Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials

A general theory of free energies for dielectric materials under isothermal conditions was given recently [1]. The present work presents more detailed formulae and results for free energies in non-magnetic isotropic materials. We shall derive expressions for the minimum, maximum and other intermediate free energy functionals which also have an extremum property. Two approaches are possible: (1) To present the developments as in mechanics [2], based explicitly on thermodynamics; and (2) To use the method given independently [3-5] for dielectric materials with memory, but with quite different methods, notation and terminology. In this work, as in [1], the fist approach will be adopted but with reference to the connection between the two methods. The intermediate free energies are analogous to quantities known in mechanics [2]. They are new and particularly physically relevant in the context of dielectrics, in that they apply to memory models that are standard for such materials, but not usually applied to viscoelastic behaviour.


Introduction
A general theory of free energies for dielectric materials under isothermal conditions was given recently [1]. The present work presents more detailed formulae and results for free energies in non-magnetic isotropic materials. We shall derive expressions for the minimum, maximum and other intermediate free energy functionals which also have an extremum property. Two approaches are possible: (1) To present the developments as in mechanics [2], based explicitly on thermodynamics; and (2) To use the method given independently [3][4][5] for dielectric materials with memory, but with quite different methods, notation and terminology. In this work, as in [1], the fist approach will be adopted but with reference to the connection between the two methods. The intermediate free energies are analogous to quantities known in mechanics [2]. They are new and particularly physically relevant in the context of dielectrics, in that they apply to memory models that are standard for such materials, but not usually applied to viscoelastic behaviour.
Applications of the results of this work, from a physical point of view, include the establishment of bounds on the level of dissipation in the material, using the fact that the total dissipation associated with the minimum(maximum) free energy is an upper(lower) bound on the actual physical dissipation. On the matter of notation, vectors and tensors are denoted by lowercase and uppercase boldface characters respectively and scalars by ordinary script. The real line is denoted by  , the non-negative reals by +  and the strictly positive reals by ++  . Similarly, −  is the set of non-positive reals and −−  the strictly negative reals.
Complex quantities arise in the frequency domain so we have complex vector spaces for which the dot product involves using the complex conjugate of objects in the dual space. The magnitude squared, denoted by 2 . refers to the dot product of objects with their complex conjugates.
Let ( ) t ψ be any free energy of the material, for the isothermal case, and D(t) the rate of dissipation. Then the first and second laws of thermodynamics can be written as [1] . Equation (3.2) gives the history and also the current value E(t) of the electric field. A given future continuation is denoted by (3.5) The relative history E t r is defined by (3.6) A relative future continuation is also defined by (3.6) We define the equilibrium free enthalpy ( ) e t  to be that given for the static history

A. Required properties of a free enthalpy
Let us state the characteristic properties of a free enthalpy, provable within a general framework [1,[6][7][8]: These will be referred to as the Graffi conditions by analogy with those for a free energy in mechanics ( [2] for example).

A Linear Memory Model
A special case of the linear model introduced in [1] is described in this section. This is the material consid-ered in [3][4][5], namely a passive, homogeneous isotropic non-magnetic dielectric. All kernels are scalar quantities. We have ( ) ( ) ( ) ( ) ( ) where (4.1b) follows from (4.1a) by partial integrations. Also (4.5d) The quantity G(s) is the relaxation function of the material. In writing the final form of (4.4), we are assuming where, using (4.4b), one sees that the polarization P(t) is given by the causal relationship Then, (4.11a) becomes is the accumulation of energy (density) transferred to the medium at the point under consideration, from the beginning of the pulse-medium interaction until time t. We can write (4.12) as is given by the integral term in (4.9a).

A. Minimal states
The fundamental definition of the state of a material with memory at time t is the history of the independent field variable and its current value (E t , E(t)). Also, different histories may be members of the same minimal state [1]. Two states A fundamental distinction between materials is that for certain relaxation functions, namely those with only isolated singularities (in the frequency domain), the set of minimal states is non-singleton, while if some branch cuts are present in the relaxation function, the material has only singleton minimal states [2,9]. For relaxation functions with only isolated singularities, there is a maximum free energy that is less than the work function W(t) and also a range of related intermediate free energies, which are discussed later. On the other hand, if branch cuts are present, the maximum free energy is W(t).
In this work, we will deal exclusively with the case where the relaxation function has only isolated singularities, so that the minimal states will be non-singleton.
For such materials, the free energy functional is positive semi-definite ( [2], page 152).
Note that the statement that are equivalent is the same as the assertion that ( ) that it goes to zero sufficiently strongly so that various integrals, introduced below, exist. The quantity G ∞ is related to the relaxation function through We deduce from (3.5), (4.1a) and the time derivative of (4.1b) that where subscripts refer to partial differentiation with respect to the first or second argument. Equation (4.7b) follows from (4.7a) by partial integrations, using (4.1a), (4.3), and the fact that The integral terms, with the negative signs included, are non-negative [1]. The total work done by the electric field up to time t is Under the assumption that ( ) = 0 is the total dissipation up to time t. We take 0 G to be equal to the vacuum permittivity 0 ∈ [3][4][5], giving We can write the Fourier transforms of ( ) The quantity ( ) χ ω + is the susceptibility, denoted by ( ) χ ω in [3][4][5]. By partial integration, one can show that giving, in particular, that Using (5.2), we see that It follows from (5.11) that 0 G G ∞ > [1]. We have

B. The complex frequency plane and the function H(ω)
We will be considering frequency domain quantities, defined by analytic continuation from integral definitions, as functions on the complex ω plane, denoted by Similarly, Ω − and Ω (−) are the lower half-planes including and excluding the real axis, respectively. G ω ′ has singularities in both Ω (+) and Ω (−) that are mirror images of each other. It goes to zero at the origin and must also be analytic there. A quantity central to our considerations is defined by It is a non-negative, even function of the frequency equivalent to the zero state (0,0), where 0 is the zero in 3  (and also the zero history), while A functional of (E t , E(t)) which yields the same value for all members of the same minimal state will be referred to as a functional of the minimal state (abbreviated to FMS) or as a minimal state variable.
E , E t t be any equivalent states.
Then, a free energy is a functional of the minimal state if It is not necessary that a free energy have this property, though it holds for the minimum and all related free energies introduced later.

Kernels and Field Variables in the Frequency Domain
For any Only real valued functions will be considered so that Where the bar denotes the complex conjugate. For f ω are respectively the Fourier cosine and sine transforms. A property of Fourier transforms which will be used later is where U(t) and ( ) 0 t φ are defined by (4.10). We see from (6.1) that W(t) can be cast in the form (4.9) by putting ( ) ( ) 12 12 , In terms of frequency domain quantities, we find that The quantity W(t)-D(t)·E(t) can be shown to obey the properties specified in subsection III A of a free enthalpy, with zero dissipation. Because of the vanishing dissipation, it must be the maximum free energy associated with the material or greater than this quantity, an observation which follows from (4.12). For relaxation functions with only isolated singu-larities, as introduced later, there is a maximum free energy which is a functional of the minimum state and is less that W(t).
Consider the scalar product on the function space of electric fields, defined by [2,5,[9][10][11][12] Recalling (5.1), we see that the quantity Also, the subscript F in the bracket notation indicates that the frequency domain version is being used. The related norm is (6.5) In the notation of (6.5), the work function is given by

Factorization of H (ω)
We consider materials such that has only a finite number of isolated singularities in Ω (+) . Thus, H(ω) has only isolated singularities in Ω (±) , which are mirror images of each other in the real axis, as as- before (5.13)). This means that it can be put in the form of a ratio of polynomials. We will take the singularities to be a finite number of simple poles. The quantity H(ω) has a finite number of zeros in Ω ± (ω), also and goes to zero quadratically at the origin. The relation (see (5.5 For the model considered later, H(ω) goes to zero at large ω so that H∞ = 0. The Fourier transforms of the history and continuation are denoted by ( ) where the notation ω ± is defined as ω ± iα, α > 0. The parameter α is assumed to tend to zero after any integrations have been carried out (see for example [2], page 551). We have The second relation follows from (3.2) and an integration by parts in the Fourier integral defining ( ) Also, based on arguments from [1] and [2], page 146, for example, we can express the constitutive equation (4.4) in the form

The Work Function
Various forms for the work function, defined by (4.11), are given for the general tensor case in [1]. Those required in this work for scalar kernels are summarized here. The analogous mechanics version of these relations may be found in [2], page 153, for example. It can be shown that • pathways leading from the minimum to the maximum free energy.
The most general free energy and rate of dissipation arising from these factorizations is given by A particular case of this linear form is the free energy proposed in [11].
Remark VII.3. The set of all free energies at time t associated with a given material is denoted by Φ(t). The boundary of this set is where one or more of the fundamental properties listed in subsection III A is breaking down; an example, which will arise later, would be where, for a given non-zero history, the rate of dissipation is zero, so that a small shift in its kernel parameters results in it becoming negative.
All the ( ) f t ψ emerge from extremum arguments. This is apparent in the case of the minimum free energy for a given state, which is obtained by finding an optimal continuation ( ) E t m s , s − ∈  yielding the maximum recoverable work from this state. Also, the maximum free energy is determined by finding the optimal history achieve the given state. It must be an equivalent state to the given history. In other words, the two histories must be in the same minimal state. The intermediate free energies are also obtained from an extremum principle, but involving an optimization of the history/continuation E t (s), s ∈  , as we shall see in section VIII.
The relevant theoretical developments motivating the results of section VIII are presented in [2], chapter 15 for mechanics; they also apply to dielectrics. We will present an abbreviated version of these arguments, and also demonstrate that the functionals which emerge are in fact free energies, with the required properties. It can be shown that all these quantities, including the intermediate functionals, are on the boundary of Φ(t) for the material ( [2], page 365), which is consistent with the fact that they emerge from an extremum principle. This property is discussed briefly below. mirror images of each other. It is real and non-negative on  , an even function of ω and therefore a function of ω 2 , in view of its analyticity about the origin. It vanishes quadratically at the origin. This non-negative quantity can be factorized in general as outlined in [2,13]. Thus, we have that The factorization (7.1) is unique up to multiplication by a phase factor e iα , where α is a constant. Relation (7.2a) reduces this arbitrariness to multiplication by a factor ±1.
The quantity H + (ω) (H − (ω)) has all its singularities in Ω (+) (Ω (−) ) and all its zeros in Ω + (Ω − ) ( [2], page 239). There are many other factorizations, obtained by interchanging some or all of the zeros of H + (ω) and H − (ω), while retaining the same singularity structure. The different factorizations are labeled by the subscript or superscript f. We have Certain types of exchanges must be excluded to ensure that (7.3b) are true [11], ( [2], page 338). Each factorization generally yields a different free energy, though there may be exceptions. All these free energies are FMSs [2]. The factorization with no exchange of zeros, which is that given by (7.1), yields the minimum free energy ψ m (t).
Remark VII.1. Each exchange of zeros, starting from these factors, can be shown to yield a free energy which is greater than or equal to the previous quantity ( [2], page 363).
Note that the zeros of ( ) f H ω ± at the origin play no part in these exchanges. Note that there are several (indeed many, for a large number of isolated singularities) different zero exchange d The minimum and maximum free energies are given also in [5]. The intermediate free energies, introduced in this work, have not been discussed previously in the context of dielectrics, but are analogous to those derived in mechanics in for example [2,[10][11][12]. e This behavior, while often used to describe memory behavior of dielectrics, is not usually applied to viscoelastic materials. The singularity structure given by (9.4) corresponds to exponential decay with sinusoidal behavior in the time domain, while viscoelastic materials are generally modeled by simple exponential decay, which in the frequency domain yields simple poles on the positive imaginary axis. The optimization problem which yields the free energy functional associated with a given factorization of H(ω) is given as follows ( [2], pages 346, 350). We minimize The resultant free energy is where E t f is the history/continuation E t RF that is the solution of the constrained minimization problem. The formulation of this problem is discussed at length in [2], chapter 15 and earlier papers. In particular, it is shown that each ( ) f t ψ has all the required properties of a free energy, which will in any case be shown later for the present context.

The Free Energy Associated with a Particular Factorization
where the quantities ( ) are particular cases of (5.1c) and (5.1d). Consider the norm , defined by (6.5). This can be written in a form similar to the quantity W(∞) in [1], from which the minimum free energy was deduced. We have, from (6.4b) and (6.5), that Let us choose a particular factorization of H(ω), as given by (7.3). Also, we define [1] where the ( ) Let us further define Note that, by virtue of Cauchy's integral formula, (8.8) The work function, given by (6.3), takes the form [1] ( ) ( ) ( ) ( ) From (4.12), (8.17) and (8.22), we deduce that the total dissipation corresponding to the minimum free energy is given by Differentiating this relation with respect to t and using (8.18a), (8.19c), gives Since K f (t), given by (8.21) can vanish for non-zero histories, D f (t) is a positive semi-definite rather than a positive definite quadratic form, as noted in remark VIII.1.
We can re-express these results in terms of relative histories [1]. Let us put By closing the contour on Ω + , it emerges that singularities only in Ω − and its inverse Fourier transform is thus non-zero only on −  . This latter quantity is the negative of the optimal continuation, yielding the minimum free energy, which is discussed for the general tensor case in [1]. It can be shown that both the constraint (8.9) (8.9), for various factorizations. These constraints are transitional between the two extremes discussed above.
We will see that all of these cases result in free energies that are on the boundary of Φ(t), defined by remark VII.3. This property is apparent for the minimum and maximum free energies. For these and all intermediate cases, we will see that the corresponding rates of dissipation are positive semi-definite rather than positive definite quadratic forms. They are zero for certain non-zero histories so that small variations in their kernels may lead to negative rates of dissipation, for these histories, in contradiction to the second law, as given by (3.1).
Thus, the fundamental physical content of the constrained optimizations is that they lead to free energies on the boundary of Φ(t).
The free energy (8.10) has the form by virtue of (8.13) and (8.15). We have the following relations ( [1,10,13,14] for example) and F e (t) is given by (4.1c). It is easy to show that F f (t) obeys the Graffi conditions listed in subsection III A. Property P2 is immediately apparent. The relation (4.12) holds by virtue of (8.17), (8.22) and (8.23). The time derivative of (4.12) yields (3.1) with D f (t) given by (8.24), which is equivalent to P3. Property P1 can be proved with the aid of (8.35b), by showing that (8.37) Remark VIII.3. It was shown in [14] (see also [2], page 340) that ( ) ft q ω − , defined by (8.4), is a function of the minimal state in the sense defined after (4.21). This result transfers to the present context without alteration. From (8.

17), we deduce that ψ f (t) is a function of the minimal state as defined by (4.22).
For the minimum free energy ψ m (t), we denote ψ int (t), introduced in (4.18), as ψ rec (t), given by is defined by (8.4) for f = 1. This is the recoverable energy defined in [3]. The quantity ( ) m t D , given by (8.23) for f = 1, is that referred to as the irrecoverable energy in [3]. Corresponding quantities , for each f, may be similarly labeled.
The optimal history/continuation, given by the inverse transform of (8.1), can be shown to have a discontinuity at the origin which is infinite if H sr = 0, while at t = ±∞, it is non-zero ( [1,2], page 354).

A Detailed Dielectric Model
We have , ω ∈  , (9.1) by virtue of (5.11). As in [3][4][5], we take the susceptibility to be modeled by a sum of Lorentz oscillators e ( ) We now briefly present double integral forms of ψ f (t), D f (t) and D(t), which are analogous to (though simpler than) the corresponding formulae in [1] for the minimum free energy in the full tensor case. Using (8.17b) and (8.28a), we can write ψ f (t) in the forms . , Also, D f (t), given by (8.24), can be expressed as , 4 . , One finds that ( ) ( ) ( ) , , The fact that the integral term is non-negative implies that [1]  The free enthalpy F f (t) corresponding to ψ f (t) may be deduced from (3.4)  zeros may also coalesce. Indeed, we cannot exclude the possibility that some of the zeros in (9.8) have a power higher than unity, even if all the singularities are simple poles. For simplicity, it is assumed that this does not happen for our choice of parameters.
Let us define for l = 1, 2, . . ., 2N −2, In this notation, (9.6) becomes ( ) with the aid of (9.14), so that which are decaying exponentials multiplying sine functions [15], so that there will be oscillatory behavior superimposed on the exponential decay. From (9.2) and (9.1), we have , ω ∈  , (9.8b) where the denominator of (9.8a) uses the notation of (9.4) while the numerator is factored to yield the zeros of H(ω); these must occur also in pairs, as in (9.4), so that l l η η ′ = − for each l. We have explicitly included the fact that H(ω) vanishes quadratically at the origin. Note that the smaller number of zeros reflects the fact that H(ω) behaves as 2 ω − for large ω.
The factorization can be carried out by inspection. We obtain The most general case of a rational function, which is considered in [2,11], can be obtained from (9.8) by allowing singularities to coalesce. As a result of this, some h ω − at large ω.
Therefore, we can also write (9.26b), (9.28c) and (9.28d) as ( ) We identify also a much larger class of factorizations of H(ω), determined by interchanging particular β l in (9.26a) with l β in (9.28a). These different factorizations are labeled by the subscript or superscript f. Thus, Observe that (9.29) also holds for the f R , a property which has been used in writing (9.31c The matrix with components Γ ij must also be hermitian and non-negative. Let us now consider the free energies ψ f (t). We can put (9.9) in the form  e .e Solutions to this equation will exist for non-zero values of e i (t). Therefore, (9.45b) is a positive semi-definite rather than a positive definite quadratic form, so that the associated matrix will have some zero eigenvalues. If one of these zero eigenvalues were to become slightly negative, then the second law would no longer hold. Thus, the free energy ψ f (t) is on the boundary of Φ(t), defined in remark VII.3. This observation is of course a special case of that after (8.24), and relates to remark VIII.1.

A. Sinusoidal histories for non-magnetic materials
Let us consider the formulae (9.44) and (9.38) for histories  . Also, using (9.30), we see that the quantity ∆ f (ω 0 ), defined by (8.34), is given by Now, by virtue of (9.29), for the f i R , it follows that We deduce the relation Using the symmetry of ∆ f (ω 0 ) with respect to ω 0 , we find that which agrees with the constant term (proportional to |A| 2 ) in (9.50).

Summary
The main results are listed below.