2.1. Remarks on Non-Equilibrium Thermodynamics with Internal Variables
The irreversibility of natural phenomena is the main reason for the difficulties arising in state thermodynamic models. Nevertheless, this condition is described well by the entropy function that is one of the more important concepts in physics [
1]. Knowledge of the change of entropy allows us to separate physical processes into two classes: reversible and irreversible. The first is connected with no changes in entropy, while in the second entropy changes occur [
1]. So entropy may be considered a measure of the irreversibility of a process and, moreover, is related to an increase of disorder for an isolated system. The branch of physics that studies these subjects is non-equilibrium thermodynamics (NET) [
1,
2]. Another problem for state thermodynamic models is the non-linearity of phenomena. However, although almost all physical phenomena show non-linearity, in many cases linear approximation is a good compromise as it gives results in accordance with the phenomena. On the other hand, there are mathematical difficulties in dealing with non-linearities that cannot be overcome analytically, but only by means of numerical methods. In line with our research on the study of some biological phenomena by means of techniques developed in the context of NET, in this paper we will refer to a linear approximation since this proved to be successful in the study of biological phenomena [
13,
14,
15,
21,
22]. As mentioned above, entropy plays a fundamental role in the whole NET. Generally, the entropy
S is considered as function of
n extensive variables
Xi (
i = 1, 2, ...,
n) [
1,
2]
By introducing generalized forces:
called affinities, and related “fluxes”:
the entropy production can be written:
Generally, the functional dependence between fluxes and affinities can be very complex and assumes a non-linear form. Here we admit, supported by a large number of phenomena, that these relations are linear:
where
Mik are called phenomenological coefficients and it can be shown that they satisfy some symmetry relations [
1,
2]. Following Kluitenberg’s theory, we will introduce a particular form of relation (5) in following sections. Here, we will make clear why we consider Kluitenberg’s theory particularly suitable for the study of biological phenomena. Apart from the non-equilibrium status of biological phenomena, there are reasons that make the theory with internal variables fit for this purpose. These are related to the connection that can arise between internal variables and processes which occur inside biological tissues not being caused by external perturbation but only by internal phenomena. We do not go into the details of the theory, which can be found at ref. [
3,
7], but we focus our attention on internal variables. Moreover, no use is made of spring-dashpot models. The heat dissipation functions for ordinary viscous fluids and for Maxwell, Kelvin (Voigt), Poynting–Thomson, Jeffreys, Prandtl–Reuss, Bingham, Saint Venant, and Hooke media may be regarded as degeneracy of the more general expression that is derived [
3,
7]. Generally, the set of variables—strain tensor, internal energy, specific volume, entropy and temperature, for example—are sufficient to characterize the state of a thermoelastic medium or fluid. But, if more complicated phenomena occur as a chemical reaction, anelastic or plastic strain, dielectric and magnetic relaxation, the aforementioned set of variables is incomplete. For instance, if we consider a fluid mixture of
n-chemical components, each with a concentration
C(K) (
K = 1, 2, ...,
n), the local thermodynamic state is completely specified by internal energy u, the volume v and the additional scalar (macroscopic) thermodynamic variables
C(K) (
K = 1, 2, ...,
n) [
2]. Thus the entropy s is assumed:
In hematology there are several examples of this kind: the ratio albumin–globulin, leukocytosis, leukemia, leukopenia. A generalization of Equation (6) can be taken into account by considering a medium for which the entropy depends on the internal energy, the tensor of total strain, and some tensorial variables Ω
ik. We shall assume that Ω
ik is a macroscopic quantity which we need in order to give a complete description of the state of the medium. Without specifying the physical nature of Ω
ik (we shall call it a hidden tensorial variable), we assume that it influences the mechanical properties of the medium and that it is a symmetrical tensor field. Hence we assume that [
3,
7]:
where Ω
ik is the tensor of total strain. A further generalization from the aforementioned ideas is the assumption that there are several microscopic phenomena which influence the mechanical properties of the medium under consideration. Besides, the thermodynamic state of the medium may be described by the internal energy, total strain and “
n” macroscopic internal variables. This also includes the possibility of scalar internal variables. So Equation (7) assumes the form:
Before proceeding to describe the theory, we must recall our definition:
We define the entropy, the variables on which the entropy depends, the quantities which are obtained by partial differentiation of the entropy (such as the temperature) and the functions of these quantities (such as free energy) as thermodynamic variables. Thermodynamic variables on which entropy depends and from which substantial time derivatives occur in the first law of thermodynamics may be called external thermodynamic variables, because the values of these parameters can be prescribed by external influences. Additional variables on which the entropy depends, and from which substantial time derivatives do not occur, may be called internal thermodynamic variables.
It can be shown [
7] that the strain tensor can be split in two parts
, which we call the elastic part, and
, the inelastic part, respectively. So we have:
Moreover it can be shown that the change of both
and
contributes to entropy production and, therefore, they represent two irreversible processes. The inelastic deformation can be due to several internal processes that occur simultaneously. Let us suppose that there occur “
n” different types of microscopic phenomena giving rise to inelastic strain, and let us further assume that:
where
is the contribution to the inelastic strain of the
h-th microscopic phenomenon. It can be shown [
7] that the expression (8) specializes as:
where we assume that partial inelastic strain tensors
(
h = 1, 2, …,
n) are related to n different microscopic phenomena. From expression (11) it is seen that
(
h = 1, 2, …,
n) plays the role of internal variables. Just the expression (11) allows the possibility of studying particular processes that occur, for example, in cancer tissues, since we can correlate every partially inelastic strain to a tumor cell [
15]. The rheological properties of these tissues are altered and those mechanical changes are revealed as emergent properties at a macroscopic level. This is the case for leukocytosis, which is an abnormal increase in the number of white blood cells; leukemia which is a neoplastic proliferation of hematopoietic stem cells; or leukopenia, which is an abnormal reduction of circulating white blood cells, especially granulocytes. In these cases, the rheological properties of the blood change, and in particular this can change the partial inelastic strain associated to each anomalous phenomena (disease). Obviously, it is very difficult to investigate these pathologies by considering directly partial inelastic strain, because a direct measure of them is very hard. However, from a rheological point of view, these diseases may be investigated by means of ultrasound waves, as we shall show in the next sections for a particular case. Here, we will show a “technique” for approaching this type of investigation [
23,
24,
25].
2.2. Rheological Differential Equation
We assume that only one microscopic phenomenon occurs inside the medium. So the Equation (11) becomes:
From which the usual equations lead:
where
T is the temperature;
ρ the mass density;
is equilibrium stress tensor; and
is affinity stress tensor. The viscous stress tensor
can be introduced:
here,
is the stress tensor that occurs in indefinite equations. Now, assuming that the inelastic strain derives from only one microscopic phenomenon, it is possible to introduce this contribution as the internal degree of freedom in the Gibbs’ relation. This assumption and the first law of thermodynamics allow an explicit form of entropy production. By considering the scalar part
τ of the stress tensor and the scalar part
ε of the strain tensor, one has for the entropy production [
7]:
From the expression thus obtained, taking into account the usual procedure of non-equilibrium thermodynamics and assuming that the cross effect among viscous flow and inelastic flow are neglected, the following phenomenological equations can be obtained [
7]:
where
are phenomenological coefficients, and we shall assume that they are constant in time. The coefficient
(volume viscosity), which has the dimension of a viscosity, is connected to irreversible processes related to the change of
, while
, which has the dimension of a fluidity, is related to change of
and the corresponding intensive variable
. However, Equations (14) and (15) are connected with irreversible changes of the strain. These, together with linear state Equations [
7]:
lead to the so-called relaxation equation for trace
τ of the stress tensor and trace ε of the strain tensor:
where:
In which b(0,0) and b(1,1) are the state coefficients related to the elasticity and inelasticity phenomena, respectively. The importance of the phenomenological and state coefficients is that they characterize the medium specifying the amount of the type of phenomena correlate for each of them. It is important to observe that their constancy is related to the time for each type of perturbation that acts on the medium. However, they vary with the change of the perturbation. For example, if the perturbation is of a harmonic type with frequency ω, then the coefficients will depend on ω which can be considered as parameter in the functional dependence of the coefficients. In this case we shall call dynamical coefficients.
It can be proved that for a fluid (such as blood) it is reasonable to assume that
is constant for each element so as to verify the basic axioms on local and instantaneous equilibrium [
1]. Thus, we assume that the mass density
is constant. It is seen from Equation (17) that sudden change in
ε(1) is impossible, while from Equation (16) it follows that sudden change in
ε(0) is possible.
2.3. Remarks on Linear Response Theory
Here we will recall how to define the Complex Dynamic Modulus, because the introduction of its fractional form is the principal ambition of this work. Since we are studying relaxation phenomena, we assume as perturbation of the system an extensive variable
f(
t) (cause) and the relative intensive variable
g(
t) as response (effect). For a linear system, it can be shown that the following convolution relation is valid [
19]:
where:
From Equation (12), and taking in account convolution theorem, it follows:
where the symbol
is the Fourier transform. We have:
where
is the transfer function. From relations (20)–(23) one has:
and therefore:
where
is the inverse Fourier transform. If we take into account the hypotheses for which the relaxation Equation (18) is valid, and we consider a harmonic deformation as input [
18]:
where
is the amplitude of the oscillation and
ω its angular frequency, it can be shown that to a harmonic input corresponds a harmonic output of the same frequency, but with different amplitude and phase, which depend on the angular frequency of input. Therefore, the output will be [
18]:
where
is a phase lag. By applying Equations (20)–(25) to this case, one has:
and the complex quantity is introduced:
with the real and imaginary parts given by:
These functions are very important for studying the aforementioned relaxation phenomena. In a physical contest,
L1 and
L2 are called storage modulus and loss modulus, respectively [
17], and it is possible to show that they are related to not-dissipative phenomena and dissipative phenomena, respectively. Moreover, these two quantities are experimentally measurable as functions of angular frequency of input.
2.4. Ultrasound Wave Approach: Summary of Previous Results
Here, we summarise our previous results showing where we can apply the fractional model, which we will obtain in the next section, and how important it is for determine an analytical expression of all thermodynamic functions. In previous papers [
21], by assuming that a longitudinal wave
perturb a medium (blood), where
u(
u1,
u2,
u3) is a vector displacement which propagates in the direction of
x-axis; A the amplitude of the wave; and
K =
K1 +
iK2 is the complex wave number, where
K1 is
in which
vs is the phase velocity and
K2 is attenuation. So, taking into account Kluitenberg’s theory and the characteristic of longitudinal waves, we obtain the following expression for phenomenological and state coefficients as functions of the wave vector
K(
ω) and for entropy production [
21]:
where
σ is the relaxation time and
L2R the relaxed value of
L2 [
19] and
Moreover, the following expression is for rheological (internal variables) functions [
22]:
where:
It is very important in our approach to take into account the well known relations between the complex wave vector
K and the complex longitudinal dynamic modulus
L* (see Equations (30) and (31)):
since we will formulate a fractional model for the modulus
L*. These relations allow us to obtain an analytical expression of the aforementioned coefficients and rheological functions, as we will see in the next section.