# Internal structure and heat conduction in rigid solids: a two temperature
approach

R.E Gonzalez-Narvaez
Centro de Investigación en Ciencias (CInC)
Universidad Autónoma del Estado de Morelos (UAEM)
Col. Chamilpa, Cuernavaca, México
<EMAIL_ADDRESS>
M. López de Haro
Instituto de Energías Renovables
Universidad Nacional Autónoma de México (U.N.A.M.)
Temixco, Morelos, México
<EMAIL_ADDRESS>
F. Vázquez
Centro de Investigación en Ciencias (CInC)
Universidad Autónoma del Estado de Morelos (UAEM)
Col. Chamilpa, Cuernavaca, México
<EMAIL_ADDRESS>

###### Abstract

A non-Fourier thermal transport regime characterizes the heat conduction in
solids with internal structure. Several thermodynamic theories attempt to
explain the separation from the Fourier regime in such kind of systems. Here
we develop a two temperature model to describe the non-Fourier regime from the
principles of non-equilibrium thermodynamics. The basic assumption is the
existence of two well separated length scales in the system, namely, one
related with the matrix dimension (bulk) and the other with the characteristic
length of the internal structure. Two Fourier type coupled transport equations
are obtained for the temperatures which describe the heat conduction in each
of the length scales. Recent experimental results from several groups on the
thermal response of different structured materials are satisfactorily
reproduced by using the coupling parameter as a fitting parameter. The
similarities and differences of the present formalism with other theories are
discussed.

##  1 Introduction

Real systems and materials are non-homogeneous either concerning their
transport properties or their inner structure (here onwards called the
microstructure). Heterogeneity may arise from compactation, packing or
sintering processes [1, 2, 3]. The effects of inhomogeneity emerge from
smaller length scales and they influence the response of the system to
external thermodynamic forces. Irreversible thermodynamic formalisms have been
formulated to include the microstructure in extended thermodynamic variable
spaces through the so called internal variables [4, 5, 6, 7, 8] and these
formalisms have been applied to the case of a rigid heat conductor [9, 10, 11,
12, 13, 14]. For a survey of published papers on this topic before $1990$, see
[16]. Theoretically and experimentally it has been found that energy transport
in solids with microstructure is dominated by non-Fourier mechanisms of heat
transport, in such a way that in order to account for these mechanisms the use
of non-Fourier models like the Guyer-Krumhansl (GK) equation and others have
been required. One may reasonably wonder whether the Fourier transport
equation is able to reproduce the non-Fourier behavior of microstructured
solids by introducing somehow the structure features in the description. If
one attempts to introduce the structure by making the heat conductivity to
depend on the position, the answer is no as it will be shown below. In this
work, however, we argue that it is possible to reproduce the non-Fourier
behavior of microstrured solids if one assumes that there exist two basic heat
transport processes which occur at different length scales: on the one hand,
heat transport in a macroscopic length scale comparable with the
characteristic length of the macroscopic system; on the other, transport
processes occurring at a microscopic scale of length determined by the
characteristic length of the microstructure. This readily leads to the
consideration of two well differentiated length scales in the system
determined, respectively, by the characteristic length of the microstructure
and by the macroscopic dimensions of the system itself. We focus on
elucidating if a two temperature model, based on Fourier transport equations
for each of the above mentioned processes, is suitable to describe the assumed
macro and micro-processes and to explain the behaviour of the measurable
temperature. The two temperature model framework is based on the fact that the
total internal energy of the system may be separated in two parts, namely, the
part associated with the macroscopic length scale processes and the part
related to processes taking place in the microscopic length scale. In our
approach, the local equilibrium hypothesis is taken into account by assuming
that the local entropy depends on the internal energy of both macro and micro-
processes. In turn, as it is usually done in two temperature modeling, each of
the internal energies is related to a quasi-temperature referred to as the
macro and the micro-temperature, respectively. The total measurable
temperature of the system is then obtained through a weighted addition of the
macroscopic and the microscopic quasi-temperatures. Thus, the measurable local
temperature will be understood as an average temperature in the control volume
as it is done in classical irreversible thermodynamics. Mention must be made
of the fact that the above statements evokes that the heat carrier population
is being separated into two parts as it is done in many heat transport
problems. For instance, this is done in transient ballistic-diffusive heat
transport where the total internal energy per unit volume is the sum of
longitudinal and transverse phononic branches [17, 18]. Also, a similar
separation is done in heat transport in common materials at the nanoscale,
where the electrons and the phonons that transport energy are out of an
equilibrium state during the heat transfer [19, 20, 21, 22, 23, 24], or in
nonlocal thermoelectric transport in thin layers [25]. In any case, however,
this point will not deserve further consideration in this work. Instead, we
will stay here within the scope of phenomenological irreversible
thermodynamics.

Our development is as follows. The time evolution equations for each one of
the temperatures will be obtained from the local equilibrium hypothesis, as
stated above, through the use of the conservation equations for the
corresponding internal energies at each length scale. Such equations lead
directly to the well known entropy balance equation from which one may
identify the "thermodynamic forces" conjugate to the respective fluxes by
examining the entropy production term [26]. We will show below that including
the micro-processes leads to the non-Fourier character of the macroprocess.
The microstructure will be incorporated into the scheme through small length
scale inhomogeneities in the spatial distribution of the thermal conductivity
of the material. Hence, the form of the thermal conductivity will in turn
characterize the microstrure. Comparison with recent theoretical and
experimental results obtained in structured solids at room temperature will
provide support to the proposed model. We will compare our scheme, in
particular, with the thermodynamic theory of internal variables [14]. An
interesting aspect to be considered is the effect of the microstructure on the
entropy production in the material, since the microstructure introduces an
additional dissipation mechanism (some authors call it the excess entropy
production [27]). Once this is done, it is then natural to explore its
relation with extremum principles.

We close this section by summarizing our purposes in this paper. The first one
is, using irreversible thermodynamics, to formulate a two temperature model to
describe heat transport in a rigid heat conductor with internal structure at
room temperature. Our aim is to have a physical interpretation of the two
introduced quasi-temperatures in terms of processes occurring at macroscopic
length scales (of the order of magnitude of the system’s size) and at
microscopic length scales (of the order of magnitude of the size of the
system’s microstructure). Secondly, we want to compare our theoretical results
with recent experimental reports on non-Fourier heat transport found in
structured solids at room temperature and to analyze cases where the Onsager
reciprocity relations are satisfied and cases where they are not. Finally, we
want to discuss the relation of the present formalism with other thermodynamic
theories used to describe structured solid heat conductors. To this end, the
paper is organized as follows. In the next section we introduce the physical
model and proceed with the irreversible thermodynamic formulation of heat
transport in such a system. This is followed by sec. 3 in which the time
evolution of the measurable temperature and its relation to the microstructure
is addressed. In both previous sections near equilibrium and far from
equilibrium situations are considered and a comparison with experimental
results in various systems is performed. Section 4 deals with the relation
between the entropy production and the microstructure. The paper is closed in
the final section with further discussion and some concluding remarks.

##  2 The physical model

Our model considers the existence of two separate spatial scales in the
system. On the one hand, the scale of the bulk of the material and on the
other, the scale of the microstructure. So we consider that there are two
interacting subsystems involved in the transport processes at the two spatial
scales. This implies that there are three sources of dissipation: i) processes
on the macroscopic spatial scale, ii) processes at the microscopic scale and
iii) interaction processes - energy exchange - between the two subsystems. To
each of the subsystems will correspond a part of the total internal energy of
the system determined by the caloric properties of each one. In agreement with
the local equilibrium hypothesis, the thermodynamic space of states will be
made up of both internal energies to which two quasi-temperatures will be
associated. The total temperature will be determined by the caloric properties
of the subsystems.It is worth mentioning that the two subsystems are analogous
to what in other two-temperature models is called a mixture of gases made up
of different heat carriers, eg, electrons and phonons in the case of
thermoelectricity. In Ref. [15], for example, a “logic” argument is used that
leads to the assumption that the respective energies of each gas are part of
the space of state variables. We do not resort to such kind of argument but
instead let the comparison with experimental results indicate whether the
assumption was adequate.

###  2.1 Thermodynamic formulation of the time evolution equations

As already mentioned, the system consists of a rigid heat conducting solid
with an internal structure in which two coupled irreversible processes take
place: on the one hand, processes that occur at a macroscopic length scale,
that is at a length scale of the same order of magnitude as the characteristic
length of the system itself $L$; on the other hand, processes that occur at a
length scale of the same order of magnitude as the characteristic length of
the internal structure $l$. The starting assumption is that the total internal
energy of the system can be divided into two parts, each one corresponding to
the above two dissipative processes, namely

$u=u_{M}+u_{m},$ (1)

where $u$ stands for the internal energy and the labels $M$ and $m$ denote
macroscopic and microscopic processes, respectively. From now on, we will
assume that $L\gg l$ and we will refer to the microstructure as the internal
structure of the system. Defining two quasi-temperatures through the known
caloric relation, $u=CT$ with $C$ the heat capacity at constant volume, we
have the following

$T_{M}=\frac{u_{M}}{C_{M}},$ (2)

and

$T_{m}=\frac{u_{m}}{C_{m}}.$ (3)

Thus, the total (measurable) temperature of the system is given by the average

$T=\frac{C_{M}T_{M}+C_{m}T_{m}}{C_{M}+C_{m}}.$ (4)

The temperatures $T_{M}$ and $T_{m}$ will be referred to as macro-temperature
and micro-temperature, respectively. Note that, for the sake of brevity, we
have omitted the prefix quasi in both instances. The macro-temperature is an
average temperature that describes the processes that occur in the bulk and
the micro-temperature describes the changes that frequently and slightly
separate the system from the average temporal evolution due to the emergence
of effects originating from the microstructure. Strictly speaking, and given
that $L\gg l$, the micro-processes take place in a medium with heterogeneous
thermal conductivity. Meanwhile, macro-processes occur in an effective
homogeneous environment. As can be seen, this approach conforms a two-
temperature model for heat transport in the rigid solid where one of the
temperatures accounts for the deviation of the system from the Fourier
behavior.

We next obtain the expression for the entropy production in the solid. We
proceed in the usual way as it is done in two temperature models within the
framework of irreversible thermodynamics [15]. The starting assumption is
that, as mentioned above, the thermodynamic space of variables is constituted
by the internal energies $u_{M}$ and $u_{m}$ in such a way that the entropy is
a function of them, namely

$S=S\left(u_{M},u_{m}\right).$ (5)

The time evolution of the total entropy of the system is then

$\frac{dS}{dt}=\left(\frac{\partial S}{\partial
u_{M}}\right)_{u_{m}}\frac{du_{M}}{dt}+\left(\frac{\partial S}{\partial
u_{m}}\right)_{u_{M}}\frac{du_{m}}{dt},$ (6)

where

$\left(\frac{\partial S}{\partial u_{M}}\right)_{u_{m}}=\frac{1}{T_{M}},$ (7)

and

$\left(\frac{\partial S}{\partial u_{m}}\right)_{u_{M}}=\frac{1}{T_{m}}.$ (8)

Additionally, we will assume that both types of processes in the solid are
conservative, that is, we will ignore the existence of energy sources. Under
those conditions

$\frac{du_{M}}{dt}=-\frac{\partial q_{M}}{\partial x},$ (9)

and

$\frac{du_{m}}{dt}=-\frac{\partial q_{m}}{\partial x}.$ (10)

For simplicity, here we have constrained the system to one dimension and
$q_{M}$ and $q_{m}$ are heat fluxes associated with energy transport in macro
and micro-processes, respectively. From Eqs. (6)-(10) it is straightforward to
get the balance equation of the total entropy, which reads

$\frac{dS}{dt}+\frac{\partial}{\partial
x}\left(\frac{1}{T_{M}}q_{M}+\frac{1}{T_{m}}q_{m}\right)=-\frac{1}{T_{M}}\left(q_{M}\frac{\frac{\partial
T_{M}}{\partial
x}}{T_{M}}\right)-\frac{1}{T_{m}}\left(q_{m}\frac{\frac{\partial
T_{m}}{\partial x}}{T_{m}}\right).$ (11)

In this last equation the right hand side is the entropy production $\sigma$,
_i.e._

$\sigma=-\frac{1}{T_{M}}\left(q_{M}\frac{\frac{\partial T_{M}}{\partial
x}}{T_{M}}\right)-\frac{1}{T_{m}}\left(q_{m}\frac{\frac{\partial
T_{m}}{\partial x}}{T_{m}}\right).$ (12)

In order to have a closed system of equations, one has to relate the fluxes
with the thermodynamic forces. Note that in this case the fluxes and forces
have the same tensorial character. In the absence of any further physical
insight, the simplest way to relate such forces and fluxes is through linear
relationships. Hence, we will assume that

$-q_{M}=L_{11}\frac{\frac{\partial T_{M}}{\partial
x}}{T_{M}^{2}}+L_{12}\frac{\frac{\partial T_{m}}{\partial x}}{T_{m}^{2}},$
(13) $-q_{m}=L_{21}\frac{\frac{\partial T_{M}}{\partial
x}}{T_{M}^{2}}+L_{22}\frac{\frac{\partial T_{m}}{\partial x}}{T_{m}^{2}},$
(14)

where the coefficients $L_{ij}$ are phenomenological coefficients with
$i,j\epsilon\left[[1,2\right]]$. It is worth pointing out that they may be
functions of the temperatures $T_{M}$ and $T_{m}$. Further, depending on the
actual physical situation, such coefficients may or may not satisfy the
Onsager reciprocity relations. Both cases will be considered. Equations (13)
and (14) are the simplest way to ensure the non-negativity of entropy
production. The proposed formalism is compatible with the second law if the
Onsager coefficients matrix is positive definite. The validity of the Onsager
relations only implies that this matrix is symmetric. So, we have expressed
positiveness of the matrix as $L_{11}>0$, $L_{22}>0$,
$L_{11}L_{22}-\left(L_{12}+L_{21}\right)^{2}/4>0$, and we have ensured that
these conditions are satisfied in all the studied examples.

###  2.2 Near equilibrium processes. Validity of Onsager reciprocity
relations.

We begin with the case of a system undergoing a slow process where it is
allowed to reach the local equilibrium state at each stage. Then, we assume
that the Onsager reciprocity relations are valid, _i.e._

$L_{12}=L_{21},$ (15)

which implies that

$\lambda_{Mm}T_{m}^{2}=\lambda_{mM}T_{M}^{2}.$ (16)

One of the necessary assumptions for the validity of the Onsager relations,
ORs, is that the state variables are well defined and independent. According
to Irreversible Thermodynamics principles, state variables satisfy both
requirements in local equilibrium states. Thus, the local equilibrium
hypothesis is a necessary condition for the validity of the Onsager relations.
The other conditions of the validity of the ORs are also satisfied by the
method used, that is, the definition of the fluxes and conjugated forces from
the second law, and a linear relationship between both groups of variables. In
the case in which the system does not transit through local equilibrium
states, as it is in the first example considered in the following section, the
first of the mentioned conditions is not ensured. In this case, the
reproduction of the experimental results as observed in the respective
figures, will give the assumed hypothesis -that is, the validity of the
Onsager relations- an acceptable degree of plausibility.

Nonlinear transport equations for the temperatures are then obtained which
read

$\displaystyle
C_{M}T_{m}^{3}\frac{dT_{M}}{dt}=\lambda_{M}T_{m}^{3}\frac{\partial^{2}T_{M}}{\partial
x^{2}}+\lambda_{mM}\left[T_{M}^{2}T_{m}\frac{\partial^{2}T_{m}}{\partial
x^{2}}-2T_{M}^{2}\left(\frac{\partial T_{m}}{\partial x}\right)^{2}\right]+$
$\displaystyle 2\lambda_{mM}T_{M}T_{m}\frac{\partial T_{M}}{\partial
x}\frac{\partial T_{m}}{\partial x},$ (17)

$C_{m}\frac{dT_{m}}{dt}=\lambda_{mM}\frac{\partial^{2}T_{M}}{\partial
x^{2}}+\lambda_{m}\frac{\partial^{2}T_{m}}{\partial
x^{2}}+\frac{\partial\lambda_{m}}{\partial x}\frac{\partial T_{m}}{\partial
x}.$ (18)

Note that by writing the constitutive relations as in Eqs. (13) and (14), we
are implying that the time evolution of the system is described by two coupled
Fourier, though nonlinear, type processes. It is worth remarking that, in this
case, the assumption of the Onsager reciprocity relations leads to nonlinear
transport equations.

As mentioned earlier, the material microstructure will be introduced in the
two temperature model by assuming that the thermal conductivity of micro-
processes, $\lambda_{m}$, is a function of position. In the following section
we will study its effects on the profile and time evolution of the total
temperature. Furthermore, we will also analyze these effects on the stationary
global entropy production.

###  2.3 Far from equilibrium processes. Non validity of the Onsager
reciprocity relations.

In this section we deal with the case in which the Onsager reciprocity
relations are not satisfied in Eqs. (13) and (14). This occurs when, for
instance, the system is thermally perturbed in a very short time scale (short
with respect its characteristic time scale) as it is the case of heat pulsed
rigid conductors in which the system is suddenly and abruptly brought out of
equilibrium. The resulting heat transport regime is of the non-Fourier type
since thermal inertia influences its response. It must then be clear that
although the Onsager relations are not satisfied it is not unreasonable to
expect that linear constitutive equations will suffice to describe suitably
the heat transport in the structured solid. Note that this situation is
different from that where non-linearity determines the behavior of the system
and may lead to non validity of the Onsager relations [37].

The first experiments on pulsed materials were performed in the 70’s in cold
crystals and recently they were replicated in some structured solids at room
temperature [38, 39, 40]. We apply Eqs. (13) and (14) to describe the thermal
response of a material at room temperature. The phenomenological coefficients
dependence on the temperatures is assumed to be $L_{11}=\lambda_{M}T_{M}^{2}$,
$L_{12}=\lambda_{Mm}T_{m}^{2}$, $L_{21}=\lambda_{mM}T_{M}^{2}$,
$L_{22}=\lambda_{m}T_{m}^{2}$. Then by combining Eqs. (13) and (14) with the
energy conservation equations (9) and (10), we obtain two coupled transport
equations for macro and micro-processes in terms of the respective
temperatures, namely

$C_{M}\frac{dT_{M}}{dt}=\frac{\partial}{\partial
x}\left(\lambda_{M}\frac{\partial T_{M}}{\partial
x}+\lambda_{Mm}\frac{\partial T_{m}}{\partial x}\right),$ (19)
$C_{m}\frac{dT_{m}}{dt}=\frac{\partial}{\partial
x}\left(\lambda_{mM}\frac{\partial T_{M}}{\partial
x}+\lambda_{m}\frac{\partial T_{m}}{\partial x}\right),$ (20)

where the coefficients $\lambda_{ij}$ are thermal conductivities of direct and
cross couplings. Note that even in the case that $\lambda_{Mm}=\lambda_{mM}$,
the phenomenological coefficientes $L_{12}$ and $L_{21}$ are different.

##  3 Microstructure and time evolution

Before we engange of the following section, we briefly discuss the transport
and fitting parameters of the two temperature model. First, we have to
differentiate between the $\lambda_{Mm}$ and $\lambda_{mM}$ coupling
parameters and the coefficients $\lambda_{M}$ and $\lambda{m}$. The former
coefficients should be considered part of the proposed model, but both
$\lambda_{M}$ and $\lambda_{m}$ are transport properties of the material.
$\lambda_{M}$ is the material bulk thermal conductivity and $\lambda_{m}$
contains the details of the microstructure of the system. Therefore, they can
be determined only through experiments or by a more microscopic theory than
the one proposed here. This way, the two temperature model involves some
fitting parameters, namely, the coupling parameters $\lambda_{Mm}$ and
$\lambda_{mM}$, which will be taken to be equal and the heat capacities
$C_{M}$ and $C_{m}$ which in the simulations will also be taken to be equal.
In this way, the model has just two fitting parameters.

###  3.1 Time evolution of the total temperature. Initial and boundary
conditions

Eqs. (19) and (20) will be solved by assuming that at one side of the system
the measurable temperature is suddenly raised to $T_{h}$ while the opposite
one is kept adiabatically isolated. This implies Dirichlet-Neumann boundary
conditions, that is

$\frac{\partial T(0,t)}{\partial x}=0,$ (21)

$T(L,t)=T_{h}-\Delta T_{h}Exp(-\frac{t}{t_{p}}),$ (22)

The initial condition of the material is assumed to be a homogeneous
distribution of the measurable temperature, i.e.,

$T(x,0)=T_{0},$ (23)

with $T_{0}$ being the room temperature. Then, the factor $\Delta T_{h}$ must
be chosen equal to $T_{h}-T_{0}$.

Eqs. (21) and (22) determine the boundary conditions for the macro and micro
temperatures since they are related with the measurable temperature through
Eq. (4). After that, Eq. (21) implies that

$\frac{\partial T_{M}(0,t)}{\partial x}=0,\frac{\partial T_{m}(0,t)}{\partial
x}=0,$ (24)

and Eq. (22)

$T_{h}-\Delta
T_{h}Exp(-\frac{t}{t_{p}})=\frac{C_{M}}{C_{M}+C{m}}T_{M}(L,t)+\frac{C_{m}}{C_{M}+C{m}}T_{m}(L,t).$
(25)

Now it is reasonable to assume that the initial sudden increase of the
measurable temperature at $x=L$ is reflected in the macro and micro
temperatures in the same manner, that is to say,

$T_{M}(L,t)=T_{M0}-\Delta T_{M0}Exp(-\frac{t}{t_{p}}),$ (26)
$T_{m}(L,t)=T_{m0}-\Delta T_{m0}Exp(-\frac{t}{t_{p}}).$ (27)

which leads to the following conditions

$T_{h}=\frac{C_{M}}{C_{M}+C{m}}T_{M0}+\frac{C_{m}}{C_{M}+C{m}}T_{m0},$ (28)
$\Delta T_{h}=\frac{C_{M}}{C_{M}+C{m}}\Delta
T_{M0}+\frac{C_{m}}{C_{M}+C{m}}\Delta T_{m0}.$ (29)

A few clarifying words on the previous assumption are necessary at this point.
The boundary condition, cf Eqs. (26) and (27), is established for the
measurable -observable- temperature, defined in Eq. (4), which at time $t=0$
rapidly increases to a $T_{h}$ value from a $T_{0}$ value. The question of how
the sudden increase in measurable temperature translates to the macro and
micro temperatures -which are not measurable- , has more than one answer. We
have given one that we think is physically reasonable. For example, assuming
that only the macro temperature, or micro temperature, is affected when the
measurable temperature increases at the boundary implies that the thermal
excitation is transferred to only one of the subsystems. This is, of course,
not reasonable. In fact, this is not the case even in ultra-fast phenomena
where a system is pulsed with a laser, the duration of the pulse being on the
order of picoseconds. In this example, although most of the injected energy is
absorbed by the one of the subsystems, the rest is absorbed by the other one.
Later, through their interaction, some of the energy is also transferred to
the second subsystem.

Now, it must be recalled that the micro temperature describes the dissipative
processes due to the presence of the internal structure of the system. As a
consequence, it behaves like a fluctuating temperature around $T_{m0}$. In
fact, the reference values $T_{M0}$ and $T_{m0}$ will be taken to be equal in
the simulations of different structured solids made later on. This condition
is a consequence of the fact that at $t=0$ the measurable temperature is
homogeneous with value $T_{0}$. In this way, we have not only determined the
boundary conditions for the macro and micro temperatures but it is reaffirmed
the microtemperature as a thermodynamic variable and not as an internal
variable. This point will be further considered in the discussion section.

Stationary profiles and the time evolution of the macro and microscopic
temperatures have been obtained by using the commercial Mathematica Wolfram
software to solve Eqs. (19) and (20) (or (17) and (18)) with conditions (24),
(26) and (27). The results are presented in the following sections.

###  3.2 On the internal structure of the solid

In this short subsection we briefly describe in general terms the assumed
internal structure of the studied solids. They are constituted by an
alternated distribution of materials with different thermal conductivities.
This distribution may be periodic or not. In this last case the materials may
be randomly distributed in the space and even to have random values of their
thermal conductivity. Figure 1 shows generic examples of the structure
introduced through $\lambda_{m}$ in the transport equations of the macro and
the micro temperatures. In Figure 1a it can be seen a superlattice-like
structure constituted by two materials with different constant thermal
conductivities. In Figure 1b the same structure with random thermal
conductivities of the second material is shown. This generic structures will
be adapted to the specific characteristics of the studied systems.

(a) Constant structure heat conductivity.

(b) Random structure heat conductivity

Figure 1: Structure characterization through the heat microconductivity
$\lambda_{m}$. The matrix heat conductivity (bulk) is 235 $W/mK$.

###  3.3 Near equilibrium processes

In this section we compare the prediction of the time evolution of the
measurable temperature obtained from the two temperature model with that
obtained by Özdemir et al. [36] through a homogenization technique. The system
is a cellular foam-like structure subjected to Dirichlet-type boundary
conditions. In Figure 2 the temperature profile for different times as
obtained in Ref. [36] is shown. We assume that inertial effects are not
present (in agreement with the same assumption already demonstrated in Ref.
[36]) and therefore that the local equilibrium hypothesis is valid along the
process towards the stationary state. We use Eqs. (17) and (18) to obtain the
temperature profiles in our structured material in order to qualitatively
compare the two behaviors. We have superimposed our results on the graphs by
Özdemir et al [36]. The fitting values of the coupling constants are in this
case $\lambda_{Mm}=\lambda_{mM}=0.1$ $W/mK$, $T_{M0}=310$ $K$, $T_{m0}=310$
$K$, $\Delta T_{M0}=\Delta T_{m0}=10$ $K$, and $t_{p}=10^{-3}$ $sec$. As it
may be observed, the temperature of the two systems shows a similar dependence
on the position for the considered times. Now we make a further comparison
with the time evolution of the temperature in the non-exposed side of the same
cellular structure obtained in [41] from a Fourier model with a boundary
condition similar to that of Ref. [36]. We have calculated the time evolution
of the temperature in the structure by using Eqs. (17) and (18). In Figure 3
the response of the structure as a function of time from [41] is shown and we
have also superimposed our result. The two curves are very close to each
other. It is worth noting the behavior of the temperature for the earliest
times. We note that the short time interval where the temperature remains
constant is reproduced well by the two temperature model.

Figure 2:  Temperature profile (dots) in the cellular foam-like structure at
different times from Özdemir et al. [36] and from the two temperature model
(solid lines). Blue: 1 Sec., Yellow: 10 Sec. Green: 50 Sec., Red: 75 Sec.
Figure 3:  Time evolution of the temperature in the cellular foam-like
structure at the non-exposed side. Dots represent the result from Lu et al.
[41] and the solid line is the result from the two temperature model.

###  3.4 Far from equilibrium processes

In this section we show that the two-temperature model proposed in this work,
Eqs. (19) and (20), describes the non-Fourier regime found in heat transport
experiments in structured solids at room temperature. Van et al. [39] analyzed
four types of materials and measured the time thermal response of the material
to a sudden increase of temperature at one side. We will focus here on the
results for pure Aluminum. These results clearly show a non-Fourier behavior
as it may be seen in Figure 4 where dots correspond to the experimental
measurements and the solid blue line to the calculated time response in the
Fourier regime obtained from

$\frac{\partial^{2}T_{M}^{F}}{\partial x^{2}}=\alpha\frac{\partial
T_{M}^{F}}{\partial t},$ (30)

with $\alpha$ the thermal diffusivity of bulk Aluminum. Note that we have
obtained the time evolution of the temperature using Eq. (30) subjected to the
same initial and boundary conditions which are used to solve the two
temperature model.

At this point, one may reasonably wonder about the physical origin of the non-
Fourier behavior given that the reported material is pure Aluminum. May one
assume the existence of some kind of internal structure? The fact is that
commercial pure Aluminum is subjected to the so called accumulative roll-
bonding process at relatively high temperature which produces a lamellar
structure characterizing the microstructure [2]. Rolled Aluminum then shows a
structure similar to that assumed here for the solid and, as we stated in our
initial assumptions, this microstructure is responsible for the non-Fourier
behavior of the material.

The solid yellow line in Figure 4 corresponds to the calculated thermal
response of the material to the excitement as derived from the two temperature
model, Eqs. (19) and (20). The microstructure in this case is a resemblance of
the inner structure of a superlattice: over a kind of matrix with the bulk
heat conductivity of Aluminum, $237$ $W/mK$, a set of slight variations of the
heat conductivity (of the order of $2$ $W/mK$) is superimposed. An example is
shown in Figure 1. The characteristic length of the microstructure is at least
one order of magnitude smaller than the size of the system. The fitting values
of the coupling constants are $\lambda_{Mm}=\lambda_{mM}=1$ $W/mK$ (see Eqs.
(19) and (20)).

The values $T_{M0}=310$ $K$, $T_{m0}=310$ $K$, $\Delta T_{M0}=\Delta
T_{m0}=10$ $K$, and $t_{p}=10^{-3}$ $sec$ have been introduced in the boundary
conditions, Eqs. (26) and (27).

Figure 4:  Time evolution of the total temperature of pure Aluminum as derived
from the Fourier model (solid blue line) and the proposed two temperature
model (solid yellow line). Dots correspond to experimental data from Van et
al. [39].

As it can be seen, the solution obtained from Eqs. (19) and (20) agrees well
with the experimental points and adequately describes the non-Fourier regime
of heat transport found by Van et al. [39]. The Fourier response (blue line)
gives an idea of how much this regime deviates from the Fourier regime. We
conclude that the non-Fourier regime described by the GK equation

$\tau\frac{\partial^{2}T}{\partial
t^{2}}-\frac{l^{2}}{\tau}\frac{\partial}{\partial
t}\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial T}{\partial
t}-D\frac{\partial^{2}T}{\partial x^{2}}=0.$ (31)

where $\tau$, $l$ and $D$ stand for the relaxation time and mean free path of
heat carriers and thermal diffusivity, respectively, is equivalent to the
description based on the two coupled Fourier processes, Eqs. (19) and (20).
This point will be discussed in more detail later in the final section.

As a second example of structured materials we examine the results obtained by
Tang et al. [38] for rapidly irradiated biological tissues. The reported
behavior of the response of the system to the irradiation is different from
that obtained for pure Aluminum. In this last case, the response does show an
inertial effect since it lags behind the response of the Fourier regime.
However, Figure 4 also indicates that there is no inertial effect in the
approximation to the stationary state which occurs asymptotically from below
the maximum temperature. In contradistinction, the response of the biological
material shows not only inertia from the beginning but also when it approaches
the stationary state from above the maximum temperature. Figure 5 illustrates
what has just been mentioned. Moreover, note that in the experiment the
stationary temperature is smaller than the stationary temperature in the
Fourier regime. As before, the solution obtained from Eqs. (19) and (20)
agrees well with the experimental points and adequately describes the non-
Fourier regime of heat transport found by Tang et al. [38]. The fitting
parameters used to obtain the calculated response, solid yellow line in Figure
5, are $\lambda_{Mm}=\lambda_{mM}=0.4$ $W/mK$ The biological tissue is
constituted by meat with heat conductivity $0.47$ $W/mK$ and insertions of fat
with $0.18$ $W/mK$. The boundary conditions are taken as in the previous case.

Figure 5:  Time evolution of the total temperature of a biological tissue as
derived from the Fourier model (solid blue line) and the proposed two
temperature model (solid yellow line). Dots correspond to experimental data
from Tang et al. [38].

Before ending this section, it is natural to wonder whether the Fourier heat
transport model can reproduce the experimental data by simply incorporating
the microstructure through the heat conductivity in Eq. (30). We have
addressed this issue explicitly for Aluminum and the thermal response overlies
on the solution of the Fourier process with constant heat conductivity (solid
blue line) in Figure 4. This reveals two important facts. On the one hand,
that the non-Fourier behavior of the structured material comes from the
influence of the micro temperature which contains the information of the
microstructure. On the other hand, the effective or bulk part of the
description of heat transport is blind to the details of the microstructure.
These conclusions will be remarked in the final section.

##  4 Microstructure and entropy production

###  4.1 Microstructure and heterogeneity

In this section we study the relationship between microstructure and
heterogeneity of the material. Heterogeneity is introduced in the spatial
distribution of thermal conductivity by assuming that it randomly varies along
the material. To be more specific, we consider that the material is
constituted by a material of $\lambda_{b}$ thermal conductivity (bulk) which
changes in a random way in certain prescribed positions within the material.
These variations are then included in $\lambda_{m}$, which is the heat
conductivity associated with the microstructure. Now, we assume that the heat
conductivity randomly changes its value at $N$ prescribed positions within the
material. The random values are generated in the interval
$(\lambda_{b}-2,\lambda_{b}+2)W/mK$. Let us denote the values generated in the
indicated way as $\lambda_{i}$. We define the heterogeneity coefficient
$C_{h}$ of the material as the mean squared deviation of the $N$ values in the
microstructure from the bulk value of the heat conductivity, $\lambda_{b}$.
That is,

$C_{h}(N)=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(\lambda_{i}-\lambda_{b})^{2}}.$ (32)

###  4.2 Entropy production

The local entropy production in the two temperature model is given by Eq. (12)
which explicitly reads

$\displaystyle\sigma\left(x,t\right)=\frac{1}{T_{M}\left(x,t\right)^{2}}\left[\lambda_{M}\left(\frac{\partial
T_{M}}{\partial x}\right)^{2}+\lambda_{Mm}\frac{\partial T_{M}}{\partial
x}\frac{\partial T_{m}}{\partial x}\right]+$
$\displaystyle\frac{1}{T_{m}\left(x,t\right)^{2}}\left[\lambda_{m}\left(x\right)\left(\frac{\partial
T_{m}}{\partial x}\right)^{2}+\lambda_{mM}\frac{\partial T_{M}}{\partial
x}\frac{\partial T_{m}}{\partial x}\right].$ (33)

Later on, Eq. (33) will be used to study the relationship between
heterogeneity and entropy production in the stationary state. But before this,
we refer to Figure 6 where the profile of the entropy production for different
times in the case of pure Aluminum can be seen. We note that the region near
the thermally excited boundary ($x=L$) is where more entropy is being
produced. Also we observe that the entropy production for the early stages of
the time evolution is greater than that for later times as the system
approaches the stationary state. This suggests that the stationary state will
obey a minimum entropy production principle, an issue that we will not further
address here but that will be object of further research.

At this stage, it is interesting to define an effective entropy production
through the expression

$\sigma_{eff}\left(x,t\right)=-q\left(x,t\right)\frac{\partial}{\partial
x}\left(\frac{1}{T}\right),$ (34)

where $T$ and $q$ stand for the total (measurable) temperature and the total
heat flux, respectively. The last is given by

$q\left(x,t\right)=q_{M}\left(x,t\right)+q_{m}\left(x,t\right)$ (35)

with $q_{M}$ and $q_{m}$ given by the constitutive equations Eqs. (13) and
(14). Therefore,

$\displaystyle\sigma_{eff}\left(x,t\right)=\left[\left(\lambda_{M}+\lambda_{Mm}\right)\frac{\partial
T_{M}}{\partial
x}+\left(\lambda_{Mm}+\lambda_{m}\left(x\right)\right)\frac{\partial
T_{m}}{\partial x}\right]\times$ $\displaystyle\frac{\partial}{\partial
x}\left(\frac{1}{aT_{M}+bT_{m}}\right),$ (36)

where $a=\frac{C_{M}}{C_{M}+C_{m}}$ and $b=\frac{C_{m}}{C_{M}+C{m}}$ (see Eq.
(4)).

A comparison of the effective entropy production and that coming from the two
temperature model in the stationary state can be seen in Figure 7, where they
are represented by the green solid line and the red solid line, respectively.
Clearly, the entropy production in the effective medium is smaller than that
of the solid with internal structure. In Figure 7 we have included the entropy
production in the Fourier process (blue solid line) and the entropy production
given by the classical expression [42]

$\sigma_{cl}\left(x,t\right)=\frac{\nabla T^{2}}{T^{2}},$ (37)

represented by the black solid line. As expected, the entropy production in
the Fourier process is the smallest (the Fourier solid does not consider the
internal structure at all) and those from the two temperature model and the
classical expression coincide. It is then found that the microstructure
introduces a new dissipation mechanism in the system. It must be considered
the fact that the microstructure involves the presence of interfaces where
entropy is produced but not heat, as it occurs in the cases we have studied.
In fact, at no point in the material there is an increase in temperature due
to heating, which would imply the presence of a source term in the energy
equation. By dissipation we are strictly understanding the processes
associated with entropy production.

In the following section the heterogeneity coefficient as obtained above is
used to examine the relation between heterogeneity and stationary entropy
production within the two temperature model.

###  4.3 Heterogeneity and entropy production

We now wonder whether a relationship exists between heterogeneity and the
total entropy production (summed over the entire spatial domain). We may only
answer this question to a limited extent due to the randomness introduced by
the fluctuating nature of the heat conductivity at the microscale. We restrict
ourselves to the case where the random values of the heat conductivity at the
microscale show an increasing or decreasing space profile. We then consider
several realizations of the conductivity profile and calculate the total
entropy production for each one of them. Finally, a linear regression analysis
to the data is performed. The result may be seen in Figures 8 and 9 for the
increasing and the decreasing cases, respectively. Both show that there exists
a positive correlation between heterogeneity as measured by the heterogeneity
coefficient and the value of the entropy production.

Figure 6:  Entropy production profile at different times for pure Aluminum.
Blue: 100, Orange: 120, Green: 150, Red: 180, Purple:220. Figure 7:
Stationary state entropy production profile for pure Aluminum. Blue: Fourier,
Red: two temperature model (Eq. 33),Green: effective (Eq. 36), Black: total
(measurable) temperature (Eq. 37). Figure 8:  Total entropy production vs.
heterogeneity coefficient for pure Aluminum. Case: increasing random thermal
conductivity as a function of position. The values of the integrated entropy
production are affected by a factor $10^{8}$. $\sigma_{T}=12.25+2.39\,C_{h}$.
Figure 9:  Total entropy production vs. heterogeneity coefficient for pure
Aluminum. Case: decreasing random thermal conductivity as a function of
position. The values of the integrated entropy production are affected by a
factor $10^{8}$. $\sigma_{T}=1.88+0.21\,C_{h}$.

##  5 Discussion and concluding remarks

###  5.1 Comparison with other theories

The two temperature model proposed in this work has some similarities and some
differences with several physical theories aimed at describing the influence
of the structure of solids on their thermal behavior. The first part of this
final section will be devoted to expose similarities and differences with some
of them related through their main assumptions although their subsequent
procedures may be quite different. Then we will discuss our results and make
some concluding remarks.

####  5.1.1 Multifield theory

This theory, which in fact encompasses several other theories, considers the
system as constituted by material patches [43], which are in fact volume
elements characterized by their position in the system, and an order parameter
which contains the information on the substructural configuration of each
patch. The order parameter is chosen such that it is an observable quantity
and its changes are considered to be responsible for non-Fourier behaviors in
heat conduction provided they are sensitive to temperature variations [44]. In
this formalism, like in ours, Fourier ’s law is not changed. Unlike ours, the
effects of the change of microscopic mechanisms are considered in the energy
conservation equation and a sort of constitutive relationship between the
geometric descriptors of the microstructure and the temperature is introduced.

####  5.1.2 Homogenization techniques

The homogenization process, used in [36] to describe the thermal behavior of
the Aluminum structure studied in Section 3.3, presents some similarities with
the two temperature scheme. It is based on the existence of a macro and a
micro length scales in the system and respective thermal processes. The
temperature at the macro level evolves in time according to energy
conservation and at the micro level it is assumed that the energy storage in
the representative volume element vanishes, which results in a stationary
state. There is not explicit coupling of both processes until the
homogenization process is implemented. Then both temperature fields are
obtained through standard multi-scale analysis.

####  5.1.3 Two temperature models

It is well known that in solid rigid conductors heat can be transported by one
or more types of carriers [17, 21, 22, 23, 24]. Experiments in superlattices
have made the wave properties of phonons evident and their effects have been
interpreted within the coherent-incoherent phonon scheme [28, 29, 30, 31]. All
the mentioned examples have in common representations in terms of two (or
more) temperatures which are widely used in the study of heat transport
problems of current scientific and applied interest. Formally, they have been
obtained within classical irreversible thermodynamics and in some cases within
extended irreversible thermodynamics [15]. In this subsection we make some
further remarks on the first main result of this work, namely, the
relationship of the two temperature model of Section 2.1 and the GK equation.

Splitting the heat flux into two parts or introducing an additional
temperature in the GK equation lead to a simplification of the heat transport
model. In the first case, the description is reduced (separated) to a Fourier
process and a Maxwell-Cattaneo process for each of the parts of the splitted
heat flux, respectively [10, 32]. In the second case, the description is
reduced to a Fourier process and a heat exchange equation [10]. Similarly, the
two temperature model of this work shows that the non-Fourier regime of heat
transport in structured materials is the result of the coupling of two Fourier
processes which take place at the macroscopic and the microscopic length
scales, respectively. The solutions of Eq. (38) and Eqs. (19) and (20) are
coincident and they describe well the experiments concerning heat transport in
the structured material as shown in Figure 2.

The GK equation, Eq. (31), contains a hierarchy of Fourier processes as it was
suggested by Van et al. [39] and also earlier by others [33, 34]. Such
hierarchy may be made apparent if the equation is rewritten in the form

$\tau\frac{\partial}{\partial t}\left(\frac{\partial T}{\partial
t}-\frac{l^{2}}{\tau}\frac{\partial^{2}T}{\partial
x^{2}}\right)+\left(\frac{\partial T}{\partial
t}-D\frac{\partial^{2}T}{\partial x^{2}}\right)=0.$ (38)

In this way, the non-Fourier regime being described by Eq. (38) is the result
of a Fourier process (second parenthesis in Eq. 38) and the time evolution of
a second Fourier process represented in the first parenthesis of the same
equation. It must be recalled that $T$ is the measurable temperature given by
Eq. (4). This interpretation of the GK equation retains some differences with
the addition of physical fields to the description. Firstly, because there is
only one temperature field and secondly, because there are no separate time
evolution equations for each of the Fourier models forming the hierarchy.
Nevertheles, it is interesting to note that the specific realization of the
first of the Fourier process is determined by the diffusivity of the system
and the second one by the phonon mean free path and the relaxation time. It is
also worth mentioning that the mean free path-relaxation time factor
characterizes the deviation of the process from Fourier behavior. This
indicates that the non-Fourier process results from two Fourier processes
occurring at different length scales. This is reminiscent of the main
assumption made here to build the two temperature model.

####  5.1.4 Internal variables thermodynamic theory

Though we named $T_{m}$ as the microtemperature, our concept maintains some
differences with the microtemperature from the internal variables theory [12].
First, the microtemperature in this framework like ours, refers to those
processes which occur at a microscopic length scale emerging from the internal
structure of the system and influencing the behavior of the measurable
temperature. However, in contradistiction with the internal variables
approach, the microtemperature in this work is not only assumed, but in fact
also shown, to be determined by the conditions at the boundaries of the
system.

As mentioned before, the microtemperature introduces a non-Fourier effect on
the time evolution of the measurable temperature which is a resemblance of the
non locality in the time (memory) introduced by the time derivatives in the
constitutive equations. This was named by Van [7] as the relocalization
effect. In this way, although the transport equation (in terms of the
measurable temperature) remains being of the parabolic type, the effect of the
microtemperature is equivalent to the presence of the time derivatives of the
heat fluxes in the constitutive equations, (Eqs. 13) and (14). In the context
of the internal variables theory, it is necessary to consider an extra
internal variable and getting the dual internal variable scheme to obtain the
relocalization effect as it was formally shown in [12]. The introduction of
the second variable leads to constitutive equations of the Cattaneo type and a
hyperbolic transport equation.

###  5.2 Further concluding remarks

We summarize in this last section the main conclusions drawn throughout the
manuscript.

The comparison of the two temperature model for structured solids with
experimental results has shown that the model reproduces well the non-Fourier
behavior in two cases, namely, when the system is externally excited in a
short time scale in such a way that it transits through non-equilibrium states
which do not satisfy the Onsager reciprocity relations and also when the
system is slowly excited in such a way that it transits through states
satisfying the Onsager relations.

The results obtained reveal two important facts. On the one hand, that the
non-Fourier behavior of the structured material comes from the influence of
the microtemperature which contains the information of the microstructure. On
the other hand, it leads to the conclusion that the Fourier description of
heat transport is blind to the details of the microstructure.

The introduction of the microtemperature in the description seems to
incorporate the important property of the inertia in the time evolution of the
system. This is equivalent to the relocalization concept established by Van
[7] and described by other non-Fourier models like those of Cattaneo and
Guyer-Krumhansl.

It must be stressed that the microtemperature of the two temperature model is
not comparable to the microtemperature of the internal variables theory, since
the former is determined through the boundary conditions on physical grounds,
see Eq. (27). Nevertheless, the concept of microtemperature in this work and
the procedures that we followed may help to clarify the physical nature of the
microtemperature in the internal variables theory [12]. This point certainly
deserves a more profound examination. Finally, we want to mention that perhaps
the closest formalism to ours is that of internal variables. It is well known,
however, that the introduction of internal variables makes it possible to
describe a system with an internal structure without making any assumptions
about the structure itself [45]. In this aspect, the scheme maintains a great
difference with the two-temperature scheme since it is necessary to include
the details of the structure to describe the behavior of the measurable
temperature. However, they bear a great similarity in the sense that the
introduction of the internal variables or the two temperatures allows to
describe memory and non-local effects (this was not formally shown in the two-
temperature scheme)

We have shown, to a limited extent, that entropy production and heterogeneity
are correlated. In the case of materials with random increasing heat
conductivity, the more heterogeneity the more entropy production observed in
the system. We have two additional comments in this respect. On the one hand,
the time behavior of the entropy production shown in Figure 6 suggests that
the stationary state obeys an extremum variational principle. This will be
further investigated in future work. On the other hand, as it may be seen in
Figure 7, the presence of the internal structure in the system produces a
further dissipation mechanism causing a bigger entropy production as compared
with the Fourier solid and the effective medium. In this context, the fact
that the entropy production from the two temperature model, (Eq.33), coincides
with that calculated from the classical expression, Eq.(37) , by using the
total measurable temperature, Eq. (4) is also remarkable.

We have left at the very end a comment on the physical nature of the non-
Fourier heat transport regime experimentally observed in solids at room
temperature. Our results suggest that the transport mechanism in such kind of
phenomenon may be different from that of hyperbolic heat conduction. In spite
of the good agreement of the predictions based on GK equation with the
experimental results, as it was shown in [39], the understanding of the non-
Fourier transport may not require the introduction of second sound and wave
behavior considerations. The GK equation predicts those behaviors at short
time scales of the order of the relaxation time of the heat flux. It seems
likely those rapid phenomena may not be determining the non-Fourier heat
transport regime observed in solids at room temperature.

### Acknowledgement

FV acknowledges fruitfull discussions with Peter Ván (Budapest), Robert Kovács
(Budapest) and Sergey Sobolev (Moscow) on related topics.

### Funding

This work was supported by PRODEP (México), IER-UNAM and CONACYT (México)
under grant 258623.

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