞ: ð3:11Þ

The time dependence of the magnetization M and the M�H relation are the subject of the next sections. The inequalities(3.9) and (3.10) are investigated to determine possible forms of the d’Alembertian inertia couple.

Remark 1. Henceforth, we let

ðWrM_MÞ � n ¼ 0; on @X; ð3:12Þ

so that (3.6) is satisfied. Incidentally, there are approaches to magnetic modelling where the extra-entropy flux does not oc-cur (see, e.g., [10]). Yet the boundary condition is placed just in the form (3.12).

Remark 2. In terms of the Gibbs free energy eW ¼ W� l0H �M we have

N ¼ dM

eWh:

Hence (3.9) becomes

dM

eWh� _M 6 0:

4. Restrictions on the evolution of M

We now have to find a function MðCÞ compatible with (3.10). Letting

w ¼ M ð4:1Þ

we can write the inequality (3.10) as

N �w 6 0; ð4:2Þ

where N and w are parameterized by M. Let

v ¼N þ �M�w; � 2 R:

The inequality (4.2) is equivalent to

v �w 6 0 ð4:3Þ

and (4.3) holds if

w ¼ �Kv þ bM� v þ mM� ðM� vÞ;

where m 2 Rþ, b 2 R and K 2 Symþ, Symþ being the space of positive semidefinite tensors. This is so because, upon substi-tution, we have

v �w ¼ �v � Kv � mjM� vj2:

Accordingly we can write the following statement.

Proposition 1. The inequality (4.2) holds if

w ¼ �KðN þ �M�wÞ þ bM� ðN þ �M�wÞ þ mM� ½M� ðN þ �M�wÞ�;

for every m 2 Rþ, every b; � 2 R, and every K 2 Symþ.By replacing w via (4.1) and taking advantage of the identity

M� ½M�; ðM� _MÞ� ¼ �jMj2M� _M;

we apply Proposition 1 to conclude that (3.10) holds if


_M ¼ ��ðKþ mjMj21ÞM� _Mþ �bM� ðM� _MÞ � KN þ bM�N þ mM� ðM�N Þ; ð4:4Þ

where K 2 Symþ and �; b; m are functions of C subject only to m P 0. At this stage the rescaled free energy W is any function ofh;M;rM. Hence, Eq. (4.4) is the most general three-dimensional evolution equation for M compatible with thermodynamics.

We are now in a position to show that the evolution equation (4.4) generalizes both Landau–Lifshitz and Gilbertequations. Indeed, as we show in Proposition 2, (4.4) provides two equations which are more general. They reduce exactly toLandau–Lifshitz and Gilbert equations in saturation conditions.

Denote by m the unit vector of M, m ¼M=jMj, and let

K ¼ a1m�mþ a2ð1�m�mÞ; a1;a2 > 0: ð4:5Þ

Of course, if a1 ¼ a2 then K reduces to a scalar times the unit tensor 1. Indeed, for any vector v the application of K as in(4.5) provides

Kv ¼ a1vk þ a2v?;

where vk and v? are the parallel and the perpendicular parts of v, relative to the pertinent unit vector m.Since

ðm�mÞM� _M ¼ 0; M� ðKuÞ ¼ a2M� u;

for any vector u, then

KðM� _MÞ ¼ a2M� _M; M� KN ¼ a2M�N :

Proposition 2. Let K be as in (4.5). The three-dimensional evolution equation (4.4) becomes

_M ¼ �KN þ s1M�M�N � s2M� _M; ð4:6Þ

or

_M ¼ ��aðN �MÞMþ �cM�N þ �kM� ðM�N Þ; ð4:7Þ

where s1; s2; �a; �c and �c are parameterized by the constants in (4.4) and by a1;a2 and jMj2.

Proof. Application of M� to (4.4) provides the expression for M� ðM� _MÞ. Substitution in (4.4) and some rearrangementsallow us to write (4.4) in the more compact form (4.6), where

s1 ¼ mþ b2

a2 þ mjMj2; s2 ¼ �ða2 þ mjMj2Þ þ b

1þ �bjMj2

a2 þ mjMj2:

By (4.5) we can write (4.6) as

_M ¼ �a1N k � a2N ? þ s1M� ðM�N ?Þ � s2M� _M

whence

_M ¼ �a1N k � ða2 þ s1jMj2ÞN ? � s2M� _M: ð4:8Þ

Vector multiplication of (4.8) by M provides the expression for M� _M. Substitution in (4.6) and some rearrangements yield

ð1þ s22jMj

2Þ _M ¼ �a1N k � ða2 þ s1jMj2ÞN ? þ s2ða2 þ s1jMj2ÞM�N ? þ s22ðM � _MÞM: ð4:9Þ

Inner multiplication by M provides M � _M. Hence (4.9) simplifies to

_M ¼ �a1N k � j1N ? þ j2M�N ?; ð4:10Þ

where

j1 ¼a2 þ s1jMj2

1þ s22jMj

2 > 0; j2 ¼ s2j1:

Now, because

N k ¼1

jMj2ðN �MÞM; N ? ¼ �

1

jMj2M� ðM�N Þ; M�N ? ¼M�N ;

we can write (4.10) formally in terms of N only as in (4.7), where �a; �k > 0 and �c 2 R are given by

�a ¼ a1

jMj2; �c ¼ s2

a2 þ s1jMj2

1þ s22jMj

2 ;�k ¼ a2 þ s1jMj2

jMj2ð1þ s22jMj

2Þ: �


It is worth looking at the simplest case of (4.4) which follows by letting b; m; � ¼ 0, or, equivalently, by letting s1; s2 ¼ 0 in(4.6), namely

_M ¼ �KN ¼ ��aðN �MÞMþ �k0M� ðM�N Þ; ð4:11Þ
where �k0 ¼ a2=jMj2. The same relation follows from (4.7) by letting �c ¼ 0 and �k ¼ �k0. As we see in Section 6, the form of K iscrucial to the splitting of (4.4) into two separate evolution equations, one governing the evolution of jMj, the other the direc-tion of M. Henceforth we examine the role played by W through N .
Eq. (4.6) is now examined in the stationary regime to prove the following statement.

Proposition 3. For any given H, the stationary states for M are solutions to the equation

hðWM �r � WrMÞ ¼ l0H; W ¼ 1h

Wðh;M;rMÞ; ð4:12Þ

subject to the constraint

r �M ¼ �r �H
and the boundary condition
WrM � n ¼ 0 on @X:

Proof. By (4.6), the stationary states solve the equation

0 ¼ �KN þ s1M� ðM�N Þ;

which can be rewritten as

0 ¼ a1N k þ ða2 þ s1jMj2ÞN ?: ð4:13Þ

Since a1;a2; s1 > 0 then (4.13) implies that N ?;N k ¼ 0 and hence N ¼ 0. The equilibrium condition (4.12) follows fromboth (4.4) and (4.11). As a consequence, existence of one (or more) solutions depends on the convexity (or non-convexity)of W with respect to M. �

Lemma 1. If f is a function which depends on rM through r�M then

r � frM ¼ �r� fr�M:

Proof. This identity follows by the observation that, in indicial notation,

@

@Mq;p¼ @

@ðr �MÞj@ðr �MÞj@Mq;p

¼ �pqj@

@ðr �MÞj;

and hence

½ðfrMÞpq�;p ¼ �pqj@f

@ðr �MÞj

" #;p

¼ �ðr� fr�MÞq;

where ; p denotes partial differentiation relative to the p-th coordinate. �

Here we assume that W depend onrM only throughr�M so that, with an abuse of notation, the additive free energy Wtakes the form

W ¼ Wðh;M;r�MÞ:

As a consequence, owing to Lemma 1, we have

dMW :¼ hðWM þr� Wr�MÞ; ð4:14Þ

while the boundary condition (3.6) holds if

ð _M� Wr�MÞ � n ¼ _M � ðWr�M � nÞ ¼ 0 on @X: ð4:15Þ

Remark 3. By virtue of Lemma 1, when W depends on rM only through r�M the stationary equation becomes

hðWM þr� Wr�MÞ ¼ l0H

subject to the boundary condition

Wr�M � n ¼ 0 on @X:


4.1. An example of free-energy function

Quite a general (rescaled) free-energy function W ¼ W=h is given by

W ¼ Fðh; jMjÞ þ 12½c1M � r �Mþ c2jr �Mj2 þ c3jM � ej2�;

where F is a non-convex function of jMj, c1 and c2 are constants, c3 is parameterized by h and e is a possibly-privileged unitdirection. Hence we have

dMW ¼ h1jMj F jMjMþ c1r�Mþ c2r�r�Mþ c3ðhÞðM � eÞe� �

: ð4:16Þ

If the material is isotropic then c3 ¼ 0. The term M � r �M in the free energy is considered for the sake of generality but doesnot seem to be motivated on physical grounds.

Landau et al. [2] regard W as though c1; c3 ¼ 0. Their position amounts to assuming that K ¼ a1 and that

Fðh; jMjÞ ¼ 12

a1ðh� hcÞM2 þ 14

a2M4;

where a1 and a2 are allowed to depend on h.We now exhibit a three-dimensional setting and let W depend on M, through a logarithmic function of M2. This depen-

dence is motivated by an investigation of one-dimensional potentials which is shown in Section 8.

5. Evolution in the three-dimensional space

The whole evolution of the system is described by the equation for _h (balance of energy) in addition to that for _M. We nowset up the corresponding scheme without any restriction on the amplitude and the direction of M. Later on, we investigatethe case when the system is saturated (jMj ¼ constant) or the direction of M is fixed in space (M=jMj ¼ constant). However,to avoid formal difficulties, we restrict attention to isotropic materials so that the free energy depends on rM throughjrMj2. For definiteness, the pertinent coefficient is taken to be proportional to the temperature h. Though with more in-volved formulae, the anisotropic case might be considered by following along the same lines.

Let the free energy W be given by

W ¼ Gðh;MÞ þ 12jhjrMj2; ð5:1Þ

where

Gðh;MÞ ¼ gðhÞ � bðuðhÞ þ 1Þ lnð1�M2=M2s ðhÞÞ � bM2=M2

s ðhÞ ð5:2Þ

and, for simplicity, uðhÞ takes the classical form u ¼ ðh� hcÞ=hc.The function MsðhÞ models the dependence of Ms on h as is the case for the relation

MsðhÞ ¼ Msð0Þð1� BhbÞ ð5:3Þ

which is often referred to as Bloch’s law (see [18,19]). Here B is the Bloch constant and is obviously dependent on the mate-rial. The Bloch exponent b equals 3=2 for bulk materials but equals roughly 1=2 for some nanoparticle specimens [20].

We let K ¼ a1 so that the evolution equation (4.6) applies with a1 ¼ a2 ¼ a. Moreover, let s1; s2 ¼ 0, namely �; b; m ¼ 0.Eq. (4.6) then reduces to

_M ¼ �aðdMW� l0HÞ:

Hence, by (5.1) we have

_M ¼ al0H� 2abðuþM2=M2

s ÞMsð1�M2=M2

s ÞMMs� ajhDM: ð5:4Þ

As a check on the validity of the evolution equation (5.4) we restrict attention to stationary ( _M ¼ 0) and uniform (DM ¼ 0)conditions. Eq. (5.4) gives

l0H ¼ 2buþM2=M2

s

Msð1�M2=M2s Þ

MMs

: ð5:5Þ

As a consequence M and H are collinear. Letting M;H be the components in the common direction we find that the differ-ential susceptibility

v ¼ dMdH


is a function of M=Ms ¼ n namely

v ¼ l0M2s

2bð1� n2Þ2

uþ ðuþ 3Þn2 � n4 ; n 2 ½0;1�: ð5:6Þ

If h > hc then u > 0. Hence, v > 0, in that 3n2 > n4, and the material is paramagnetic. Also, for small values of n we have

v ’ l0M2s

2bhc

h� hc; ð5:7Þ

whence the differential susceptibility v varies with h as ðh� hcÞ�1, which is the content of the Curie–Weiss law (see [2]).If h < hc then, by (5.7), v < 0 for small values of n whereas, by (5.6), v > 0 as n2 approaches 1 or the body is almost sat-

urated. This is the typical behaviour of ferromagnetic materials. Again, the differential susceptibility varies with h asðh� hcÞ�1 for small values of n. If h! 0 then u! �1 and (5.4) reduces to

_M ¼ al0Hþ 2ab

M2s

M:

Similar, though more involved, models and conclusions are obtained by letting u be given, e.g., as in Section 8.2.It is of interest to look at the function HðMÞ, or HðnÞ, as h > hc or h < hc . By (5.5) we have

l0Ms

2bH ¼ ðuþ n2Þn

1� n2 : ð5:8Þ

Fig. 1 shows the right-hand side of (5.8) as h ¼ 1:2hc and h ¼ :8hc .The evolution of the material is completed by accounting also for the balance of energy. Now, by (5.1) and (5.2) we

have

g ¼ �g0 þ bhc

lnð1�M2=M2s Þ þ 2

bM2

M3s

uþM2=M2s

1�M2=M2s

M0s �

12jjrMj2;

e ¼ wþ hg ¼ 12ðl0H2 þ �0E2Þ þ g � hg0 þ lðh;M;Ms;M

0sÞ;

where a prime stands for the derivative with respect to the temperature h and

lðh;M;Ms;M0sÞ ¼ �b

M2

M2s

þ 2bhM2

M3s

½h� hcð1�M2=M2s Þ�M

2

hcM3s ð1�M2=M2

s ÞM0

s:

The balance of energy in the form (2.4) then yields

½�hg00 þ lh þ lMs M0s þ lM0s

M00s � _hþ ðlM � l0HÞ � _M� J � Eþr � q� r ¼ 0; ð5:9Þ

where J and q are to be viewed as given by the pertinent constitutive functions and r is a possible given function (heat sup-ply). Eqs. (5.4) and (5.9) constitute the system of evolution equations for the two fields hðx; tÞ;Mðx; tÞ.

Remark 4. Ferrimagnetic materials, like ferromagnets, hold a spontaneous magnetization below the Curie temperature hc

and are paramagnetic above hc . However, the amount of spontaneous magnetization in ferrimagnetic materials, such asferrites and magnetic garnets, is smaller than in ferromagnets. This is so because a ferrimagnetic material consists of

Fig. 1. On the left, plot of ðl0Ms=2bÞH versus n in the paramagnetic phase, h ¼ 1:2hc . On the right, in the ferromagnetic phase, h ¼ 0:8hc .


different sublattices with opposed but unequal magnetic moment, whereas in antiferromagnetic materials the magneticmoments of the two sublattices are equal and opposed. For instance, in ferrite the sublattices are given by two families ofions, Fe2þ and Fe3þ. In addition, in magnetic garnets there is a temperature below hc , called magnetic compensation point, atwhich the spontaneous magnetization vanishes. They also exhibit a third critical temperature corresponding to the angularmomentum compensation point [21]. The modelling of such materials requires the occurrence of two distinct vector phasevariables, say M1 and M2, each obeying an evolution law like (4.11).

6. Evolution of direction and amplitude

Also for a more direct comparison with some models of magnetization in matter, we now examine two representations ofthe field M. Let Ms > 0 be the saturation value of magnetization. We can write

Mðx; tÞ ¼ Ms/ðx; tÞmðx; tÞ; / 2 ½0;1�; jmj ¼ 1: ð6:1Þ

Hence, m is the unit vector of M. The saturation magnetization Ms is here regarded as a (temperature-independent) constant.As shown by measurements on nanoparticles (see [20]), the approximation is reasonable in many circumstances.

In saturation conditions,

Mðx; tÞ ¼ Msmðx; tÞ; / ¼ 1:

If the magnetization has a constant direction we can represent M as

Mðx; tÞ ¼ Msnðx; tÞe; n 2 ½�1;1�; jej ¼ 1;

where e is the fixed unit vector. More generally, for a variable direction we can write

Mðx; tÞ ¼ Msnðx; tÞrðx; tÞ; n 2 ½�1;1�; r 2 U; ð6:2Þ

where U is a solid cone,

U ¼ fr : jrj ¼ 1; r 2 U) �r R Ug:

If U is the set of possible directions then the two representations are equivalent by letting

/ ¼ jnj; r ¼ ðsgnnÞm:

We now proceed by representing M as Ms jnjm, regarding n and m as independent quantities and looking for the separateevolution equations

_m ¼ mðCÞ; _n ¼ nðCÞ:

By the representation (6.2) we have

_M ¼ Msð _nrþ jnj _mÞ; rM ¼ Msðrn� rþ jnjrmÞ; jnj _m ¼ n _r; jnjrm ¼ nrr;

and hence

Wn ¼ MsðWM � rþ WrM � rÞ; Wrn ¼ MsWrMr;

Wm ¼ MsðWMjnj þ WrM � rjnjÞ; Wrm ¼ MsWrMjnj;

and

_M � dMW ¼ Msðr � dMWÞ _nþMsðjnjdMWÞ � _m:

Now,

Msr � dMW ¼ MshðWM �r � WrMÞ � r ¼ h½Wn �r � ðMsWrMrÞ� ¼ hðWn �r � WrnÞ ¼: dnW;

MsjnjdMW ¼ MshðWM �r � WrMÞjnj ¼ h½Wm �r � ðMsWrMjnjÞ� ¼ hðWm �r � WrmÞ ¼: dmW:

Hence,

_M � dMW ¼ dnWnþ dmW � m

and (3.9) becomes

ðdnW� l0MsH � rÞnþ ðdmW� l0MsjnjHÞ � m 6 0: ð6:3Þ

Since nr ¼ jnjm then

ndnW ¼m � dmW:


In view of Proposition 1, a sufficient condition for the validity of (6.3) is that the evolution functions n and m take the form

n ¼ �xðdnW� l0MsH � rÞ; ð6:4Þm ¼ ��a1Nk � �a2N? þ �bm� Nþ �mm� ðm� NÞ; ð6:5Þ

where the decomposition Nk, N? is relative to m and

N ¼M þ ��m� _m; M ¼ dmW� l0MsjnjH: ð6:6Þ

Also, x; �a1; �a2; �m are non-negative valued functions of C and �b; �� are real valued. Because jmj ¼ 1, m satisfies the constraint

_m �m ¼ m �m ¼ 0

and hence it follows from (6.5) that

0 ¼ �a1Nk �m:

This condition holds by merely requiring that �a1 ¼ 0. Hence, letting �a2 ¼ �a we can write (6.5) in the form

m ¼ ��a½ð1�m�mÞM þ ��m� _m� þm� ð�bNþ �mm� NÞ: ð6:7Þ

By means of (6.7) we can now prove the following statement.

Proposition 4. If n – 0 then the evolution equation for m can be expressed in terms of M in the form

_m ¼ �cm�M þ �km� ðm�M Þ; ð6:8Þ

where

�c ¼�bþ ��½�b2 þ ð�aþ �mÞ2�ð1þ ��bÞ2 þ ��2ð�aþ �mÞ2

; �k ¼�aþ �m

ð1þ ��bÞ2 þ ��2ð�aþ �mÞ2:

Proof. Since jmj ¼ 1 then m � _m ¼ 0, j _mj ¼ jm� _mj, and

m� ðm� _mÞ ¼ � _m:

Also,

ð1�m�mÞM ¼ �m� ðm�M Þ:

As a consequence, by (6.7) the evolution equation becomes

ð1þ ��bÞ _m ¼ ��ð�aþ �mÞm� _mþ �bm�M þ ð�aþ �mÞm� ðm�M Þ: ð6:9Þ

Apply m� to (6.9), derive the expression of m� _m and then replace it in (6.9). Upon some algebraic rearrangements we ob-tain (6.8). �

As expected, the evolution equation (4.4) follows from (6.4) and (6.7) once we make appropriate identifications. They aregiven by

x ¼ a1

M2s

; �a ¼ a2

M2s jnj

2 ;�b ¼ b

Msjnj; �m ¼ m; �� ¼ �M3

s jnj3

while K has the form (4.5). At saturation (jnj ¼ 1), the constancy of a1;a2; b; m; � is equivalent to that of x; �a; �b; �m; ��.As we shall see in a moment, the term dmW, which occurs in M , accounts for exchange and anisotropic interactions. If

such effects are neglected then (6.8) reduces to

_m ¼ �c0m�H� k0m� ðm�HÞ; ð6:10Þ

where

c0 ¼ l0Msjnj�c; k0 ¼ l0Msjnj�k:

Two particular cases are of interest, namely the magnetic saturation, which occurs if jnj ¼ 1, and the fixed alignment of M,which means that the magnetic direction r ¼ sgnn m is constant. They are now examined by allowing for non-isothermalconditions and assuming that the temperature is below the critical temperature hc .

6.1. Magnetic saturation

In magnetically-saturated bodies, jnj ¼ 1, and hence

M ¼ dmW� l0MsH:


Also, dmW ¼ MsdMW so that

M ¼ MsðdMW� l0HÞ ¼: �l0MsHeff ;

which ascribes to

Heff ¼ H� 1l0

dMW ð6:11Þ

the role of effective magnetic field.Let h < hc be constant in space and time. We look for equilibrium configurations by applying the evolution equation (6.8).

If _M ¼ 0, that is _m ¼ 0, then we have

�cm�M þ �km� ðm�M Þ ¼ 0:

Hence the equilibrium condition becomes

l0Msm�Heff ¼ 0; ð6:12Þ

where Heff is the effective field given by (6.11). Eq. (6.12) traces back to Brown [5] and ascribes to- dMW=l0 the meaning ofthe field arising from exchange and anistropic interactions. Indeed, the standard form of Heff ([5, p. 48])

Heff ¼ H� 1l0Ms

fm þ2

l0Msr � ðArmÞ;

follows from (3.11) and (6.11) by letting

W ¼ f ðMÞ þ rM � ArM;

where A is a symmetric fourth-order tensor.Let f depend on M in the anisotropic form

f ðMÞ ¼ 12l0M � QM

where Q 2 Symþ. Hence we have

Heff ¼ H�MsQmþ 2l0

Msr � ðArmÞ: ð6:13Þ

The last term, ð2=l0ÞMsr � ðArmÞ, accounts for exchange interactions and penalizes magnetization inhomogeneities. If theexchange interaction is isotropic in space (for instance such is the case for a cubic cell, see [2]) then A ¼ ðl0A=2Þ1, where A isthe so-called exchange constant which depends on the lattice geometry (body-centred, face-centred cubic crystals).

Denote by Dx;Dy;Dz and ex; ey; ez the eigenvalues and the eigenvectors of Q. Hence we can express f as a function ofMx ¼ Msmx;My ¼ Msmy;Mz ¼ Msmz, the components of M relative to the eigenvector basis, in the form

f ðMÞ ¼ 12l0M2

s ðDxm2x þ Dym2

y þ Dzm2z Þ;

where the coefficients Dx;Dy;Dz > 0 characterize the anisotropy of the body. Of course, m2x þm2

y þm2z ¼ 1. If Dx;Dy;Dz are dis-

tinct values then the energy term f accounts for biaxial anisotropy,

f ðMÞ ¼ 12l0M2

s ½Dz þ ðDx � DzÞm2x þ ðDy � DzÞm2

y �:

If, instead, Dx ¼ Dy ¼ D? and Dz – D? then f accounts for uniaxial anisotropy with easy direction ez,

f ðMÞ ¼ 12l0M2

s ½Dz þ ðD? � DzÞð1�m2z Þ�:

In such a case the body is transversely isotropic (relative to the z-axis) and the effective field takes the form

Heff ¼ H�MsðDzmzez þ D?m?Þ;

where mz ¼m � ez and m? ¼m�mzez. In particular, for strongly anisotropic materials D? is negligible with respect to Dz andhence

Heff ’ H� DzðM � ezÞez: ð6:14Þ

In transversely-isotropic materials the evolution equation (6.8) accounts also for the damping switching [6], namely theswitching of a component, say mz, generated by the z-component of the magnetic field opposite to the initial value mzð0Þ.

Finally, in isotropic bodies Dx ¼ Dy ¼ Dz ¼ D > 0 and A ¼ ðl0A=2Þ1 so that f simplifies to the constant

f ðMÞ ¼ 12l0M2

s D

and the effective field reduces to


Heff ¼ HþMsADm ¼ Hþ ADM:

Accordingly, when the body is magnetically isotropic and subject to uniform fields (rm ¼ 0) the effective field reducesmerely to the applied field H.

Section 7 is devoted to the relation of these particular cases with known models appeared in the literature.

6.2. Fixed alignment

If the direction e ¼ ðsgnnÞm of M is constant, say r ¼ e, then M ¼ Msne and _M ¼ Ms_ne. Accordingly, m is constant except

when sgnn changes. Such is the case usually assumed for an isotropic magnet in a magnetic field H with constant direction.Indeed, because dmW ¼ ðsgnnÞdrW, by (6.8) we have

0 ¼ �cr� ðdrW� l0MsnHÞ þ �ksgnn r� ½r� ðdrW� l0MsnHÞ�

whence

r� ðdrW� l0MsnHÞ ¼ 0:

As a consequence, when anisotropic and exchange interactions are neglected (drW ¼ 0) we have r�H ¼ 0, which means thatH and M are parallel (one-dimensional setting). The same result holds also for magnets with strong uniaxial anisotropy pro-vided that the direction of the applied field H coincides with the easy direction of the body. If such is the case then it followsfrom (6.14) that Heff is approximately parallel to r. In addition, by (6.4) we have

_n ¼ �xðdnW� l0HÞ; x > 0; ð6:15Þ

where H ¼ MsH � r, which governs the evolution of the (magnetic intensity) component n. If, as a special case,

W ¼ 12

aðh� hcÞjnj2 þ14

bjnj4

then (6.15) becomes

_n ¼ �x½aðh� hcÞnþ bn3 � l0H�

which is in fact the non-isothermal model by Landau et al. [2].In Section 8 known models of free energy are shown to follow as particular cases and, morevorer, a new free energy is

established.

7. Relation to the Gilbert–Landau–Lifshitz model

It is natural to contrast (6.8) with known evolution equations of micromagnetics which apply to magnetically-saturatedbodies. Landau and Lifshitz [1] started with the evolution equation

_l ¼ �cl�H

for a magnetic spin momentum l of an electron in a magnetic field H, c > 0 being the absolute value of the gyromagneticratio. In the continuum limit they wrote

_M ¼ �cM�H

where H is the magnetic field in matter. Dissipation was modelled by adding a torque which pushes M toward the field H.The torque was taken in the form �KM� ðM�HÞ, where K > 0, so that

_M ¼ �cM�H�KM� ðM�HÞ ð7:1Þ

Later on, Gilbert [3,4] wrote the evolution equation with a dissipative term TM� _M, T > 0, in the form

_M ¼ �cM�HþTM� _M: ð7:2Þ

In both cases, taking the inner product with M gives

M � _M ¼ 0; ð7:3Þ

which means that jMj is constant in time. Hence (7.1) and (7.2) apply to magnetically-saturated bodies. Since jMj is constant,we can replace M with Msm in (7.1), (7.2) to get

_m ¼ �cm�H� km� ðm�HÞ; ð7:4Þ_m ¼ �~cm�Hþ sm� _m; ð7:5Þ

where k ¼ KMs and s ¼TMs, s being the dimensionless Gilbert damping constant. Eqs. (7.4) and (7.5) are formally equiv-alent. For instance, replacing _m in the right-hand side of (7.5) with the whole right-hand side we obtain


_m ¼ �c�m�H� k�m� ðm�HÞ; ð7:6Þ

where

c� ¼~c

1þ s2 ; k� ¼ s~c1þ s2 :

Incidentally, comparison of (7.5) and (7.6) shows that s ¼ k=c and ~c ¼ ðc2 þ k2Þ=c. This in turn is consistent with the fact thats and k are dissipation coefficients. If k! 0 then s! 0 and ~c! c so that (7.4) and (7.5) reduce to

_m ¼ �cm�H:

Remark 5. Let H be constant. By (7.4) we have

ddtðm �HÞ ¼ kjm�Hj2 P 0:

The component of m along H increases until m and H are collinear. Hence the Landau–Lifshitz equation (7.1), in the approx-imation of a constant magnetic field, models a magnetization M, with constant jMj, that tends to orient itself parallel to H. Inaddition, letting eH be the unit vector of H, so that H ¼ jHjeH, we have

ddtðm � eHÞ ¼ kjm� eHj2jHj;

which shows that the rate dðm � eHÞ=dt is proportional to jHj.

Recently more involved evolution equations have been considered by replacing H, in (7.5), with an effective magneticfield Heff . A generalization of the evolution equation (7.5) is performed in [8] (see also [9]) by replacing H with

Heff ¼ fDmþ gðe �mÞeþH; ð7:7Þ

where e is the unit vector of the easy axis of magnetization, H is the sum of the stray field and of the external field (see also[7,11]). The model (7.7) is a particular case of (6.13) as it follows by letting e ¼ ez, g ¼ �MsDz, MsA ¼ 1

2 l0f1. In such a case the(Gilbert) evolution equation takes the form

_m� sm� _m ¼ �cm� ½fDmþ gðm � eÞeþH�; ð7:8Þ

which reduces to (6.12) in stationary conditions. As a check we see that m � _m ¼ 0. Hence we can write (7.8) also in the form

_m ¼ c�f�m� ½fDmþ gðm � eÞeþH� þ sm� ½m� ðfDmþ gðm � eÞeþHÞ�g: ð7:9Þ

The evolution equations (7.4), (7.5) and (7.8) are particular cases of (6.8) and (6.10) and hence are compatible with ther-modynamics. Eq. (7.4) coincides with (6.10) once the identifications k ¼ k0 and c ¼ c0 are made.

The same identifications hold for the Gilbert equation (7.6) with c� and k� in place of c0 and k0. The generalized Gilbertequation, in the form (7.9), follows from (6.8) by letting the temperature h be constant, the rescaled free energy take the form

W ¼ g2l0jM � ej2 � f

2l0jr �Mj2

and

c� ¼ l0Msjnj�c; s ¼ �k=�c:

8. Potentials for one-dimensional models

Henceforth we look for appropriate functions W to model ferromagnetism and the associated phase transition. For sim-plicity we restrict attention to the one-dimensional case and apply (4.4) by letting b; c ¼ 0. Denote by M;H 2 R the signifi-cant components of M;H. Eq. (4.4) for M, likewise (6.15) for n ¼ M=Ms, then becomes

_n ¼ �a½hðWn �r � WrnÞ � h�; ð8:1Þ

where h ¼ l0H. In stationary conditions, n ¼ constant, we have

h ¼ hðWn �r � WrnÞ: ð8:2Þ

Also we let

Wðh; n;rnÞ ¼ 12

chjrnj2 þ Vðn; uÞ;


where c is a positive constant and u is a suitable increasing function, of the temperature h, which vanishes at the Curie pointhc . The function V is often referred to as the potential. Hence (8.1) and (8.2) become

_n ¼ �aðVn � h� chDnÞ; ð8:3Þh ¼ Vn � chDn: ð8:4Þ

This setting allows us to obtain an immediate connection with other approaches. For instance, a standard free energy ap-plied in the literature (see, e.g., [12]) corresponds to letting u ¼ ðh� hcÞ=hc and

V ¼ n2½aðh� hcÞ þ bn2�; ð8:5Þ

where a; b are positive constants. In such a case, Eqs. (8.3) and (8.4) become

_n ¼ �2a½aðh� hcÞ þ 2bn2�nþ ahþ 12achDn;

and

h ¼ �chDnþ ½2aðh� hcÞ þ 4bn2�n: ð8:6Þ

The potential (8.5) traces back to Ginzburg [2]. The function
eU ¼ Vðn; uÞ � hn
is considered in [22], the potential eU being identified with the Lagrangian density.Equilibrium conditions are sought for homogeneous configurations,rn ¼ 0. Hence the equilibrium conditions are the sta-

tionary points of the potential. The potential (8.5), as well as similar ones in the literature, does not provide a reasonable setof equilibrium values and a satisfactory scheme for phase transition which occurs when u ¼ 0. Indeed, it is a well-knowndrawback of the potential (8.5) that it does not allow for saturation. By (8.6), for large values of h in homogeneous config-urations we have

n ’ ð4bÞ�1=3h�2=3h;

as though the material had a permittivity proportional to h�2=3. This motivates the search for schemes where the saturation isallowed.

8.1. Weiss model of ferromagnetism

We now review the Weiss model of ferromagnetism (see, e.g., [23] and refs therein) and show how it allows for a poten-tial VW ðh; nÞ.

Let L be the Langevin function, on R, defined by

LðxÞ ¼ coth x� 1x:

The function L is strictly increasing and odd and moreover LðxÞ ! �1 as x! �1. As a consequence the inverse L�1 mapsð�1;1Þ into R. Weiss model relates the magnetization M and the magnetic field H in the form

H þ bMlh

¼L�1ðM=MsÞ ð8:7Þ

where Ms is the maximum value of the magnetization, that is M 2 ð�Ms;MsÞ, b is the molecular-field parameter, and l ¼ k=mis the ratio of Boltzmann’s constant over the magnitude of the (atomic) magnetic moment. By (8.7) we have

H ¼ lhL�1ðM=MsÞ � bM: ð8:8Þ

Letting n ¼ M=Ms we can write H ¼HðnÞ, where H is defined by

HðnÞ ¼ lhL�1ðnÞ � bMsn; n 2 ð�1;1Þ:

If h! 0þ then

H ¼ �bM;

which is the classical law of diamagnetism or paramagnetism according as b > 0 or b < 0. Now, ferromagnetism indicatesthat there is a temperature hc such that

H0ð0Þ > 0 as h > hc; H0ð0Þ 2 ð�bMs;0Þ as h 2 ð0; hcÞ:

Consistently, it is expected that H0ð0Þ ¼ 0 at h ¼ hc , namely

lhcðL�1Þ0ð0Þ � bMs ¼ 0: ð8:9Þ


The requirement (8.9) induces a relation between b;l, and hc. In this connection, letting

z ¼ H þ bMlh

we have

ðL�1Þ0ðnÞ ¼ 1L0ðzðnÞÞ :

Since L�1ð0Þ ¼ 0 then zð0Þ ¼ 0. Moreover, L0ð0Þ ¼ 1=3. As a consequence

ðL�1Þ0ð0Þ ¼ 1L0ð0Þ ¼ 3:

Hence (8.9) provides

hc ¼bMs

3l:

This means that the parameter l is related to the transition temperature hc by

l ¼ bMs

3hc:

Upon substitution we can write Eq. (8.8) in the form

1bMs

H ¼ uþ 13

L�1ðnÞ � n ð8:10Þ

where u ¼ ðh� hcÞ=hc . Eq. (8.10) provides the value of the magnetic field H in terms of the magnetization M, in that n ¼ M=Ms.The equilibrium (8.10) may be viewed as the stationary condition for the potential VW such that

V 0WðnÞ ¼ bMsuþ 1

3L�1ðnÞ � bMsn� H:

The integration gives

VWðnÞ ¼ bMsuþ 1

3KðnÞ � 1

2bMsn

2 � Hn; ð8:11Þ

where K is the integral of L�1.To evaluate the residual magnetization Mr we can go back to (8.10) and require that the right-hand side vanishes at Mr .

We find that nr ¼ Mr=Ms satisfies

nr ¼LðdnrÞ; d ¼ 3hc

h: ð8:12Þ

To obtain the coercive field Hc , by (8.10) we look for the maximum of the right-hand side. We find that the coercive field Hc ,at nc ¼ Mc=Ms, is given by

L0ðL�1ðncÞÞ ¼1d; Hc ¼ bMs½

1dL�1ðncÞ � nc�: ð8:13Þ

In essence, the advantage of the Weiss description is that the saturation effect is allowed and the condition nr ! 1, ash! 0, holds. However, both the residual magnetization Mr and the coercive field Hc are provided in an implicit way, through(8.12) and (8.13).

8.2. A logarithmic potential

Still we let Ms be the saturation value of the magnetization, M 2 ð�Ms;MsÞ, and n ¼ M=Ms 2 ð�1;1Þ while h ¼ l0H. Wedenote by nr the residual or remnant (relative) magnetization and by hc the coercive field. By definition, nr is the (positive)value of n at h ¼ 0 whereas hc is the local maximum of h.

We now examine the properties of the magnetization curve associated with the Lagrangian density

eU ¼ �bðuþ 1Þ lnð1� n2Þ � bn2 � hn; ð8:14Þ

where b is a positive constant and u is a function of h such that

uðhÞ 2 ð�1; 0Þ as h 2 ð0; hcÞ; uð0Þ ¼ �1; uðhcÞ ¼ 0:


The potential

V lnðnÞ ¼ eUðnÞ þ hn ¼ �bðuþ 1Þ lnð1� n2Þ � bn2

is similar to the Weiss potential VW in (8.11). In this sense �bðuþ 1Þ lnð1� n2Þ replaces bMsKðuþ 1Þ=3.For definiteness we can take uðhÞ in the form

uðhÞ ¼ � 1� hhc

� �q� �p

½1þ ahhc

� �q

�p; a 2 ð0;1� ð8:15Þ

and p; q 2 Q. Since

u0 ¼ pqhq�1h�qc ½1� aþ 2aðh=hcÞq�f½1� ðh=hcÞq�½1þ aðh=hcÞq�gp�1

it is apparent that u is a monotonic, increasing function of h as h 2 ½0; hc�. Furthermore, it is increasing as h 2 Rþ if p ¼ 1. Thestandard form of u, namely u ¼ ðh� hcÞ=hc , is a particular case of (8.15) that corresponds to a ¼ 0, p ¼ q ¼ 1 or to a ¼ p ¼ 1,q ¼ 1=2.

In stationary conditions ð _n ¼ 0Þ Eq. (8.1) amounts to the vanishing of eUn whence

hðnÞ ¼ 2bðuþ 1Þ n

1� n2 � 2bn: ð8:16Þ

Let h 2 ð0; hcÞ and hence u 2 ð�1;0Þ. As h ¼ 0, Eq. (8.16) provides the three solutions

n ¼ 0; n ¼ �ffiffiffiffiffiffijuj

p; n ¼

ffiffiffiffiffiffijuj

p
and hence the residual magnetization nr is given by
nr ¼ffiffiffiffiffiffijuj

p: ð8:17Þ

Look at the maximum and the minimum of hðnÞ. Observe that

12b

h0 ¼ uþ ðuþ 3Þn2 � n4

ð1� n2Þ2:

Since u 2 ð�1;0Þ then h0 vanishes if

n4 � ð3� jujÞn2 þ juj ¼ 0:

There are then four solutions for n. Two of them, ni;�ni, such that

n2i ¼

12ð3� juj �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3� jujÞ2 � 4juj

qÞ;

are found to belong to ð�1;1Þ. The remaining two solutions, no, �no, such that

n2o ¼

12ð3� juj þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3� jujÞ2 � 4juj

qÞ;

are found to be outside ½�1;1�. Hence we have

hc ¼2bni

1� n2i

ðn2i � jujÞ: ð8:18Þ

Borrowing e.g. from [24], we say that the function

nrðhÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffijuðhÞj

p; h 2 ð0; hcÞ;

is concave, monotonic decreasing and subject to nrð0Þ ¼ 1, nrðhcÞ ¼ 0. Now, by (8.15) we see that nrð0Þ ¼ 1, nrðhcÞ ¼ 0 and,moreover, a 6 1 makes sgnu constant. We now ascertain monotonicity and concavity of nrðhÞ. Letting x ¼ h=hc we have
ffiffiffiffiffiffi
jujp

¼ sðxÞ :¼ ½ð1� xqÞð1þ axqÞ�p=2:

Upon evaluation of s0 and s00 we conclude that monotonicity holds for p; q P 0 and that concavity holds if and only if a ¼ 1� �and q > 1; p 6 2, for small values of �, or a ¼ 1 and q > 1=2; p 6 2.

Remark 6. If

p ¼ 2; q ¼ 34;

we have

nr ¼ 1þ ða� 1Þðh=hcÞ3=4 � aðh=hcÞ3=2:


Since q ¼ 3=4 is allowed only if a ¼ 1 then we have

nr ¼ 1� ðh=hcÞ3=2; ð8:19Þ

which is exactly Bloch’s law (see (71.7) of [25] or [18]) for the dependence of the spontaneous magnetization on the tem-perature. The standard choice

u ¼ ðh� hcÞ=hc ð8:20Þ

corresponds to a ¼ 0; p ¼ q ¼ 1 or to a ¼ p ¼ 1; q ¼ 1=2, which are limit cases of (8.15).

Remark 7. At the absolute zero, u ¼ �1 and the function hðnÞ shows a crucial behaviour. For, as u ¼ �1 we have

n2i ¼ n2

o ¼ 1

and hence the magnetization curve hðnÞ degenerates into three straight lines,

n ¼ �1; n ¼ 1; h ¼ �2an:

As h > hc we have u > 0 and hence, as expected, h vanishes only if n ¼ 0.

In conclusion, the logarithmic potential V ln provides a good approximation of VW around n ¼ 0, allows for the saturationeffect, yields the residual magnetization and the coercive field in an explicit way and provides a concave function nrðhÞ.

8.3. The logarithmic potential for crystals of iron

For definiteness we look for the particular function u of (8.15) for crystals of iron. Although iron may be viewed as fer-rimagnetic, we apply the ferromagnetic model since no compensation point occurs.

Our purpose is to determine the parameters a; p; q by means of the experimental data for iron (see [24]) namely the curveof the residual magnetization nr versus the absolute temperature h.

Let p; q > 0. First we observe that by

s0ðxÞ ¼ 12

pq½1þ ða� 1Þxq þ ax2q�ðp�2Þ=2xq�1ða� 1� 2axqÞ

we have

s0ðxÞ ! �1 or s0ðxÞ ! 0;

as x! 1�, according as p < 2 or p > 2. Because the experimental data show that s0ð1�Þ ¼ �1we find a further reason for thecondition p < 2.

Assume p; q are fixed, p 2 ð0;2Þ. Since

s2=pðxÞ ¼ 1� xq þ aðxq � x2qÞ;

letting

y ¼ s2=p; zðxÞ ¼ 1� xq; vðxÞ ¼ xq � x2q

Fig. 2. The dots represent given values of ðh=hc ; nrÞ. The curve interpolates the dots.


we can write

yðxÞ ¼ zðxÞ þ avðxÞ:

The linear dependence of y on a allows us to find a as the least squares solution.Otherwise, once a set of points ðxi; siÞ are given, the joint derivation of the parameters a; p; q may be performed e.g. by the

program PLOT (MacOSX). Fig. 2 shows the dots extracted from Bozorth [24]. The corresponding parameters turn out to bep ¼ 0:6838; q ¼ 1:7298; a ¼ 0:5777: The curve gives the corresponding function sðxÞ.

As a check of consistency, if we let p ¼ 0:6838; q ¼ 1:7298 and evaluate the corresponding optimal value of a we find thata ¼ 0:568. If, instead, we take q ¼ 1:7298; a ¼ 0:5777 then we find that p ¼ 0:688. In both cases the values (of a or p) are veryclose to those given by the joint derivation.

9. Conclusions

The thermodynamic analysis of the modelling of magnetic materials is based on h;E, H, M, rh;rM;rrM as the set ofindependent variables and on the statement of the second law through an inequality involving an extra entropy flux k.We have found that the second law inequality is satisfied if hk ¼ WrM

_M and _M is subject to (3.10).The evolution equation is governed by the rescaled free energy Wðh;M;rMÞ=h. We find that the logarithmic function

(8.14), with a < b, has two equilibrium solutions. The corresponding values of the order parameter is (�1) times the residualmagnetization. The choice (5.1) for W where G has the logarithmic form (5.2) provides the model for the paramagnetic–fer-romagnetic transition. Also, the Curie–Weiss law is obtained in the approximation of small values of M.

Evolution equations arisen within micromagnetics have been considered, namely the Landau–Lifshitz equation (7.1), theGilbert equation (7.2) and recent improvements (7.8). The generality of the present thermodynamic approach allows us toframe also the micromagnetic equations as evolution equations for M, generated by an appropriate form of the free energypotential, as a particular form of (4.4).

It is a positive feature of the present approach that the same logarithmic potential (5.2) provides a satisfactory descriptionof both the ferromagnetic (h < hc) and the paramagnetic (h > hc) behaviour and fits the curve of residual magnetization, as afunction of the absolute temperature, for crystals of iron.

Acknowledgement

The research leading to this paper has been supported by the Italian MIUR through the Project PRIN 2005 ‘‘Mathematicalmodels and methods in continuum physics”.

References

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International Journal of Engineering Science · phase-transition fronts in deformable ferromagnets. As a particular case, namely in stationary conditions, the evolution equation is