1.1.2 Basic Thermodynamics for magnetized media. Thermodynamic potentials

We consider now a small volume $dV$ of magnetic material which is subject to an external magnetic field $\mathbf{H}_$a and is in contact with a thermal bath at constant temperature $T$. We introduce the quantity $\mathcal{M}=\textbf{M}dV$ such that $\mu_0\mathcal{M}$ is the net magnetic moment present in the volume $dV$. We assume that no volume changes due to thermal expansion and magnetostriction occur. The First Law of thermodynamics states that for any transformation between two equilibrium states $A$ and $B$, it happens that:

$$\Delta U=U_B-U_A=\Delta L + \Delta Q \quad,$$ (1.3)

where $\Delta U$ is the variation of the internal energy $U$, $\Delta L$ is the work performed on the system and $\Delta Q$ is the heat absorbed by the system. The magnetic work, under constant external magnetic field $\mathbf{H}_$a, has the following form:

$$\Delta L=\mu_0\mathbf{H}_ \cdot \Delta\mathcal{M}\quad.$$ (1.4)

The Second Law of thermodynamics for isolated systems states that, for any transformation between equilibrium states $A$ and $B$, the following inequality is satisfied [7]:

$$\Delta S=S_B-S_A\geq 0 \quad,$$ (1.5)

where $S$ is the entropy. In Eq. (1.5) the equal sign holds in case of reversible transformations. In this respect, reversible transformations occur when the system passes through a sequence of thermodynamic equilibrium states. The second law (1.5) has to be interpreted as follows. Referring to our magnetic body, let us imagine that it is prepared in a certain initial state $A$ by using appropriate constraints which allow to keep fixed, for instance, the magnetic moment of the body. Then, the constraints are partially or totally removed and the system is left isolated (no work, no heat is exchanged with the system). In this situation, the system relaxes toward a new equilibrium state $B$, and therefore the magnetic moment approaches a new value too. The remarkable fact is that the new equilibrium state $B$ will be necessarily characterized by a value of the entropy $S_B$ greater than $S_A$. The Second Law of thermodynamics can be also written for non-isolated systems in the following way [7]:

$$\Delta S\geq \frac{\Delta Q}{T} \quad.$$ (1.6)

where the equal sign still holds in case of reversible transformations. Moreover, to study transformations occurring at constant temperature, appropriate thermodynamic potentials can be introduced. For instance, the Helmholtz free energy $F(\mathcal{M},T)$ can be defined by means of suitable Legendre transformation [6]:

$$F=\min_S [U-TS] \quad.$$ (1.7)

The inequality (1.6) leads to suitable inequality involving the Helmholtz free energy $F$. In fact, for constant temperature, the variation of $F$ between two equilibrium states $A$ and $B$ can be written as:

$$\Delta F=\Delta U-T \Delta S \quad.$$ (1.8)

Now, by taking into account that $T \Delta S\geq \Delta Q$, according to the second law (1.6), and the first law (1.3), one obtains:

$$\Delta F\leq \Delta L \quad.$$ (1.9)

where the equal sign holds for reversible transformations. In addition, if no work is done on the system, the latter equation becomes

$$\Delta F=F_B-F_A\leq 0 \quad,$$ (1.10)

meaning that, if the system is prepared in a certain equilibrium state with certain constraints, the removal of constraints implies that the Helmholtz free energy has to decrease towards a minimum. Another thermodynamic potential is the Gibbs free energy $G(\mathbf{H}_$a$,T)$, which, for constant temperature and constant external field $\mathbf{H}_$a can be written as [6]:

$$G=\min_\mathcal{M}[F-\mu_0\mathcal{M}\cdot\mathbf{H}_ ] \quad.$$ (1.11)

By following very similar line of reasoning to the one done for the Helmholtz free energy, one can easily derive that, for constant external field and temperature, the transformation between equilibria $A$ and $B$, induced by the removal of the constraints, satisfies the following inequality:

$$\Delta G=G_B-G_A\leq 0 \quad,$$ (1.12)

meaning that also the Gibbs free energy has to decrease towards a minimum. The Gibbs free energy is very useful as far as experiments are considered where one can somehow control the external field, since it is instead very difficult controlling the magnetic moment $\mu_0\mathcal{M}$. In addition, for reversible transformations at constant temperature, one can easily derive that:

$$dF=\delta L=\mu_0\mathbf{H}_ \cdot \delta\mathcal{M}\quad,$$ (1.13)

$$dG=-\mu_0\mathcal{M}\cdot d\mathbf{H}_ \quad.$$ (1.14)

This leads to the following relationship holding for equilibrium states:

$$\frac{1}{\mu_0}\left[\frac{\partial F}{\partial \mathcal{M}}\right]_T=\mathbf{H}_ \quad,\quad \left[\frac{\partial G}{\partial \mathbf{H}_\text{a}}\right]_{T}=-\mu_0\mathcal{M}\quad.$$ (1.15)

We observe that the Gibbs free energy (1.11) depends by definition only on $(\mathbf{H}_$a$,T)$. This means that the value of $\mathcal{M}$ has to be expressed through the equation of state:

$$\mathcal{M}=\mathcal{M}(\mathbf{H}_ ,T) \quad,$$ (1.16)

which is well-defined at thermodynamic equilibrium. In other words, at thermodynamic equilibrium, for given $(\mathbf{H}_$a$,T)$, the state variable $\mathcal{M}$ is uniquely determined. If we consider now the case of a ferromagnetic body, this property is not fulfilled anymore, that is, a given value of $(\mathbf{H}_$a$,T)$, is not sufficient to determine uniquely the state variable $\mathcal{M}$. In fact, we deal with a system whose free energy has many local minima corresponding to metastable equilibria [20]. This framework is known as non-equilibrium thermodynamics and is not yet well-established from theoretical point of view. Nevertheless, many contributions in this sense have been developed. In this respect, the presence of many metastable state can be taken into account, as result of a deeper analysis in the framework of non-equilibrium thermodynamics, by the following generalized Gibbs^1.1 free energy

$$G(\mathbf{H}_ ,T,\mathcal{M})=F(\mathcal{M},T)-\mu_0\mathbf{H}_ \cdot\mathcal{M}\quad.$$ (1.17)

We observe that the free energy (1.17) coincides with the Gibbs free energy (1.11) at thermodynamic equilibrium. The explicit dependance on $\mathcal{M}$ expresses somehow the distance of the system from thermodynamic equilibrium when the state variable assumes the particular value $\mathcal{M}$, as if it were an external constraint. In this framework, one can determine the (metastable) equilibrium condition by imposing that the free energy (1.17) is stationary^1.2 with respect to $\mathcal{M}$:

$$\left[\frac{\partial G}{\partial \mathcal{M}}\right]_{\mathbf{H}_... ...\partial F}{\partial \mathcal{M}}\right]_{T}-\mu_0\mathbf{H}_\text{a}=0 \quad.$$ (1.18)

In the latter equation, the first of Eqs. (1.15) has been used. It is important to underline that, from this analysis, one cannot say which metastable state the system will reach, given an initial state. The only way to determine this information is to introduce dynamics. Therefore, an appropriate dynamic equation must be considered to describe the evolution of the system. The above considerations can be extended to the case of an inhomogeneous system, where the state variables and the thermodynamic potentials are also space-dependent functions, under the hypothesis that the body is in local thermodynamic equilibrium^1.3. Therefore, the state functions can be well-defined within each elementary volume and thermodynamic relations which are valid for homogeneous bodies, can be written point-wise as balance equations. Moreover, the thermodynamic potentials become functionals of the state variables which, in turn, are space functions. In the following sections we analyze the contributions to the free energy functional for ferromagnetic bodies. In this respect, the role of the state variable will be played by the magnetization vector field $\textbf{M}$ and the equilibrium condition will be computed by imposing that the variational derivative of the free energy functional $G(\textbf{M},\mathbf{H}_$a$)$, with respect to $\textbf{M}$, vanishes according to Eq. (1.18). Finally, in section 1.3 we shall introduce the appropriate dynamic equation which is necessary to describe the evolution of the system, as seen before.