# Local structure and its implications for the relaxor ferroelectric Cd2Nb2O7

Daniel Hickox-Young  Department of Materials Science and Engineering,
Northwestern University, Evanston, IL 60208, USA  Geneva Laurita  Department
of Chemistry and Biochemistry, Bates College, Lewiston, ME 04240, USA  Quintin
N. Meier  Université Grenoble Alpes, CEA, LITEN, 17 rue des Martyrs, 38054
Grenoble, France  Daniel Olds  National Synchrotron Light Source II,
Brookhaven National Laboratory, Upton, NY 11973, USA  Nicola A. Spaldin
Materials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zürich,
Switzerland  Michael R. Norman<EMAIL_ADDRESS>Materials Science Division,
Argonne National Laboratory, Lemont, IL 60439, USA  James M. Rondinelli
<EMAIL_ADDRESS>Department of Materials Science and Engineering,
Northwestern University, Evanston, IL 60208, USA  Northwestern-Argonne
Institute of Science and Engineering (NAISE), Northwestern University,
Evanston, IL 60208, USA

###### Abstract

The relaxor ferroelectric transition in Cd2Nb2O7 is thought to be described by
the unusual condensation of two $\Gamma$-centered phonon modes,
$\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$. However, their respective roles have
proven to be ambiguous, with disagreement between ab initio studies, which
favor $\Gamma_{4}^{-}$ as the primary mode, and global crystal refinements,
which point to $\Gamma_{5}^{-}$ instead. Here, we resolve this issue by
demonstrating from x-ray pair distribution function measurements that locally,
$\Gamma_{4}^{-}$ dominates, but globally, $\Gamma_{5}^{-}$ dominates. This
behavior is consistent with the near degeneracy of the energy surfaces
associated with these two distortion modes found in our own ab initio
simulations. Our first-principles calculations also show that these energy
surfaces are almost isotropic, providing an explanation for the numerous
structural transitions found in Cd2Nb2O7, as well as its relaxor behavior. Our
results point to several candidate descriptions of the local structure, some
of which demonstrate two-in/two-out behavior for Nb displacements within a
given Nb tetrahedron. Although this suggests the possibility of a charge
analog of spin ice in Cd2Nb2O7, our results are more consistent with a
Heisenberg-like description for dipolar fluctuations rather than an Ising one.
We hope this encourages future experimental investigations of the Nb and Cd
dipolar fluctuations, along with their associated mode dynamics.

##  I Introduction

Relaxor ferroelectrics are a fascinating class of materials [1] whose
dielectric properties depend strongly on frequency, with dynamics described by
a Vogel-Fulcher relation, a relation also found for the magnetic response in
spin-ice materials [2, 3, 4]. They are typically chemically complex solid
solutions of perovskites, with the relaxor behavior thought to be driven by
the dynamics of local polar clusters. Yet the disorder inherent in these
materials inhibits unambiguous interpretation of the data, leading to
differing ideas about the origin of relaxor behavior. Therefore, there has
been a search for stoichiometric materials that exhibit relaxor behavior that
would aid in identifying intrinsic behavior and help settle this debate.

A well-known example is the valence-precise pyrochlore Cd2Nb2O7, which
undergoes a relaxor-type ferroelectric transition between 204 K and 196 K from
a cubic $Fd\overline{3}m$ phase above 204 K (Figure 1a) to a polar
orthorhombic phase (point group $mm2$ [5]), well-described as having $Ima2$
symmetry [6, 7], below 196 K. The intermediate phase between 204 K and 196 K
has been challenging to identify because of its limited temperature range, but
is believed to have an $mmm$ point group [5], implying from group-subgroup
relations that it has $Imma$ symmetry [8]. This is inconsistent with density
functional theory (DFT) studies that find that the structural mode that
defines $Imma$, $\Gamma_{5}^{+}$, is completely stable [9, 7] and so should
not condense.

Figure 1: (a) NbO6 and Cd2O sublattice positions in the ideal
$Fd\overline{3}m$ structure and subsequent (b) $\Gamma_{4}^{-}$ and (c)
$\Gamma_{5}^{-}$ mode distortions that lead to the low temperature
$Ima2$-$\alpha$ polar structure. Adapted from Laurita et al. [7].

Comparison of the $Ima2$ structure with the cubic reference using group theory
identifies $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$ as the dominant distortion
modes describing the symmetry-lowering process (visualized in Figure 1b).
Phonons calculated from first principles for the cubic phase likewise exhibit
two soft phonon modes at $\Gamma$ with irreps matching the $\Gamma_{4}^{-}$
and $\Gamma_{5}^{-}$ distortion modes present in the $Ima2$ structure.
However, the exact nature of the ferroelectric phase transition, and the role
of the two distortion modes, is unclear, and differing scenarios have been
proposed [10].

Furthermore, DFT identifies $\Gamma_{4}^{-}$ as the more unstable mode,
suggesting it is the primary driving mode, although the energy difference
between the two is rather small [9, 7]. However, $\Gamma_{5}^{-}$
displacements are found to be dominant in global structural refinements, and
exhibit a classic order-parameter-like behavior as a function of temperature,
as opposed to the weak temperature dependence of $\Gamma_{4}^{-}$ [6]. We
therefore denote the $\Gamma_{5}^{-}$-dominant phase revealed by the global
refinement as $Ima2$-$\alpha$ and the $\Gamma_{4}^{-}$-dominant phase
suggested by DFT as $Ima2$-$\beta$ in the remainder of the paper.

At temperatures below 100 K, two further transitions occur to monoclinic
phases, with likely space group $Cc$ [11, 9]. Refinements to date have not
been performed for these two monoclinic phases. Again, DFT studies for the
suggested $Cc$ space group indicate that $\Gamma_{4}^{-}$ should be dominant.
Moreover, the phase transition picture is further complicated by the presence
of several additional strain and displacive modes [6], whose role in the
stabilization of the various phases is poorly understood.

In this study, we find from an analysis of x-ray data a local structure that
is dominated by $\Gamma_{4}^{-}$, but reduces to a dominant $\Gamma_{5}^{-}$
for the global structure. This finding is further supported by our DFT study.
Such ‘averaging out’ of the polar $\Gamma_{4}^{-}$ distortions over long
distances is typical behavior for a relaxor ferroelectric. Intriguingly, some
of our local structural descriptions exhibit two-in/two-out behavior of the Nb
displacements, as previously suggested by Malcherek based on simulations of
diffuse scattering data [12]. Although those data suggest Cd2Nb2O7 might be a
long sought charge analog of pyrochlore spin ices [13], our results are more
consistent with a Heisenberg-like rather than Ising description of the dipolar
fluctuations.

##  II Methods

###  II.1 Density functional theory

First-principles calculations were performed using DFT as implemented in the
Vienna ab initio simulation package (VASP) [14, 15] using a plane-wave basis
set with a 750 eV energy cutoff. The Perdew-Burke-Ernzerhof (PBE) exchange-
correlation functional revised for solids (PBEsol) [16] was employed, along
with the projector augmented wave (PAW) potentials to treat the separation of
the core and valence electrons [17].

Energy surfaces were generated by perturbing the high-symmetry cubic
experimental structure (as refined in Ref. [7]) along the soft
eigendisplacements of the force constant matrix. These form two triply
degenerate sets of orthogonal displacement vectors (exhibiting
$\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$ symmetry, respectively), each set of
displacements forming a three-dimensional distortion space. These 3D
distortion spaces were studied by perturbing the crystal structure along
linear combinations of the basis eigenvectors and then relaxing the primitive
cell size and shape until the forces and stress tensor were converged to 2 meV
Å-1 and 5 meV Å-2, respectively.

###  II.2 Sample preparation

A polycrystalline sample of Cd2Nb2O7 was prepared by mixing a stoichiometric
ratio of CdO (Alfa Aesar, 99.998%) and Nb2O5 (Alfa Aesar, 99.998%). The sample
was pressed into a pellet and fired in air at 500 ∘C for 10 h to prevent Cd
volatilization, followed by a subsequent grinding, pressing, and heating at
1000 ∘C for 12 h.

###  II.3 X-ray scattering

X-ray pair distribution function (XPDF) measurements were performed at the
National Synchrotron Light Source II, Brookhaven National Laboratory, on the
Pair Distribution Function (PDF) beamline. The sample was ground into a fine
powder and loaded into a 1 mm Kapton capillary for measurement. Data were
collected upon cooling with a wavelength of 0.1665 Å every 20 K from 300–80 K.
2D scattering data were collected on a Perkin-Elmer flat panel amorphous Si-
based area detector run at 4 Hz, with each dataset corresponding to the sum of
60 second data collection (240 frames). Empty Kapton capillaries were measured
under identical conditions and used for background subtraction. GSAS-II [18]
was utilized to integrate the 2D data to 1D diffraction patterns. Corrections
to obtain $S(Q)$ and subsequent Fourier transform with $Q_{max}=22$ Å-1 and an
$r$-grid of 0.01 Å were performed using the program PDFgetX3 [19] to obtain
the x-ray pair distribution function $G(r)$. These parameters were chosen to
optimize the $r$-resolution while minimizing Fourier termination ripples
satisfactorily for all temperatures across the series. Least-squares
refinement of the XPDF data were performed with PDFgui [20].

##  III Results and Discussion

###  III.1 Previous DFT results

An important finding of previous DFT studies is that a number of space groups
have almost identical energies (Figure 2), with the $\Gamma_{4}^{-}$-dominated
ones having an energy only a few meV per formula unit below those of the
$\Gamma_{5}^{-}$-dominated ones, which are in turn also close in energy [7].
This implies almost isotropic energy surfaces for both modes, which are also
coupled to one another. The Landau theory for this has been written out in our
previous study [21], where we also illustrated the existence of various Higgs
and Goldstone modes, and their anti-phase variants (Leggett modes), that
should exist as a consequence. The former are also found to play an important
role in the related pyrochlore Cd2Re2O7 [22, 23].

Figure 2: Calculated energies for Cd2Nb2O7 in various space groups relative to
the cubic phase. The bars colored in blue have a dominant $\Gamma_{4}^{-}$
component, the ones in red a dominant $\Gamma_{5}^{-}$ component, and the
variant in green is driven by an $X_{4}$ mode. The four structures on the
right without bars are associated with modes that are stable, and therefore do
not lower the energy. Updated from Laurita et al. [7].  Figure 3: (a,b,)
Contour maps of the energy surfaces for the three-dimensional (a)
$\Gamma_{4}^{-}$ and (b) $\Gamma_{5}^{-}$ distortion modes. (c) Energy
reduction relative to the cubic phase as a function of distortion amplitude
along the (a,0,0) (solid curves) and (a,a,0) (dot-dashed curves) order
parameter directions for $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$. The inset
shows the two-dimensional energy surfaces defined by the order parameter
(a,b,0) for $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$. (d) Energy reduction
relative to the cubic phase with constant distortion amplitude, r, varying the
order parameter direction for $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$ through
one half of a great circle, that is (a,a,b) with 2a2+b2=r2, following the path
shown by the black curve on the sphere in the inset.

###  III.2 Local versus global structure

This brings up the question of the nature of the phase transitions at 204 K
and 196 K (it is below the latter temperature that relaxor behavior is most
apparent). It has been proposed that in the ferroelectric phase, there is a
mixture of order-disorder and displacive behavior, with experimental results
indicating that Nb is primarily displacive, whereas Cd is primarily order-
disorder [24, 6, 12]. Alternatively, $\Gamma_{5}^{-}$ may be more displacive-
like while $\Gamma_{4}^{-}$ may be more order-disorder-like. This highlights
the fact that the DFT analysis is for a homogeneous material, whereas relaxor
behavior implies local inhomogeneities.

To address these points, we note that while crystallographic analysis (such as
Rietveld refinement) can provide a detailed description of global order and
phase transitions as a function of temperature, local probes are necessary to
reveal any local distortions or mid-range ordering that might average out in a
global refinement. Techniques such as IR and Raman spectroscopy are valuable
for describing the coordination environment of the cations, but are unable to
describe any structural ordering between next-nearest neighbors, such as Cd-
Cd/Nb-Nb ($M-M$) and Cd-Nb ($M-M^{\prime}$) interactions. The pair
distribution function (PDF) is obtained from a combination of Bragg and
diffuse scattering events when a material is irradiated, and is a histogram of
all atom-atom interactions in a material, presented as a function of
$r$-range. This technique is similarly useful for describing coordination
environments (typically captured in peaks below 3 Å), and can additionally
provide insight on mid- to long-range atomic correlations. Local ordering is
often hidden as structural disorder in the form of enlarged atomic
displacement parameters in a Rietveld refinement, but can have meaningful
impact on the observed behavior of a material, such as relaxor behavior. In
this study, ‘local’ correlations were fit over a range of 1.7 - 5.7 Å, ‘mid-
range’ correlations were fit over a range of 1.7 - 12.0 Å, and ‘long-range’ or
‘average’ correlations were fit over a range of 5.7 - 30 Å. For our study,
x-ray scattering was chosen as a probe due to the high neutron absorption
cross-section of Cd (2520 barn) [25]. It should be noted that this necessary
choice unfortunately de-emphasizes contributions from oxygen, which is a poor
scatterer of x-rays in comparison to the heavy metals in Cd2Nb2O7; however,
meaningful information about Cd-Cd/Nb-Nb ($M-M$) and Cd-Nb ($M-M^{\prime}$)
interactions can be obtained, as well as some information on Cd-O and Nb-O
bond interactions. Data were collected from room temperature to 80K to
elucidate any local behavior in the cubic and orthorhombic phases, and as such
our analysis will not experimentally address the lower temperature monoclinic
phases mentioned above.

###  III.3 Local $\Gamma_{4}^{-}$ displacements: DFT simulations

Before analyzing the coupling and competition between the two dominant
distortion modes, we consider the energy landscape associated with each mode
independently. Using the eigendisplacements of the force constant matrix as a
basis set, we compute the energy surfaces for pure $\Gamma_{4}^{-}$ and pure
$\Gamma_{5}^{-}$ distortions in three dimensions (Figure 3). Corresponding
order-parameter directions and symmetries for the modes are given in Table 1.
We find that both energy surfaces are remarkably isotropic, resulting in a 3D
version of the well known ‘Mexican hat’ potential that describes the
corresponding 2D case. Pure displacements associated with the order parameter
direction along the cubic axes are slightly more energetically favorable than
off-axis directions for $\Gamma_{5}^{-}$, but for the most part a displacement
of a given magnitude lowers the energy to the same degree regardless of the
order parameter direction (Figure 3c). The particularly high degree of
$\Gamma_{4}^{-}$ isotropy is consistent with polarization measurements near
the ferroelectric transition temperature, which found that the polarization
direction is easily rotated by an electric field [5]. Note that these
calculations describe the 0 K state, and we might expect thermal energy to
further reduce the weak anisotropy. Indeed, the barrier between energy minima
is less than 0.5 meV/f.u. for $\Gamma_{5}^{-}$ and less than 0.1 meV/f.u. for
$\Gamma_{4}^{-}$ (Figure 3d), implying fluctuations between order parameter
directions should readily take place. A similar behavior for the energy
surface of $\Gamma_{4}^{-}$ has been remarked on for perovskite relaxors [26].

Table 1: Order parameter directions for space groups within the manifold of $\Gamma_{4}^{-}$, $\Gamma_{5}^{-}$ modes (generated using 27). $\Gamma_{4}^{-}$ | $\Gamma_{5}^{-}$ | Space Group
---|---|---
(a,0,0) |  | $I4_{1}md$
| (a,0,0) | $I\overline{4}2d$
(a,0,0) | (0,b,0) | $Fdd2$
(a,a,0) | (0,b,-b) | $Ima2$
(a,b,0) | (0,c,d) | $Cc$
(a,a,b) | (0,c,-c) | $Cm$
Figure 4: (a) Goodness of fit ($R_{w}$) as a function of temperature for a
series of candidate space groups with a fit $r_{max}$ of 5.7 Å, 12.0 Å, and 30
Å. The fit against the cubic $Fd\overline{3}m$ structure is shown as black
circles, space groups with primarily $\Gamma_{4}^{-}$ distortion modes are
shown as blue symbols, space groups with primarily $\Gamma_{5}^{-}$ distortion
modes are shown as red symbols, and the space group with a primary $\Gamma$5+
distortion mode ($Imma$) is shown as yellow symbols. (b) Local fits (1.7–5.7
Å) of the XPDF data against the $I4_{1}md$ and $I\overline{4}2d$ space groups
at 240 K, 180 K, and 120 K. The $M$-O correlations are indicated by the red
short-dash lines, and the $M$-$M^{\prime}$ correlations by the blue long-dash
lines.

Whereas the energy scales are comparable, we nonetheless observe that the
$\Gamma_{4}^{-}$ mode is both more unstable (i.e., it has a more negative
curvature of the energy versus displacement around zero amplitude) and has a
lower energy minimum than $\Gamma_{5}^{-}$, consistent with previous first
principles investigations [9, 7]. Until now, this has been perceived as
inconsistent with experimental measurements, where $\Gamma_{5}^{-}$
displacements dominate the global refinement [6]. However, the isotropic
energy surfaces of the two modes imply that long-range characterization
techniques may be blind to the local mode dynamics along the energy surfaces
[21]. We also observe that the $\Gamma_{4}^{-}$ energy surface is
significantly more isotropic than the $\Gamma_{5}^{-}$ one (Figure 3c,d),
implying that the $\Gamma_{4}^{-}$ distortions exhibit a greater degree of
disorder at higher temperatures. Conversely, the larger anisotropy of the
$\Gamma_{5}^{-}$ energy surface enables $\Gamma_{5}^{-}$ to condense over
longer length scales than $\Gamma_{4}^{-}$, consistent with the dominance of
the $\Gamma_{5}^{-}$ mode in global refinements of the crystallographic
structure.

###  III.4 Local $\Gamma_{4}^{-}$ displacements: X-ray scattering

To investigate this further, we performed XPDF measurements as a function of
temperature and fit the results with a variety of space groups suggested from
the calculated energy surface as possible descriptions of the local symmetry
breaking. Eight candidate models with decreasing symmetry/ordering were chosen
to fit the data, falling into four categories: (1) the high temperature cubic
$Fd\overline{3}m$ structure, (2) five candidate structures dominated by the
$\Gamma_{4}^{-}$ distortion mode, (3) two candidate structures dominated by
the $\Gamma_{5}^{-}$ distortion mode, and (4) an orthorhombic $Imma$ structure
previously suggested as the intermediate phase between $Fd\overline{3}m$ and
$Ima2$ (for (2) and (3), see Table 1). All structures were derived from our
previous work on Cd2Nb2O7 [7], except for the $\Gamma_{4}^{-}$-dominated $Cc$
structure which was taken from Ref. [9] and further relaxed with DFT. To
minimize the number of refined parameters in the least-squares analysis of the
PDF, atomic positions were fixed to the theoretical atomic positions obtained
from the DFT analysis. Due to the minimal x-ray scattering power of oxygen in
the presence of heavier cations, the oxygen atomic displacement parameters
(ADPs) were modeled as being isotropic (Uiso) with a fixed value of 0.01 Å2.
For these fits, the scale, lattice parameters, and isotropic Cd and Nb ADPs
were allowed to refine, and a correlated motion correction ($\delta$-2) was
applied to the fits to account for peak sharpening at low-$r$. Goodness of fit
(reported as $R_{w}$) as a function of temperature for each candidate space
group is shown in Figure 4.

Figure 5: Fits of the XPDF data collected at 300 K against proposed structures
for Cd2Nb2O7: (a) cubic $Fd\overline{3}m$, (b) orthorhombic $Fdd2$, (c)
orthorhombic $Ima2$-$\alpha$, (d) orthorhombic $Ima2$-$\beta$, and (e)
monoclinic $Cc$. Fit ranges are indicated by the range shown in each panel,
$M$-O correlations of interest are shown as short red dashes, and
$M$-$M$/$M$-$M^{\prime}$ correlations of interest are shown as long blue
dashes.

From a long-range perspective (correlations up to 30 Å), the space groups
dominated by $\Gamma_{4}^{-}$ distortions better fit the data at high
temperature, with a cross-over to structures dominated by the $\Gamma_{5}^{-}$
distortion mode around 140 K (bottom graph of Figure 4a). This length scale
provides a description of the data closer to the crystallographic structure
(i.e. encompassing several degrees of atom-atom interactions), but does not
capture any microstructural effects that might be at play, such as the role of
nanodomains in the material. Such analysis would be better suited to
microscopy studies [28], which are outside of the scope of the present work.

It should be noted that, whereas the XPDF fits provide insight into the
prevalence of $\Gamma_{4}^{-}$ versus $\Gamma_{5}^{-}$-type local distortion
patterns, it is important to be conservative in stating which specific
structure is the best description. For instance, the $Cm$ space group has nine
refined parameters as opposed to three in $Fd\overline{3}m$, five in $Fdd2$,
seven in $Ima$2, and eight in $Cc$. Nonetheless, since all
$\Gamma_{4}^{-}$-based structures have a lower $R_{w}$ than both of the
$\Gamma_{5}^{-}$ space groups, we can say that the $\Gamma_{4}^{-}$ distortion
pattern better describes the data over this fit range.

At 300 K, both cubic and $\Gamma_{4}^{-}$-type models equally describe the
data out to 30 Å, with the highest symmetry cubic $Fd\overline{3}m$ having
larger atomic displacement parameters (ADPs) and the lowest symmetry $Cm$
structure having small distortions in the atomic positions, suggesting that
the cubic and $\Gamma_{4}^{-}$-type models are equivalent descriptions at this
length scale. However, both the enlarged anisotropic ADPs in the ideal
$Fd\overline{3}m$ model and atom off-centering in the $\Gamma_{4}^{-}$ models
indicate that distortions are present and cations are not located at the
center of their coordination environments. In the mid-range region
(correlations up to 12 Å, middle graph of Figure 4a), the trend is similar to
the long-range fits, with $Ima2$-$\beta$ providing a better description of the
mid-range structure upon cooling to 120 K, whereas $Ima2$-$\alpha$ becomes the
better description of correlations at this length scale for lower
temperatures. In the local range (correlations up to 5.7 Å, top graph of
Figure 4a), the space groups with primarily $\Gamma_{4}^{-}$ distortion modes
provide a better fit of the data at all temperatures. $Cc$ has the lowest
$R_{w}$ at 300 K and exhibits similar $R_{w}$ values to $Cm$, $Ima2$-$\beta$,
and $Fdd2$ upon cooling.

To better understand the nature of the distortions, comparisons of the fits to
the $I4_{1}md$ ($\Gamma_{4}^{-}$ only) and $I\overline{4}2d$ ($\Gamma_{5}^{-}$
only) structures above (240 K), near (180 K), and below (120 K) the
ferroelectric transition were performed. The local fits (1.7–5.7 Å) to the
XPDF are shown in Figure 4b. At all temperatures, $I4_{1}md$ provides a better
fit than $I\overline{4}2d$ of the Cd-O peaks (Cd coordination environment) and
the $M-M^{\prime}$ peak at approximately 5.2 Å, which is the next-nearest Cd-
Nb correlation. Moreover, at all temperatures, $Fdd2$ further improves the fit
of all $M$-O peaks. $Ima2$-$\beta$ has a similar improvement in the $M$-O
local peaks, in addition to a better fit of the mid-range $M-M$ and
$M-M^{\prime}$ peaks. This is illustrated for 300 K data in Figure 5.

###  III.5 Intermediate phase scenario

Neither our first-principles results nor our experimental data provide strong
evidence of an intermediate phase between 204 K and 196 K belonging to point
group $mmm$ before the relaxor ferroelectric transition occurs. We calculate
no unstable or low-lying phonon modes in the high-symmetry structure which
break the requisite symmetries, and XPDF data do not favor the $Imma$
structure at any length scale. Based on the above results, we can offer a
possible scenario for the intermediate phase. Our DFT results indicate that
$Cm$, $Cc$, $Ima2$-$\beta$, $Fdd2$, and $I4_{1}md$ structures have almost
identical energies (Figure 2). The last two only differ in that $I4_{1}md$ has
no $\Gamma_{5}^{-}$ component. To understand the proposed scenario, note that
$Fdd2$, $Ima2$ and $Cc$ define a great circle on each of the $\Gamma_{4}^{-}$
and $\Gamma_{5}^{-}$ energy surfaces ($\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$
are each three-dimensional irreps, see Figure 3), with $Fdd2$ corresponding to
$\phi=0^{\circ}$, $Ima2$ to $\phi=45^{\circ}$, and $Cc$ to an intermediate
angle, as can be seen from Table 1 [21]. This is exactly analogous to
Cd2Re2O7, where the respective space groups are $Ima2$, $I4_{1}22$ and $F222$.
For the latter, one has a proposed sequence of transitions from
$Fd\bar{3}m\rightarrow I4m2\rightarrow I4_{1}22\rightarrow F222$ [29], which
corresponds to various positions around the brim of the ‘Mexican hat’ energy
surface (in this case, from $\Gamma_{3}^{-}$ which is a two-dimensional
irrep). The analogous series for Cd2Nb2O7 would be $Fd\bar{3}m\rightarrow
Fdd2\rightarrow Ima2\rightarrow Cc$. There are several points in favor of this
scenario, in that it explains the experimental absence of the (0,0,10) Bragg
peak in the temperature range of 196K to 204K [8], since this is allowed for
$Ima2$ but not for $Fdd2$. It also explains why optical studies indicate a
ferroelectric polarization vector along the cubic axis [5] as opposed to
$Ima2$ where it is along (1,1,0) instead. $Fdd2$ is also found to be the
global space group upon sulfur doping [7] and has been proposed to be the
local structure for Cd2Nb2O7 itself [12]. The claim of a centrosymmetric space
group ($mmm$ point group) in this temperature range [5] could then be a domain
averaging effect, similar to the phenomenon observed in cubic BaTiO3 [30].

###  III.6 Cd and Nb displacements

Our results help to explain the challenges associated with identifying
possible intermediate space groups. The isotropic energy surfaces (Figure 3)
imply that local distortions along multiple order parameter directions may
occur simultaneously, a result corroborated by the number of space groups with
comparable goodness of fit in the XPDF data. The observation that local fits
differ significantly from the mid- and long-range fits also supports the
conclusion that local structural disorder (in the form of uncorrelated atomic
displacements) plays a large role in the relaxor ferroelectric transition of
Cd2Nb2O7.

The combination of DFT and our XPDF data implies a distinction between the
role of the two primary distortion modes. $\Gamma_{4}^{-}$ displacements
dominate the local structure even at high temperatures, whereas
$\Gamma_{5}^{-}$ displacements condense over longer length scales at lower
temperatures, consistent with previous global refinements [6].

Figure 6:  (a,b) Displacement pattern of the relaxed $Fdd2$ structure relative
to the high-symmetry cubic structure ($Fd\overline{3}m$) for the (a) Nb
sublattice and (b) Cd-O sublattice. (c,d) Displacement pattern of the relaxed
$\Gamma_{4}^{-}$-dominated $Ima2$-$\beta$ structure relative to the high-
symmetry cubic structure ($Fd\overline{3}m$) for the (c) Nb sublattice and (d)
Cd-O sublattice.

Visualizing the $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$ displacement patterns
(Figure 6) also allows us to comment on the relative role of Cd and Nb in the
phase transitions. As Nb displacements (likely due to the second-order Jahn-
Teller effect typical of $d^{0}$ ions) are central to the ferroelectric
behavior, we turn to a detailed description of these. As commented on by
Malcherek [12], $Fdd2$ is a possible description of the local structure. This
is not inconsistent with our analysis, in that $Fdd2$ produces one of the
lowest $R_{w}$ values at the local scale below 270K (Figure 4a) and
corresponds to a local energy minimum for the $\Gamma_{4}^{-}$ mode (see Table
1 and Figure 3). The $Fdd2$ displacements correspond to the classic two-
in/two-out behavior characteristic of spin moments in pyrochlore spin ices.
That is, for a given tetrahedron of Nb, two of the Nb displacements point
inwards, two point outwards (Figure 6), with the Nb displacements along the
Nb-Nb bond directions as opposed to along the local trigonal axis as in spin
ices (here, the polar displacement along $c$ drops out, since all Nb ions are
shifted by this). A similar picture has been advocated for Bi displacements in
Bi2Ti2O7 [13]. This implies that some of the same physics found in spin ices
might be relevant for the dielectric response for Cd2Nb2O7. In particular,
violation of the ice rules leads to the creation of monopole-antimonopole
pairs (i.e., three-in/one-out–three-out/one-in), whose dynamics can explain
the Vogel-Fulcher relation for the relaxation rate. We note that monopole-
antimonopole pairs have also been invoked in the context of cubic perovskite
relaxors, where they represent eight-in–eight-out instead [31].

However, as noted above, assignment of a specific local space group on the
basis of our XPDF data is inadvisable. For example, XPDF fits against
$Ima2$-$\beta$ ($\Gamma_{4}^{-}$-dominated) can produce $R_{w}$ values even
lower than $Fdd2$. This can be attributed in part to the larger number of
structural parameters in $Ima2$-$\beta$ than $Fdd2$, but nevertheless we
cannot justify ascribing the local structure to one or the other. Similarly,
the energy difference between $Fdd2$ and $Ima2$-$\beta$ as computed with DFT
is below 1 meV/f.u., meaning neither can be called more energetically
favorable. Therefore, while a charge ice description of Cd2Nb2O7 is
attractive, our results indicate a more Heisenberg-like description of the
dipolar fluctuations than the Ising-like one implied by a charge ice picture.
That is, our data depict an isotropic, $\Gamma_{4}^{-}$-dominated energy
surface (Figure 3), implying that Nb atoms displace within their local
coordination environments along various order parameter directions within the
$\Gamma_{4}^{-}$ distortion space.

We now turn to Cd displacements. Cd has two short bonds with oxygen along its
local trigonal axis (O(2) ions) and six long bonds with oxygen in the NbO6
octahedra (O(1) ions). To improve bonding with the O(1) ions, Cd displaces
normal to the local trigonal axis (Figure 6), as is seen in many pyrochlores.
For instance, diffuse scattering in La2Zr2O7 finds a pronounced annular
displacement of La normal to the local trigonal axis [32], consistent with
what we find for Cd in Cd2Nb2O7, and Cd likely remains disordered in the
ferroelectric phase [24]. As such, Cd displacements are unlikely to drive the
symmetry-lowering either locally or over longer ranges.

##  IV Conclusion

We have presented evidence that Cd2Nb2O7 is locally characterized by the
ferroelectric mode $\Gamma_{4}^{-}$ but globally described by the non-polar
mode $\Gamma_{5}^{-}$. This observation relates this pyrochlore to perovskite
relaxor ferroelectrics whose relaxor behavior is attributed to local polar
clusters [1]. Recently, further insights into relaxor behavior in perovskites
have been achieved by contrasting XPDF, which emphasizes higher $Z$ ions, with
neutron PDF where oxygens play a significant role [33]. A neutron PDF analysis
in Cd2Nb2O7 would be possible with Cd isotopically-enriched samples. This
could be complemented by a DFT determination of the various parameters in the
Landau free energy that has been critical in studies of structural phase
transitions, such as in the perovskite YMnO3 where the amplitude of the
structural order parameter is order-disorder-like but the phase behaves
displacively [34, 35, 36], a study we hope to report on in the future.

In addition, we have presented candidate descriptions of the local structure,
some of which ($Fdd2$ and $I4_{1}md$) are dominated by two in/two out
displacements of Nb ions in a given Nb tetrahedron, which might imply that
Cd2Nb2O7 is a charge ice analog of the pyrochlore spin ices that have been
much studied in the past two decades. Our results, however, are consistent
with a richer energy landscape that is more Heisenberg-like than Ising-like.
Therefore, inspired by recent work in spin ices [2, 3, 4], we believe that the
dynamics of the dielectric response of Cd2Nb2O7 is worth re-investigating.

###### Acknowledgements.

D.H.Y and J.M.R. were supported by the National Science Foundation (NSF) under
award number DMR-2011208. G.L. acknowledges support for this work from Bates
College, and from the National Science Foundation (NSF) through DMR 1904980.
M.R.N. was supported by the Materials Sciences and Engineering Division, Basic
Energy Sciences, Office of Science, U.S. Department of Energy. N.A.S. was
supported by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program project HERO grant (No. 810451)
and by ETH Zurich. Q.N.M. was supported by the Swiss National Science
Foundation under Project No. P2EZP2_191872. This research used the Pair
Distribution Function (PDF) beamline of the National Synchrotron Light Source
II, a U.S. Department of Energy (DOE) Office of Science User Facility operated
for the DOE Office of Science by Brookhaven National Laboratory under Contract
No. DE-SC0012704. Calculations were performed using the Department of Defense
High Performance Computing Modernization Program (DOD-HPCMP).

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#  Second Law and Liu Relations:
The No-reversible-direction Axiom - Revisited

W. Muschik111Corresponding author<EMAIL_ADDRESS>
Institut für Theoretische Physik
Technische Universität Berlin
Hardenbergstr. 36
D-10623 BERLIN, Germany

Abstract: A thermodynamic process is governed by balance equations in field-
formulated thermodynamics. Especially the balance equation of entropy takes a
prominent role: it introduces the Second Law in the form of a dissipation
inequality via the non-negative entropy production. Balance equations and
dissipation inequality are independent of the considered material which is
described by additional constitutive equations which need the introduction of
a state space. Inserting these constitutive equations into the balance
equations results in the balance equations on state space which include
process directions. Why do not appear the latter in the Liu Relations which
describe material as the equations on state space do ? The answer is given by
the axiom: process directions which are in the kernel of the balance equations
on state space do not enter the entropy production. A toy example concerning
heat conduction in compressible fluids is added for elucidation.
Keywords: balance equations on state space; dissipation inequality; process
directions; No-reversible-direction Axiom; Coleman-Mizel formulation of the
2nd law; Liu Relations.

##  1 Introduction

###  1.1 Basic concepts

Non-equilibrium thermodynamics in field formulation222statistical and
stochastic thermodynamics are out of scope here launches out with balance
equations333the other description is: thermodynamics of discrete systems[1, 2,
3]. The minimal concept consists of the four balance equations for spinless
materials

mass: $\displaystyle\ d_{t}\varrho+\varrho\nabla\mbox{\boldmath{$v$}}:{\bf 1}\
=\ 0,$ (1) momentum: $\displaystyle\ \varrho
d_{t}\mbox{\boldmath{$v$}}+\nabla\cdot{\bf P}-\mbox{\boldmath{$k$}}\ =\
\mbox{\boldmath{$0$}},\ \ {\bf P}^{\top}={\bf P},$ (2) internal energy:
$\displaystyle\ \varrho
d_{t}\varepsilon+\nabla\cdot\mbox{\boldmath{$q$}}+\nabla\mbox{\boldmath{$v$}}:{\bf
P}-r\ =\ 0,$ (3) entropy: $\displaystyle\ \varrho
d_{t}s+\nabla\cdot\mbox{\boldmath{$\Phi$}}-\gamma-\sigma\ =\ 0,\ \ \sigma\geq
0,$ (4)

(mass density $\varrho$, material velocity $v$, symmetric pressure tensor
${\bf P}$, external force density $k$, specific internal energy $\varepsilon$,
heat flux density $q$, energy supply $r$, specific entropy $s$, entropy flux
density $\Phi$, entropy supply $\gamma$, non-negative entropy production
$\sigma$, $d_{t}:=\partial_{t}+\mbox{\boldmath{$v$}}\cdot\nabla$)444⊤means
transposed, a:=b marks a setting by definition, a is defined by b..

The entropy balance equation (4)1 results by taking the 2nd law (4)2 into
account in a dissipation inequality

$\varrho d_{t}s+\nabla\cdot\mbox{\boldmath{$\Phi$}}-\gamma\ \geq\ 0,$ (5)

and the question arises, how to achieve the compatibility of the balance
equations (1) to (3) with the dissipation inequality (5)?

According to (1) to (3), the wanted (basic) fields span the (large) state
space [4, 5]

${\cal
Z}\ni\mbox{\boldmath{$z$}}:=(\varrho,\mbox{\boldmath{$v$}},\varepsilon).$ (6)

The other quantities in (1) to (3) and (5) are constitutive fields which
depend on the basic fields

${\bf M}:=({\bf
P},\mbox{\boldmath{$q$}},s,\mbox{\boldmath{$\Phi$}})(\mbox{\boldmath{$z$}})$
(7)

or are given by the system’s environment

${\bf N}:=(\mbox{\boldmath{$k$}},r,\gamma).$ (8)

Consequently, the differential operators $d_{t}$ and $\nabla$ act on a
constitutive field $\boxplus$ by applying the chain rule

$\nabla\cdot\boxplus=\frac{\partial\boxplus}{\partial\mbox{\boldmath{$z$}}}:\nabla\mbox{\boldmath{$z$}},\quad
d_{t}\boxplus=\frac{\partial\boxplus}{\partial\mbox{\boldmath{$z$}}}\cdot
d_{t}\mbox{\boldmath{$z$}}.$ (9)

Inserting (9) into the balance equations (1) to (3) results in the so-called
balance equations on the state space [6].

A special non-systematical derived example of the dissipation inequality (5)
is the Clausius-Duhem inequality [7, 8] which results from the setting

$\mbox{\boldmath{$\Phi$}}:=\frac{\mbox{\boldmath{$q$}}}{\Theta},\qquad\gamma:=\frac{r}{\Theta}$
(10)

(temperature555$\Theta$ may be the local equilibrium temperature by
presupposing local equilibrium or a non-equilibrium temperature such as the
contact temperature [9, 10, 11, 12]. $\Theta$) which connects constitutive
fields of the dissipation inequality (5) with those of the internal energy
balance (3) by physical intuition.

###  1.2 Toy example: Heat conducting fluid

The balance equations (1) to (4) do not contain the temperature, and
consequently, the temperature is not a state space variable (6). Introducing
the specific free energy

$\psi:=\varepsilon-\Theta s,$ (11)

the internal energy balance (3) becomes666$[\otimes]^{s}$ is the symmetric
part of $\otimes$

$\varrho d_{t}\psi+\varrho\Theta d_{t}s+\varrho
sd_{t}\Theta+\nabla\cdot\mbox{\boldmath{$q$}}+[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}-r\ =\ 0.$ (12)

By the setting (11), the temperature $\Theta$ becomes a state variable,
whereas the specific free energy $\psi$ and the specific entropy $s$ are
constitutive equations on the state space replacing the internal energy
$\varepsilon$.

Taking the dissipation inequality (5) into accout, (12) results in an other
shape of the dissipation inequality which refers in contrast to (5) not to the
entropy density but to the internal energy density

$\varrho
d_{t}\psi-\Theta\nabla\cdot\mbox{\boldmath{$\Phi$}}+\Theta\gamma+\varrho
sd_{t}\Theta+\nabla\cdot\mbox{\boldmath{$q$}}+[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}-r\leq 0.$ (13)

Multiplication with $1/\Theta$ generates a dissipation inequality with regard
to the entropy density

$\frac{\varrho}{\Theta}(d_{t}\psi+sd_{t}\Theta)+\nabla\cdot(\frac{\mbox{\boldmath{$q$}}}{\Theta}-\mbox{\boldmath{$\Phi$}})-\mbox{\boldmath{$q$}}\cdot\nabla\frac{1}{\Theta}+(\gamma-\frac{r}{\Theta})+\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}\leq 0.$ (14)

Introducing expressions

$\mbox{\boldmath{$\Omega$}}\ :=\
\frac{\mbox{\boldmath{$q$}}}{\Theta}-\mbox{\boldmath{$\Phi$}},\qquad\kappa\
:=\ \gamma-\frac{r}{\Theta}$ (15)

which are zero, if the Clausius-Duhem relations (10) for the entropy flux
density $\Phi$ and the entropy supply $\gamma$ are valid. The dissipation
inequality (14) becomes

$-\frac{\varrho}{\Theta}(d_{t}\psi+sd_{t}\Theta)-\nabla\cdot\mbox{\boldmath{$\Omega$}}+\mbox{\boldmath{$q$}}\cdot\nabla\frac{1}{\Theta}-\kappa-\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}\geq 0.$ (16)

The state space and the constitutive fields which belong to (16) and (12) are

$\mbox{\boldmath{$z$}}=(\varrho,\Theta,\nabla\Theta,[\nabla\mbox{\boldmath{$v]^{s}$}}),\quad{\bf
M}=(\psi,s,\mbox{\boldmath{$q$}},{\bf P},r,\mbox{\boldmath{$\Phi$}},\gamma).$
(17)

Before this example is continued in sect.3, some general remarks on the
program and on the balance equations on state space in matrix formulation are
made.

###  1.3 The program

Now the question arises again, how to solve the balance equations on state
space [(1) to (3) with (9) inserted] by taking into consideration one of the
dissipation inequalities (5) or (14) [13]? Using the cumbersome procedure of
first solving the balance equations and then checking up whether the
dissipation inequality is satisfied at each position for all times, or better
generating a class of materials [4] which always satisfies the dissipation
inequality by construction? Here, the second case is considered by using an
easy, but general toy-model which lightens the understanding.

Because the differential operators of the balance equations on state space are
linear, the resulting balance equations are linear in the so-called higher
derivatives
$\mbox{\boldmath{$y$}}^{\top}=(d_{t}\mbox{\boldmath{$z$}},\nabla\mbox{\boldmath{$z$}})$
which are time and position derivatives of the state space variables $z$ which
are not included in the state space itself [4]. Consequently, the higher
derivatives represent process directions in the state space. By using the
evident, but nevertheless axiomatic statement

”process directions which are in the kernel (null-space) of the balance
equations (on

state space) do not appear in the entropy production” 777more details in
sect.8,
the no-reversible-directions axiom can be established [6, 14]:

the local entropy production does not depend on the process directions.
These statements allow to introduce a process independent Lagrange multiplier
which connects the balance equations directly to the dissipation inequality
resulting in the Liu Relations which do not include the process direction
anymore [15]. The Liu Relations represent constraints on the constitutive
equations with regard to the dissipation inequality which can be exploited
without taking process directions into account. How this sketched program
works in detail is performed below by considering a set of general balance
equations and the special Clausius-Duhem example presented in sect.1.1.

##  2 Balance Equations on State Space

Local balance equations can be written in different shapes

$\mbox{$\displaystyle\frac{\partial}{\partial
t}$}(\rho\Psi)+\nabla\cdot(\rho\mbox{\boldmath{$v$}}\Psi+\mbox{\boldmath{$\Phi$}})-\Sigma=0\
\longrightarrow\ \partial_{t}u_{A}+\partial_{j}\Phi^{j}_{A}=r_{A},\qquad
A=1,2,....,M.$ (18)

Here $\Psi$ or $u_{A}$ is the balanced basic field which may be of arbitrary
rank, $\rho$ the mass density, $\Phi$ is the non-convective flux of $\Psi$ and
$\Phi^{j}_{A}$ the total flux of $u_{A}$, $v$ the material velocity and
$\Sigma$ or $r_{A}$ the sum of production and supply of $\Psi$ or $u_{A}$,
respectively. Among the $M$ equations is that of the entropy $A:=S$

$u_{S}\ =\ \rho s,\quad\Phi^{j}_{S}\ =\ v^{j}\rho s+\Xi^{j},\quad r_{S}\ =\
\gamma+\sigma\ \geq\ \gamma$ (19)

(specific entropy $s$, non-convective entropy flux $\Xi^{j}$, entropy supply
density $\gamma$, entropy production density $\sigma\geq 0$). The inequality
(19)4 characterizes the Second Law.

The fields in (18)2 depend on the considered material which is characterized
by a state space ${\cal Z}\ni\mbox{\boldmath{$z$}}$ which is spanned by its
components $z$. These material dependent fields are the constitutive equations

$u_{A}\ =\ u_{A}(\mbox{\boldmath{$z$}}),\quad\Phi^{j}_{A}\ =\
\Phi^{j}_{A}(\mbox{\boldmath{$z$}}),\quad r_{A}\ =\
r_{A}(\mbox{\boldmath{$z$}})$ (20)

which are inserted into the balance equations (18)2 and the dissipation
inequality (19)4. Applying the chain rule results in the so-called balance
equations on state space (21) and the dissipation equation on state space (22)
[4, 5]

$\displaystyle\frac{\partial
u_{A}(\mbox{\boldmath{$z$}})}{\partial\mbox{\boldmath{$z$}}}\partial_{t}\mbox{\boldmath{$z$}}+\frac{\partial\Phi^{j}_{A}(\mbox{\boldmath{$z$}})}{\partial\mbox{\boldmath{$z$}}}\partial_{j}\mbox{\boldmath{$z$}}$
$\displaystyle=$ $\displaystyle r_{A}(\mbox{\boldmath{$z$}}),$ (21)
$\displaystyle\frac{\partial
u_{S}(\mbox{\boldmath{$z$}})}{\partial\mbox{\boldmath{$z$}}}\partial_{t}\mbox{\boldmath{$z$}}+\frac{\partial\Phi^{j}_{S}(\mbox{\boldmath{$z$}})}{\partial\mbox{\boldmath{$z$}}}\partial_{j}\mbox{\boldmath{$z$}}$
$\displaystyle\geq$ $\displaystyle\gamma(\mbox{\boldmath{$z$}})$ (22)

which are linear in the higher derivatives

$\mbox{\boldmath{$y$}}^{\top}\ :=\
\Big{(}\partial_{t}\mbox{\boldmath{$z$}},\partial_{j}\mbox{\boldmath{$z$}}\Big{)}.$
(23)

Introducing the matrices

$\displaystyle{\bf A}(\mbox{\boldmath{$z$}})\ :=\ \Big{(}\frac{\partial
u_{A}(\mbox{\boldmath{$z$}})}{\partial\mbox{\boldmath{$z$}}},\frac{\partial\Phi^{j}_{A}(\mbox{\boldmath{$z$}})}{\partial\mbox{\boldmath{$z$}}}\Big{)},$
$\displaystyle\mbox{\boldmath{$B$}}(\mbox{\boldmath{$z$}})\ :=\
\Big{(}\frac{\partial
u_{S}(\mbox{\boldmath{$z$}})}{\partial\mbox{\boldmath{$z$}}},\frac{\partial\Phi^{j}_{S}(\mbox{\boldmath{$z$}})}{\partial\mbox{\boldmath{$z$}}}\Big{)},$
(24) $\displaystyle\mbox{\boldmath{$C$}}(\mbox{\boldmath{$z$}})\ :=\
r_{A}(\mbox{\boldmath{$z$}}),\hskip 71.13188pt$ $\displaystyle
D(\mbox{\boldmath{$z$}})\ :=\ \gamma(\mbox{\boldmath{$z$}}),$ (25)

the balances on state space (21) and (22) write

${\bf A}(\mbox{\boldmath{$z$}})\mbox{\boldmath{$y$}}\ =\
\mbox{\boldmath{$C$}}(\mbox{\boldmath{$z$}}),\qquad\mbox{\boldmath{$B$}}(\mbox{\boldmath{$z$}})\mbox{\boldmath{$y$}}\
\geq\ D(\mbox{\boldmath{$z$}}).$ (26)

The wanted basic fields are
$u_{A}\Big{(}\mbox{\boldmath{$z$}}(t,\mbox{\boldmath{$x$}})\Big{)}$, whereas
$\Phi_{A}^{j}\Big{(}\mbox{\boldmath{$z$}}(t,\mbox{\boldmath{$x$}})\Big{)}$ and
$r_{A}\Big{(}\mbox{\boldmath{$z$}}(t,\mbox{\boldmath{$x$}})\Big{)}$ are given
as functions of the basic fields or from the environment of the considered
system.

##  3 Example Continued: Higher derivatives

The balance equations (1) and (2) and the entropy balance (4) are now
transformed into the matrix shape (26) which is linear in the following eight
higher derivatives

$\mbox{\boldmath{$y$}}^{\top}\ =\
\Big{(}d_{t}\varrho,\nabla\varrho,d_{t}\mbox{\boldmath{$v$}},d_{t}\Theta,d_{t}\nabla\Theta,\nabla\nabla\Theta,d_{t}[\nabla\mbox{\boldmath{$v$}}]^{s},\nabla[\nabla\mbox{\boldmath{$v$}}]^{s}\Big{)}$
(27)

which follow from the linear balance equations and from the state space (17)1.
Thus, $y$ is a (8,1)-matrix and consequently, $\bf A$ is a (2,8)-matrix
because two balances, (1) and (2), are taken into account, whereas the balance
of the specific free energy (12) is taken into consideration by the
dissipation inequality (16) as demonstrated by replacing $d_{t}s$ in (12).
Because the dissipation inequality is scalar, $B$ is a (1,8)-matrix according
to (26)2.

In detail, the matrix formulation of the balance equations (1) and (2) is

$\displaystyle{\bf A}\mbox{\boldmath{$y$}}:=\hskip 284.52756pt$
$\displaystyle=\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&(\partial{\bf P}/\partial\varrho)&\varrho&0&0&(\partial{\bf
P}/\partial\nabla\Theta)&0&(\partial{\bf
P}/\partial[\nabla\mbox{\boldmath{$v$}}]^{s})\\\
\end{array}\right)\mbox{\boldmath{$y$}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left(\\!\\!\\!\begin{array}[]{c}-\varrho[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
1}\\\ -(\partial{\bf
P}/\partial\Theta)\cdot\nabla\Theta+\mbox{\boldmath{$k$}}\\\
\end{array}\right)$ (32) $\displaystyle=:$ $C$ (33)

and that of the dissipation inequality (16)

$\displaystyle\small\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}:=$
$\displaystyle\small-\\!\\!\begin{array}[]{cccccccc}\Big{(}\\!\frac{\varrho}{\Theta}(\partial\psi/\partial\varrho)\\!&\\!\partial\mbox{\boldmath{$\Omega$}}/\partial\varrho\\!&0\\!&\frac{\varrho}{\Theta}[\partial\psi/\partial\Theta\\!+\\!s]\\!&\\!\frac{\varrho}{\Theta}(\partial\psi/\partial\nabla\Theta)\\!&\\!\partial\mbox{\boldmath{$\Omega$}}/\partial\nabla\Theta\\!&\\!\frac{\varrho}{\Theta}(\partial\psi/\partial[\nabla\mbox{\boldmath{$v$}}]^{s})\\!&\\!\partial\mbox{\boldmath{$\Omega$}}/\partial[\nabla\mbox{\boldmath{$v$}}]^{s}\\!\Big{)}\end{array}\\!\\!\\!\mbox{\boldmath{$y$}}\hskip
8.5359pt$ (35) $\displaystyle\small\hskip
113.81102pt\geq\Big{(}\partial\mbox{\boldmath{$\Omega$}}/\partial\Theta-\mbox{\boldmath{$q$}}\Big{)}\cdot\nabla\frac{1}{\Theta}+\kappa+\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}=:D.$ (36)

$C$ is a (2,1)-matrix and $D$ a (1,1)-matrix.

Before the example can be continued in sect.10, some more general
considerations are needed.

##  4 Material Axioms

The constitutive equations (20) generating the balance equations on state
space (26) cannot be arbitrary. ${\bf A}(\mbox{\boldmath{$z$}})$,
$\mbox{\boldmath{$C$}}(\mbox{\boldmath{$z$}})$,
${\mbox{\boldmath{$B$}}}(\mbox{\boldmath{$z$}})$ and
$D(\mbox{\boldmath{$z$}})$ have to satisfy material axioms which describe
constraints which materials have to satisfy. These axioms are:
1\. The Second Law (26)2 and its additionals,
2\. Transformation properties by changing the observer,
3\. Material symmetry,
4\. State spaces $\cal Z$ that guarantee finite speed of wave propagation.
Here, we are interested in #1. More hints can be found in the literature [2,
4, 16, 17].

##  5 Process Directions

The state space variables $\mbox{\boldmath{$z$}}(t,\mbox{\boldmath{$x$}})$
depend on time and position according to (21) and (22). For each event
$(t,\mbox{\boldmath{$x$}})$, the higher derivatives (23) represent a process
direction in the state space $\cal Z$. Having solved the balance equations
(26)1 (or (21)) for given initial conditions and geometrical constraints, we
know the higher derivatives
$\mbox{\boldmath{$y$}}^{\top}(t,\mbox{\boldmath{$x$}})$, and we have to check,
whether or no these higher derivatives satisfy the dissipation inequality
(26)2. Only if it is satisfied, the balance equations (26)1 can be attached to
a thermodynamical process, depending on the chosen constitutive equations
${\bf A}(\mbox{\boldmath{$z$}})$,
${\mbox{\boldmath{$C$}}}(\mbox{\boldmath{$z$}})$,
$\mbox{\boldmath{$B$}}(\mbox{\boldmath{$z$}})$ and $D(\mbox{\boldmath{$z$}})$.
This procedure is cumbersome, because if the dissipation inequality (26)2 is
not satisfied by the chosen constitutive equations, we have to start the
procedure again with renewed constitutive equations until (26)2 is satisfied.
This is the so-called global procedure which requires the solution of the
balance equations.

Does a more specific procedure exist which can determine possible constitutive
equations in advance without solving (26)1 so that (26)2 is satisfied ? To
answer this question, the space of the higher derivatives (of all process
directions) at an arbitrary, but fixed event
${(t_{0},\mbox{\boldmath{$x$}}_{0})}$ is introduced

${\cal Y}(t_{0},\mbox{\boldmath{$x$}}_{0})\ \ni\
\mbox{\boldmath{$y$}}^{\top}(t_{0},\mbox{\boldmath{$x$}}_{0}).$ (37)

That means, the global consideration of the balance equations (26)1 is changed
into a local procedure at arbitrary, but fixed
$(t_{0},\mbox{\boldmath{$x$}}_{0})$. The balance equations (26) are replaced
by a local set of identical balance equations with different values of the
higher derivatives

$\displaystyle\mbox{\boldmath{$y$}}^{k\top}\in{\cal
Y}(t_{0},\mbox{\boldmath{$x$}}_{0}):$ (38) $\displaystyle{\bf
A}\Big{(}\mbox{\boldmath{$z$}}(t_{0},\mbox{\boldmath{$x$}}_{0})\Big{)}\mbox{\boldmath{$y$}}^{k}=\mbox{\boldmath{$C$}}\Big{(}\mbox{\boldmath{$z$}}(t_{0},\mbox{\boldmath{$x$}}_{0})\Big{)},\quad\mbox{\boldmath{$B$}}\Big{(}\mbox{\boldmath{$z$}}(t_{0},\mbox{\boldmath{$x$}}_{0})\Big{)}\mbox{\boldmath{$y$}}^{k}\geq
D\Big{(}\mbox{\boldmath{$z$}}(t_{0},\mbox{\boldmath{$x$}}_{0})\Big{)},$ (39)

searching for higher derivatives $\mbox{\boldmath{$y$}}^{k}$ which satisfy
(39) at $(t_{0},\mbox{\boldmath{$x$}}_{0})$ and then along the complete
process $(t,\mbox{\boldmath{$y$}}(t))$.

For the following, two statements (I and II) are considered which exclude each
other [6]:

I. All local solutions888irreversible and reversible ones of the local balance
equations (39)1 satisfy the dissipation inequality (39)2

$\displaystyle\bigwedge_{k}\Big{\\{}\mbox{\boldmath{$y$}}^{k}\in{\cal
Y}_{>}(t_{0},\mbox{\boldmath{$x$}}_{0})|{\bf
A}\mbox{\boldmath{$y$}}^{k}=\mbox{\boldmath{$C$}}\Big{\\}}$
$\displaystyle\longrightarrow$
$\displaystyle\Big{\\{}\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{k}>D\Big{\\}},$
(40)
$\displaystyle\bigwedge_{m}\Big{\\{}\mbox{\boldmath{$y$}}^{m}_{rev}\in{\cal
Y}_{=}(t_{0},\mbox{\boldmath{$x$}}_{0})|{\bf
A}\mbox{\boldmath{$y$}}^{m}_{rev}=\mbox{\boldmath{$C$}}\Big{\\}}$
$\displaystyle\longrightarrow$
$\displaystyle\Big{\\{}\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{m}_{rev}=D\Big{\\}}.$
(41)

The process directions are divided into irreversible
$\mbox{\boldmath{$y$}}^{k}$ and reversible $\mbox{\boldmath{$y$}}^{m}_{eq}$
ones. The linear combined process direction, 1$>\alpha>$0,

${\bf
A}\Big{(}\alpha\mbox{\boldmath{$y$}}^{k}+(1-\alpha)\mbox{\boldmath{$y$}}^{m}_{eq}\Big{)}\
=\ \mbox{\boldmath{$C$}}\ \longrightarrow\
\mbox{\boldmath{$B$}}\Big{(}\alpha\mbox{\boldmath{$y$}}^{k}+(1-\alpha)\mbox{\boldmath{$y$}}^{m}_{eq}\Big{)}\
>\ D$ (42)

satisfies the balance equations and belongs to an irreversible process
according to (42)2 and (40)2. No additional reversible process directions
belonging to the subspace ${\cal
Y}_{=}(t_{0},\mbox{\boldmath{$x$}}_{0})\subset{\cal
Y}(t_{0},\mbox{\boldmath{$x$}}_{0})$ can be created by a linear combination of
$y^{k}$ and $\mbox{\boldmath{$y$}}^{m}_{eq}$.

The second statement is:
II. According to (40) and (41), there are local solutions of the balance
equations (on state space) $\mbox{\boldmath{$y$}}^{k}$ and
$\mbox{\boldmath{$y$}}^{m}_{eq}$ which satisfy the dissipation inequality.
Additional process directions $\mbox{\boldmath{$y$}}^{j}_{\Box}$ are now
presupposed which do not satisfy (40) and (41) in common, representing local
solutions of the balance equations which do not satisfy the dissipation
inequality (39)2 and which therefore do not represent a thermodynamical
process

$\bigwedge_{j}\Big{\\{}\mbox{\boldmath{$y$}}^{j}_{\Box}\in{\cal
Y}_{<}(t_{0},\mbox{\boldmath{$x$}}_{0})|{\bf
A}\mbox{\boldmath{$y$}}^{j}_{\Box}=\mbox{\boldmath{$C$}}\Big{\\}}\
\longrightarrow\
\Big{\\{}\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{j}_{\Box}<D\Big{\\}}.$
(43)

By presupposing (43), the space of the higher derivatives (37) is

${\cal Y}(t_{0},\mbox{\boldmath{$x$}}_{0})\ =\ {\cal
Y}_{>}(t_{0},\mbox{\boldmath{$x$}}_{0})\cup{\cal
Y}_{=}(t_{0},\mbox{\boldmath{$x$}}_{0})\cup{\cal
Y}_{<}(t_{0},\mbox{\boldmath{$x$}}_{0}).\\\ $ (44)

As proved in the next section 6, from statement #II follows that additional
reversible process directions beyond those in ${\cal
Y}_{=}(t_{0},\mbox{\boldmath{$x$}}_{0})$ can be created from ${\cal
Y}_{>}(t_{0},\mbox{\boldmath{$x$}}_{0})\cup{\cal
Y}_{<}(t_{0},\mbox{\boldmath{$x$}}_{0})$. This strange result paves the way to
the axiom of no-reversible process directions.

##  6 Reversible Process Directions in Non-equilibrium?

From (40)2 and (43)2 follows

$\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{k}>D>\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{j}_{\Box}\quad\longrightarrow\quad\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{k}-D\
>\ 0\ >\ \mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{j}_{\Box}-D.$ (45)

The inequality (45)2 is transformed to an equality by introducing two positve
constants

$\alpha>0,\
\beta>0:\qquad\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{k}-D+\alpha(\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{j}_{\Box}-D)\
\doteq\
\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{j}_{\Box}-D+\beta(\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{k}-D).$
(46)

This results in

$\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{k}(1-\beta)+\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}^{j}_{\Box}(\alpha-1)\
=\ D(-\beta+\alpha).$ (47)

Setting

$-\beta+\alpha\ \doteq\ 1\ \longrightarrow\ \alpha-1=\beta$ (48)

creates a reversible process direction

$\mbox{\boldmath{$B$}}\Big{(}(1-\beta)\mbox{\boldmath{$y$}}^{k}+\beta\mbox{\boldmath{$y$}}^{j}_{\Box}\Big{)}\
=\ D$ (49)

which is according to (40)1 and (43)1 also a local solution of the balance
equations

${\bf
A}\Big{(}(1-\beta)\mbox{\boldmath{$y$}}^{k}+\beta\mbox{\boldmath{$y$}}^{j}_{\Box}\Big{)}\
=\ (1-\beta)\mbox{\boldmath{$C$}}+\beta\mbox{\boldmath{$C$}}\ =\
\mbox{\boldmath{$C$}},\quad 0<\beta<1,\quad(2>\alpha>1).$ (50)

Consequently, if not all local solutions of the balance equations satisfy the
dissipation inequality (#II), a reversible process direction (49) can be
constructed by linear combination of non-reversible ones (${\cal
Y}_{>}(t_{0},\mbox{\boldmath{$x$}}_{0})\cup{\cal
Y}_{<}(t_{0},\mbox{\boldmath{$x$}}_{0})$) which are a local solutions of the
balance equations (50). This strange result is discussed in the next section
7.

##  7 Coleman-Mizel’s Shape of the Second Law

The dissipation inequality (26)2 representing the Second Law can be
differently interpreted:

(#I) If all process directions satisfy the dissipation inequality, arbitrary
constitutive equations are not possible, they are restricted by the
dissipation inequality.

(#II) If the constitutive equations ${\bf A}(\mbox{\boldmath{$z$}})$,
${\mbox{\boldmath{$C$}}}(\mbox{\boldmath{$z$}})$,
$\mbox{\boldmath{$B$}}(\mbox{\boldmath{$z$}})$ and $D(\mbox{\boldmath{$z$}})$
are given, the dissipation inequality (26)2 excludes those process directions
which do not satify the balance equations (26)1.

The Second Law states nothing about these two each other excluding cases.
Consequently, an amendment to the Second Law is required for deciding which
case, #I or #II, is valid [14, 18]. Here, this decision is given by an axiom
which excludes the situation described in section 6:

A reversible process direction cannot be generated by non-reversible ones.
(51)

Therefore the second inequality of (45)1 must not be valid, that means

${\cal Y}_{<}(t_{0},\mbox{\boldmath{$x$}}_{0})\ =\ \emptyset,\quad\mbox{for
all }(t_{0},\mbox{\boldmath{$x$}}_{0})$ (52)

is valid and consequently process directions of negative entropy production do
not exist. Consequently, all process directions are of non-negative entropy
production. Especially, there are by choice of ${\bf
A},\mbox{\boldmath{$C$}},\mbox{\boldmath{$B$}},D$999representing the choice of
an admissible material no solutions of the balance equations of negative
entropy production. Therefore #I, (40) and (41), is true

All local solutions of the balance equations $\displaystyle\hskip
11.38092pt\mbox{\bf have to satify the dissipation inequality}.$ (53)

After having excluded by (52) all process directions of negative entropy
production, the dissipation inequality represents a constraint with regard to
the constitutive equations (24) and (25) and is not excluding any process
direction of $\Big{(}{\cal Y}_{>}(t_{0},\mbox{\boldmath{$x$}}_{0})\cup{\cal
Y}_{=}(t_{0},\mbox{\boldmath{$x$}}_{0})\Big{)}$ which satisfy the Second Law:
Consequently, the constitutive equations, cannot be independent of each other,
but must have the property, that the entropy production density is not
negative for all local -and therefore also for all global- solutions of the
balance equations. This is the Coleman-Mizel (CM)-formulation of the Second
Law [19] which presupposes the validity of (40) and (41) ad-hoc. Taking (52)
into account, the CM-formulation of the Second Law follows and adopts its
physical interpretation:

The constitutive equations enforce that there are no solutions of the balance
equations of negative entropy production. (54)

##  8 No-reversible Process Direction Axiom

Taking a special material into consideration for which the process-direction-
independent ${\bf A},\mbox{\boldmath{$C$}},\mbox{\boldmath{$B$}}$ and $D$ are
given and for which (53) is valid. According to (42), a lot of processes of
different process directions are possible, resulting in the fact that the
matrix $\bf A$ of the constitutive equations has a kernel ${\cal K}$,
$\Big{(}{\cal Y}_{>}(t_{0},\mbox{\boldmath{$x$}}_{0})\cup{\cal
Y}_{=}(t_{0},\mbox{\boldmath{$x$}}_{0})\Big{)}\supset{\cal
K}\ni\mbox{\boldmath{$y$}}_{ker}$

${\bf A}\mbox{\boldmath{$y$}}_{ker}\ =\
\mbox{\boldmath{$0$}}\quad\longrightarrow\quad\mbox{\boldmath{$y$}}\ =\
\mbox{\boldmath{$y$}}_{0}+\mbox{\boldmath{$y$}}_{ker}\quad\longrightarrow\quad{\bf
A}\mbox{\boldmath{$y$}}\ =\ {\bf A}\mbox{\boldmath{$y$}}_{0}\ =\
\mbox{\boldmath{$C$}}.$ (55)

Here $y$ and $\mbox{\boldmath{$y$}}_{0}$ are local solutions of the balance
equations according to (26)1, and with them also

${\bf
A}\Big{(}\alpha\mbox{\boldmath{$y$}}+(1-\alpha)\mbox{\boldmath{$y$}}_{0}\Big{)}\
=\ \mbox{\boldmath{$C$}}$ (56)

satisfies the balance equation.

Introducing the entropy production $\sigma\geq 0$, the dissipation inequality
becomes according to (40)

$\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}\ =\
\mbox{\boldmath{$B$}}(\mbox{\boldmath{$y$}}_{0}+\mbox{\boldmath{$y$}}_{ker})\
=\ D+\sigma(\mbox{\boldmath{$y$}})\ =\
D+\sigma(\mbox{\boldmath{$y$}}_{0}+\mbox{\boldmath{$y$}}_{ker})\geq\ D.$ (57)

According to (55)1, the kernel of ${\bf A}$ is not present in the balance
equations and therefore it should also not be present in the entropy
production

$\framebox[170.71652pt]{$\sigma(\mbox{\boldmath{$y$}})\ =\
\sigma(\mbox{\boldmath{$y$}}_{0}+\mbox{\boldmath{$y$}}_{ker})\
\stackrel{{{}_{\bf ax}}}{{=}}\ \sigma(\mbox{\boldmath{$y$}}_{0}).$}$ (58)

Although being clear from a view of physics, (58)2 is an axiom which is more
stringent than the verbal formulations (51) and (53) as demonstrated below at
(63). Consequently, from (57)2 and (58)1,2 follows

$\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}-\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}_{0}\
=\ D+\sigma(\mbox{\boldmath{$y$}})-(D+\sigma(\mbox{\boldmath{$y$}}_{0}))\ =\
0\ =\ \mbox{\boldmath{$B$}}(\mbox{\boldmath{$y$}}-\mbox{\boldmath{$y$}}_{0})\
=\ \mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}_{ker},$ (59)

that means, $B$ is perpendicular to the kernel of ${\bf A}$.

Consider two arbitrary local solutions of the balance equations (55)3,
$\mbox{\boldmath{$y$}}^{1}$ and $\mbox{\boldmath{$y$}}^{2}$, and additionally
(55)1,2

$\mbox{\boldmath{$C$}}-\mbox{\boldmath{$C$}}\ =\ \mbox{\boldmath{$0$}}\ =\
{\bf A}(\mbox{\boldmath{$y$}}^{1}-\mbox{\boldmath{$y$}}^{2})\ =\ {\bf
A}(\mbox{\boldmath{$y$}}^{1}_{0}-\mbox{\boldmath{$y$}}^{2}_{0})\
\longrightarrow\ \mbox{\boldmath{$y$}}^{1}_{0}-\mbox{\boldmath{$y$}}^{2}_{0}\
=:\ \mbox{\boldmath{$y$}}^{12}_{ker}\ \in\ \cal{K},$ (60)

resulting according to (55)2, (60)3 and (59)4 in

$\displaystyle\mbox{\boldmath{$y$}}^{1}-\mbox{\boldmath{$y$}}^{2}$
$\displaystyle=$
$\displaystyle\mbox{\boldmath{$y$}}^{12}_{ker}+\mbox{\boldmath{$y$}}^{1}_{ker}-\mbox{\boldmath{$y$}}^{2}_{ker},$
(61)
$\displaystyle\mbox{\boldmath{$B$}}(\mbox{\boldmath{$y$}}^{1}-\mbox{\boldmath{$y$}}^{2})$
$\displaystyle=$
$\displaystyle\mbox{\boldmath{$B$}}(\mbox{\boldmath{$y$}}^{12}_{ker}+\mbox{\boldmath{$y$}}^{1}_{ker}-\mbox{\boldmath{$y$}}^{2}_{ker})\
=\ 0\ =\ \sigma(\mbox{\boldmath{$y$}}^{1})-\sigma(\mbox{\boldmath{$y$}}^{2}),$
(62)

according to (57)2. Consequently, the entropy production does not locally
depend on the process direction according to (62)3. If one process direction
at $(t_{0},\mbox{\boldmath{$x$}}_{0})$ is reversible,
$\mbox{\boldmath{$y$}}_{rev}\in{\cal Y}_{=}(t_{0},\mbox{\boldmath{$y$}}_{0})$,
all other process directions at $(t_{0},\mbox{\boldmath{$x$}}_{0})$ are also
reversible, and an equilibrium state is present. This results in the verbal
formulation of the Axiom of No-reversible Process Directions [20] which
represents an amendment of the Second Law:

Except in equilibria, reversible process directions in state space do not
exist (63)

##  9 Liu Relations

Introducing a suitable vector
$\mbox{\boldmath{$\lambda$}}(\mbox{\boldmath{$y$}})$ which for the present
depends on the process direction $y$, and by use of (57)2 and (55)3, the
following equality is valid

$(\mbox{\boldmath{$B$}}-\mbox{\boldmath{$\lambda$}}{\bf
A})\mbox{\boldmath{$y$}}\ =\
D+\sigma(\mbox{\boldmath{$y$}})-\mbox{\boldmath{$\lambda$}}(\mbox{\boldmath{$y$}})\mbox{\boldmath{$C$}}.$
(64)

Now $\lambda$ can be chosen specially, depending on the entropy production

$\mbox{\boldmath{$\lambda$}}(\mbox{\boldmath{$y$}})\mbox{\boldmath{$C$}}\
\doteq\ D+\sigma(\mbox{\boldmath{$y$}}),$ (65)

taking into account that $C$ and $D$ do not depend on the higher derivatives
$y$. Because the entropy production does not depend on the process direction
according to (62)3, $\lambda$ is also independent of the process direction.

The choice of a process-direction-independent $\lambda$ in the scalar product
of the LHS of (65) is not unequivocal, but always possible, and by taking (65)
into account (64) results in

$(\mbox{\boldmath{$B$}}-\mbox{\boldmath{$\lambda$}}{\bf
A})\mbox{\boldmath{$y$}}\ =\
D+\sigma-\mbox{\boldmath{$\lambda$}}\mbox{\boldmath{$C$}}\ =\ 0$ (66)

which is valid for all higher derivatives resulting in

$\framebox[56.9055pt]{$\mbox{\boldmath{$B$}}\ =\
\mbox{\boldmath{$\lambda$}}{\bf
A}$}\quad\mbox{and}\quad\framebox[62.59596pt]{$\mbox{\boldmath{$\lambda$}}\mbox{\boldmath{$C$}}\
\geq\ D$}.$ (67)

Thus, the process-direction-independent Liu Relations (67) [15, 21] are
generated by the setting (65) and taking into account that the entropy
production is independent of the process direction.

Because ${\bf A}$ has no right-hand inverse, $\lambda$ is not determined by
the given $B$ and ${\bf A}$, but the constitutive quantities $B$ and ${\bf A}$
are not independent of each other: there exists a $\lambda$, so that the Liu
Relations (67) are satisfied. Because there are less balance equations than
higher derivatives101010see: eqs.(1) to (3) (r=3 balances) in comparison with
eq.(27) (c=8 higher derivatives), the types (number of rows $r$ and of columns
$c$) of the following matrices are

${\bf A}:(r,c),\ r<c,\quad\mbox{\boldmath{$y$}}:(c,1),\quad
D:(1,1),\quad\mbox{\boldmath{$B$}}:(1,c),\quad\mbox{\boldmath{$\lambda$}}:(1,r),\quad\mbox{\boldmath{$C$}}:(r,1).$
(68)

A tutorial example concerning the use of the Liu Relations can be found in
[4].

The Liu Relations represent constraints on the constitutive properties. They
bring together constitutive quantities of the balance equation with those of
the dissipation inequality. These restrictions have to be inserted into the
balance equations and the dissipation inequality (26), resulting by use of
(66) in a differential equation which takes the dissipation inequality into
account

$\mbox{\boldmath{$B$}}\mbox{\boldmath{$y$}}\ =\
\mbox{\boldmath{$\lambda$}}\mbox{\boldmath{$C$}}.$ (69)

An other kind of exploitation the dissipation inequality regards the $\lambda$
as a Lagrange multiplier introducing the balance equations into the
dissipation inequality [15]. The dissipation inequality (64)2 can be replaced
by

$\mbox{\boldmath{$B$}}\mbox{\boldmath{$\omega$}}-\mbox{\boldmath{$\lambda$}}({\bf
A}\mbox{\boldmath{$\omega$}}-\mbox{\boldmath{$C$}})\ =\ D+\sigma\ \geq\ D.$
(70)

Here, $\omega$ is an arbitrary process direction which not necessarily has to
satisfy the balance equations and/or the dissipation inequality (40) and/or
(41). The usual dictum is: add the balance equations as constraints with a
Lagrange factor to the dissipation inequality resulting in arbitrary process
directions.

##  10 Example Continued:
Liu Relations + Entropy Production

The Liu Relations (67) belonging to the example (33) and (36) are as follows

$\displaystyle\mbox{\boldmath{$\lambda$}}{\bf
A}=\Big{(}\begin{array}[]{cc}\lambda^{1}&\lambda^{2}\end{array}\Big{)}\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&(\partial{\bf P}/\partial\varrho)&\varrho&0&0&(\partial{\bf
P}/\partial\nabla\Theta)&0&(\partial{\bf
P}/\partial[\nabla\mbox{\boldmath{$v$}}]^{s})\\\ \end{array}\right)=$ (74)
$\displaystyle=\Big{(}\begin{array}[]{cccccccc}\lambda^{1}&\lambda^{2}(\partial{\bf
P}/\partial\varrho)&\lambda^{2}\varrho&0&0&\lambda^{2}(\partial{\bf
P}/\partial\nabla\Theta)&0&\lambda^{2}(\partial{\bf
P}/\partial[\nabla\mbox{\boldmath{$v$}}]^{s})\end{array}\Big{)}=\mbox{\boldmath{$B$}}=$
(76)
$\displaystyle\small-\\!\\!\begin{array}[]{cccccccc}\Big{(}\\!\frac{\varrho}{\Theta}(\partial\psi/\partial\varrho)\\!&\\!\partial\mbox{\boldmath{$\Omega$}}/\partial\varrho\\!&0\\!&\frac{\varrho}{\Theta}[\partial\psi/\partial\Theta\\!+\\!s]\\!&\\!\frac{\varrho}{\Theta}(\partial\psi/\partial\nabla\Theta)\\!&\\!\partial\mbox{\boldmath{$\Omega$}}/\partial\nabla\Theta\\!&\\!\frac{\varrho}{\Theta}(\partial\psi/\partial[\nabla\mbox{\boldmath{$v$}}]^{s})\\!&\\!\partial\mbox{\boldmath{$\Omega$}}/\partial[\nabla\mbox{\boldmath{$v$}}]^{s}\\!\Big{)},\end{array}$
(78)

and

$\displaystyle\mbox{\boldmath{$\lambda$}}\mbox{\boldmath{$C$}}=\Big{(}\begin{array}[]{cc}\lambda^{1}&\lambda^{2}\end{array}\Big{)}\left(\begin{array}[]{c}-\varrho[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
1}\\\ -(\partial{\bf
P}/\partial\Theta)\cdot\nabla\Theta+\mbox{\boldmath{$k$}}\\\
\end{array}\right)=-\lambda^{1}\varrho[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
1}-\lambda^{2}\Big{(}(\partial{\bf
P}/\partial\Theta)\cdot\nabla\Theta-\mbox{\boldmath{$k$}}\Big{)}\geq$ (82)
$\displaystyle\geq
D=\Big{(}\partial\mbox{\boldmath{$\Omega$}}/\partial\Theta-\mbox{\boldmath{$q$}}\Big{)}\cdot\nabla\frac{1}{\Theta}+\kappa+\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}.\hskip 34.14322pt$ (83)

From the matrix elements (1,3) and (1,1) of $\mbox{\boldmath{$\lambda$}}{\bf
A}=\mbox{\boldmath{$B$}}$ follows

$\lambda^{2}\ =\ 0,\qquad\lambda^{1}\ =\
-\frac{\varrho}{\Theta}(\partial\psi/\partial\varrho).$ (84)

The matrix elements (1,4), (1,5) and (1,7) of $\mbox{\boldmath{$\lambda$}}{\bf
A}=\mbox{\boldmath{$B$}}$ result in

$s\ =\ -(\partial\psi/\partial\Theta),\qquad 0\ =\
(\partial\psi/\partial\nabla\Theta),\qquad 0\ =\
(\partial\psi/\partial[\nabla\mbox{\boldmath{$v$}}]^{s}),$ (85)

and by taking (84)1 into account, (1,2), (1,6) and (1,8) are

$0\ =\ \partial\mbox{\boldmath{$\Omega$}}/\partial\varrho,\qquad 0\ =\
\partial\mbox{\boldmath{$\Omega$}}/\partial\nabla\Theta,\qquad 0\ =\
\partial\mbox{\boldmath{$\Omega$}}/\partial[\nabla\mbox{\boldmath{$v$}}]^{s}.$
(86)

The entropy production follows from the dissipation inequality (83) by taking
(84) into account

$\frac{\varrho^{2}}{\Theta}(\partial\psi/\partial\varrho)[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
1}-\Big{(}\partial\mbox{\boldmath{$\Omega$}}/\partial\Theta-\mbox{\boldmath{$q$}}\Big{)}\cdot\nabla\frac{1}{\Theta}-\kappa-\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}\ \geq\ 0.$ (87)

Especially for Clausius-Duhem materials which satisfy

$\mbox{\boldmath{$\Omega$}}\ =\ 0,\qquad\kappa\ =\ 0$ (88)

according to (15), the entropy production (87) becomes

$\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:\Big{(}\varrho^{2}\frac{\partial\psi}{\partial\varrho}{\bf
1}-{\bf P}\Big{)}+\mbox{\boldmath{$q$}}\cdot\nabla\frac{1}{\Theta}\ \geq\ 0.$
(89)

The calculation above takes the balance equation of the internal energy (12)
only indirectly into account by inserting the dissipation inequality (5) into
(12), thus generating a special shape of the dissipation inequality (16) which
is used together with the balances of mass (1) and momentum (2). Now the
question arises, how the only implicit use of the energy balance equation (12)
restricts the result compared to its full consideration ? This question is the
item of the next section.

##  11 Example Continued:
Liu Relations + Class of Materials

Now the Liu Relations are generated by use of all balance equations (1) to (3)
and the dissipation inequality (5). The balance equation of the internal
energy is now completely taken into account, that means, the (2,8)-matrix
${\bf A}$ in (33) has to be supplemented by (12) in matrix formulation
resulting in a third row of (33), whereas the balance equations of mass and
momentum are taken into account by the first two rows. Thus, the balance
equations write ${\bf
A}^{\ast}\mbox{\boldmath{$y$}}=\mbox{\boldmath{$C$}}^{\ast}$,
(3,8)(8,1)=(3,1). The state space (17)1 is chosen as above in sect.1.2
resulting in the set of higher derivatives (27).

The matrix formulation of (3) according to the chosen state space (17)1 is

$\displaystyle\begin{array}[]{cccccccc}\Big{(}\varrho\partial\varepsilon/{\partial\varrho}&\partial\mbox{\boldmath{$q$}}/{\partial\varrho}&0&\varrho\partial\varepsilon/{\partial\Theta}&\varrho\partial\varepsilon/{\partial\nabla\Theta}&\partial\mbox{\boldmath{$q$}}/{\partial\nabla\Theta}&\varrho\partial\varepsilon/{\partial[\nabla\mbox{\boldmath{$v$}}]^{s}}&\partial\mbox{\boldmath{$q$}}/{\partial[\nabla\mbox{\boldmath{$v$}}]^{s}}\Big{)}\end{array}\\!\\!\mbox{\boldmath{$y$}}\
=$ (91) $\displaystyle=\
-\partial\mbox{\boldmath{$q$}}/\partial\Theta\cdot\nabla\Theta-\nabla[\mbox{\boldmath{$v$}}]^{s}:{\bf
P}+r.\hskip 19.91684pt$ (92)

The first two rows of ${\bf A}^{\ast}$ and $\mbox{\boldmath{$C$}}^{\ast}$ can
be found in (33) and the third row in (92). The dissipation inequality is now
(5) instead of (16). Its matrix formulation according to the chosen state
space (17)1 is

$\displaystyle\begin{array}[]{cccccccc}\Big{(}\varrho\partial
s/{\partial\varrho}&\partial\mbox{\boldmath{$\Phi$}}/{\partial\varrho}&0&\varrho\partial
s/{\partial\Theta}&\varrho\partial
s/{\partial\nabla\Theta}&\partial\mbox{\boldmath{$\Phi$}}/{\partial\nabla\Theta}&\varrho\partial
s/{\partial[\nabla\mbox{\boldmath{$v$}}]^{s}}&\partial\mbox{\boldmath{$\Phi$}}/{\partial[\nabla\mbox{\boldmath{$v$}}]^{s}}\Big{)}\end{array}\\!\\!\mbox{\boldmath{$y$}}\
=$ (94) $\displaystyle=:\ \mbox{\boldmath{$B$}}^{\ast}\mbox{\boldmath{$y$}}\
\geq\ \gamma-\partial\mbox{\boldmath{$\Phi$}}/\partial\Theta\cdot\nabla\Theta\
=:\ D^{\ast}.\hskip 19.91684pt$ (95)

The corresponding Liu Relations (67) are

$\mbox{\boldmath{$\lambda$}}^{\ast}{\bf A}^{\ast}\ =\
\mbox{\boldmath{$B$}}^{\ast},\qquad\mbox{\boldmath{$\lambda$}}^{\ast}\mbox{\boldmath{$C$}}^{\ast}\
\geq\ D^{\ast}$ (96)

written in components as done in (78) and (83), (96)1 yields

$\displaystyle\lambda^{1\ast}+\lambda^{3\ast}\varrho\frac{\partial\varepsilon}{\partial\varrho}$
$\displaystyle=$ $\displaystyle\varrho\frac{\partial s}{\partial\varrho},$
(97) $\displaystyle\lambda^{2\ast}\frac{\partial{\bf
P}}{\partial\varrho}+\lambda^{3\ast}\frac{\partial\mbox{\boldmath{$q$}}}{\partial\varrho}$
$\displaystyle=$
$\displaystyle\frac{\partial\mbox{\boldmath{$\Phi$}}}{\partial\varrho},$ (98)
$\displaystyle\lambda^{2\ast}\varrho$ $\displaystyle=$ $\displaystyle 0,$ (99)
$\displaystyle\lambda^{3\ast}\varrho\frac{\partial\varepsilon}{\partial\Theta}$
$\displaystyle=$ $\displaystyle\varrho\frac{\partial s}{\partial\Theta},$
(100)
$\displaystyle\lambda^{3\ast}\varrho\frac{\partial\varepsilon}{\partial\nabla\Theta}$
$\displaystyle=$ $\displaystyle\varrho\frac{\partial
s}{\partial\nabla\Theta},$ (101)
$\displaystyle\lambda^{2\ast}\frac{\partial{\bf
P}}{\partial\nabla\Theta}+\lambda^{3\ast}\frac{\partial\mbox{\boldmath{$q$}}}{\partial\nabla\Theta}$
$\displaystyle=$
$\displaystyle\frac{\partial\mbox{\boldmath{$\Phi$}}}{\partial\nabla\Theta},$
(102)
$\displaystyle\lambda^{3\ast}\varrho\frac{\partial\varepsilon}{\partial[\nabla\mbox{\boldmath{$v$}}]^{s}}$
$\displaystyle=$ $\displaystyle\varrho\frac{\partial
s}{\partial[\nabla\mbox{\boldmath{$v$}}]^{s}},$ (103)
$\displaystyle\lambda^{2\ast}\frac{\partial{\bf
P}}{\partial[\nabla\mbox{\boldmath{$v$}}]^{s}}+\lambda^{3\ast}\frac{\partial\mbox{\boldmath{$q$}}}{\partial[\nabla\mbox{\boldmath{$v$}}]^{s}}$
$\displaystyle=$
$\displaystyle\frac{\partial\mbox{\boldmath{$\Phi$}}}{\partial[\nabla\mbox{\boldmath{$v$}}]^{s}},$
(104)

and (96)2 becomes by taking (33), (92) and (95) into account

$\displaystyle-\lambda^{1\ast}\varrho[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
1}-\lambda^{2\ast}\Big{(}(\partial{\bf
P}/\partial\Theta)\cdot\nabla\Theta-\mbox{\boldmath{$k$}}\Big{)}+\lambda^{3\ast}\Big{(}-\partial\mbox{\boldmath{$q$}}/\partial\Theta\cdot\nabla\Theta-[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}+r\Big{)}\geq$
$\displaystyle\geq\gamma-\partial\mbox{\boldmath{$\Phi$}}/\partial\Theta\cdot\nabla\Theta,\hskip
17.07182pt$ (105)

resulting in the entropy production by taking (99) into account

$\displaystyle-\lambda^{1\ast}\varrho[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
1}-\partial(\lambda^{3\ast}\mbox{\boldmath{$q$}}-\mbox{\boldmath{$\Phi$}})/\partial\Theta\cdot\nabla\Theta+\mbox{\boldmath{$q$}}(\partial\lambda^{3\ast}/\partial\Theta)\cdot\nabla\Theta-\hskip
56.9055pt$
$\displaystyle-\lambda^{3\ast}([\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P})+\lambda^{3\ast}r-\gamma\geq 0$ (106)

The Liu procedure does not determine $\lambda^{1*}$ and $\lambda^{3*}$. But
physical understanding yields according to (15)2 that $\lambda^{3*}$ is a
reciprocal temperature and $\lambda^{1*}$ is then determined by (97).
Consequently, the class of materials which fit the balance equations (1) to
(3) and the dissipation inequality (4) is charactrized by (97) to (106). A
special sub-family is the Clausius-Duhem family which is defined by
$\mbox{\boldmath{$\Omega$}}=\mbox{\boldmath{$0$}}$ and $\kappa=0$ in (15).
Consequently, the entropy production belonging to this sub-family is according
to (106), (17)1 and (11)

$\displaystyle-\Big{(}\varrho\frac{\partial
s}{\partial\varrho}-\frac{1}{\Theta}\varrho\frac{\partial\varepsilon}{\partial\varrho}\Big{)}\varrho[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
1}-\mbox{\boldmath{$q$}}\frac{1}{\Theta^{2}}\cdot\nabla\Theta-\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}$ $\displaystyle\geq$ $\displaystyle 0,$ (107)
$\displaystyle\Big{(}\frac{\partial\varepsilon}{\partial\varrho}-\Theta\frac{\partial
s}{\partial\varrho}\Big{)}\varrho^{2}\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
1}+\mbox{\boldmath{$q$}}\cdot\nabla\frac{1}{\Theta}-\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf
P}$ $\displaystyle\geq$ $\displaystyle 0,$ (108)
$\displaystyle\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:\Big{(}\frac{\partial\psi}{\partial\varrho}\varrho^{2}{\bf
1}-{\bf P}\Big{)}+\mbox{\boldmath{$q$}}\cdot\nabla\frac{1}{\Theta}$
$\displaystyle\geq$ $\displaystyle 0.$ (109)

A comparison with (89) demonstrates that the abridged procedure in sect.10
yields the same result for the entropy of Clausius-Duhem materials, a fact
which was not expected. Introducing the pressure $p\doteq(1/3)\mbox{Tr}{\bf
P}$, the entropy production follows from (108)

${\bf P}\ =\ p{\bf 1}+\mbox{\boldmath{$\Pi$}}\ \longrightarrow\ \sigma\ =\
\frac{1}{\Theta}(\nabla\cdot\mbox{\boldmath{$v$}})\Big{(}\frac{\partial\psi}{\partial\varrho}\varrho^{2}-p\Big{)}+\mbox{\boldmath{$q$}}\cdot\nabla\frac{1}{\Theta}-\frac{1}{\Theta}[\nabla\mbox{\boldmath{$v$}}]^{s}:{\bf\Pi}\
\geq\ 0.$ (110)

Consequently, the entropy production of incompressible Clausius-Duhem
materials without internal friction is those of pure heat conduction

$\Big{(}\nabla\cdot\mbox{\boldmath{$v$}}=0\Big{)}\ \wedge\
\Big{(}{\bf\Pi}={\bf 0}\Big{)}\ \longrightarrow\ \sigma\ =\
\mbox{\boldmath{$q$}}\cdot\nabla\frac{1}{\Theta}\ \geq\ 0,$ (111)

a hint at the efficacy of the Liu Relations (96).

##  12 Summary

For describing materials, a state space is necessary on which the linear
differential operators of the balance equations111111$d_{t}$ and $\nabla$ act
by applying the chain rule. Thus, time and position derivatives of the state
space variables121212the so-called higher derivatives appear in the balance
equations which for discrimination are called balances on state space and
which are linear in the higher derivatives which can be interpreted as process
directions in state space. Because there are less balance equations than
higher derivatives, these ones are not determined by the balances on state
space, rather higher derivatives exist forming a kernel of the balances on
state space which are linear in the higher derivatives. Due to this linearity,
higher derivatives can be combined linearily representing different process
directions. According to the dissipation inequality, there are three kinds of
process directions: irreversible, reversible and not-satisfying131313the
dissipation inequality ones.

The axiomatical statement ”A reversible process direction cannot be generated
by non-reversible ones” results in the Coleman-Mizel formulation of the Second
Law ”All local solutions of the balance equations (on state space) have to
satisfy the dissipation inequality”. A second axiomatical statement ” Process
directions which are in the kernel of the balance equations (on state space)
do not enter the entropy production” results in the Axiom of No-reversible
Process Directions ”Except in equilibria, reversible process directions in
state space do not exist”, that is, reversible process directions appear only
in equilibrium and the entropy production in non-equilibrium does not depend
on the process direction, it is a state function. Consequently, reversible
process directions do not exist in non-equilibrium.

If the entropy production were dependent on the process direction, the Liu
Relations could not be derived by the here presented procedure. These results
are based on the obvious, but axiomatic fact that process directions which do
not appear in the balance equations also do not enter the entropy production.
Using this axiom, the Liu Relations can be derived from the balance equations
and the dissipation inequality. These relations are independent of the process
directions and connect the balance equations directly with the dissipation
inequality, thus generating constraints for the choice of the material. The
advantage of the Liu procedure with respect to the Coleman-Noll technique [22,
21] is that the Liu Relations do not contain higher derivatives which by using
the Coleman-Noll technique have to be removed by setting the brackets to zero
which are factors of the higher derivatives.

Ethical Approval: not applicable

Funding: not applicable

Availability of data and materials: not applicable

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