EXISTENCE AND UNIQUENESS FOR A THREE DIMENSIONAL MODEL OF FERROMAGNETISM

Transcription

1 1 EXISTENCE AND UNIQUENESS FOR A THREE DIMENSIONAL MODEL OF FERROMAGNETISM V. BERTI and M. FABRIZIO Dipartimento di Matematica, Università degli Studi di Bologna, P.zza di Porta S. Donato 5, I-4126, Bologna, Italy berti@dm.unibo.it fabrizio@dm.unibo.it C. GIORGI Dipartimento di Matematica, Università degli Studi di Brescia, Via Valotti 9, I Brescia, Italy giorgi@ing.unibs.it A three-dimensional model describing the behavior of a ferromagnetic material is proposed. Since the passage between paramagnetic and ferromagnetic state at the Curie temperature is a second order phase transition, the phenomenon is set in the context of the Ginzburg-Landau theory, identifying the order parameter with the magnetization M. Accordingly, M satisfies a vectorial Ginzburg-Landau equation, interpreted as a balance law of the internal order structure. Thermoelectromagnetic effects are included by coupling Maxwell s equation and the appropriate representation of the heat equation. The resulting differential system is completed with initial and boundary conditions and its well-posedness is proved. Keywords: Ferromagnetic materials, phase transitions, well-posedness. 1. Introduction The model we propose describes the paramagnetic-ferromagnetic transition induced by variations of the temperature. The phenomenon of ferromagnetism occurs in some metals like iron, nickel, cobalt, when an external magnetic field yields a large magnetization inside the material, due to the alignment of the spin magnetic moments. Below a critical value θ c of the temperature, called Curie temperature, even if the external magnetic field is removed, the spin magnetic moments stay aligned. However, when the temperature overcomes the threshold value θ c, the material returns to the

2 2 paramagnetic state, since the residual alignment of the spins disappears. The passage from the paramagnetic to the ferromagnetic state can be interpreted as a second order phase transition according to Landau s terminology (see Refs. 2,7), since no latent heat is released or absorbed. Therefore, we set the phenomenon into the general framework of the Ginzburg-Landau theory. The first step is the identification of a suitable order parameter. Indeed, each phase transition can be interpreted as a passage from a less ordered structure to a more ordered one and vice versa. In this context, the order parameter is the variable measuring the internal order of the material. In ferromagnets, the internal order is given by the alignment of the spin magnetic moments. Accordingly, the order parameter can be identified with the magnetization vector, which means that the internal order is due both to the number of the oriented spin and to their direction. The vectorial structure of the order parameter leads to a three-dimensional model for ferromagnetism. Like other phase transitions models, the magnetization satisfies a Ginzburg-Landau equation which can be interpreted as a balance law of the internal order (see Ref. 4). Moreover, since the transition is induced by variations of the temperature we couple an appropriate representation of the heat equation deduced from the energy balance and consistent with the principles of thermodynamics. Finally, the evolution equations are completed with Maxwell s equations describing the variations of the electromagnetic field inside the ferromagnet. For the resulting differential system, endowed with boundary and initial conditions, existence and uniqueness of solutions have been proved (see Ref. 1). 2. Evolution equations We denote by R 3 the domain occupied by the ferromagnetic material. Maxwell s equations rule the variations of the electromagnetic field inside E = Ḃ, H = Ḋ + J, (1) B =, D = ρ e, (2) where E, H, D, B, J, ρ e are respectively the electric field, the magnetic field, the electric displacement, the magnetic induction, the current density and the free charge density. We assume the electromagnetic isotropy of the material and the following constitutive equations D = εe, B = µh + M, J = σe, (3)

3 3 where ε, µ, σ are respectively the dielectric constant, the magnetic permeability, the conductivity, and M is the magnetization vector. Moreover, as usual in ferromagnetic models, we will neglect the displacement current εė. Accordingly, from (1) (3) we deduce µḣ + Ṁ = 1 H, (µh + M) =. (4) σ In order to model the paramagnetic-ferromagnetic transition at the Curie temperature, we have to specify the law relating the variation of the magnetization to the magnetic field. Like other macroscopic models for phase transitions, this relation can be deduced by a balance law of the order inside the material. In particular, the order of the internal structure in a ferromagnetic material is due both to the position and to the direction of the magnetic spins. This leads to the choice of the magnetization vector as the variable which measures amount of order of the internal structure. Two different approaches are allowed. A possible choice 1 consists in identifying the order parameter of the transition with the components of the magnetization M. A different model 4 is based on the decomposition M = ϕm, ϕ, m = 1. (5) In the first case the phase transition is characterized by a vector valued order parameter. Thus, we define the internal structure order as the vectorvalued measure whose representation is K i (A) = ρk dx, A, A where ρ is the mass density. The vector k, called specific internal structure order, accounts for the internal order of the spins determined by their orientation. Moreover, we assume a representation of the external structure order vector in the form K e (A) = Pn A ds + ρσ dx, A, A A where A denotes the boundary of A and n A is its unit outward normal vector. Here P is a second order tensor such that Pn A provides the specific flux of the structure order through A and σ is the structure order supply. In particular, the tensor P(x, t) describes the distribution of the internal order in a neighborhood of x at the instant t, while σ represents a source of internal order inside the domain. In the sequel, we assume σ =. Moreover, we suppose that the mass density is constant and for simplicity we let ρ = 1.

4 4 The structure order balance on every sub-body A reads K i (A) = K e (A), from which we obtain a.e. in the local equation k = P. (6) The previous equation gives the relation between the magnetization and the magnetic field, provided that constitutive equations for k and P are chosen. We assume k = γṁ + θ cf ( M ) + θg ( M ) H (7) P = ν M, (8) where γ, ν are positive constants and θ > is the absolute temperature. Since the paramagnetic-ferromagnetic transition is a typical second order phase transition, a possible choice of the functions F, G is (see Ref. 3) F ( M ) = 1 4 M M 2, G( M ) = 1 2 M 2. (9) With this assumption, the function W = θ c F + θg admits the minimum M = when θ θ c. Conversely, below the Curie temperature, the minimum is attained when the modulus of the magnetization is M = θ c θ θ c (, 1). As we will see later, W represents the part of the free energy depending on the magnetization. Therefore, the minimum M of the free energy is such that M continuously as θ θ c, which is the typical situation occurring in second-order phase transitions. It is worth noting that the dependence of F, G on M implies that these functions characterize the transition in isotropic ferromagnets, when the magnetization has no preferred direction at equilibrium. Substitution of (7)-(9) into (6) leads to the Ginzburg-Landau equation γṁ = ν M θ c( M 2 1)M θm + H. (1) The model based on the decomposition (5), requires the introduction of two distinct evolution equations for the modulus ϕ and the direction m of the magnetization. The variable ϕ, identified with the order parameter, can be interpreted as a phase field, and it satisfies the scalar Ginzburg-Landau equation γ ϕ = ν ϕ θ c F (ϕ) θg (ϕ) + H m. (11)

5 5 Such equation can be deduced by a balance law on the internal structure, written in the local form as where, like in the previous model k = p, k = γ ϕ + θ c F (ϕ) + θg (ϕ) H m p = ν ϕ. Equation (11) needs to be coupled with an evolution equation for the versor m. For example, in the classical Landau-Lifshitz model, 6 m satisfies ṁ = τm H λm (m H), τ, λ >, (12) where the term τm H allows to explain the precession of m around H, and λm (m H) accounts for dissipation. Henceforth, we consider the model based on equations (4), (1). Similar arguments can be used to deal with the case described by (4), (11), (12). In order to provide a coherent model able to include both thermal and electromagnetic effects, it is essential to deduce the appropriate representation of the heat equation. As known, the thermal balance law is expressed by the equation h = q + r, where h is the rate at which heat is absorbed, q is the heat flux vector and r is the heat source. Moreover, denoting by e the internal energy, from the first law of thermodynamics, we deduce ė = P i + h, where P i is the internal power. Hence ė = P i q + r. (13) The latter provides the heat equation governing the evolution of the temperature, once we assume constitutive equations for e, q and give the representation of the internal power P i. By following the approach proposed in Ref. 4, equation (6) is regarded as a field equation able to yield a power (and then an energy variation) P i M, related the internal structure order of the material. Therefore, by keeping the electromagnetic effects into account, the internal power is given by the sum P i = P i M + P i el,

6 6 where Pel i is defined as P i el = Ḋ E + Ḃ H + J E = εė E + µḣ H + Ṁ H + σ E 2. The representation of the internal power PM i, is obtained by multiplying (6) by Ṁ, i.e. k Ṁ + ν M Ṁ = ν ( MT Ṁ), where the superscript T denotes the transpose of a tensor. The previous identity leads to the identification of the internal and external power as P i M = k Ṁ + ν M Ṁ, Pe M = ν ( M T Ṁ). Therefore, (7) and (9) imply P i M = γ Ṁ 2 + ν M Ṁ + [θ c( M 2 1) + θ]m Ṁ H Ṁ. Now, we choose the following constitutive equations for the internal energy and the heat flux e = c(θ) ε E µ H 2 + θ c 4 ( M 2 1) 2 + ν 2 M 2 (14) q = k(θ) θ, k(θ) >. (15) Substitution into (13) leads to the evolution equation for the temperature c (θ) θ σ E 2 γ Ṁ 2 θm Ṁ = (k(θ) θ) + r. (16) Let us prove the consistence of this model with the second law of thermodynamics, written by means of the Clausius-Duhem inequality as ( q ) η + r θ θ, where η is the entropy density. By introducing the free energy ψ = e θη and using (13), from the previous inequality we obtain ψ + η θ + q θ θ Pi. Substituting the expression of the internal power we deduce ψ + η θ + q θ θ γ Ṁ 2 ν M Ṁ [θ c ( M 2 1) + θ]m Ṁ εė E µḣ H σ E 2. (17) This inequality leads to the following representations ψ = µ 2 H 2 + ε 2 E 2 + ν 2 M 2 + θ c 4 ( M 2 1) 2 + θ 2 M 2 + α(θ) (18) η = ψ θ = 1 2 M 2 α (θ), (19)

7 7 where α satisfies the equation α(θ) θα (θ) = c(θ). Substitution of (15), (18), (19) into (17) provides the reduced inequality k(θ) θ θ 2 σ E 2 γ Ṁ 2, which holds along any process and proves the thermodynamical consistence of the model. In order to specialize the differential equation describing the evolution of the temperature, we suppose that heat conductivity and specific heat depend on θ according the polynomial laws k(θ) = k + k 1 θ, c(θ) = c 1 θ + c 2 2 θ2, k, k 1, c 1, c 2 >. In addition, we restrict our attention to processes for which the fields E, Ṁ, θ are small enough so that the quadratic terms E 2, Ṁ 2, θ 2 are negligible with respect to other contributions in (16). After dividing by θ, the energy balance (16) reduces to c 1 t (ln θ) + c 2 θ M Ṁ = k (ln θ) + k 1 θ + ˆr and for the sake of simplicity, we assume that ˆr = r/θ is a known function of x, t. 3. Well posedness of the problem In the previous section we have deduced the differential equations ruling the evolution of the ferromagnetic material, namely γṁ = ν M θ c( M 2 1)M θm + H (2) c 1 t (ln θ) + c 2 θ M Ṁ = k (ln θ) + k 1 θ + ˆr (21) µḣ + Ṁ = 1 σ H (22) (µh + M) =. (23) This system is completed by the initial and boundary conditions M(x, ) = M (x), θ(x, ) = θ (x), H(x, ) = H (x), x (24) M n =, θ n =, ( H) n =, on. (25) Concerning the boundary conditions, we have assumed that the magnetization satisfies the Neumann boundary condition typical of phase transitions.

8 8 The same homogeneous Neumann condition is required for the temperature. Finally, we observe that, in the quasi-steady approximation, Maxwell s equation (1) 2 reduces to σe = H, so that (25) 3 implies the continuity of the tangential component of the electric field across. In view of (1) 1 and (23), if we impose the following conditions on the initial data (µh + M ) n =, (µh + M ) =, (26) then at any subsequent t > (see Ref. 5) hold. (µh + M) n =, and (µh + M) = a.e. in. Definition 3.1. A triplet (M, θ, H) such that M L 2 (, T, H 2 ()) H 1 (, T, L 2 ()) θ L 2 (, T, H 1 ()), θ >, ln θ L 2 (, T, H 1 ()) c 1 ln θ + c 2 θ H 1 (, T, H 1 () ) H L 2 (, T, H 1 ()) H 1 (, T, H 1 () ) satisfying (24)-(25) almost everywhere, is a weak solution of (2)-(25) if γṁ + ν M + [θ c( M 2 1) + θ]m H =, a.e. in [ c 1 t (ln θ)ω + c 2 θω M Ṁω + k (ln θ) ω ] +k 1 θ ω ˆrω dx = (µḣ w + Ṁ w + 1σ ) H w dx = µ H + M =, a.e. in for any ω, w H 1 () and a.e. t (, T ). Existence of weak solutions is proved by introducing a suitable approximation of the logarithmic nonlinearities. More precisely, we denote by ln ε the Yosida approximation of the logarithm function defined as ln ε τ = τ ρ ε(τ), τ R, ε

9 9 where ρ ε (τ) is the unique solution of equation ρ ε (τ) + ε ln ρ ε (τ) = τ. Then we consider the approximated problem obtained by replacing into (21) the logarithm function with its regularization ln ε. Definition 3.2. A triplet (M ε, θ ε, H ε ) such that M ε L 2 (, T, H 2 ()) H 1 (, T, L 2 ()) θ ε L 2 (, T, H 1 ()), c 1 ln ε θ ε + c 2 θ ε H 1 (, T, H 1 () ) ln ε θ ε L 2 (, T, H 1 ()) H ε L 2 (, T, H 1 ()) H 1 (, T, H 1 () ) and satisfying (24)-(25) almost everywhere, is a weak solution of the approximated problem if Ṁ ε + ν M ε + [θ c ( M ε 2 1) + θ]m ε H ε =, a.e. in (27) [ c 1 t (ln ε θ ε )ω + c 2 θε ω M ε Ṁ ε ω + k (ln ε θ ε ) ω ] +k 1 θ ε ω ˆrω dx = (28) ( µḣε w + Ṁε w + 1 ) σ H ε w dx = (29) µ H ε + M ε = a.e. in (3) for any ω, w H 1 () and a.e. t (, T ). Lemma 3.1. Let ˆr L 2 (, T, H 1 () ) and M H 1 (), θ, H L 2 (), such that (26) hold. For every ε > and T >, there exists a solution (M ε, θ ε, H ε ) in the sense of Definition 3.2. Proof. The proof is based on the Galerkin procedure. The advantage of introducing a family of approximating problems stands in the properties of the Yosida regularization. The details of the proof can be found in Ref. 1. Existence of solutions to (2)-(25) is proved, by showing that the solutions (M ε, θ ε, H ε ) of the approximated problem converge to a solution (M, θ, H) of the original problem as ε. In the sequel we denote by the usual L 2 () norm.

10 1 Theorem 3.1 (Existence). Let ˆr L 2 (, T, H 1 () ) and M H 1 (), θ, H L 2 (), such that (26) hold. For every T >, problem (2)-(25) admits a solution (M, θ, H) in the sense of Definition 3.1. Proof. We test (27) with Ṁε, (28) with θ ε, (29) with H ε. Adding the resulting equation and integrating over (, t), we obtain 1 [ν M ε 2 + c 1 I ε (θ ε )dx + c 2 θ ε 2 + µ H ε 2 + θ ] c 2 2 M ε 4 L 4 t + [γ Ṁε 2 + k 1 θ ε 2 + 1σ ] H ε 2 + k ln ε(θ ε ) θ ε 2 dx dτ C + c 1 I ε (θ ) dx, (31) 2 where C > depends on the norms of the initial data and I ε (τ) = τ s ln ε(s)ds = τ ln ε (τ) τ ln ε (s)ds, τ R. In order to prove an estimate independent of ε, we exploit the inequality ln ε τ ln τ, τ > which leads to θ I ε (θ ) θ ln ε θ + ln ε s ds 2θ. 2 Thus, from (31) we deduce t [ M ε 2 H + θ ε 2 + H 1 ε 2 + Ṁε 2 + θ ε 2 + H ε 2] dτ C. Now let us test (28) with c 1 ln ε (θ ε ) + c 2 θ ε thus obtaining 1 d 2 dt c 1 ln ε θ ε + c 2 θ ε 2 + k c 1 2 (ln ε θ ε ) 2 + k 1 c 2 θ ε 2 ( λ c 1 ln ε θ ε + c 2 θ ε 2 + M ε 2 H 1 Ṁε 2 + ˆr 2 H 1 () + θ ε 2), (32) where λ is a positive constant. By recalling (32), Gronwall s inequality implies c 1 ln ε θ ε + c 2 θ ε 2 + and comparison with (28) yields t t (ln ε θ ε ) 2 dτ C c 1 t ln ε θ ε + c 2 θε 2 H 1 () dt C.

11 11 The previous a priori estimates allow to pass to the limit as ε into (27)-(3) and to obtain a solution (M, θ, H) of (2)-(25), satisfying the inequality t ( M 2 H + 1 θ 2 + H 2 + Ṁ 2 + θ 2 + H 2) dτ C. Theorem 3.2 (Uniqueness). Let (M, θ, H ) H 1 () L 2 () L 2 () satisfying (26) and ˆr L 2 (, T, H 1 () ). Then problem (2)-(24) admits a unique solution (M, θ, H). Proof. Let (M 1, θ 1, H 1 ), (M 2, θ 2, H 2 ) be two solutions, with the same initial data and sources. We introduce the differences M = M 1 M 2, θ = θ 1 θ 2, ξ = ln θ 1 ln θ 2, H = H 1 H 2, which solve the differential problem γṁ = ν M θ c( M 1 2 1)M 1 + θ c ( M 2 2 1)M 2 θ 1 M 1 + θ 2 M 2 + H (33) c 1 ξ + c2 θ = M1 Ṁ 1 M 2 Ṁ 2 + k ξ + k 1 θ (34) µḣ + Ṁ = 1 H. (35) σ By integrating (34) and (35) over (, t), we obtain c 1 ξ + c 2 θ = 1 2 ( M 1 2 M 2 2 ) + µh + M + 1 σ t t (k ξ + k 1 θ) dτ = (36) Hdτ =. (37) Let us multiply (33) by M, (36) by k ξ + k 1 θ and (37) by H. Adding the resulting equations and integrating over, we deduce [ 1 d γ M t 2 t ] 2 2 dt σ Hdτ + (k ξ + k 1 θ) dτ +ν M 2 + µ H 2 + (c 1 ξ + c 2 θ)(k ξ + k 1 θ)dx J 1 + J 2, (38) where [ J 1 = θc ( M 1 2 1)M 1 + θ c ( M 2 2 ] 1)M 2 θ 1 M 1 + θ 2 M 2 Mdx J 2 = 1 M (M 1 + M 2 )(k ξ + k 1 θ)dx. 2

12 12 By means of Young s, Hölder s inequalities and the embedding inequalities w L p C w H 1, p = 1,..., 6, w H 1 (), w L C w H 2, w H 2 (), the integrals J 1, J 2 can be estimated as J 1 + J 2 ν 2 M 2 + µ 2 H (k c 1 ξ 2 + k 1 c 2 θ 2 ) + ζ(t) M 2, where ζ(t) = C(1 + θ 1 (t) 2 H 1 + M 1(t) 2 H 2 + M 2(t) 2 H 2). Substitution into (38) and Gronwall s inequality prove M =, θ =, H =. References 1. V. Berti, M. Fabrizio and C. Giorgi, J. Math. Anal. Appl. 335, 661 (29). 2. M. Brokate, J. Sprekels Hysteresis and phase transitions (Springer, New York, 1996). 3. M. Fabrizio, Internat. J. Engrg. Sci. 44, 529 (26). 4. M. Fabrizio, C. Giorgi, A. Morro, Math. Methods Appl. Sci. 31, 627 (28). 5. M. Fabrizio, A. Morro, Electromagnetism of Continuous Media (Oxford University Press, 23). 6. L.D. Landau, E.M. Lifshitz, Phys. Z. Sowietunion 8, 153 (1935). 7. L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Electrodynamics of continuous media (Pergamon Press, Oxford, 1984).