# Ethane-xenon mixtures under shock conditions

Rudolph J. Magyar  Seth Root  Kyle Cochrane  Thomas R. Mattsson  Dawn G.
Flicker  Sandia National Laboratories, Albuquerque, New Mexico, 87185, USA.

###### Abstract

Mixtures of light elements with heavy elements are important in inertial
confinement fusion and planetary science. We explore the physics of molecular
scale mixing through a validation study of equation of state (EOS) properties.
Density functional theory molecular dynamics (DFT-MD) at elevated-temperature
and pressure is used to obtain the thermodynamic state properties of pure
xenon, ethane, and various compressed mixture compositions along their
principal Hugoniots. To validate these simulations, we have performed shock
compression experiments using the Sandia Z-Machine. A bond tracking analysis
correlates the sharp rise in the Hugoniot curve with the completion of
dissociation in ethane. The DFT-based simulation results compare well with the
experimental data along the principal Hugoniots and are used to provide
insight into the dissociation and temperature along the Hugoniots as a
function of mixture composition.

density functional theory, DFT, hydrocarbons, xenon, shock compression

###### pacs:

62.50.Ef, 71.15.Pd, 71.30.+h

††preprint: Z-GROUP: pre-print to be submitted to Physics Rev. B

##  I Introduction

Understanding the behavior of mixtures under intense dynamic loading
conditions is needed for designing inertial confinement fusion (ICF) targets
and developing planetary formation models.Lorenzen  _et al._ (2009);
H.F.Wilson and Militzer (2010); Ostanin  _et al._ (2006) The complicated
computer simulations of these dynamic events rely on accurate equation of
state models (EOS) that span a broad range of temperatures, pressures, and
densities. During impact/implosion situations or in planetary mix layers where
convection occurs, pure materials will often dynamically mix, and a new model
for the mixed material is required. The most extreme limit of mixing is full
homogenization on the molecular scale. We examine the case of thermally
equilibrated and homogenized molecular mixing. This is the final stage of a
dynamic mix and where one would expect the most significant interspecies
interactions resulting in the largest deviations from models based on isolated
species.

Even in the case of binary mixtures, it is often impractical and expensive to
perform experiments on many mixture compositions. Computer simulations can
more expeditiously be performed on a much larger set of mixture compositions
at a vastly reduced cost. However, the physics of mixtures even at the
molecular scale is complex and the reliance on these simulations must be
tempered with validation studies. We report here a detailed validation study
using density functional theory - quantum molecular dynamics (DFT-QMD)
simulations of warm dense matter mixtures and experiments.

As a specific example, we study the miscible cryogenic liquid mixtures of
xenon and ethane (C2H6). This combination is amenable to experiments because
mixtures can be attained with modest pressures and liquid nitrogen cooling.
The mixture is also a good proxy for the polymer liners and heavier elements
that might be used in fuel capsules on the National Ignition Facility and
other Inertial Confinement Fusion (ICF) targets.Biener  _et al._ (2012); Clark
_et al._ (2010) Previous studies of the isolated species have found good
agreement between experiment and theory. Since we expect significant
interspecies interactions for molecular scale mixtures, it is not immediately
obvious that computations of the mixtures would be as accurate as pure species
simulations. We perform DFT-QMD analysis of various ethane-xenon mixture
ratios and calculate their Hugoniots. The results are compared to Hugoniot
data obtained from flyer plate experiments using Sandia’s Z-Machine.

Once validated, simulations can provide additional insight into the physical
properties of materials. For shock compression experiments, temperature is
difficult to measure. This quantity however comes naturally in the DFT-QMD
simulations, and a validation study of the pressures and compression ratios
adds support that the predicted temperatures are reliable. An important
feature in the Hugoniot data of reactive systems is the onset and completion
of molecular dissociation. Often at the onset of dissociation, a change in
slope or a plateau is observed in the Hugoniot data. Dissociation increases
the pressure and free electron density causing the Hugoniot to show
significant steepening upon completion where small changes in the shock
compressed density result in precipitous increases in the pressure.Dick
(1970); Mattsson  _et al._ (2010); Sherman  _et al._ (2012); Root  _et al._
(2013) However, a possible competing effect is exotic bonding such as the
formation of xenon-hydride under pressure.Somayazulu  _et al._ (2010) Through
a bond tracking analysis we can measure the stoichiometric ratios of various
products to characterize the interspecies chemistry along the Hugoniot.

This paper is organized into 3 sections. In the first, we discuss some of the
details and convergence criteria used in the DFT-QMD simulations. The second
section describes the details of the experimental setup on the Z-Machine. and
the third section summarizes the combined results of simulations and
experiments.

##  II Density functional / quantum molecular dynamics simulations

In DFT/QMD, a number of nuclei are moved on the Born-Oppenheimer potential-
energy surface of thermally excited electrons. The DFT method is outlined in
Refs. 11; 12; 13. The DFT/QMD simulations were performed with VASP 5.2, Kresse
and Hafner (1993, 1994); Kresse and Furthmüller (1996) a plane-wave, periodic-
boundary-conditions code that employs projector augmented-wave (PAW) core
functions.Blöchl (1994); Kresse and Joubert (1999) We use stringent
convergence criteria.Mattsson  _et al._ (2005) The simulations are performed
in the NVT ensemble (fixed number of atoms and fixed volume/density at
prescribed temperature). We employ velocity scaling as the thermostat for the
simulations; however, additional simulations using an Nose-Hoover thermostat
show negligible difference for Hugoniot states. Complex k-point sampling with
the Balderashi mean-value point is applied because its accuracy and efficiency
for disordered structures at high temperature. We run Mermin’s finite
temperature formulation of DFT with ground-state exchange correlation
functionals,Mermin (1965) shown to be critical for high energy-density
applications.Desjarlais (2003) We report the results for only AM05, Armiento
and Mattsson (2005); Mattsson  _et al._ (2008) which is particularly well
suited to describe compressed solids and liquids.Root  _et al._ (2010) Results
within the local density approximation (LDA) are comparable.

A direct route to compare experimental shock data to DFT-MD simulations is
through the calculation of Hugoniot states. The hydrostatic Hugoniot condition
can be expressed as $2(E-E_{ref})=(P+P_{ref})(V_{ref}-V)$ where E is the
internal energy per mass, P is the system pressure, and V is the specific
volume and is related to the inverse of the mass density, $V=1/\rho$. The
subscript ref refers to the reference state, which is at the initial
conditions of the experiment.

Each simulation was allowed to equilibrate at a constant temperature and
density for multiple picoseconds or until the block averaged Allen and
Tildesley (1990) standard deviation of the mean was less than $1\%$. At each
density, we used two temperatures to approximate the Hugoniot relation; one
temperature such that the pressure and energy were too high and the other too
low. We then interpolated between them to obtain the Hugoniot pressure,
energy, and temperature. For higher compression points along the Hugoniot, an
alternative approach is possible in which several densities at fixed
temperatures are performed and used to locate the Hugoniot state.

Throughout, we reserve $x$ for the mass mixing ratio defined as

$x=\frac{n_{ethane}M_{ethane}}{n_{Xe}M_{Xe}+n_{ethane}M_{ethane}}$ (1)

where n is the number density of molecules (atoms) and M is the molecular mass
of each species. This form is convenient because EOS tables are often given in
terms of mass densities.

Figure 1:  Snapshot of the liquid ethane reference state nuclear positions
within a super-cell at density $\rho=0.571$ $g/cm^{3}$at 163 K from a DFT/QMD
AM05 calculation. Carbon atoms are brown and hydrogen atoms are white.

The reference state ($E_{ref}$, $P_{ref}$, and $V_{ref}$) for ethane, Fig. 1,
was chosen to closely match experimental initial conditions of 0.571
$g/cm^{3}$and 163 Kelvin determined from cryomixture data.Filipe  _et al._
(2000) For the 50/50 molar mixing $x=0.19$, the reference state at 163K and P
$<10$ kBar is $\rho_{ref}=1.676$ $g/cm^{3}$and is shown in Fig. 2. For the
50/50 mass mixing $x=0.5$, the reference state at 163K and P $<20$ kBar is
$\rho_{ref}=0.960$ $g/cm^{3}$. The densities are taken to be given for the
reference state and we do not perform calculations to optimize over density at
ambient pressure. The results for pure xenon are reported in a previous
validation study.Root  _et al._ (2010)

A computer code was created to position atoms within a super cells to
represent the Xe-Ethane mixture. The center of mass positions of the ethane
molecules were chosen to uniformly fill a super cell with $2^{3}$, $3^{3}$,
and $4^{3}$ ethane molecules. To achieve a desired mix ratio $x$, a number of
ethane molecules were randomly substituted with Xe. It was found that the
largest super cell ($4^{3}$)was impractical for more than a few simulations.
The $3\times 3\times 3$ was converged with respect to simulation size through
a scaling test of the pressure at 1 MBar. The cell size was scaled to achieve
a given density. Note that changing the density was done adiabatically by
gradually scaling the super cell, running a dynamic DFT-MD simulation to
equilibrate the bond lengths, and then scaling further. A sudden large density
scaling would strain bonds, increasing the system energy too dramatically and
result in premature bond breaking.

We determined that 27 ethane molecules in each super-cell is sufficient by
simulating 64 molecules at 0.8, 1.1, 1.2, 1.5, and 1.8 $g/cm^{3}$and comparing
to 27 molecule results for pressure and energy. The large super-cell
simulations gave the same Hugoniot points to within $\leq 0.5\%$ in pressure.
The number and type of atoms in each super-cell varies according to the mix
ratio. The 50/50 mass mix ($x=0.5$) had 5 xenon and 22 ethane molecules (5 Xe,
44 C 132 H) for a total of 181 atoms. The 50/50 molar mix ($x=0.19$, see Fig.
2 had 13 Xenon and 13 Ethane (13 Xe, 26 C and 78 H ) for a total of 117 atoms.

We address the degree of dissociation using a bond tracking analysis on the
nuclear positions.Root  _et al._ (2013) Two elements are considered bonded if
they are within a certain bonding radius for at least as long as a given
persistence time. For the results quoted here, we choose a persistence time of
100 fs which corresponds to an inverse vibrational frequency time scale for
the C-C bond. The bond lengths we target are slightly longer than the
equilibrium bond lengths as we expect elongation with temperature. We use Xe-
Xe 0.25 Å, Xe-C 2.0 Å, Xe-H 2.0 Å, C-C 1.68 Å, C-H 1.28 Å, and H-H 0.8 Å.
Since Xe-Xe bonding is unlikely, we chose an arbitrarily small bond cut-off.

Figure 2: Same as Fig. 1 except now including Xenon with mass ratio x=0.19 and
density $\rho=1.676$ $g/cm^{3}$. Xenon atoms are grey.

Results from the DFT simulations are listed in Tables 1-3 and are compared to
the experimental data in Section IV.

Table 1: DFT-MD results for the liquid ethane Hugoniot Density | Pressure | Energy | Temperature
---|---|---|---
($g/cm^{3}$) | (GPa) | (eV/atom) | (K)
0.571 | 0.4674 | -5.264 | 163
0.800 | 2.290 | -5.238 | 204
1.00 | 7.438 | -5.137 | 530
1.10 | 11.23 | -5.054 | 618
1.20 | 16.92 | -4.918 | 947
1.30 | 25.30 | -4.705 | 1487
1.40 | 35.44 | -4.471 | 2175
1.50 | 46.98 | -4.232 | 2876
1.60 | 55.71 | -3.953 | 3142
1.70 | 64.50 | -3.677 | 3386
1.80 | 78.20 | -3.259 | 4043
1.906 | 110.3 | -2.586 | 7000
1.929 | 134.7 | -2.019 | 10000
1.942 | 175.0 | -1.040 | 15000
1.95 | 187.4 | -0.754 | 16669
1.964 | 215.8 | -0.056 | 20000
1.993 | 300.8 | 2.044 | 30000
2.000 | 320.1 | 2.526 | 32139
2.019 | 390.9 | 4.288 | 40000
Table 2: DFT-MD Hugoniot points for the 50/50 mass mixture ($x=0.5$) of liquid xenon-ethane Density | Pressure | Energy | Temperature
---|---|---|---
($g/cm^{3}$) | (GPa) | (eV/atom) | (K)
0.960 | 0.893 | -5.151 | 163
1.250 | 2.068 | -5.098 | 214
1.500 | 4.270 | -5.076 | 226
1.600 | 5.875 | -5.043 | 308
1.750 | 9.055 | -4.971 | 495
2.000 | 16.20 | -4.790 | 1003
2.250 | 28.25 | -4.465 | 1972
2.500 | 42.81 | -4.041 | 2919
2.750 | 58.30 | -3.560 | 3730
2.997 | 77.51 | -2.948 | 5000
3.000 | 77.78 | -2.939 | 5021
3.050 | 83.12 | -2.770 | 5451
3.140 | 117.0 | -1.753 | 10000
3.191 | 186.2 | 0.289 | 20000
3.231 | 260.2 | 2.492 | 30000
3.307 | 337.9 | 4.866 | 40000
3.360 | 421.4 | 7.418 | 50000
3.378 | 508.0 | 10.03 | 60000
Table 3: DFT-MD Hugoniot points for the 50/50 molar mixture ($x=0.19$) of liquid xenon-ethane Density | Pressure | Energy | Temperature
---|---|---|---
($g/cm^{3}$) | (GPa) | (eV/atom) | (K)
1.676 | 0.780 | -4.708 | 163
2.500 | 3.257 | -4.664 | 167
2.600 | 3.997 | -4.646 | 209
2.800 | 5.983 | -4.594 | 310
3.050 | 8.876 | -4.508 | 500
3.250 | 10.64 | -4.445 | 800
3.500 | 16.23 | -4.264 | 1221
3.750 | 21.92 | -4.062 | 1834
4.000 | 29.58 | -3.783 | 2579
4.500 | 44.98 | -3.173 | 3652
5.000 | 67.10 | -2.266 | 5902
5.250 | 81.74 | -1.655 | 7409
5.514 | 128.0 | 0.198 | 15000
5.550 | 142.8 | 0.786 | 17463
5.609 | 156.1 | 1.327 | 20000
5.742 | 219.0 | 3.855 | 30000
5.850 | 286.2 | 6.576 | 40000
6.089 | 442.1 | 13.10 | 60000

##  III Experimental Approach and Results

To compare with the DFT simulations and to examine pure ethane and ethane-
xenon mixtures at extreme conditions, we performed a series of shock and
reshock experiments using Sandia’s Z-Machine.M. E. Savage et al. (2007) The
Z-Machine is capable of accelerating solid aluminum flyers up to velocities of
40 km/s.Lemke  _et al._ (2005) This technique has been used to accurately
measure the Hugoniot and second shock states for several cryogenic fluids such
as deuterium, M. D. Knudson et al. (2001); Knudson  _et al._ (2004) xenon,Root
_et al._ (2010) and CO2Root  _et al._ (2013). The target cell consists of two
z-cut $\alpha$ quartz windows on the front and back of the target cell. A
copper spacer is placed between the quartz windows. The rear window consists
of two quartz windows bonded together using Epotek 301-2 to form a top-hat.
The combination of the copper spacer and the smaller diameter quartz window in
the top-hat set the cryogenic liquid sample thickness. The quartz windows are
anti-reflected coated to reduce Fresnel reflections at the interfaces. The
target cells are connected to a liquid nitrogen cryostatRoot  _et al._ (2010,
2013) and cooled to temperature. A schematic view of the cryotarget and
experimental configuration is shown in Figure 3. Further details of the
cryotarget and system can be found in Ref. 23.

Velocity interferometry (VISAR) Barker and Hollenbach (1972) is the primary
diagnostic for the shock experiments. The Sandia VISAR system consists of two
dual velocity per fringe (vpf) interferometers, which allows up to four
different vpf settings to be used on a single target. This eliminates 2$\pi$
ambiguity in the data analysis. Typical uncertainty in the velocity records is
$<0.5$%. As shown in the experimental schematic view (Fig. 3), the 532 nm
laser used for the VISAR passes through the target cell and reflects off the
aluminum flyer. The velocity of the aluminum flyer is measured up to impact
with the quartz front window. The shock front generated in the quartz window
at impact is reflectiveKnudson and Desjarlais (2009) and the shock velocity in
the quartz is measured directly using the VISAR. The shock transmitted into
the ethane or the ethane-xenon mixture has a reflective shock front allowing
the shock velocity in the cryogenic liquid sample to be measured directly.
Lastly, the shock front in the rear quartz top-hat is also reflective, from
which the state in the quartz and the reshock state in the liquid sample can
be determined accurately.

Figure 3: Schematic view of the target cell configuration for the ethane and
the ethane-xenon mixture shock compression experiments

The target cells were filed with high purity ($>$99.99%) ethane gas (Matheson
TriGas) to 16.8 PSI and cooled to the final temperature. The initial density
of liquid ethane was determined from the fit to the experimental data in
Reference 32 with an uncertainty of 0.5%. The index of refraction of liquid
ethane was determined from the data in Reference 33. Ethane-xenon gas mixtures
were supplied by Matheson TriGas and the molar ratio content was verified
using mass spectrometry. The initial density of the ethane-xenon mixtures is
determined from a linear fit to the experimental density data in Reference 25.
The index of refraction for the mixture was calculated using the Lorentz -
Lorentz mixing rule:

$\frac{n_{12}^{2}-1}{n_{12}^{2}+2}=\Phi_{1}\frac{n_{1}^{2}-1}{n_{1}^{2}+2}+\Phi_{2}\frac{n_{2}^{2}-1}{n_{2}^{2}+2}$
(2)

with

$\Phi_{i}=y_{i}m_{i}/\rho_{i}/\sum_{j}{y_{j}m_{j}/\rho_{j}}.$

where $y$ is the molar fraction, $m$ is the atomic mass, and $\rho$ is the
density.Mehra (2003) The initial density and index of refraction values for
liquid xenon were taken from the literature.Leadbetter and Thomas (1965) Since
the shock velocity is measured directly, the measured shock velocities can be
integrated with respect to time to ensure the distance traveled is consistent
with the sample size.

The principal Hugoniot state is derived using a Monte Carlo impedance matching
with a cubic fit to the published quartz dataKnudson and Desjarlais (2009);
Root  _et al._ (2013) and a quartz release model based on an effective
Grünesein $\Gamma$ parameter.Knudson and Desjarlais (2013) For the experiments
on the mixtures, the uncertainty in the molar concentration contributes
primarily to the uncertainty in initial density (up to $\approx$4%), while the
effect on the index of refraction is small, $<0.05$%. The small effect on the
index is primarily caused by xenon and ethane having similar indices of
refraction at the experimental temperatures: at 161.5 K $n_{Xe}$ = 1.392 and
$n_{ethane}$ = 1.343. Tables 4 \- 6 list the experimental observables
($U_{S}^{Quartz}$ and $U_{S}^{Liquid}$) and the derived Hugoniot states.

Table 4: Experimental data of the principal Hugoniot for shock compressed liquid ethane. Shot | Quartz $U_{S}$ | $\rho_{0}^{Ethane}$ | $T_{0}$ | $U_{P}$ | $U_{S}$ | $\rho$ | P
---|---|---|---|---|---|---|---
| (km/s) | (g/cm3) | (K) | (km/s) | (km/s) | (g/cm3) | (GPa)
Z2331 | 17.32$\pm$0.03 | 0.571 | 162.0 | 14.00 $\pm$ 0.05 | 20.07 $\pm$ 0.03 | 1.888 $\pm$ 0.017 | 160.4 $\pm$ 0.8
Z2226 | 18.17$\pm$0.04 | 0.570 | 161.5 | 14.89 $\pm$ 0.06 | 21.3 $\pm$ 0.07 | 1.897 $\pm$ 0.024 | 181.1 $\pm$ 1.1
Z2277 | 20.52$\pm$0.04 | 0.572 | 163.5 | 17.40 $\pm$ 0.06 | 24.6 $\pm$ 0.06 | 1.952 $\pm$ 0.021 | 244.5 $\pm$ 1.4
Table 5: Principal Hugoniot experimental data for shock compressed liquid Xe-ethane $\sim$50/50 mass mix ($x=0.5$). Shot | Quartz $U_{S}$ | $\rho_{0}^{MIX}$ | $T_{0}$ | $U_{P}$ | $U_{S}$ | $\rho$ | P
---|---|---|---|---|---|---|---
| (km/s) | (g/cm3) | (K) | (km/s) | (km/s) | (g/cm3) | (GPa)
Z2527N | 23.25 $\pm$ 0.07 | 0.958$\pm$0.021 | 161.5 | 18.61 $\pm$ 0.13 | 25.75 $\pm$ 0.07 | 3.457 $\pm$ 0.061 | 459.1 $\pm$ 8.1
Z2527S | 24.23 $\pm$ 0.08 | 0.958$\pm$0.021 | 161.5 | 19.59 $\pm$ 0.14 | 27.08 $\pm$ 0.09 | 3.464 $\pm$ 0.067 | 508.1 $\pm$ 9.0
Table 6: Experimental data of the principal Hugoniot for shock compressed mixtures of a 50/50 molar mixture ($x=0.19$) of liquid ethane and liquid xenon with an initial density of $\rho$=1.676 $g/cm^{3}$ .  Shot | Quartz $U_{S}$ | $\rho_{0}^{MIX}$ | $T_{0}$ | $U_{P}$ | $U_{S}$ | $\rho$ | P
---|---|---|---|---|---|---|---
(km/s) | (g/cm3) | (K) | (km/s) | (km/s) | (g/cm3) | (GPa)
Z2226 | 18.15 $\pm$ 0.05 | 1.676$\pm$0.073 | 163.5 | 12.18 $\pm$ 0.16 | 17.12 $\pm$ 0.09 | 5.814 $\pm$ 0.142 | 349.4 $\pm$ 11.3
Z2277 | 20.56 $\pm$ 0.04 | 1.676$\pm$0.073 | 163.5 | 14.29 $\pm$ 0.18 | 19.62 $\pm$ 0.08 | 6.171 $\pm$ 0.122 | 469.6 $\pm$ 15.0
Z2295 | 20.95 $\pm$ 0.03 | 1.676$\pm$0.073 | 163.0 | 14.63 $\pm$ 0.18 | 20.08 $\pm$ 0.07 | 6.173 $\pm$ 0.109 | 491.9 $\pm$ 15.7

##  IV Discussion and Results

Figure 4 plots the experimental observables $U_{S}^{Quartz}$ and
$U_{S}^{Liquid}$ for the four samples: pure xenon, 50/50 molar mixture
($x=0.19$), 50/50 mass mixture ($x=0.5$), and pure ethane. The results show
that the xenon $U_{S}^{Xenon}$ has the lowest velocity as a function of
$U_{S}^{Quartz}$. The observed $U_{S}^{Liquid}$ increases as the amount of
ethane increases. The molar and mass mixture observable data are bounded by
the pure ethane and xenon data as expected. A large change in the slope
parameter occurs in the linear fits to the data between the molar and mass
mixtures. That the change in the slope parameter occurs between the mass and
molar mix ratios suggests that the high pressure response is governed more by
the differing masses of the elements rather than the relative chemical
interactions.

Figure 5 shows the compiled Hugoniot data sets for the pure and mixed liquids
including the pure xenon, which was shown in Ref. 23. Tables 1-3 show the
calculated values for points along the Hugoniot. The temperatures given in
clean values such as 40000K were fixed when a density scaling procedure is
performed. These are often the higher temperature points where the sharp rise
in the Hugoniot makes temperature scaling inaccurate. Note that for ethane, we
have both density and temperature scaling points interwoven suggesting that
both methods are equally accurate in an intermediate region. The results from
the DFT simulations show good agreement with the experimental data. For the
pure ethane and the two mixtures, the Hugoniot is observed to undergo a sharp
steepening after the complete dissociation of the ethane. The DFT simulation
results tend to show a slightly stiffer Hugoniot than the experimental results
at the highest pressures. This has been observed in other comparisons with
experiments,Root  _et al._ (2010, 2013) which is likely caused by the
computational limits on the number of plane waves that can be used to describe
the energy states.

Figure 4: Experimental measurements: quartz shock velocity ($U_{S}^{Quartz}$)
vs. sample shock velocity ($U_{S}$). The data for the pure xenon is from Ref.
23. The lines are linear fits to the data. Figure 5: Hugoniot data from the
Z-experiments and the DFT simulations for the pure ethane, mass mixture, molar
mixture, and pure xenon. All DFT data start from a reference state density for
the system at T=163K and 0 bar pressure. The initial states for the
experimental data are listed in Tables 4 \- 6 Error bars for the experimental
data are on the order of the symbol unless indicated otherwise.

Experimentally, measuring temperature is difficult. However, temperature is a
known quantity in the DFT simulations. Figure 6 shows the DFT Hugoniot data in
pressure - temperature space. For a given pressure, the shock temperature in
pure xenon is noticeably higher than the pure ethane or ethane-xenon mixtures.
Dissociation of the ethane absorbs energy generated in the shock, thus keeping
the temperature lower. The addition of xenon means fewer dissociating ethane
molecules to absorb the shock energy and higher temperature for a given
pressure. Because of xenon’s higher mass it moves slower on average at a given
temperature than ethane. This results in less frequent collisions and
resulting chemistry for a given temperature.

The temperature data (Fig. 6) also provides indication of complete
dissociation of the ethane molecules around 100 GPa where the slope in the P-T
data increases. The mass mixture is mostly ethane and still exhibits the
discreet regions of differing slopes in P-T before the sharp up turn. This is
not evident in the molar results suggesting that the chemistry is highly
affected by the 1-to-1 ratio of xenon to ethane. The inclusion of Xe also has
a mollifying effect on the sharp upturn observed in ethane $\rho$-P Hugoniot
(Fig. 5 ), which is a signature of dissociation completion. Both the mass and
molar mixture exhibit a sharp upturn but at slightly higher pressures and the
post-dissociation Hugoniot is softer than the pure ethane.

Figure 6: DFT Hugoniot calculations in pressure - temperature space.

In Fig. 7, we juxtapose the principal Hugoniot for shocked liquid ethane with
the decomposition pathway of the ethane determined from the DFT simulations.
Ethane decomposition starts at approximately 29 GPa and corresponding density
of 1.3 $g/cm^{3}$. In terms of compression, $\mu=1-\rho_{0}/\rho$; the onset
of decomposition is $\mu=0.56$. The decomposition begins with the removal of H
from the ethane molecule. Initially, the trace molecules observed are
primarily C2H5 and C2H4, but can include other C-H molecules. As the shock
pressure increases, the number of liberated H and the number of trace
molecules increases. By 1.7 $g/cm^{3}$, most molecules have decomposed with
only a few (1% to 2%) CH and $CH_{2}$ present at any given time. At pressures
above 55 GPa and $\rho=$ 1.7 $g/cm^{3}$, the trace CXHY molecules begin to
dissociate rapidly into C and H atoms. Complete dissociation - no C2H6 and no
trace C-C or C-H molecules occurs by 110 GPa and a density of 1.9
$g/cm^{3}$($\mu=0.70$).

Figure 7: The percent of total atoms in the system that are in each molecule
type. The xenon atom percentage is not shown. Trace refers to all possible C-C
and C-H combinations excluding C2H6. Juxtaposed on the plot is the DFT
Hugoniot (dashed-line). Complete dissociation is observed by 1.9 g/cm3.

Figure 8 plots the decomposition pathways for the mass mix
($\rho_{0}=0.960$$g/cm^{3}$, $x=0.50$). The onset of appreciable decomposition
starts after 9 GPa and a density of 1.75 $g/cm^{3}$, which corresponds to a
compression of $\mu=0.45$. The remainder of trace molecules grows rapidly with
shock pressure obtaining a maximum at 43 GPa. Above this pressure, atomic C
begins to form. Decomposition is completed by 117 GPa and
$\rho=3.14$$g/cm^{3}$. The compression at this state is $\mu=0.69$. At this
point, the Hugoniot steepens significantly as shown in Figure 8.

Figure 9 shows a super-cell of the molar mixture ($x=0.19$) near complete
dissociation. The carbon atoms appear to form cross-linking chains under
compression with the majority of carbon atoms participating in chain
formation. This is because the bonds shown are for instantaneously close
carbons. The persistence time of these chains is shorter than the 100fs
window. However, the separate species co-location suggests the tendency for H
and C to demix under these conditions. Figure 10 shows the decomposition
pathway for the molar mixture ($\rho_{0}=1.676$$g/cm^{3}$, $x=0.19$). The
additional xenon increases the pressure for complete dissociation to
$\approx$126 GPa and a density of 5.5$g/cm^{3}$; this corresponds to a
compression of $\mu$=0.69 for complete dissociation. While the addition of
xenon causes the complete dissociation pressure and temperature to increase
slightly with increased xenon, the end compression state $\mu$ is the nearly
same for ethane and mixtures of xenon-ethane. In neither the mass ($x=0.5$)
nor the molar ($x=0.19$) mixtures did we observe a xenon-hydride as observed
in static high pressure experiments Somayazulu  _et al._ (2010).

The similar compression ratios indicates that complete dissociation for the
pure ethane and ethane-xenon mixture systems has a strong dependence on the
density. Interestingly, the ratio $\mu=0.69$ for ethane is also similar to the
compression ratios for other C-C bonded systems. Polystyrene completely
dissociates at $\mu$=0.62;Wang  _et al._ (2011) polyethylene is completely
dissociated at $\mu$=0.62;Cochrane  _et al._ (2012) and poly
4-methyl-1-pentene (PMP) is completely dissociated at $\mu$=0.68.Mattsson  _et
al._ (2010) This suggests that a limiting compression exists for C-C bonded
systems.

Figure 8: Decomposition of the 50/50 mass mixture ($\rho_{0}$=0.96g/cm3,
$x=0.50$) along the Hugoniot. The dashed line indicates the DFT calculated
Hugoniot. Figure 9: Same as Fig. 1 except now including Xenon with mass ratio
x=0.19 and density $\rho=5.30$ $g/cm^{3}$at $T=10,000K$. Xenon atoms are grey.
Figure 10: Decomposition of the 50/50 molar mixture
($\rho_{0}$=1.676$g/cm^{3}$, $x=0.19$) along the Hugoniot. The dashed line
indicates the DFT calculated Hugoniot.

##  V Conclusion

We have performed series of experimental measurements and DFT simulations on
pure ethane and xenon - ethane mixtures to better understand the response of
atomic mixtures at high pressures and temperatures. The experimental
measurements provided accurate Hugoniot data for ethane and xenon mixtures
that were used to validate DFT simulations. Examination of the DFT simulations
showed that ethane dissociation begins with the removal of H. As the pressure
state increases, the ethane eventually completely dissociates into C and H
atoms. The addition of xenon increased the pressure and temperature for
complete dissociation, but that the ultimate compression for dissociation of
the system remained the same regardless of the xenon concentration.

These results suggest that the primary role of introducing xenon into the
mixture is to raise the temperature needed for dissociation and the onset of
the pressure up-turn of the $\rho-P$ Hugoniot relative to the pure ethane
result. This result is dominated by kinematic considerations and inter-species
chemistry plays a minor role in the data. Since the upturn occurs at the
similar compression ratios for each species, the state of complete
dissociation is likely caused by a limiting compression rather than purely
kinetic reasons.

Hydrodynamics simulations of high energy density physics phenomena require a
detailed knowledge of the equation of state of the constituents as well as
their mixtures. In this paper, we analyzed validate high-fidelity DFT/QMD
simulations of mixtures under shock loading conditions. DFT-MD is seen to
provide provide accurate descriptions of Hugoniot state properties of
dissimilar pure species as well as homogeneous mixtures of them.

###### Acknowledgements.

We thank Dr. John Benage for valuable discussions on this work. We thank Dr.
Joel Kress at Los Alamos National Laboratory for valuable discussions on mix
rules and DFT/QMD simulations. We thank Jesse Lynch and Nicole Cofer for
assembling the cryo-targets and we thank Andrew Lopez, Keegan Shelton, and
Jose Villalva for operating the cryogenics systems on Z. The NNSA Science
Campaigns supported this work. Sandia National Laboratories is a multi-program
laboratory managed and operated by Sandia Corporation, a wholly owned
subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy s
National Nuclear Security Administration under contract DE-AC04-94AL85000.

##  Appendix A Reshock States

Although we did not calculate the reshock states using DFT, the target design
shown in Fig. 3, with the rear quartz top-hat, does permit the measurement of
the reshock state in the ethane and mixture samples. Using the method
described in Ref. 9 the reshock state can be determined. The method utilizes a
fit to the $U_{S}$-$U_{P}$ data to determine the initial state of the sample
prior to reshock because some attenuation can exist in the shock velocity as
the shock traverses the sample. Since we only have a few experimental points
for the principal Hugoniot of ethane and the mixtures, we include the DFT data
in the $U_{S}-U_{P}$ fits. We assume a 0.5% uncorrelated uncertainty in the
DFT $U_{S}$-$U_{P}$ data points. The final state pressure $P$ and $U_{P}$ are
known to a high degree of accuracy because of the quartz Hugoniot.Knudson and
Desjarlais (2009) The linear $U_{S}$-$U_{P}$ parameters along with the
correlation between the fit parameters are listed in Table 7. The reshock
states are listed in Tables 8 and 9.

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Table 7: Linear fit parameters to the $U_{S}$-$U_{P}$ experimental data and DFT results. $U_{S}=C_{0}+S_{1}U_{P}$. The term $\sigma_{C_{0}}\sigma_{S_{1}}$ is the off-diagonal term in the covariance matrix of the fit parameters | Range (km/s) |  $C_{0}$ (km/s) | $S_{1}$ | $\sigma_{C_{0}}\sigma_{S_{1}}\times 10^{3}$
---|---|---|---|---
Ethane |  $U_{S}>$ 15.0 | 1.045$\pm$0.254 | 1.349 $\pm$ 0.017 | -4.1634
Mass Mix |  $U_{S}>$ 13.0 | 1.069$\pm$0.195 | 1.336 $\pm$ 0.014 | -2.6083
Molar Mix |  $U_{S}>$ 10.0 | 0.929$\pm$0.127 | 1.313 $\pm$ 0.014 | -1.7045
Table 8: Reshock data for liquid ethane. Shot | Ethane $U_{S}$ | Quartz $U_{S}$ | $\rho_{2}$ | $P_{2}$
---|---|---|---|---
| (km/s) | (km/s) | (g/cm3) | (GPa)
Z2277 | 24.26$\pm$0.06 | 18.36$\pm$0.04 | 2.864 $\pm$ 0.064 | 514.8 $\pm$ 2.7
Z2226 | 21.14$\pm$0.06 | 16.41$\pm$0.06 | 2.722 $\pm$ 0.068 | 397.9 $\pm$ 3.4
Z2331 | 19.81$\pm$0.05 | 15.54$\pm$0.05 | 2.684 $\pm$ 0.064 | 351.3 $\pm$ 2.6
Table 9: Reshock data for the ethane-xenon mixtures. Shot | Mix Ratio | Mix $U_{S}$ | Quartz $U_{S}$ | $\rho_{2}$ | $P_{2}$
---|---|---|---|---|---
| $x$ | (km/s) | (km/s) | (g/cm3) | (GPa)
Z2226 | 0.19 | 16.86$\pm$0.07 | 17.72$\pm$ 0.04 | 6.347$\pm$0.430 | 474.5$\pm$2.5
Z2277 | 0.19 | 19.12$\pm$0.08 | 19.88$\pm$ 0.05 | 6.332$\pm$0.400 | 617.9$\pm$3.7
Z2295 | 0.19 | 19.51$\pm$0.07 | 20.19$\pm$ 0.03 | 6.380$\pm$0.406 | 640.3$\pm$2.3
Z2527 (N) | 0.5 | 25.19$\pm$0.07 | 21.90$\pm$ 0.09 | 4.379$\pm$0.185 | 771.8$\pm$7.4
Z2527 (S) | 0.5 | 26.52$\pm$0.09 | 22.88$\pm$ 0.08 | 4.405$\pm$0.185 | 853.5$\pm$7.0


cross-circuits are then used to properly partition the set of edges of a map
so that homomorphisms will act independently on each subset of the partition.

######  Lemma 36.

Let $M$ be a connected map, and let $N$ be a connected submap of $M$ with the
same signed genus. Then,

  1. (i)

Each facial walk $\kappa$ in $N$ is a prefacial cross-circuit of $M$.

  2. (ii)

Conversely, for each prefacial cross-circuit $\kappa$, there exists a submap
$N(\kappa)$ with the same signed genus as $M$ such that $\kappa$ is a facial
walk of $N(\kappa)$.

  3. (iii)

Each edge in $E(M)\setminus E(N)$ belongs to a unique $M_{\kappa}\setminus
E(\kappa_{\ltimes})$ for some facial walk $\kappa$ in $N$.

###### Proof.

For (i) and (iii) it suffices to prove that if we add the edges $e$ of
$E(M)\setminus E(N)$ to $N$ one by one (so that the resulting map is always
connected), each edge $e$ either subdivides an existing face or adds a new
vertex. Then, a small inductive argument over the facial walks of the
successive maps establishes the result.

As the orientability of the submaps generated by adding the edges $e$ remains
constant, so does the signed genus and the Euler genus (see Lemma 5). Then,
all these submaps have the same genus, which, by Lemma 4, implies that edge
$e$ is either a dual link (hence dividing a face into two) or a bridge (adding
a new vertex), as desired.

Part (ii) follows from removing in $M$ the edges and vertices in $M_{\kappa}$
that do not belong to $\kappa$; this operation creates a submap $N(\kappa)$ of
$M$ satisfying the conditions of the statement. ∎

######  Lemma 37.

Let $\kappa$ be a separating cross-circuit of a connected map $M$.

  1. (i)

A homomorphism $h_{\kappa}:M_{\kappa}\to M_{\kappa}$ fixing the crosses of
$\kappa$ induces a homomorphism $h:M\to M$ with the following properties:

     * •

$\kappa$ is a separating cross-circuit of $h(M)$,

     * •

$h(M)\ltimes\kappa$ consists of the map $M^{\kappa}$ and the connected map
$h_{\kappa}(M_{\kappa})$,

     * •

$h$ is the identity on $M\setminus[E(M_{\kappa})\setminus E(\kappa)]$.

  2. (ii)

The cross-circuit $\phi_{z}$ corresponding to a face $z$ in $M$ is a facial
walk of either $M_{\kappa}$ or $M^{\kappa}$. Conversely, a facial walk of
$M_{\kappa}$ (resp. $M^{\kappa}$) other than $\kappa_{\ltimes}$ (resp.
$\kappa^{\ltimes}$) defines a facial walk of $M$.

###### Proof.

(i) Since the crosses of $\kappa$ are fixed by $h_{\kappa}$, the rifflings
associated with the vertex gluings that $h_{\kappa}$ defines in $M_{\kappa}$
do not involve any cross from $\kappa_{\ltimes}$. Further, we may assume that
no cross from $\kappa_{\ltimes}$ is removed when gluing duplicate edges (in
the notation of Definition 16 at least one of the crosses $a$ and $b$ does not
belong to $\kappa_{\ltimes}$; if one of them belongs, say $a$, we can say that
the edge containing $a$ still belongs to the map). Similarly, we may consider
the identity map on $M^{\kappa}$, and conclude that no vertex or duplicate
edge gluings involve crosses from $\kappa^{\ltimes}$.

As $\kappa$ is separating, Lemma 34(ii)(i) give that
$\mathbf{sg}(M)=\mathbf{sg}(M\ltimes\kappa)=\mathbf{sg}(M_{\kappa}\sqcup
M^{\kappa})$. Thus, we can reverse the cutting operations that form
$M\ltimes\kappa$ by gluing the appropriate vertices (in the notation of
Definition 32 these vertex gluings correspond to riffling back all the crosses
$(\alpha_{2}^{\prime}\alpha_{0}c_{i},\alpha_{1}^{\prime}\alpha_{2}^{\prime}c_{i-1})$,
indices modulo $\ell$) and then gluing the edges of $\kappa_{\ltimes}$ with
the edges of $\kappa^{\ltimes}$. This creates a homomorphism $h:M\to M$
satisfying the stated properties.

(ii) Cutting around a non-self-intersecting cross-circuit $\kappa$ creates, by
Lemma 33, two additional faces in $M\ltimes\kappa$, and these precisely
involve those crosses added in the process of cutting around; the remaining
facial walks not involving the added crosses to $M\ltimes\kappa$ are
unchanged. As in addition additional separating hypothesis on $\kappa$,
$M\ltimes\kappa=M_{\kappa}\sqcup M^{\kappa}$ and the result follows. ∎

A face $z$ of a map $M$ that is a face of $M_{\kappa}$ (in the sense that the
pair of cross cycle permutations of $\phi_{z}$ defines a facial walk of both
$M$ and $M_{\kappa}$) is said to be contained in $\kappa$ (and $\kappa$
contains $z$).

######  Observation 38.

A prefacial cross-circuit $\kappa$ of odd length is either a facial walk or
contains an odd number of odd-degree faces. If $\kappa$ has even length, it
contains an even number of odd-degree faces. The argument to prove the odd-
length statement is the following (similar for even length): $M_{\kappa}$ is
plane, and has an odd-degree face defined by $\kappa_{\ltimes}$. Considering
the dual map $M_{\kappa}^{*}$, there must be an odd number of other faces of
odd degree in $M_{\kappa}$, each of which by Lemma 37(ii) is a face of $M$
contained in $\kappa$ (by the handshaking lemma, the number of odd-degree
vertices is even).

Fixed a prefacial cross-circuit $\kappa$ of a map $M$, there is a partial
order $\preceq$ on contractible cross-circuits of $M_{\kappa}$ defined by:
$\lambda\preceq\mu$ if $\lambda$ determines a cross-circuit in $M_{\mu}$ and
all the crosses in $M_{\lambda}$ belong to $M_{\mu}$ (perhaps with the
exception of $\lambda_{\ltimes}$). This order has maximum element
$\kappa_{\ltimes}$ and, for each face $z$ of $M_{\kappa}$, one of its minimal
elements is $\phi_{z}$.

We next use the partial order $\preceq$ and bring together the previous
results in this section to establish when a whole plane map can be sent (in a
map-homomorphic way) to its surrounding prefacial cross-circuit. When no
confusion may arise, we shall use $M\to\lambda$ to indicate that a map $M$ is
mapped to the map induced by the edges of a prefacial cross-circuit $\lambda$.

######  Lemma 39.

Let $\kappa$ be a prefacial cross-circuit of a connected map $M$. Assume that
there exists a face $z$ of $M_{\kappa}$ such that:

  1. (i)

$z$ is different from the face defined by $\kappa_{\ltimes}$ in $M_{\kappa}$;

  2. (ii)

the degree of $z$ is at least the degree of the face traversed by
$\kappa_{\ltimes}$, and both faces have degree of the same parity;

  3. (iii)

every other face of $M_{\kappa}$ has even degree;

  4. (iv)

$\ell(\kappa)\leq\ell(\lambda)$ for every prefacial cross-circuit $\lambda$ of
$M_{\kappa}$ such that $\phi_{z}\preceq\lambda\preceq\kappa$.

Then there is an endomorphism of $M_{\kappa}$ that maps $M_{\kappa}$ to the
map induced by the edges of $\kappa$ (equivalently $\kappa_{\ltimes}$) and it
is the identity over $\kappa$ and $\kappa_{\ltimes}$.

The following facts are used in the proof of the preceding lemma.

######  Observation 40.

Under the hypotheses of Lemma 39 it holds that:

  * •

$M_{\kappa}$ and all its connected submaps are plane, and hence orientable.
Thus, we can indicate a single permutation cycle associated to faces,
vertices, and contractible cross-circuits.

  * •

Any cross-circuit of $M_{\kappa}$ that is a cycle in the underlying graph of
$M_{\kappa}$ is a contractible cross-circuit (as $M_{\kappa}$ is plane).

  * •

Any prefacial cross-circuit $\lambda$ such that
$\phi_{z}\preceq\lambda\preceq\kappa$ has the same parity as $\kappa$ and
$\phi_{z}$ (this follows from assumptions (ii) and (iii) by a parity argument
involving the fact that if an even contractible cross-circuit contains an odd
degree face, then it contains at least two of them (Observation 38)).

###### Proof of Lemma 39..

We show the result inductively on the number of edges and vertices inside
$M_{\kappa}$. The base case occurs when $\phi_{z}=\kappa$, and so $M_{\kappa}$
is a plane cycle; the statement follows by using the identity map as a map
homomorphism. Suppose now that $\phi_{z}\neq\kappa$; the induction argument
follows a case analysis.

_Case 1: There is no prefacial cross-circuit $\lambda$ with
$\phi_{z}\preceq\lambda\preceq\kappa$ and $\phi_{z}\neq\lambda\neq\kappa$._
Let $z^{\prime}$ be a face different from $z$ containing crosses of $\kappa$
in $M_{\kappa}$ (it exists as $\phi_{z}\neq\kappa$). We shall assume that
$\phi_{z^{\prime}}\neq\kappa$, otherwise $\kappa$ would only contain
$z^{\prime}$ and not $z$ as well. Let
$\phi_{z^{\prime}}=(\;\;c_{0}\;\;\;\cdots\;\;\;c_{s}\;\;)$ such that
$c_{0}\in\kappa$ and $c_{1}\notin\kappa$. Note that $z^{\prime}$ must contain
crosses in $\kappa$ and outside $\kappa$: if all the crosses were in $\kappa$,
it would be $\kappa=\phi_{z^{\prime}}$ as $\kappa_{\ltimes}$ is a cycle in the
underlying graph of the plane map; if all the crosses were outside $\kappa$,
one could find a prefacial cross-circuit $\lambda$ between $\phi_{z}$ and
$\kappa$ (using the connectedness of $M$ and $M_{\kappa}$.).

_Case 1.1: If the vertex adjacent to $c_{2}$ belongs to $\kappa$ and is the
same as the vertex of $c_{0}$_, then $c_{2}=c_{0}$; otherwise we can find a
prefacial cross-circuit $\lambda$ strictly shorter than $\kappa$ containing
$z$ but, as there are no other prefacial circuits between them, it must be
$\lambda=\phi_{z}$, which contradicts assumption (ii). Hence, we conclude
$c_{2}=c_{0}$ and $z^{\prime}$ is a face of length $2$. By removing/gluing the
edge closing $z^{\prime}$, we obtain a map $M_{\kappa}^{\prime}$ with no new
prefacial cross-circuits between $\phi_{z}$ and $\kappa$ satisfying the
hypotheses of Lemma 39. By induction, $M_{\kappa}^{\prime}$ can be mapped to
$\kappa$ by a homomorphism. Thus, the composition of first removing/gluing the
edge from $M_{\kappa}$ to $M_{\kappa}^{\prime}$ and then mapping
$M_{\kappa}^{\prime}$ to $\kappa$ gives us the result.

_Case 1.2: If the vertex adjacent to $c_{2}$ belongs to $\kappa$ and is
different than the vertex of $c_{0}$,_ then $\kappa$ is of the form
$(\;\;c_{0}\;\;\;c_{1}^{\prime}\;\;\;\cdots\;\;\;c_{i}^{\prime}\;\;\;c_{2}\;\;\;c_{i+2}^{\prime}\;\;\;\cdots\;\;\;c_{\ell(\kappa)-1}^{\prime}\;\;)$
and it can be partitioned into the crosses
$\\{c_{1}^{\prime},\ldots,c_{i}^{\prime}\\}$ and
$\\{c_{2},c_{i+2}^{\prime},\ldots,c_{\ell(\kappa)-1}^{\prime},c_{0}\\}$. Then,
either the prefacial cross-circuit formed by $\alpha_{2}\alpha_{0}c_{1}$ and
the crosses $\\{c_{1}^{\prime},\ldots,c_{i}^{\prime}\\}$, or the one formed by
$c_{1}$ and
$\\{c_{2},c_{i+2}^{\prime},\cdots,c_{\ell(\kappa)-1}^{\prime},c_{0}\\}$
contains face $z$. Hence, this prefacial cross-circuit must be $\phi_{z}$ as
there are no prefacial cross-circuits between $\kappa$ and $\phi_{z}$ (and it
is not $\kappa$). Since the vertices of $c_{0}$ and $c_{2}$ are distinct, the
length of $\phi_{z}$ is strictly smaller than that of $\kappa$, contradicting
assumption (ii).

_Case 1.3: If the vertex adjacent to $c_{2}$ does not belong to $\kappa$,_
then we glue the vertices of $c_{0}$ and $c_{2}$ along the face $z^{\prime}$
via $M_{\kappa}^{(c_{0}\;c_{2})}$, and remove/glue the edge of $c_{2}$. We
obtain a new map $M_{\kappa}^{\prime}$ with $\kappa_{\ltimes}$ in the outer
face, one edge and one vertex less, and all the faces having even length (as
$z^{\prime}$ had even length and we have shrink it by two units). Further, we
have not created any prefacial cross-circuit between $\kappa$ and $\phi_{z}$,
as it would correspond to a (perhaps several) contractible cross-circuit
between $\kappa$ and $\phi_{z}$ in $M_{\kappa}$. No other face length has been
altered (in particular, the length of $z$). Thus, the assumptions of Lemma 39
are met, and we can apply the result by induction. The result follows by
composing the vertex gluing $M_{\kappa}^{(c_{0}\;c_{2})}$ with gluing the now
duplicate edge containing $c_{2}$ and the mapping from $M_{\kappa}^{\prime}$
to $\kappa$. (Note that one of the glued vertices does not belong to $\kappa$,
and neither does the edge where $c_{2}$ belongs, thus the condition that the
mapping is the identity over $\kappa$ is preserved.)

_Case 2: There exists a prefacial cross-circuit $\lambda$ with
$\phi_{z}\preceq\lambda\preceq\kappa$ and $\phi_{z}\neq\lambda\neq\kappa$._
Let $\lambda_{0}$ be the prefacial cross-circuit that is the “closest” to
$\phi_{z}$ having the same length as $\kappa$ (perhaps being $\kappa$ itself),
that is,

  * •

$\lambda_{0}\neq\phi_{z}$ and $\phi_{z}\prec\lambda_{0}\preceq\kappa$;

  * •

$\ell(\kappa)=\ell(\lambda_{0})$ and $\ell(\lambda_{0})<\ell(\lambda)$ for
every $\lambda$ with $\phi_{z}\prec\lambda\preceq\lambda_{0}$.

The prefacial cross-circuit $\lambda_{0}$ induces a graph cycle in
$M_{\kappa}$; otherwise, by choosing $\lambda_{0}^{\prime}$ to be the cross-
circuit of the (minimal) graph cycle all whose crosses belong to $\lambda_{0}$
and that contains face $z$, we would have
$\phi_{z}\preceq\lambda_{0}^{\prime}\preceq\kappa$ and
$\ell(\lambda_{0}^{\prime})<\ell(\kappa)$, which contradicts assumption
(iv).666Note that we are not claiming that such $\lambda_{0}^{\prime}$ is
inside $\lambda_{0}$; for instance, if $\lambda_{0}^{\prime}$ is a cycle and
$\lambda_{0}$ is that cycle with a pendant edge inside it, then $\lambda_{0}$
is inside $\lambda_{0}^{\prime}$.

_Case 2.1: If $\lambda_{0}\neq\kappa$,_ then $M_{\lambda_{0}}$ is a strict
submap of $M_{\kappa}$ with the same signed genus. Indeed, both maps are plane
and $\lambda_{0}$ induces a graph cycle, and so cutting around $\lambda_{0}$
in $M_{\kappa}$ is the same as removing all the edges and vertices outside
$\lambda_{0}$ in $M_{\kappa}$. Thus, by induction, there is an endomorphism
$M_{\lambda_{0}}\to\lambda_{0}$ that maps $\phi_{z}$ to $\lambda_{0}$, and is
the identity over $\lambda_{0}$. Lemma 37 allows us to extend this
endomorphism to a mapping $h_{\lambda_{0}}:M_{\kappa}\to M_{\kappa}$
(restricted to $M_{\lambda_{0}}$ is the endomorphism
$M_{\lambda_{0}}\to\lambda_{0}$, and is the identity elsewhere). Since
$\lambda_{0}\neq\kappa$, the image $h_{\lambda_{0}}(M_{\kappa})$ is a strict
submap of $M_{\kappa}$ and has $\lambda_{0}$ as a facial walk. We can apply
induction to obtain an endomorphism
$h_{\kappa}:h_{\lambda_{0}}(M_{\kappa})\to\kappa$ that maps $\lambda_{0}$ to
$\kappa$, and is the identity over $\kappa$. The composition
$h_{\lambda_{0}}\circ h_{\kappa}:M_{\kappa}\to\kappa$ has the desired
properties.

_Case 2.2: Assume now that $\lambda_{0}=\kappa$._ We perform a similar
procedure of gluing a vertex and an edge (as in Case 1.3), or simply glue an
edge (as in Case 1.1). Let $\lambda_{1}$ be a prefacial cross-circuit with
$\phi_{z}\prec\lambda_{1}\prec\kappa$ (it exists by the assumption of Case 2)
and such that there is no prefacial cross-circuit $\lambda_{2}$ with
$\lambda_{1}\prec\lambda_{2}\prec\kappa$ (can be assumed as the the number of
prefacial walks is finite). We have $\ell(\lambda_{1})>\ell(\kappa)$ (by the
properties of $\lambda_{0}$ and that $\lambda_{0}=\kappa$ in our case). A
small reduction argument (as done for Case 1) shows that $\lambda_{1}$ is of
the form
$(\;\;c_{0}\;\;\;c_{1}\;\;\;\cdots\;\;\;c_{i}\;\;\;c_{i+1}^{\prime}\;\;\;\cdots\;\;\;c_{t}^{\prime}\;\;)$
where the $c_{j}$’s are crosses of $\kappa$ (assuming we start with $c_{0}$
for convenience) and $c_{i+1}^{\prime},\ldots,c_{t}^{\prime}$ are not. The
vertex of $c_{i+1}^{\prime}$ belongs to $\kappa$, yet the vertex adjacent to
$c_{i+2}^{\prime}$ does not (otherwise it would be
$\ell(\lambda_{1})\leq\ell(\kappa)$). The face of $c_{i+1}$ is also the face
of $\alpha_{0}\alpha_{2}c_{i+1}^{\prime}$ (in fact,
$\phi\alpha_{0}\alpha_{2}c_{i+1}^{\prime}=c_{i+1}$), otherwise there would
exist another prefacial cycle between $\lambda_{1}$ and $\kappa$. Moreover,
the face of $c_{i+1}$ is not $z$ as it is not inside $\lambda_{1}$. Now, we
glue the vertices adjacent to $c_{i+2}$ and $c_{i+2}^{\prime}$ by the riffling
$M_{\kappa}^{(\alpha_{0}\alpha_{2}c_{i+1}\;\phi c_{i+1})}$. Observe that, for
this new map:

  * •

Condition (ii) holds, as $\kappa$ and $\phi_{z}$ have not been modified.

  * •

The parities of the prefacial cross-circuits between $\phi_{z}$ and $\kappa$
are the same as $\kappa$ (Observation 40).

  * •

Condition (iii) holds since we do a gluing through a face outside $z$ of two
vertices at distance two in a face. Further, the parity of the prefacial
cross-circuits between $\phi_{z}$ and $\kappa$ remains the same (using
Observation 40 on both the new and the old map).

  * •

Some prefacial cross-circuits between $\phi_{z}$ and $\kappa$ may have been
removed or modified by reducing their size. The reduction would be of
(exactly) two units, and thus the new prefacial cross-circuits between
$\phi_{z}$ and $\kappa$ satisfy condition (iv) with $\leq$ (the hypothesis on
Case 2 asked for a strict inequality between their lengths).

Therefore, we can apply Lemma 39 inductively to the new map
$M_{\kappa}^{(\alpha_{0}\alpha_{2}c_{i+1}\;\phi c_{i+1})}$. Since the two
mappings are the identity over $\kappa$ (we are using here Lemma 37 for the
vertex gluing that we have performed) the composition of gluing a vertex and
afterwards performing the inductive step is also a mapping that is the
identity over $\kappa$, thus the result follows. ∎

###  4.4 Characterization of map cores

We bring together the notions and results of the previous sections on cores in
the following theorem.

######  Theorem 41.

A connected map $M$ is a core if and only if each of its prefacial cross-
circuits $\lambda$ either is a facial walk, or contains more than one face of
odd degree, or contains no face whose degree is greater than or equal to
$\ell(\lambda)$ and has the same parity as $\ell(\lambda)$.

###### Proof.

We argue with the contrapositive in both implications. From left to right,
assume that there is a prefacial cross-circuit $\lambda$ of $M$ such that: (i)
is not a facial walk, (ii) contains at most one face of odd degree, (iii)
contains a face $z$ whose degree is at least $\ell(\lambda)$ and has the same
parity as $\ell(\lambda)$.

If $\lambda$ has odd length, by (ii) and (iii), $z$ is the only face of odd
degree inside $\lambda$, and any prefacial cross-circuit $\mu$ with
$\phi_{z}\preceq\mu\preceq\lambda$ has odd length (see Observation 40). If
$\lambda$ has even length, then all the faces inside $\lambda$ have even
degree (otherwise, by Observation 38, there would be at least two faces of odd
degree inside $\lambda$).

Let $\kappa$ be a prefacial cross-circuit such that
$\phi_{z}\prec\kappa\preceq\lambda$ and
$\ell(\kappa)=\min\\{\ell(\mu):\phi_{z}\preceq\mu\preceq\lambda\\}$ (perhaps
being $\lambda$ itself). The conditions of Lemma 39 are met with face $z$ and
prefacial cross-circuit $\kappa$. There is thus a homomorphism
$h_{\kappa}:M_{\kappa}\to\kappa$ that is the identity over $\kappa$. Since
$\kappa$ is a prefacial cross-circuit, by Lemma 36, we can conclude that the
submap $M^{\prime}$ obtained by removing the interior of $\kappa$ (which is
the largest submap of $M$ where $\kappa$ is a facial walk) has the same signed
genus as $M$. Further, $\kappa\neq\phi_{z}$ which implies that $M^{\prime}$ is
a proper submap of $M$. Lemma 37 (applied to $h_{\kappa}$) then gives a
homomorphism $h:M\to M^{\prime}\subsetneq M$; therefore, $M$ is not a core.

Now, we prove the implication from right to left using the contrapositive
again. Assume that $M$ is not a core; we want to find a prefacial cross-
circuit satisfying conditions (i)–(iii) stated above.

Let $N$ be the core of $M$, and let $h:M\to N$ be a homomorphism, whose
restriction to $N$ is the identity (see Proposition 27(ii)). Since $N$ is a
proper submap of $M$ with the same signed genus, Lemma 36 asserts that there
is a facial walk of $N$ that is a prefacial cross-circuit $\kappa$ of $M$ but
not a facial walk of $M$. Consider again the submap $M^{\prime}$ of $M$
(defined as above), and let $h^{\prime}:M\to M^{\prime}$ be the mapping
obtained from $h$ as follows:

  * •

Vertices $u,v$ are glued if $h$ glues them and either they are in the interior
of $\kappa$, or $u$ is in the interior of $\kappa$ and $v$ is in $\kappa$.

  * •

Edges $e,e^{\prime}$ are glued if $h$ glues them and either they are in the
interior of $\kappa$, or $e$ is in the interior of $\kappa$ and $e^{\prime}$
belongs to $\kappa$.

Once we cannot perform any of the two previous operations, we stop and obtain
$h^{\prime}$. Observe that we never glue two vertices or two edges of $\kappa$
as $h$ is the identity over $N$.

We now construct a layered and labelled graph $T$ by matching the above
described gluing instructions; the layers will represent the state, at that
stage of the process, of the faces in the interior of $\kappa$, which are
denoted by $\\{f_{1},\ldots,f_{r}\\}$. In the first layer of $T$, we place $r$
vertices labelled $f_{1},\ldots,f_{r}$ (one vertex per face). Now, if a vertex
gluing subdivides a face (into two new faces), we place two new vertices (in
the following layer) adjacent to the vertex that represents the original face
(the parent); the edges are directed, always pointing towards the new
vertices, which are both labelled with the label of the parent. If we glue two
edges, the two faces that they bound are merged into one; this is represented
by a vertex in the new layer, which is adjacent to the parent vertices by
directed edges (again pointing towards the “new”). This new vertex takes the
label of the face with the largest degree among the two that we are merging
(if there is a tie between two faces of degree $2$, we choose the label of one
of them arbitrarily). For the remaining vertices (representing faces,
including those that have not been subdivided), we place a new vertex in the
new layer with the same label as the parent, and with a directed edge pointing
from the old vertex to the new one.

The preceding procedure ensures that all the vertices of $T$ bear one of the
labels of the original faces inside $\kappa$, that is,
$\\{f_{1},\ldots,f_{r}\\}$. Further, from one layer of $T$ to the next, the
process involve some subdivision of faces (when two vertices are glued) and
some destruction of faces (when two edges are glued); at the end, only one
face remains, represented by a unique vertex in the last layer of $T$, whose
label indicates the original face that has not been destroyed/merged-into-
another-one. We next discuss how the parity of the degree of the faces evolve
in this process.

When a face of odd degree is subdivided, a new face of odd degree and a face
of even degree appear; if we subdivide a face of even degree, two faces of odd
degree or two faces of even degree appear. Then, no faces of odd degree are
created when subdividing even degree faces, as there is only one face at the
end of the process, and when an odd degree face is created, actually two are
created, and none of them can be removed (we can only remove faces of degree
$2$). Further, the vertices of any path in $T$ traversing different layers
(from higher to lower layers) have the property that if their label is the
same, the sequence of degrees of their associated faces is weakly decreasing.
Let $f_{i}$ be the label of the unique vertex in the last layer, and consider
a path with all vertices labelled $f_{i}$ that goes from the first to the last
layer; we conclude that:

  * •

the parity of $f_{i}$ is the same as $\kappa$ (it is maintained through the
process, and such facial walk is modified until it becomes $\kappa$);

  * •

the degree of the other faces in $M$ is even, and the degree of $f_{i}$ is
larger or equal than $\ell(\kappa)$.

Thus, $\kappa$ is a prefacial cross-circuit of $M$ satisfying conditions
(i)–(iii); this finishes the contrapositive of the implication from right to
left and completes the proof of the whole statement. ∎

#### Application of the characterization to four classes of maps.

A bouquet is a map with one vertex, all edges being loops. Dually, a quasi-
tree is a map with one face, also known as a unicellular map. (Bouquets and
quasi-trees are necessarily connected.) Theorem 41 says that: a bouquet is a
core if and only if it has no duplicate edges; a quasi-tree is a core if and
only if either it is plane and consists of one edge or one isolated vertex, or
it is not plane and has no vertices of degree one. (These two facts can be
shown directly.) For the other two classes of connected maps, we exploit the
parity conditions for $\kappa$ in Theorem 41 to conclude the following. A
connected map with no vertices of degree one and all faces of odd degree is a
core. Any plane connected core map satisfies exactly one of the following
properties: (i) consists on a single edge or a single isolated vertex (when
all faces have even degree and thus the underlying graph is bipartite by
Observation 40), or (ii) is a cycle of odd length (thus having precisely two
faces, both of odd degree), or (iii) has at least four odd degree faces (four
instead of three using Observation 38).

#### Graph cores vs. map cores.

Since a map homomorphism is a graph homomorphism, any embedding of a graph
core is a map core. However, there are map cores for which the underlying
graph is not a graph core, for example, the quasi-tree embedding of a vertex
with two loops on a torus. Further, the characterization of quasi-trees as map
cores (stated in the previous paragraph) gives a sharp contrast between map
cores and graph cores. Indeed, the fact that any connected graph has a quasi-
tree embedding in some surface, possibly non-orientable [4, Page 123, Third
paragraph], yields that any connected graph without vertices of degree one can
be the underlying graph of a map core by embedding it in an appropriate
surface.

On the other side, identifying two vertices by a graph homomorphism can be
seen as gluing two vertices by a map homomorphism, using an appropriate
embedding (placing a face between the two vertices to be glued); to deal with
parallel edges, the graph is embedded so that they are duplicated. Therefore,
any graph homomorphism can be viewed as the composition of several map
homomorphisms (perhaps by changing the embeddings of the corresponding
underlying graphs).

This discussion prompts a natural question on whether map cores can be used to
identify graph cores: for every connected graph $\Gamma$ that is not a core,
is it possible to find a map $M$ with underlying graph $\Gamma$ such that $M$
is not a (map) core?

We answer this question in the negative. Let $\Gamma$ be the graph resulting
from identifying two copies of the complete graph $K_{5}$ by a vertex (it has
9 vertices and no additional edges). The core of $\Gamma$ is $K_{5}$. In fact,
all graph homomorphisms from $\Gamma$ leading to a subgraph (not necessarily a
core) are obtained by mapping one of the copies of $K_{5}$ into the other.
Now, suppose that there exists a map $M$ with underlying graph $\Gamma$ that
is not a map core. We can apply the arguments used in the proof of Theorem 41
to conclude that there exist a prefacial cross-circuit $\kappa$ (which is not
a facial walk) and a homomophism from $M$ to a strict submap of $M$ that maps
the interior of $\kappa$ to $\kappa$. In particular, this homomorphism is a
graph homomorphism from $\Gamma$ to a subgraph that must be $K_{5}$.
Therefore, the underlying graph of the interior of $\kappa$ should contain
$K_{5}$, which is a contradiction as the interior of $\kappa$ is a plane map.

###  4.5 The poset of map cores

Having now described how to tell whether a map is a core, we introduce and
study the poset of map cores, comparing its properties with those of the poset
of graph cores.

For fixed $g\in\mathbb{Z}$, let $\mathcal{M}(g)$ and $\mathcal{M}_{c}(g)$ be
the sets of, respectively, maps and cores of signed genus $g$. For
$M,M^{\prime}\in\mathcal{M}(g)$, we say that $M\leq M^{\prime}$ if there
exists a map homomorphism from $M$ to $M^{\prime}$, that is, if $M\to
M^{\prime}$. The associated strict order relation, $M<M^{\prime}$, holds if
$M\to M^{\prime}$ and $M^{\prime}\not\to M$. The relation $\leq$ is defined
analogously on $\mathcal{M}_{c}(g)$; indeed, it is obtained from the preorder
$(\mathcal{M}(g),\leq)$ upon quotienting out by the equivalence relation
$\cong$ defined by $M\cong M^{\prime}$ if $M\leq M^{\prime}$ and
$M^{\prime}\leq M$.

######  Lemma 42.

The relation $\leq$ defines a preorder on $\mathcal{M}(g)$ and a partial order
on $\mathcal{M}_{c}(g)$.

###### Proof.

The preorder part follows by Proposition 21 and the fact that the composition
of homomorphisms between maps in $\mathcal{M}(g)$ is again a homomorphism. For
cores, Proposition 26 ensures that the preorder is in fact a partial order,
since we are considering maps up to isomorphism. ∎

The first property that we establish is that the poset
$(\mathcal{M}_{c}(g),\leq)$ is connected, by first showing that the preorder
$(\mathcal{M}(g),\leq)$ is connected. Recall that a preorder $(P,\leq)$ is
connected if for every pair of elements $a,b$ there is a sequence
$a=c_{0},c_{1},\dots,c_{k-1},c_{k}=b$ such that $c_{i}\leq c_{i+1}$ or
$c_{i+1}\leq c_{i}$ for each $i=0,1,\dots,k-1$.

######  Lemma 43.

Given connected maps $M$ and $M^{\prime}$ of the same signed genus and with at
least one edge, $M^{\prime}$ can be obtained from $M$ by a sequence of vertex
gluings, vertex splittings (Definition 31 with the extra condition of Lemma
7(2)(i)), edge splittings, and duplicate edge gluings.

###### Proof.

Vertex gluing and vertex splitting under the conditions indicated are inverse
operations, as are duplicate edge gluing and edge splitting, and all these
operations preserve signed genus and connectivity. A connected map with more
than one face has some edge that is a dual link; dually, a connected map with
more than one vertex has some edge that is a link. For a dual link
$e\equiv(\;c\;\;\alpha_{2}\alpha_{0}c\;)\;(\;\alpha_{2}c\;\;\alpha_{0}\alpha_{2}c\;)$,
crosses $c$ and $\alpha_{2}c$ belong to different faces. Splitting the vertex
containing crosses $c$ and $\alpha_{2}c$ by riffling $c$ and $\tau c$ has the
effect of adding a vertex and removing a face (so that all the crosses of $e$
now belong to a single face), while the signed genus and the connectivity are
preserved, so the number of faces is reduced. This procedure can be repeated
until the map becomes a quasi-tree. If this quasi-tree has more than one
vertex, by contracting links we can reduce the number of vertices, while
preserving signed genus (Lemma 4(ii)) and connectivity, thus producing no new
faces; the quasi-tree can thus be made simultaneously into a bouquet by a
sequence of such link contractions. Such a map with one vertex and one face is
called by Tutte a unitary map. If the map has at least two edges, contracting
a link
$e\equiv(\;c\;\;\alpha_{2}\alpha_{0}c\;)\;(\;\alpha_{2}c\;\;\alpha_{0}\alpha_{2}c\;)$
can be realized by a sequence of vertex gluings/vertex splittings (Definition
31 with the extra condition of Lemma 7(2)(i)) and duplicate edge gluings/edge
splittings as follows: (1) glue the two vertices incident with $e$ by riffling
crosses $c$ and $\phi c$, making $e$ become a loop; then (2) split the vertex
incident with this loop by riffling $c$ and $\alpha_{2}\alpha_{0}c$, thereby
creating a leaf vertex; then (3) glue this leaf vertex to the vertex
containing $\phi^{-1}c$ by riffling $\alpha_{0}\alpha_{2}c$ and $\phi^{-1}c$,
thereby creating a pair of duplicate edges; and finally (4) glue these
duplicate edges.

We have just seen how any connected map with at least one edge can be taken to
a unitary map by a sequence of vertex gluings/vertex splittings and duplicate
edge gluings/edge splittings. The _orientable canonical map_ of genus $g$ is
the unitary map whose vertex permutation has a cycle of the form777When $g=0$
the canonical map is an isolated vertex and the vertex permutation consists of
a pair of empty cycles; as Tutte does not allow edgeless maps, he defines the
canonical map of genus zero to be the map consisting of a single link.
$(\;\;c_{1}\;\;\;c_{2}\;\;\;\alpha_{0}\alpha_{2}c_{2}\;\;\;\alpha_{0}\alpha_{2}c_{1}\;\;\cdots\;\;c_{2g-1}\;\;\;c_{2g}\;\;\;\alpha_{0}\alpha_{2}c_{2g}\;\;\;\alpha_{0}\alpha_{2}c_{2g-1}\;\;),$
and the _non-orientable canonical map_ of genus $g$ has vertex permutation
with a cycle of the form

$(\;\;c_{1}\;\;\;\alpha_{0}c_{1}\;\;\;c_{2}\;\;\;\alpha_{0}c_{2}\;\;\cdots\;\;c_{g}\;\;\;\alpha_{0}c_{g}\;\;).$

As remarked after Definition 31, Tutte defines an operation of vertex
splitting [19, Figs. X.7.1 and X.7.2] related to ours in following our vertex
splitting operation by the insertion of a link; this makes his operation
inverse to that of contracting a link. Thus, to realize Tutte’s vertex
splitting operation we take the inverse sequence of four operations above that
realized the operation of link contraction. In other words, start with the
inverse of step (4) above (split an edge to make a duplicate edge), follow
this by the inverse of step (3) (split an endpoint of the duplicate edge to
make a leaf vertex), then carry out the inverse of step (2) (glue the leaf
vertex to the vertex incident to it), finishing with the inverse of step (1)
(split the endpoint of the loop). Tutte showed [19, Theorem X.37] that, using
vertex splittings in his sense and link contractions, one can transform any
unitary map into a canonical map while maintaining the signed genus.888The
crosses belonging to a _cross cap_ (twisted loop appearing as a pair
$c,\alpha_{0}c$ in a cycle of the vertex permutation) or to a _handle_ (two
interlaced non-twisted loops $a,b,\alpha_{0}\alpha_{2}a,\alpha_{0}\alpha_{2}b$
in a cycle of the vertex permutation) appear consecutively in the canonical
map, or, to use Tutte’s term, they are _assembled_ : Tutte in this way gives a
combinatorial proof of the classification theorem for compact surfaces. Hence
the same is true when using vertex gluings and splittings in our sense, along
with edge splittings and duplicate edge gluings, as these operations can
simulate the two operations used by Tutte in the way described above.

Finally, then, we compose the sequence of operations that reduces $M$ to the
canonical map of its signed genus with the inverse of the sequence of
operations reducing $M^{\prime}$ to the same canonical map, which takes the
canonical map to $M^{\prime}$. In this way we move from $M$ to $M^{\prime}$
via the canonical map of their signed genus by a sequence of operations of the
type described in the lemma statement. ∎

######  Theorem 44.

The poset $(\mathcal{M}_{c}(g),\leq)$ is connected.

###### Proof.

By Lemma 43, for each pair of maps $M,M^{\prime}\in\mathcal{M}(g)$ with at
least one edge there is a sequence of maps
$M_{0}=M,M_{1},M_{2},\dots,M_{k}=M^{\prime}$ such that there is either a
homomorphism from $M_{i}$ to $M_{i+1}$ (when $M_{i+1}$ is obtained from
$M_{i}$ by vertex gluing or duplicate edge gluing) or from $M_{i+1}$ to
$M_{i}$ (when $M_{i+1}$ is obtained from $M_{i}$ by vertex splitting or edge
splitting); all the operations are the ones described in Lemma 43. With the
additional observation that the single vertex can be mapped everywhere in the
case of $g=0$, this shows that the preorder $(\mathcal{M}(g),\leq)$ is
connected; then, by considering the corresponding homomorphisms between their
cores, we obtain that the poset $(\mathcal{M}_{c}(g),\leq)$ is also connected.
∎

We recall some further definitions from poset theory. Let $a,b$ be elements of
a poset $(P,<)$ such that $a<b$. The poset $P$ is _dense_ between $a$ and $b$
if for any $c,d\in P$ with $a\leq c<d\leq b$, there exists an element $e$ such
that $c<e<d$; the pair $a$, $b$ forms a _gap_ in $P$ if there is no element
$c$ with $a<c<b$ (alternatively, $b$ covers $a$, i.e., $b$ is an immediate
successor of $a$). In the poset of (cores of) graphs with order $G\leq H$
defined by the existence of a graph homomorphism $G\to H$, each comparable
pair of cores (with the exception of the one between the vertex and $K_{2}$)
defines an interval that is a dense total order (actually, such interval is
universal, in the sense that every countable partial order can be seen as one
of its suborders [5]). The following theorem shows that the partial order of
map cores is drastically different from that of graph cores.

######  Theorem 45.

In the partial order $(\mathcal{M}_{c}(g),\leq)$, there is no pair of cores
$N_{1},N_{2}$ with $N_{1}<N_{2}$ and a dense total order between them.

###### Proof.

Suppose for a contradiction that there is a dense total order between two
cores $N_{1},N_{2}\in\mathcal{M}_{c}(g)$, and assume first that both are
connected. Thus, there are infinitely many cores in between $N_{1}$ and
$N_{2}$, and there must exist two cores $N_{3},N_{4}\in\mathcal{M}_{c}(g)$
with $N_{1}<N_{3}<N_{4}<N_{2}$ such that every core between them (including
them) have the same number of odd-degree faces999$N_{2}$ has a finite number
of odd-degree faces, and the total number of odd-degree faces is non-
decreasing when taking homomorphic images: removal of parallel edges preserves
their degree, while vertex gluing always produces one odd-degree face from an
odd-degree face, and either zero or two odd-degree faces from an even-degree
face., and there is a dense total order between them. Let $\mathcal{N}_{3,4}$
be the set of cores between $N_{3}$ and $N_{4}$.

Claim: There exist two cores $N_{5},N_{6}\in\mathcal{N}_{3,4}$ with
$N_{5}<N_{6}$ such that:

  * •

there is a dense total order of cores from $\mathcal{N}_{3,4}$ in between
$N_{5}$ and $N_{6}$, and

  * •

for each pair of cores $N^{\prime}<N^{\prime\prime}$ in between $N_{5}$ and
$N_{6}$ (including them) in the total order, there is a homomorphism
$N^{\prime}\to N^{\prime\prime}$ that induces a one-to-one correspondence
between faces of odd degree and preserves their degree.

###### Proof of the claim.

Let $N,\widetilde{N}\in\mathcal{N}_{3,4}$ with $N<\widetilde{N}$, and let
$h:N\to\widetilde{N}$. The image under $h$ of each odd-degree face of $N$ is a
face that induces a prefacial cross-circuit $\kappa$ in $\widetilde{N}$ of
length smaller or equal than its degree (here we use that $h(N)$ is a submap
of $\widetilde{N}$ and Lemma 36; note that the image of a face does not induce
a prefacial cross-circuit, but rather a union of prefacial cross-circuits, we
select one of them). Each of these $\kappa$’s contains exactly one face
$z_{\kappa}$ of odd degree in $\widetilde{N}$ as, by Observation 38, it should
contain at least one, and it cannot contain more since the number of odd faces
in $\widetilde{N}$ would be strictly larger than that in $N$, a contradiction
with $N,\widetilde{N}\in\mathcal{N}_{3,4}$. Since $\widetilde{N}$ is a core,
by Theorem 41, either $\phi_{z_{\kappa}}=\kappa$, or the degree of
$z_{\kappa}$ is strictly smaller than $\ell(\kappa)$. As there are infinitely
many cores in $\mathcal{N}_{3,4}$, totally ordered, these degrees stabilize
between two cores, which are the desired $N_{5},N_{6}$. These cores
$N_{5},N_{6}$ can be further assumed to contain a dense total order between
them, as there is a dense total order between $N_{3}$ and $N_{4}$. ∎

Let $\mathcal{N}_{5,6}$ be the set of cores totally ordered in between $N_{5}$
and $N_{6}$. We next prove that there are two cores in $\mathcal{N}_{5,6}$
with finitely many cores in between them, which contradicts the assumption
that there is a dense total order between $N_{1}$ and $N_{2}$.

Let $h:N\to N^{\prime}$ where $N<N^{\prime}$ are two cores in
$\mathcal{N}_{5,6}$. Each even-length facial walk $\lambda$ of $N$ turns into
several even-length facial walks in $h(N)\subseteq N^{\prime}$ whose total
length is at most $\ell(\lambda)$ (it may happen that the face completely
disappears); this follows from the fact that no even-degree face can be
subdivided (applying the sequence of vertex and duplicate edge gluings) to
create two odd-degree faces. These even-degree faces in $h(N)$ become
prefacial cross-circuits $\kappa$ in the core $N^{\prime}$. If $\kappa$ is not
a facial walk, by Theorem 41, the faces in the interior of $\kappa$ have even
degree strictly smaller than $\ell(\kappa)$. Therefore, when we consider a
homomorphism from $N_{5}$ into a core in $\mathcal{N}_{5,6}$, its image is
obtained by either reducing the number of faces (gluing duplicated edges) or
subdividing even-degree faces into smaller even-degree faces (gluing
vertices); one can place within these faces a plane submap with all the faces
of strictly smaller even degree (we use that the codomain is another core, and
Theorem 41 where the prefacial cross-circuit $\kappa$ is a part of a facial
walk from $N_{5}$). Since there is an absolute minimum on the degree of a
face, this process cannot be performed infinitely many times.101010When fixing
the image of a homomorphism, the number of faces themselves is finite, even
though we could fill a given face with arbitrarily many configurations that
are cores using faces of strictly smaller degree; this arbitrary number will
be fixed again, and all those inner faces will have strictly smaller degree.
Thus, this procedure cannot be repeated infinitely many times once that
arbitrary number is fixed and a particular collection of cores is chosen.
Hence, at some point, the number and the degree of the even-degree faces is
stabilized. Since we have also stabilized the number of odd-degree faces and
their respective degree, and there is only a finite number of maps with such
parameters, we conclude that the interval between $N_{5}$ and $N_{6}$ is not
dense. This finishes the proof when the cores $N_{1},N_{2}$ are connected.

Suppose now that the cores $N_{1},N_{2}$ are not connected; we argue similarly
on the connected components as they are cores and a map homomorphism can only
merge them. Since the signed genus is fixed and the Euler genus is additive
along connected components, up to plane connected components, we can find two
cores $N_{3},N_{4}$ with $N_{1}<N_{3}<N_{4}<N_{2}$ such that the profile of
connected components with different signed genuses (different from $0$) is the
same, and it is the same with all the cores in-between. Regarding the plane
connected components, we next use an argument on the number of odd-degree
faces.

As mentioned in the applications of Theorem 41, each of the plane connected
components either has at least three odd-degree faces or it is a plane odd
cycle or it is bipartite (all faces of even length). In the latter case, the
core is an edge, which can be mapped anywhere else. We also know that there
cannot be more than one plane odd cycle among the connected components since
otherwise one could be mapped into another and the whole map would not be a
core. Furthermore, a homomorphism between cores of $\mathcal{N}_{3,4}$ cannot
merge one plane connected component into a non-plane one, since that would
strictly increase the number of odd-degree faces of the non-plane component
(each plane component besides, possibly, the odd cycle, has at least three
odd-degree faces, while merging two connected components can reduce the total
number of odd-degree faces by at most two). Thus, as the overall number of
odd-degree faces is bounded in $N_{4}$, the number of plane connected
components is also bounded for each of the cores in $\mathcal{N}_{3,4}$.
Therefore, there exist $N_{5},N_{6}\in\mathcal{N}_{3,4}$ with $N_{5}<N_{6}$
that have the same profile of connected components and odd-degree faces (the
genuses of the components, and the number of odd-degree faces on each of the
connected components), and with a dense total order between them. Applying the
same reasoning as in the connected case, we find two cores in
$\mathcal{N}_{5,6}$ with finitely many cores in between them. ∎

######  Corollary 46.

In the preorder $(\mathcal{M}(g),\leq)$, there is no pair of maps
$M_{1},M_{2}$ with $M_{1}<M_{2}$ and a dense total order between them.

###### Proof.

This follows from the fact that a dense total order between two maps would
imply, by Proposition 27 part (iv), a dense total order between their cores;
here we also use that if $M_{1}<M_{2}$ then their cores are different. ∎

Let us now exhibit an infinite chain of map cores. For the remainder of this
section we use $C_{i}$ to denote the plane map of a graph cycle of length $i$.
Let $T_{i,j}$ be the plane map obtained by adding to $C_{4}$ a path of length
$i$ between the first and the third vertex, and a path of length $j$ between
the second and the fourth vertex (each of these paths subdivides one of the
faces induced by the $C_{4}$ on the plane). Thus, $T_{i,j}$ has four faces,
two of length $i+2$ and two of length $j+2$. Since $T_{2k+1,\,2s+1}$ has all
faces of odd degree and no vertices of degree one, then $T_{2k+1,\,2s+1}$ is a
core for all $k,s\geq 0$. By folding one edge appropriately, we have
$T_{2k+1,\,2s+1}\to T_{2k+1,\,2(s-1)+1}$ for $s\geq 1$ and $k\geq 0$;
symmetrically, $T_{2k+1,\,2s+1}\to T_{2(k-1)+1,\,2s+1}$ for $s\geq 0$ and
$k\geq 1$. As $T_{i,3}$ has a face of length $5$, we conclude that
$C_{5}\to\cdots\to T_{2s+1,3}\to T_{2(s-1)+1,3}\to\cdots\to T_{3,3}\to
T_{1,3}$ is an infinite (ordered) chain of cores.

We can also find an infinite chain of gaps by using that a plane connected map
that is a core is bipartite (an edge or an isolated vertex) or a cycle of odd
length or has at least four faces of odd degree, and that the codomain of a
map homomorphism must have at least as many odd-degree faces as the domain.

######  Theorem 47.

Let $C_{i}$ denote the plane cycle of length $i$. For each $k\geq 1$,
$C_{2k+3}\to C_{2k+1}$ and there is a gap between $C_{2k+3}$ and $C_{2k+1}$.

For each signed genus $g$, we obtain an infinite antichain in
$\mathcal{M}_{c}(g)$ and $\mathcal{M}(g)$ by considering maps with an
increasing number of odd-degree faces, and an increasing degree for each of
these faces. In [11] the authors consider, for each graph $H$, the family
$\mathcal{G}(H)=\\{G\>|\>G\text{ is a graph and }G\to H\\}$ and show the
existence of a graph $H_{0}$ for which $\mathcal{G}(H_{0})$ contains an
infinite antichain [11, Theorem 6] (the argument uses only planar graphs); we
next present a similar construction to find arbitrarily large antichains in
the posets given by $\mathcal{F}(H)=\\{M\>|\>M\text{ is a map and }M\to H\\}$.

######  Theorem 48.

For every even $n$, and odd $k\geq 3$, there exist plane maps $B$ and
$A_{1},\ldots,A_{n}$ such that

  * •

the underlying graphs of $A_{1},\ldots,A_{n}$ are graph cores, and $B$ is a
map core,

  * •

each of $B$ and $A_{1},\ldots,A_{n}$ have underlying graph of odd girth
$2k+1$,

  * •

$A_{i}\to B$ for each $i\in[n]$, and $A_{i}\nrightarrow A_{j}$ for distinct
$i,j\in[n]$.

###### Proof.

We slightly adapt the arguments from [11, Theorem 6]; see Figures 10 and 11
for illustrations of the constructions.

$c_{1}$$b_{2k-1}$$b_{k-1}=v$$b_{1}$$c_{2}$$c_{2k-1}$$a_{1}=u$$a_{2}$$a_{2k-1}$$d$$U$$T_{1}$$T_{2}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{n+1}$$v_{n+2}$$v_{n+3}$$v_{n+4}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{n+1}$$v_{n+2}$$v_{n+3}$$v_{n+4}$…$T_{n}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{n+1}$$v_{n+2}$$v_{n+3}$$v_{n+4}$
Figure 10: Copy of the graph $U$, and the dimaps $T_{i}$.
$A_{1}$$b_{1}$$b_{1}$$b_{1}$$b_{1}$$b_{1}$$c_{1}$$c_{1}$$c_{1}$$c_{1}$$c_{1}$$u$$u$$u$$u$$u$$u$$d$$d$$d$$d$$d$$v$$v$$v$$v$$v$$v$…
Figure 11: Exchanging edges of $T_{1}$ by copies of $U$ to create $A_{1}$. The
map $B$ is obtained by gluing the first $u$ and the last $v$, along the
exterior face.

Let $U$ be a graph with vertices $\\{a_{1},a_{2},\allowbreak\dots,\allowbreak
a_{2k-1},b_{1},b_{2},\allowbreak\ldots,\allowbreak b_{2k-1},\allowbreak
c_{1},c_{2},\ldots,c_{2k-1},d\\}$ where there is a cycle
$(a_{1},a_{2},\allowbreak\dots,\allowbreak
a_{2k-1},b_{1},b_{2},\allowbreak\ldots,\allowbreak b_{2k-1},\allowbreak
c_{1},c_{2},\ldots,c_{2k-1},a_{1})$ and $d$ is adjacent to
$b_{1},c_{1},a_{1}$. Now use the label $u$ for $a_{1}$ and $v$ for $b_{k-1}$.
A plane embedding of $U$ is a map core since all the faces have odd degree. It
is also a graph core as each odd cycle should be mapped to an odd cycle, and a
cycle of length the odd girth should be mapped to a cycle of length the odd
girth; two of the cycles cannot be mapped to a single one as otherwise the
third odd-girth-length cycle would be shortened.

Let $T_{i}$, $1\leq i\leq n$, be a path with $n+4$ directed edges, all of them
oriented in one direction but the $i+2$-th edge that is oriented in the
opposite direction. Let $G_{i}$ be the graph obtained by replacing each edge
of $T_{i}$ by a copy of $U$, placing vertex $u$ as the tail of the edge and
$v$ as the head. Now, consider the plane embeddings of the graphs $G_{i}$,
denoted by $A_{i}$, where all the vertices $b_{1}$ of the respective triangles
are “facing up” except the $b_{1}$ corresponding to the copy of $U$ given by
the backward edge, which is “facing down”. The plane map $B$ is formed by
gluing the first vertex $u$ with the last vertex $v$. Thus, $B$ consists of
$n+4$ copies of $U$ with all but one vertex $b_{1}$ to the outside of the
circle, and one of the vertices $b_{1}$ to the inside. It is clear that
$A_{i}\to B$, but by the argument in [11, Theorem 6] all the graphs $G_{i}$
are graph cores and $A_{i}\nrightarrow A_{j}$ (indeed, assume not, since $U$
is a core, each copy of $U$ in $A_{i}$ should be mapped to a copy of $U$,
vertices $u$ in different copies should be mapped among them as well as
vertices $v$, since $v$ is located slightly to one side along an odd-girth-
length cycle; we would thus have a graph homomorphism between $T_{i}$ and
$T_{j}$, a contradiction) and thus $A_{i}$ are map cores. Further, for $n$
even, each face of $B$ has odd degree, and thus it is a map core. This follows
from the fact that each face of a plane embedding of $U$ has odd degree, and
so the path $uv$ has odd length on one side and even length on the other; in
particular, by joining an even number of $U$’s together, and merging the first
$u$ and the last $v$ as in $A_{i}$, we obtain two faces, both of odd degree
(the facial walk of one face is obtained as the addition of an odd number of
odd-length paths plus an even-length path, and the facial walk of the other
face is obtained as the addition of an odd number of even-length paths plus an
odd-length path). ∎

In [11, Theorem 6], the graph codomain of the antichain is the odd cycle of
length $2k+1$, which is a (graph) core and does not depend on $n$, but only on
$k$. However, since the number of odd faces of $A_{1},\ldots,A_{n}$ increases
with $n$, when using the same construction, the map core $B$, codomain of the
antichain, also depends on $n$. Thus, we can find arbitrarily large
antichains, but not an infinite one. This raises the following question:

> Is there a map $B$ for which $\mathcal{F}(B)$ contains an infinite
> antichain?

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