# Collective and non-collective molecular dynamics in a ferroelectric nematic
liquid crystal studied by broadband dielectric spectroscopy

Aitor Erkoreka  Department of Physics, Faculty of Science and Technology,
University of the Basque Country UPV/EHU, Bilbao, Spain  Alenka Mertelj  Jožef
Stefan Institute, Ljubljana, Slovenia  Mingjun Huang  Satoshi Aya  South China
Advanced Institute for Soft Matter Science and Technology (AISMST), School of
Molecular Science and Engineering, South China University of Technology,
Guangzhou, China  Guangdong Provincial Key Laboratory of Functional and
Intelligent Hybrid Materials and Devices, South China University of
Technology, Guangzhou, China  Nerea Sebastián  Jožef Stefan Institute,
Ljubljana, Slovenia  Josu Martinez-Perdiguero<EMAIL_ADDRESS>Department of Physics, Faculty of Science and Technology, University of the
Basque Country UPV/EHU, Bilbao, Spain

(25 August 2023)

###### Abstract

A great deal of effort has been recently devoted to the study of dielectric
relaxation processes in ferroelectric nematic liquid crystals, yet their
interpretation remains unclear. In this work, we present the results of
broadband dielectric spectroscopy experiments of a prototypical ferroelectric
nematogen in the frequency range 10 Hz–110 MHz at different electrode
separations and under the application of DC bias fields. The results evidence
a complex behavior in all phases due to the magnitude of polar correlations in
these systems. The observed modes have been assigned to different relaxation
mechanisms based on existing theoretical frameworks.

##  I Introduction

Ever since the existence of ferroelectric liquid crystals (LCs) was predicted
and confirmed by Meyer et al. in the 1970s,Meyer  _et al._ (1975) it was long
thought that polar order and high fluidity were mutually exclusive
properties.Terzis, Photinos, and Samulski (1997); Takezoe, Gorecka, and Čepič
(2010) The nomenclature of ferroelectric LCs was historically used to
designate tilted smectic phases formed by chiral mesogens (SmC*), in which
molecules are arranged in layers with their long molecular axis parallel to
one another, and tilted at an angle with respect to the layer normal. Meyer el
al. showed that if molecules possess a transverse component of the dipole
moment, symmetry allows spontaneous polarization in the direction
perpendicular to the tilt plane due to the chiral center.Meyer  _et al._
(1975) Due to the molecular chirality, the tilt plane additionally turns
around the layer normal from layer to layer, and with it, the direction of
polarization, resulting in a helical superstructure. The unwinding of this
helix results in a macroscopic polarization with typical values of the order
of nC$/$cm2.Terzis, Photinos, and Samulski (1997); Kremer and Schönhals (2003)
However, the recent discovery of the ferroelectric nematic (NF) phase has
completely changed this notion, Chen  _et al._ (2020); Sebastián, Čopič, and
Mertelj (2022) with polar ordering appearing in mesophases with no positional
ordering.

The conventional nematic (N) mesophase, in which rod-like molecules only
possess orientational order, is nonpolar with as many dipoles pointing “up” as
“down”, i.e. N phases possess inversion symmetry. The direction of preferred
molecular orientation is denoted as the director $\mathbf{n}$, and then
$\mathbf{n}$ and $-\mathbf{n}$ are equivalent. In the NF phase there is a
preferential direction for the dipoles, which breaks the inversion symmetry
and gives rise to very large polarization values ($\mu$C$/$cm2).Nishikawa  _et
al._ (2017); Chen  _et al._ (2020, 2022); Brown  _et al._ (2021) In some
compounds, another new intermediate phase between the N and NF phases has been
found, which has been described as an antiferroelectric splay nematic phase.
Mertelj  _et al._ (2018); Sebastián  _et al._ (2020); Sebastián, Čopič, and
Mertelj (2022); Chen  _et al._ (2023) Still subject of debate, this phase has
been referred to with different denominations such as M2,Nishikawa  _et al._
(2017) NX, Brown  _et al._ (2021) SmZA Chen  _et al._ (2023) and NS (splay
nematic) Mertelj  _et al._ (2018) (throughout this work we will use the
latter). Due to the novelty of these phases, research on this area has
attracted considerable attention and their complete understanding is
considered of fundamental importance in condensed matter physics and
chemistry. Moreover, the properties of NF materials have a wide array of
potential high-impact applications such as in a new generation of LC displays,
light modulators, memory devices, energy storage, etc.

Among the interesting properties of the NF phase are the extremely large
values of the dielectric permittivity (between $10^{3}$ and $10^{5}$) that
have been reported.Li  _et al._ (2021); Manabe, Bremer, and Kraska (2021);
Yadav  _et al._ (2023a); Erkoreka  _et al._ (2023) While Vaupotič et al. argue
that these measurements reflect the actual material properties,Vaupotič  _et
al._ (2023) Clark et al. have expressed caution about interpreting these
results.Clark  _et al._ (2022) In any case, their theoretical models predict a
thickness-dependent dielectric response, which has already been experimentally
observed and quantified in the prototypical NF RM734 material.Erkoreka  _et
al._ (2023) Broadband dielectric spectroscopy (BDS) is a powerful technique to
shed light on the molecular dynamics of these systems. In classical
ferroelectric smectics, BDS evidences two dielectrically active collective
modes corresponding to the fluctuations in the orientation of the polarization
vector (Goldstone mode, fluctuations of the tilt direction) and to
fluctuations in the amplitude of the polarization (soft mode, fluctuations of
the tilt angle).Kremer and Schönhals (2003); Gouda, Skarp, and Lagerwall
(1991) The Goldstone mode shows a negligible temperature dependence while, on
the contrary, the soft mode is characterized by a critical slowing down of the
relaxation frequency and a diverging amplitude near the transition to the
higher temperature non-tilted phase (SmA).Kremer and Schönhals (2003);
Lagerwall (1999); Gouda, Skarp, and Lagerwall (1991)

A similar thickness effect to that recently reported for the NF phaseErkoreka
_et al._ (2023) was already observed in some SmC∗ compounds, which was
explained by deviations from the ideal helicoidal structure arising due to
strong polar anchoring on the measurement cells.Haase and Wróbel (2003).
However, in ferroelectric nematics, it is assumed that both the polarization
and director are spatially uniform and parallel to the cell plates. As long as
a thickness dependence of unknown origin is observed in the spectra, any
investigation on the relaxation mechanisms of these materials must proceed
with caution. In particular, the effect of DC bias fields could offer new
insights into these issues. In this work, we present the results of a set of
systematic BDS experiments in the N, NS and NF phases of the prototypical
material DIO at different sample thicknesses and DC bias fields. BDS studies
in NF materials are often circumscribed to frequencies below $10$ MHz, which
not only limits the range of relaxation processes that can be observed but can
also hinder the interpretation of the lower frequency ones. Thus, an extended
characterization up to $110$ MHz has been carried out. On the other hand, the
results of the thickness-dependent dielectric response have been analyzed in
the N, NS and NF phases. Additionally, we have extended the previously
reported analysis of the effect of DC bias fields in the NS phase Brown  _et
al._ (2021) to the NF phase, thereby addressing the effect of a DC bias
applied while cooling from the antiferroelectric NS phase on the molecular
dynamics of the system.

##  II Materials and Methods

###  II.1 Material

The studied material is the ferroelectric nematogen DIO
(2,3′,4′,5′-tetrafluoro-[1,1′-biphenyl]-4-yl
2,6-difluoro-4-(5-propyl-1,3-dioxan-2-yl) benzoate), which was synthesized via
literature methods. Its molecular structure, phase sequence and transition
temperatures are shown in Fig. 1. DIO molecules have a large dipole moment of
$10.3$ D directed $14^{\circ}$ away from the long molecular axis.Li  _et al._
(2022)

Figure 1:  Chemical structure of DIO along with its transition temperatures.

###  II.2 Broadband dielectric spectroscopy

The complex dielectric permittivity
$\varepsilon(f)=\varepsilon^{\prime}(f)-i\varepsilon^{\prime\prime}(f)$ was
measured in the frequency range $f=$ 10 Hz–110 MHz. Low- and intermediate-
frequency (up to 3 MHz) measurements were performed with an Alpha-A impedance
analyzer from Novocontrol Technologies GmbH. The oscillator voltage was set to
0.03 Vrms for measurements without bias voltage. To obtain a better signal-to-
noise ratio in the measurements under DC bias, the field was set to 0.01
V${}_{\mathrm{rms}}/\mu\mathrm{m}$ , always well below the Fréedericksz
threshold. The thickness dependence experiments were performed in custom-made
gold-coated glass cells in a parallel plate configuration with no surface
treatment because, although polyimide layers are useful to obtain a proper
alignment, they act as an additional large capacitance in series with the LC
cell, and can lead to other undesired effects such as charge
accumulation.Brown  _et al._ (2021); Vaupotič  _et al._ (2023); Clark  _et
al._ (2022) Cells of different thicknesses were made using silica spheres with
the final thickness determined by optical interferometry. The material was
introduced into the cells by capillary action in the N phase at 115∘C to avoid
sample degradation. The temperature during the measurements was controlled
with a hot stage (Linkam). Qualitative polarizing optical microscopy (POM)
observations with high intensity light were possible despite the high optical
density of the gold layers. In order to perform high-frequency measurements
(up to 110 MHz), a dedicated setup comprising an HP 4294A impedance analyzer
was employed. In this case, circular gold-plated brass electrodes were used,
adjusting the electrode separation with rod-shaped silica spacers. The sample
was then placed at the end of a modified HP 16091A coaxial test fixture, and a
Quatro Cryostat from Novocontrol was employed for temperature control. The
sample was first heated up to 115∘C at 5∘C/min to the N phase and stabilized
for 5 minutes at this temperature. Measurements were then carried out on
cooling at 0.25∘C/min. The complex dielectric permittivity was determined by
dividing the measured capacitance by the capacitance of the empty cells. The
stray capacitance of the measurement circuit was carefully taken into account
in all cases. For a quantitative analysis of the dielectric relaxation
processes, the data were fitted to the Havriliak-Negami (HN) formula with a
conductivity term:

$\varepsilon(f)=\sum_{k}\frac{\Delta\varepsilon_{k}}{\left[1+\left(i\frac{f}{f_{k}}\right)^{\alpha_{k}}\right]^{\beta_{k}}}+\varepsilon_{\infty}-i\frac{\sigma}{\varepsilon_{0}(i\,2\pi
f)^{\lambda}}\mathrm{,}$ (1)

where $\Delta\varepsilon_{k}$, $f_{k}$, $\alpha_{k}$ and $\beta_{k}$ are
respectively the dielectric strength, relaxation frequency and broadness
exponents of mode $k$, $\varepsilon_{\infty}$ is the high-frequency dielectric
permittivity, $\sigma$ is a measure of the conductivity, and $\lambda$ is an
exponent between 0 and 1.

##  III Results and discussion

First, the main features of the dielectric spectra of DIO from 1 kHz up to 110
MHz will be discussed. This frequency range allows the identification of all
the relevant dielectrically active modes of this material. Fig. 2 shows the
imaginary component of the complex dielectric permittivity of DIO as measured
on cooling in a $d=40\;\mu$m thick cell. In the N phase, we can discern two
dielectrically active modes, referred to in this work as mL and mH, where L
and H stand for low- and high-frequency respectively. At 84∘C, the NS phase is
reached, marked by a small discontinuity in $\varepsilon^{\prime\prime}$. In
the NS phase, mL and mH are still present, but a new intermediate-frequency
mode appears, denoted by mI. Lastly, the NF phase is reached at 68∘C,
characterized by a very large increase in $\varepsilon^{\prime\prime}$ where
another two relaxation processes are observed: mNF,L being the most prominent
one and mNF,H at higher frequencies. The sample then crystallizes at 56∘C.

Figure 2:  Three-dimensional plot of the imaginary component of the complex
dielectric permittivity as a function of frequency and temperature for a 40
$\mu$m thick cell. The dark black lines indicate the transition temperatures
of the various phases. The dashed lines are guides for the eyes and color
coded according to Fig. 3.

In order to study the aforementioned relaxation processes more carefully, the
experimental data were fitted to HN relaxations (see Equation 1). The obtained
dielectric strengths and frequencies of maximum absorption are presented in
Fig. 3 (fit examples and parameters can be found in Fig. S1 and Table S1).
These results agree with those reported by Brown et al.,Brown  _et al._ (2021)
with the addition that we are able to observe further high-frequency
processes. It is worth noticing that the permittivity values in both the N and
NS phases are quite large if compared to regular nematics, which can be
expected in highly polar molecules such as DIO. Regarding the molecular
alignment, the cell for high-frequency measurements does not allow for direct
visual inspection. However, because the measured permittivity value is one
order of magnitude larger than the one obtained in cells for which planar
alignment can be deduced from POM (as will be shown below), we concluded that
this cell favors homeotropic-like alignment in the N phase. Such homeotropic-
like alignment in this kind of cells is consistent with the strong tendency
towards out-of-plane alignment already observed for RM734 in the same cells.
Erkoreka  _et al._ (2023) According to the Nordio-Rigatti-Segre theory,Kremer
and Schönhals (2003); Luigi Nordio, Rigatti, and Segre (1973) mL can be
associated with the rotation of individual molecules around their short axis,
i.e. fluctuations of the longitudinal component of the dipole moment
$\mu_{\mathrm{L}}$ around the short axis of the mesogenic unit, and is also
known as $\delta$-process. The absorption frequency of this mode follows the
Arrhenius equation
$f_{a}=f_{\infty}\exp\left(-E_{\mathrm{A}}/k_{\mathrm{B}}T\right)$, where
$f_{\infty}$ is a prefactor, $E_{\mathrm{A}}$ is the activation energy,
$k_{\mathrm{B}}$ is the Boltzmann constant and $T$ is the temperature. A fit
to this function yields an activation energy of 110 kJ/mol (see Fig. S2). This
is a high activation energy, which is expected for the $\delta$-process since
the nematic potential hinders this kind of rotation. As a comparison, the
activation energy of this mode in 7CB, a conventional nematic LC, is $91.1$
kJ/mol.Kremer and Schönhals (2003) However, it is important to point out that
mL involves at least some degree of collective behavior. This is evident from
the large amplitude of this process, which can not be uniquely explained by
the large dipole moment of DIO molecules. Furthermore, the polar correlations
become even more pronounced upon the transition to the antiferroelectric NS
phase, where this process exhibits soft mode behavior. This mode, mL,
corresponds to the low-frequency process observed in RM734 in equivalent
measurements, see Fig. S3.Erkoreka  _et al._ (2023); Sebastián  _et al._
(2020) In RM734, with direct N-NF transition, mL exhibits strong softening
already in the N phase, for which it has been shown that the growth of polar
order is coupled to splay fluctuations.Sebastián  _et al._ (2020) In the case
of DIO, it has been proposed that, in the NS phase, the periodicity at which
polarization changes sign is 180 Å.Chen  _et al._ (2023) Moreover, Emelyanenko
et al. have theoretically modeled all three phases of DIO, and they have
suggested that the size of these polar domains increases with increasing
temperature in the NS phase and diverges at the NS-N transition.Emelyanenko
_et al._ (2022) Consequently, it is plausible that short-range correlations
between dipoles start to occur already in the N phase accounting for the
relatively large $\varepsilon^{\prime}$ observed already in the non-polar
phase. This has recently been proposed from optical and electrooptical
experiments as well.Yadav  _et al._ (2023b) The mode mH, on the contrary, can
be attributed to the rotation of individual molecules around their long axis
(called $\beta$-process), which involves fluctuations in the transverse
component of the dipole moment $\mu_{T}$ projected along the direction of the
probe electric field. It is also a temperature-activated process, with an
activation energy of 36 kJ/mol (see Fig. S2). This lower activation energy
indicates that this mode is not as influenced by the molecular environment as
the $\delta$-process. In the NS phase, mI also exhibits soft mode behavior
near the transition to the NF phase, which suggests it might be related to
some kind of collective fluctuation. It should be noted that near the N-NS
transition mI begins to be discernible even in the N phase, however, its
accurate deconvolution is not possible. For completeness, the dashed line in
Fig. 3 marks the range of temperatures at which the mode is discerned.

Figure 3:  Dielectric strengths ($\Delta\varepsilon$) and frequencies of
maximum absorption ($f_{a}$) of the observed modes in a 40 $\mu$m thick cell
as a function of temperature obtained from fits to the HN formula (Equation
1).

Finally, two modes appear upon the transition to the NF phase. The most
prominent one is mNF,L, which shows a negligible temperature dependence in
both strength and relaxation frequency. This process could be the Goldstone
mode that is expected to appear in the NF phase. In the continuous
phenomenological model (CPM) developed by Vaupotič et al., Vaupotič  _et al._
(2023) the minimization of the free energy for a system with coupled
polarization and director fluctuations yields three dielectrically active
collective modes: two so-called phason modes related to the fluctuations in
the direction of the director and of the polarization, respectively, and one
soft mode (SM) related to the fluctuations in the amplitude of the
polarization (so-called amplitudon). In order to differentiate the two phason
modes more clearly, we will refer to the slow process as Goldstone mode (GM,
equation 12 in Vaupotič  _et al._ (2023)) and to the fast process as optic
mode (OM, equation 13 in Vaupotič  _et al._ (2023)). While the director and
polarization fluctuate in phase in the GM, they fluctuate in counter phase in
the OM. Additionally, the GM (associated primarily with director orientation
fluctuations) dominates the dielectric spectrum, while the authors expect the
OM (dominated by fluctuations in the direction of polarization) to appear in
the MHz regime. The SM, on the other hand, is characterized by a diverging
amplitude and slowing down of the relaxation frequency near the transition to
the ferroelectric phase. These theoretical results could be consistent with
our data, since the dominant mNF,L could be attributed to the GM and mNF,H to
the OM, being mL the SM in the NS phase. A similar interpretation was proposed
by Yadav et al. for mL (P1 in their paper).Yadav  _et al._ (2023a) However, we
have to note that in our measurements the amplitude of mNF,H is small and it
is also compatible with the $\beta$-relaxation (or a mixture of both effects).
A visual representation of the proposed relaxation mechanisms can be seen in
Fig. 4.

Figure 4:  Schematic representation of the proposed non-collective and
collective modes. Non-collective molecular rotational modes contributing to
dielectric relaxations can be described as (a) molecular rotation around the
short ($\delta$-process) and long ($\beta$-process) molecular axis, and (b)
the precessional motion about the director. (c) Collective fluctuations in the
direction of the nematic director $\mathbf{n}$ and the polarization
$\mathbf{P}$. Hydrodynamic and optic (non-hydrodynamic) modes for the
fluctuations in the direction of $\mathbf{n}$ and $\mathbf{P}$. Representation
for thin cells, where green ellipsoid accounts for the local director
orientation and black arrows correspond to the local polarization direction,
in correspondence with the theoretical model proposed in the work by Vaupotič
et al.Vaupotič  _et al._ (2023) Arrow representation in the right visualizes
that the hydrodynamic GM mode is dominated by director fluctuations in phase
with the smaller polarization fluctuations, while the higher frequency OM is
dominated by fluctuations of polarization with $\mathbf{n}$ and $\mathbf{P}$
in counter phase. This is analogous to the hydrodynamic and optic modes
reported in the case of suspensions of ferroelectric nanoparticles in nematic
liquid crystals.Čopič  _et al._ (2007); Sebastian  _et al._ (2018) (d)
Representation of the fluctuations in the amplitude of $\mathbf{P}$, i.e. soft
mode (amplitudon). (f) Dipole moment of DIO is directed $14^{\circ}$ away from
the long molecular axis.Li  _et al._ (2022) In the uniaxial state dipoles are
randomly oriented as represented by the cone.Mandle  _et al._ (2021)
Collective precession and reorientation around the molecular short axis allow
for illustration of the hydrodynamic and optic modes depicted in (c).

In order to systematically study the cell thickness effects in this material,
we prepared four different glass cells with sputtered gold electrodes and
thicknesses ranging from 5.1 to 57 $\mu$m. Fig. 5 shows the real component of
the measured dielectric permittivity as a function of temperature at 100 Hz
for all sample thicknesses. Starting from the high-temperature N phase, we
observe that in this type of cells the alignment reached is not hometropic-
like and birefringent domains are observed, suggesting a random planar
orientation of the sample (see Fig. S4). The transition to the NS ($\sim
86^{\circ}$C) phase is marked by a small bump in $\varepsilon^{\prime}$,
especially noticeable in the 14.7 $\mu$m thick cell, where the difference in
value can be attributed to a slightly different alignment. On cooling, the
transition to the NF phase (69∘C) is marked by a very large increase of
$\varepsilon^{\prime}$. In this phase, a strong thickness dependence is
readily observed, in which the thicker the measurement cell, the larger the
permittivity. As a matter of fact, $\varepsilon^{\prime}$ is increased by
about an order of magnitude from 5.1 $\mu$m to 57 $\mu$m, pointing towards a
linear dependence on the sample thickness $d$. In these runs, the NF phase is
stable within a temperature interval of $\sim 20^{\circ}$C except in the 5.1
$\mu$m thick cell, in which the sample crystallized at a slightly lower
temperature. As expected, the dielectric response of the crystal (Cr) phase
does not exhibit a thickness dependence, and the values of
$\varepsilon^{\prime}$ agree for all thicknesses.

Figure 5:  Temperature dependence of the real component of the measured
complex dielectric permittivity for various cell thicknesses at 100 Hz.

The relaxation processes measured in these cells can be identified in the
dielectric absorption spectra presented in Fig. 6 for the three LC phases, as
well as in the temperature evolution of their frequencies and amplitudes shown
in Fig. 7 for all thicknesses, as obtained from the fits to the HN formula
(Equation 1) (only the amplitudes of two thicknesses are shown for clarity).
Like in the results presented above with gold-plated brass electrodes, in the
N phase, the mL and mH processes can be observed. The larger strength of mL in
the 14.7 $\mu$m thick cell is attributed to the alignment effect previously
inferred from Fig. 5. In fact, a greater relaxation amplitude is expected if
the alignment is somewhat more homeotropic-like, since the fluctuations in
$\mu_{\mathrm{L}}$ projected along the direction of the probe electric field
would be greater in that case. At higher frequencies the onset of mH appears,
although its relaxation frequency is far beyond the accessible frequency range
in this type of cells. At low frequencies, another absorption process of great
amplitude takes place, which appears as a straight line due to the limited
frequency range, and is related to conductivity and electrode polarization
effects (see also Fig. S5). In the NS phase, we again observe mL and mI. In
the N and NS phases, aside from slight differences in strength, the
temperature evolution of the modes is equivalent. We observe a small but
systematic thickness dependence in both the slope and value of the mI
frequencies, which might an issue worth investigating in the future.

Figure 6:  Spectrum of the imaginary component of the dielectric permittivity
measured in the N, NS and NF phases for various sample thicknesses. Figure 7:
Dielectric strengths ($\Delta\varepsilon$) and frequencies of maximum
absorption ($f_{a}$) of the observed modes as a function of temperature
obtained from fits to the HN formula (Equation 1) for various sample
thicknesses. In the upper graph, only two thicknesses are shown for clarity.

In the NF phase, only mNF,L is seen in the spectra. In addition to the strong
thickness effect on the relaxation strength already inferred from Fig. 5, the
dependence on the thickness of the relaxation frequency is also evident. Such
thickness effect is summarized in Fig. 8, which shows the dielectric strength
and frequency of maximum absorption of mNF,L as a function of $d$. As can be
inferred from Figs. 5, 6 and 7, the dielectric strength qualitatively shows a
linear dependence on d, while the relaxation frequency is inversely
proportional to $d$. This is the same behavior reported by our group for
RM734.Erkoreka  _et al._ (2023) In that work, the results were analyzed on the
basis of three different mechanisms. Clark et al. proposed that the GM that is
expected to be present in ferroelectric nematics would be responsible for the
large values of the dielectric permittivity and for the thickness effect.Clark
_et al._ (2022) The minimization of the free energy of the NF system renders
the polarization spatially uniform and parallel to the electrodes. The GM
would then correspond to a reorientation of the polarization vector upon the
application of an external electric field. This phenomenon, coupled with the
high fluidity of these materials, would screen the electric field in the bulk
via the charging of the interfacial layers that bound the sample in typical
measurement cells, giving rise to a so-called polarization-external
capacitance Goldstone reorientation mode (PCG). The authors argue that the
measurement results should therefore be interpreted as apparent values, and
they predict a Debye-type relaxation process, with $\Delta\varepsilon\propto
d$ and $f_{r}\propto 1/d$,Clark  _et al._ (2022) in line with our results (see
Fig. 8).

Figure 8:  Thickness dependence of the dielectric strength (circles) and
frequency of maximum absorption (squares) of mNF,L at 60∘C. The dashed lines
are fits to the experimental data.

On the other hand, the CPM model developed by Vaupotič et al. includes
thickness dependent terms and predicts a relaxation frequency with a $1/d$
dependence (Equation 12 in the corresponding paper).Vaupotič  _et al._ (2023)
Although they neglect them based on their qualitative observations, their
inclusion would also be in line with our experimental results and would
suggest that the dominant term in their model would be associated with
flexoelectricity. Barthakur et al. recently measured a very large
flexoelectric effect in RM734 which could explain this dominance.Barthakur
_et al._ (2023) Unfortunately, to the best of our knowledge, no measurements
of flexoelectric coefficients have been performed for DIO, which could help
setting a common ground for this characteristic effect of the NF phase. It is
also interesting to note that Emelyanenko et al. have theoretically studied
the flexoelectric effect in DIO starting from an intermolecular interaction of
a particular symmetry.Emelyanenko  _et al._ (2022) In fact, they argue that it
is an effect of great importance, since it appears to be the generator of both
the splayed structure and the improper polarization present in the NS phase.
Returning to the CPM model, the lower relaxation frequency of thicker cells
might be suggesting that the director has more freedom to reorient (less
energy cost, lower frequency) than in thinner cells.

As discussed in a previous article,Erkoreka  _et al._ (2023) it is well known
that the electrode polarization (EP) effect can create a thickness dependent
dielectric response and give rise to extremely large permittivity
values.Emmert  _et al._ (2011); Chassagne  _et al._ (2016) This phenomenon
arises due to the accumulation of ions on the electrode surface, and is
present in all kinds of substances at low enough frequencies. These charged
layers screen the electric field in the bulk, manifesting itself as a
thickness dependent relaxation process that can potentially mask the intrinsic
response of the material. Several works predict that an EP affected spectrum
will show a permittivity value proportional to $d$ below a given frequency,
the latter being inversely proportional to $d$.Chassagne  _et al._ (2016);
Sawada  _et al._ (1998); Klein  _et al._ (2006); Kondratenko  _et al._ (2018)
This behavior is again compatible with the present results (see Fig. 8). EP
should not be ignored in NF materials such as DIO or RM734, as they can be
expected to contain larger ion concentrations than conventional nematicsBasnet
_et al._ (2022); Sebastián  _et al._ (2023) due to their large molecular
polarity. However, one should note that EP occurs at the time-scales of ion
dynamics in the sample Chassagne  _et al._ (2016); Sawada  _et al._ (1998);
Klein  _et al._ (2006); Kondratenko  _et al._ (2018) and, thus, the present
observations would imply that the latter change drastically at the transition
to the NF phase, where $f_{r}$ is in the kHz range.

Additional insights into molecular mechanisms can be obtained from the
dielectric characterization under DC bias fields. Experiments have been
performed for different cell thicknesses. However, we limit this part of the
study to thin cells ($\sim 5\;\mu$m) due to the large electroconvective
currents and sample degradation observed when large DC bias fields (up to 1
VDC/$\mu$m) were applied in thicker cells (see Fig. S6). After each
measurement, the bias field was removed and the resulting spectrum was
compared to the original one to ensure that the sample was not degraded. In
the first set of experiments the cell was heated up to 110∘C and then cooled
at 1∘C/min to the desired temperature for DC bias application. Fig. 9 shows
the DC bias field dependence of the two modes of the NS phase at 76∘C. On the
one hand, it is observed that, while the strength of mL increases, that of mI
decreases. Comparing this with the results of Fig. 3, in which the strength of
mL is larger than that of mI, we conclude that we are certainly inducing a
homeotropic-like alignment. Additionally, it would be consistent with the
previous assignment of mL to the SM, since a DC bias would increase the
magnitude of the amplitude fluctuations. On the other hand, the relaxation
frequency of mL stays practically constant, while that of mI slightly
increases. It is interesting to note, however, that these features only start
to happen from a threshold field of around 0.08 VDC/$\mu$m. These results
agree with those of Brown et al.Brown  _et al._ (2021) It is also worth
noticing that Nishikawa et al. reported, for fields above a threshold of 0.4
V/µm, field induced second harmonic generation signal indicating the growth of
oriented polar domains.Nishikawa  _et al._ (2017)

An interesting open question relates to the origin of these polar nematic
phases and what drives different materials to exhibit or not the intermediate
antiferroelectric phase. We note here that for RM734 (Iso-N-NF phase sequence)
and DIO (Iso-N-NS-NF), the molecular dipole moment has a component
perpendicular to the long molecular axis. A comparison of the relaxation
processes observed in these two materials can be found in Fig. S3.
Interestingly, for material $\mathbf{1}$ in an article by Manabe et
al.,Manabe, Bremer, and Kraska (2021) which shows a direct isotropic to NF
transition, the dipole is almost along the molecular long axis. In this
context, and in combination with devoted models taking into account the
intermediate phase, broadband dielectric spectroscopy could prove to be
essential. An off-axis dipole moment implies that the nematic director and
polarization do not need to be parallel. The two order parameters give rise to
the in-phase and counter-phase fluctuations of the director and polarization
as described by the CPM model,Vaupotič  _et al._ (2023) which can be
rationalized as the collective precession around the cone and reorientations
around the molecular short axis (Fig. 4). Through the N-NF phase transition of
RM734, it has been shown that the splay mode is coupled with the magnitude of
the polarization.Sebastián  _et al._ (2020) If the phase is modulated, as is
the NS phase, the twist mode, as depicted in Fig. S7, can also be seen
dielectrically. For a periodicity of $p=18$ nm Chen  _et al._ (2023) and
diffusivities $K_{2}/\gamma$ in the range 10-11 to 10-12 m2/s, a relaxation
frequency $f=(K_{2}/\gamma)(2\pi/p)^{2}$ of $0.1-1$ MHz is expected, where
$K_{2}$ is the twist elastic constant and $\gamma$ is the rotational
viscosity. However, experimental phase structural resolution together with a
devoted theoretical model are needed for further understanding dielectrically
accessible molecular correlations and their importance in the formation of the
NS phase.

Figure 9:  Effect of a DC bias on the dielectric spectrum of the NS phase at
76∘C in a 5.1 $\mu$m thick cell. Top: Dielectric loss spectrum at 0 VDC/$\mu$m
and 0.18 VDC/$\mu$m. Bottom: Dielectric strengths ($\Delta\varepsilon$, solid
symbols) and frequencies of maximum absorption ($f_{\mathrm{a}}$, empty
symbols) of the observed modes as a function of the external DC bias field.

In the NF phase at 65∘C, upon progressive application of the DC bias field,
the amplitude of mNF,L slightly increases (from approximately $7\,000$ to
$10\,000$) and then, above 0.3 VDC/$\mu$m, it gets partially suppressed and
its relaxation frequency decreases (see Fig. 10). The suppression of the mode
reveals a second process of lower strength and slightly higher frequency
(termed m${}_{\mathrm{NF,L}}^{*}$). This behavior can be explained within the
CPM model, in which the NF phase spectrum would be dominated by the GM.
However, when a DC bias field is applied, the dominant GM is suppressed and
shifted to lower frequencies, and the SM, corresponding to fluctuations in the
magnitude of the polarization, becomes visible at slightly higher frequencies.
This was observed by Vaupotič et al. with a homologue of RM734.Vaupotič  _et
al._ (2023) In our case we would identify mNF,L as the dominant GM and
m${}_{\mathrm{NF,L}}^{*}$ as the SM. On the other hand, the reason for the
slight increase of $\Delta\varepsilon$ at low DC bias fields (below 0.3
VDC/$\mu$m) might be the following: before the polarization vector and
director are completely reoriented, the effect of the DC bias field, as
suggested by Vaupotič et al., is to shrink the layer of LC material over which
the polarization rotates to become parallel to the electric field.Vaupotič
_et al._ (2023) This would increase the capacitance of the dielectric layer
and, thus, the strength of the mode should increase. This is precisely in line
with the PCG mode proposed by Clark et al.Clark  _et al._ (2022) Lastly,
returning to Fig. 10, a high-frequency process of negligible DC bias field
dependence appears (m${}_{\mathrm{NF,H}}^{*}$). This could be the continuation
of mI in the NF phase. The reorientation of the director under sufficiently
strong bias fields was confirmed by POM (see Fig. S8).

Figure 10:  Effect of a DC bias on the dielectric spectrum of the NF phase at
65∘C in a 5.1 $\mu$m thick cell. Top: Dielectric loss spectrum at 0
VDC/$\mu$m, 0.24 VDC/$\mu$m and 1.1 VDC/$\mu$m. Bottom: Dielectric strengths
($\Delta\varepsilon$, solid symbols) and frequencies of maximum absorption
($f_{a}$, empty symbols) of the observed modes as a function of the external
DC bias field.

Finally, we studied the effect of an DC bias field applied while cooling on
the dielectric spectra of the NS and NF phases. To minimize sample degradation
due to long-time bias application, the cooling rate was set to 1∘C/min and we
chose a 0.5 VDC/$\mu$m bias field to be well above the thresholds observed in
Figs. 9 and 10. Fig. 11 shows the temperature dependence of the relaxation
processes frequency and strength in a 5.1 $\mu$m-thick cell, where the results
with no DC bias are also plotted for comparison. In the NS phase, mL shows a
much larger strength under an applied DC bias field, as expected from Fig. 9,
while its relaxation frequency is practically unaltered. In both cases, the SM
behavior is present. This again suggests that mL is indeed the high-
temperature branch of the SM. Under a DC bias field, mI is strongly suppressed
and slightly shifted in frequency. A dedicated model for the NS phase should
be developed in order to fully understand the origin of this mode. In the NF
phase, two modes are present under an applied DC bias field. The lower
frequency process is the suppressed mNF,L (GM), which dominates the spectra at
0 VDC, while the higher frequency one is the previously mentioned
m${}_{\mathrm{NF,L}}^{*}$ (SM). It is interesting to note that cooling the
sample under a DC bias field does not significantly shift the GM in frequency.
Near the NS–NF transition, m${}_{\mathrm{NF,L}}^{*}$ shows soft mode behavior,
as expected. In particular, it is interesting to note that the temperature
dependence of the low-temperature branch of the SM is steeper than that of the
high-temperature one. This is in fact predicted by the CPM model, although the
high-temperature phase considered by the authors was the conventional N phase.
Vaupotič  _et al._ (2023) Away from the NS–NF transition,
m${}_{\mathrm{NF,L}}^{*}$ shows a negligible temperature dependence, just like
the mNF,L.

Figure 11:  Dielectric strengths ($\Delta\varepsilon$) and frequencies
($f_{a}$) of maximum absorption of the observed modes obtained from fits to
the HN formula (Equation 1) with no bias field (solid symbols) and with a bias
field of 0.5 VDC/$\mu$m (empty symbols) in a 5.1 $\mu$m thick cell. The dashed
lines are guides for the eyes.

##  IV Conclusions

In summary, we have shown that the ferroelectric nematic compound DIO exhibits
several relaxation processes of great complexity, which could be traced back
to the magnitude of polar correlations in all three nematic phases. By
studying the dielectric spectra in a broad range of frequencies and
temperatures, as well as under the application of external electric fields, we
were able to assign the observed modes to different dipolar fluctuations. The
highest frequency mode accessible in our experiments, denoted by mH, can be
associated with the rotation of individual molecules around their long axis or
$\beta$-process, and is present throughout the whole phase sequence. The
lowest frequency mode in the N and NS phases, called mL, is related to the
flip-flop rotation of molecules or $\delta$-process, but it involves some
degree of collective behavior, even in the paraelectric N phase. In the NS
phase, the dipolar correlations grow and are responsible for the amplitude
fluctuations (SM). An intermediate-frequency process appears in this phase,
called mI, which also exhibits SM behavior and might be associated with the
twist mode. Finally, the GM and OM predicted by the CPM model are observed in
the NF phase. However, the GM is overlapped with the SM, which has a much
lower amplitude, but can be made visible by applying a sufficiently strong DC
bias field. The non-ideal nature of this GM, which appears at a non-zero
frequency, is thought to be the origin of the large permittivity values and
the thickness effect. In view of the present results, a realistic value for
$\varepsilon^{\prime}$ might be of the order $10^{2}$–$10^{3}$ due to the
large dipole moment of DIO, as observed when suppressing polarization
reorientation processes by applying a sufficiently strong DC bias field.

##  V Acknowledgements

A.E. and J.M.-P. acknowledge funding from the Basque Government Project
IT1458-22. A.E. thanks the Department of Education of the Basque Government
for a predoctoral fellowship (grant no. PRE_2022_1_0104). A.M. and N.S.
acknowledge funding from the Slovenian Research and Innovation Agency (grant
no. P1-0192). M.H. and S.A. acknowledge the National Key Research and
Development Program of China (no. 2022YFA1405000) and the Recruitment Program
of Guangdong (no. 2016ZT06C322). We are grateful to Ander García Díez from the
Basque Center for Materials, Applications and Nanostructures (BCMaterials) for
the preparation of the gold electrodes.

##  VI Author Declaration

###  VI.1 Conflict of Interest

The authors have no conflicts to disclose.

##  VII Data Availability

The data that support the findings of this study are available within the
article and its supplementary material.

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#  Computational studies on electrochemical performances of doped and
substituted $Ti_{3}C_{2}O_{2}$ MXene

Mandira Das<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Guwahati, Guwahati-781039, Assam, India.  Subhradip Ghosh
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Guwahati, Guwahati-781039, Assam, India.

###### Abstract

Using Density functional theory (DFT) in conjunction with a solvation model we
have investigated the phenomenon of eletrode- electrolyte interaction at the
electrode surface and its consequences on the electrochemical properties like
the charge storage and total capacitance of doped and substituted oxygen
functionalised Ti3C2 supercapcitor electrode. We have studied nitrogen doped,
nitrogen substituted and molybdenum substituted Mxenes in acidic electrolyte
H2SO4 solution. By considering nitrogen doping at different sites, we found
that the greatest capacitance is obtained for doping at functional sites. Our
results agree well with the available experiment. We also found that the
enhancement in capacitances due to nitrogen doping is due to amplifications in
the pseudocapcitances. We propose that the primary mechanism leading to the
enhanced value of the capacitances due to nitrogen doping is surface redox
activity. The performances for substituted systems, on the other hand, are
degraded in comparison to the pristine ones. This suggests that better storage
capacities in Ti3C2O2 electrode can be obtained by doping only. We provide
insights into the reasons behind contrasting behaviour in doped and
substituted systems and suggest ways to further improve the capacitances in
doped system.

##  I Introduction

To alleviate the problem of ever increasing demand for efficient and clean
energy it is imperative to design energy storage devices that meet the desired
benchmarks that is a combination of high energy density, power density and
long life cycle. Electrochemical capacitors or supercapacitors generated quite
a bit of excitement lately due to the combination of the last two parameters
found in them Ren  _et al._ (2017); Masaki  _et al._ (2019); Yang  _et al._
(2020). The supercapacitors can be categorised based upon the mechanism of
energy storage: the electrostatic interaction giving rise to electrical double
layer capacitance (EDLC) and the surface redox reaction leading to
pseudocapacitance.While the EDL capacitors’ electrode materials, inspite of
often having large accessible surface area, can store only a few electrons per
atom, pseudocapacitor electrodes can store order of magnitude higher number of
electrons because of the fast and reversible redox processes. However, the
accessible surfaces for the hitherto standard pseudocapacitor electrodes like
RuO2, MnO2 and Co3O4 are limited Sugimoto  _et al._ (2004); Wu and Julia
(2004); Yang  _et al._ (2013)hindering exploitation of the advantages offered.
This limitation can be alleviated with the discovery of new two-dimensional
(2D) nanomaterials which could maintain the high surface area while having the
potential of redox reactions near the surface of the electrode material.

Accordingly, the electrochemical performances of several 2D electrodes were
examined Yu  _et al._ (2015); Chen  _et al._ (2018); Zhang  _et al._ (2014);
Gao  _et al._ (2014); Xu  _et al._ (2018).The reasonable gains obtained there
were amplified once electrochemical performances of MXenes, the new members in
the 2D family, were investigated. MXenes with chemical composition Mn+1Xn; M a
transition metal, X either carbon or nitrogen,$n$ an integer, offered the
necessary tunability as a plethora of new materials could have been possible
by changing the chemical composition.That their surfaces are functionalized
with F,O or OH Naguib  _et al._ (2012a) during the process of exfoliating from
MAX phases, added more possibilities towards improvement in electrochemical
performances of MXene electrodes Ji  _et al._ (2015a). Functionalised Ti3C2,
the first discovered MXene Naguib  _et al._ (2011), has been investigated
extensively for it’s electrochemical performance Lukatskaya  _et al._ (2013);
Ghidiu  _et al._ (2014); Rakhi  _et al._ (2015); Lin  _et al._ (2016); Shao
_et al._ (2020). The capacitances obtained in Ti3C2 are much higher than those
obtained from Carbon-based electrodes like Graphene Lukatskaya  _et al._
(2013). The capacitance of few other MXene electrodes also turned out to be
substantial Halim  _et al._ (2016); Shan  _et al._ (2018); Guan  _et al._
(2020). In order for systematic exploration of new electrode materials for
supercapacitors it is important to understand the origin of the
supercapacitances in the already discovered ones. Specifically, it is
desirable to investigate the prevalent mechanism - EDL or Redox reaction, that
leads to the superior electrochemical performance of a given electrode
material. Density functional theory (DFT) Kohn and Sham (1965) based first
principles electronic structure calculations, in this regard, were found to
have provided significant insights Kang  _et al._ (2011); Ji  _et al._
(2015b); Zhang  _et al._ (2016a); Zhan  _et al._ (2018); Das and Ghosh (2022).

Doping and substitution with heteroatoms is a proven effective tool to improve
physical properties of materials. The electrochemical performances of several
2D electrode materials like graphene Gopalkrishna  _et al._ (2013), boron
nitride Zhang  _et al._ (2019) and molybdenum disulfide Lei  _et al._ (2019)
could be significantly improved by manipulating their electron transport
processes through doping. Significant enhancement in the capcitance was
observed for nitrogen doped Graphene Jeong  _et al._ (2011); Jiang  _et al._
(2012); Sun  _et al._ (2012); Kaur  _et al._ (2018). Very recently
electrochemical performances of nitrogen doped Ti3C2 MXenes were investigated
Wen  _et al._ (2017); Chengjie  _et al._ (2020). In Ref Wen  _et al._ , 2017,
improvement of capacitance by 460$\%$ was obtained at low concentration of
doped nitrogen. It degraded upon increase in the nitrogen concentration. This
significant increase was attributed to contributions from both EDL and redox
mechanisms. A combination of DFT calculations and experimental measurements
were reported in Ref Chengjie  _et al._ , 2020. They examined the influence of
site occupancy of the dopant on the capacitance and inferred that nitrogen
doping, in general increases the value of the capacitance although the gain
was not as substantial as reported in Ref Wen  _et al._ , 2017. Their results
indicated that EDL mechanism is dominant for nitrogen substitution at the
carbon site while the redox mechanism prevails for doping at functional or
surface sites. DFT calculations for Ti3CN monolayer Zhang  _et al._
(2016b)that is a 50$\%$ substitution of nitrogen at carbon lattice site
suggested that nitrogen substitution may lead to significant increase in the
charge storage capacity contributed mainly due to surface redox reaction.

In the investigations on the electrochemical performances of nitrogen doped
and substituted functionalised Ti3C2, done so far, understanding of the origin
of the supercapacitative behaviour is done only qualitatively, often in an
indirect way. Direct computation of various contributions to the total
capacitance providing robust ground to the qualitative understanding is still
lacking. In this work, we aim at looking into this unexplored area for both
doped and substituted Ti3C2 using DFT in conjunction with thermodynamic
solvation models for accurate description of the electrochemical phenomena at
the surface of the electrode in an aqueous solution. We have systematically
investigated the effects on doping nitrogen at three possible sites in oxygen
functionalised Ti3C2 (Ti3C2O2) by computations of the different contributions
to the total capacitance. Further we examined the effects of substitution at
Ti and C sites by substituting 66$\%$ of Ti with Mo (Mo2TiC2O2)and 50$\%$ C
with nitrogen (Ti3CNO2)respectively. This systematic exploration yields a
substantial understanding of the impacts of doping and substitution at
different sites on the supercapacitative properties of Ti3C2O2.

##  II Methodology and Calculation Details

Total Capacitance($C_{T}$) of a Supercapacitor has contributions from the
electrical part ($C_{E}$) and the Quantum capacitance ($C_{Q}$). The
electrical part of the total capacitance is due to contributions from parallel
combination of the EDLC ($C_{EDL}$) and the redox or pseudocapacitance
($C_{redox}$). $C_{E}$, therefore depends on the electrode-electrolyte
interactions at the interface. The Quantum capacitance, on the other hand, is
solely dependent on the electrode material. It originates due to imperfect
screening of the electric fields by the metal electrodes and is substantial
for materials at the nano-scale. Since $C_{Q}$ is in parallel with $C_{E}$
Luryi (1988), a non-negligible contribution from $C_{Q}$ is found to affect
the overall capacitance Mead (1961); Saad  _et al._ (2004).

$\frac{1}{C_{T}}=\frac{1}{C_{Q}}+\frac{1}{C_{E}}$ (1)

Quantum Capacitance is related to the electronic structure of the electrode
material the following way,

$\displaystyle{C_{Q}}^{diff}=e^{2}\int_{-\infty}^{+\infty}D(\xi)F_{T}(\xi-U)d\xi$
(2) $\displaystyle{C_{Q}}^{int}=\frac{1}{eV}\int_{0}^{V}{C_{Q}}^{diff}dV$ (3)

$C_{Q}^{diff}$ and $C_{Q}^{int}$ are the differential and integrated Quantum
Capacitance respectively, $D(\xi)$ is the densities of states and
$F_{T}(\xi-U)$ is the broadening function. Integrated Quantum Capacitance
gives a better representation as it provides the charge storage capacity of
the electrode when charged up to a certain voltage.

The interaction of solid electrode and liquid electrolyte leading to $C_{E}$
is simulated by the Joint Density Functional Theory (JDFT). JDFT is an ab
initio approach that connects the quantum DFT of material with the classical
DFT of liquid Petrosyan  _et al._ (2005). It can accurately address the
microscopic behavior of the electrostatic potential near the electrode
surface. Here, an ab initio electrochemistry calculation is done by explicitly
including the atoms making up the environment and performing thermodynamical
averaging over the locations of those atoms.The free energy $A$ of an explicit
quantum-mechanical system with its nuclei at fixed locations and in
thermodynamic equilibrium with a liquid can be expressed asPetrosyan  _et al._
(2007),

$A=\min_{n(r),\big{\\{}N_{\alpha}(r)\big{\\}}}\Bigg{\\{}G[n(r),{N_{\alpha}(r)},V(r)]-\\\
\int d^{3}rV(r)n(r)\Bigg{\\}}$ (4)

$G[n(r),\big{\\{}N_{\alpha}(r)\big{\\}},V(r)]$ is an universal functional of
$n(r)$, the electron density of the explicit system,
$\big{\\{}N_{\alpha}(r)\big{\\}}$, the densities of the nuclei of the various
atomic species in the environment , and $V(r)$, the electrostatic potential
due to the nuclei of the explicit system.
$G[n(r),\big{\\{}N_{\alpha}(r)\big{\\}},V(r)]$ depends only on the nature of
the environment and that its dependence on the explicit system is only through
the electrostatic potential of the nuclei included in $V(r)$ and $n(r)$, the
electron density of the explicit system. The functional
$G[n(r),\big{\\{}N_{\alpha}(r)\big{\\}},V(r)]$ can be separated into three
partsPetrosyan  _et al._ (2007)

$G[n(r),\big{\\{}N_{\alpha}(r)\big{\\}},V(r)]\equiv
A_{KS}[n(r)]+\Omega_{lq}[\big{\\{}N_{\alpha}(r)\big{\\}}]+\\\ \Delta
A[n(r),\big{\\{}N_{\alpha}(r)\big{\\}},V(r)]$ (5)

$A_{KS}[n(r)]$ and $\Omega_{lq}[\big{\\{}N_{\alpha}\big{\\}}]$ are the
standard universal Kohn-Sham electron-density functional of the explicit
solute system in isolation(including its nuclei and their interaction with its
electrons) and the ’classical’ density functional for the liquid solvent
environment in isolation, respectively. $\Delta
A[n(r),\big{\\{}N_{\alpha}(r)\big{\\}},V(r)]$ depicts the coupling between the
solvent and the solute.The solvation effect and the electrolyte response are
then approximated by implicit solvation modelsLetchworth-Weaver and Arias
(2012). The free energy in this approximation is given as

$\tilde{A}=\min_{n(r)}(A_{KS}[n(r),\big{\\{}Z_{I},R_{I}\big{\\}}]+\Delta\tilde{A}[n(r),\big{\\{}Z_{I},R_{I}\big{\\}}])$
(6)

The effects of the liquid environment appears in the new term,

$\Delta\tilde{A}[n(r),\big{\\{}Z_{I},R_{I}\big{\\}}]\equiv\min_{N_{\alpha}(r)}(\Omega_{lq}[N_{\alpha}(r)]+\\\
\Delta A[n(r),N_{alpha}(r),\big{\\{}Z_{I},R_{I}\big{\\}}])$ (7)

$Z_{I}$ and $R_{I}$ are the charges and positions of the surface nuclei.
Minimization of equation (6) leaves a functional in terms of only the
properties of the explicit system and incorporates all of the solvent effects
implicitly. There are many advanced solvation models for JDFT which include
non-local effectsKornyshev and Sutmann (1998), nonlinear dielectric
responseGunceler  _et al._ (2013), spherically averaged liquid susceptibility
ansatz (SaLSA)Sundararaman  _et al._ (2015), and charge-asymmetric nonlocally
determined local-electric (CANDLE)Sundararaman and Goddard III (2015)
solvation model.

In this work, we have considered a non-ideal Faradaic process as implemented
by Gogotsi $et$ $al.$ Zhan  _et al._ (2018) to compute the capacitances due to
EDL and surface redox reactions for $Ti_{3}C_{2}O_{2}$ in $H_{2}SO_{4}$
electrolyte. We have generalised it for our systems. The redox reaction for
N-doped $Ti_{3}C_{2}O_{2}$ (N-$Ti_{3}C_{2}O_{2}$) can be written as ,

${N-Ti_{3}C_{2}O^{q}_{2}}+2H^{+}(aq)\\\
\longrightarrow{N-Ti_{3}C_{2}O_{2-2x}(OH)^{q^{{}^{\prime}}}_{2-2x}}+2(1-x)H^{+}(aq)$
(8)

where $x$ is surface H coverage (between 0 and 1) and $q$ is the net charge on
the electrode.The applied voltage controls both the $q$ and $x$ during the
charging process. The free energy of the charged electrode system with
coverage ($x$) and applied electrode potential$(\phi)$ is given by

$G(x,\phi)=E(x)+xE_{ZPE}+Q(V(x,\phi))\phi\\\
+E_{EDL}(V(x,\phi))+(1-x)\mu_{H^{+}}$ (9)

$E(x)$ is the total energy of the solvated electrode with H coverage of $x$ in
zero surface charge. $E_{ZPE}$ is zero point energy difference of the
electrode between no H and full H coverage.The term, $Q\phi$ , is the
electrical work to move the charge $Q$ (net charge on electrode) form zero
potential (in the bulk electrolyte) to the electrode with the potential
$\phi$.$E_{EDL}$ is the energy of the induced EDL by the electrode charge
$Q$.The last term is the chemical potential of the solvated proton in the
electrolyte. $V(x,\phi)$ is the relative potential with respect to PZC (Point
of zero charge) at coverage $x$ and electrode potential $\phi$,

$V(x,\phi)=\phi-\phi_{PZC}(x)$ (10)

Once $V(x,\phi)$ is known , the charge $Q$ and $E_{EDL}$ can be obtained by ,

$Q(x,\phi)=\int_{V=0}^{V=\phi-\phi_{PZC}(x)}C_{EDL}dV$ (11)
$E_{EDL}(x,\phi)=\int_{V=0}^{V=\phi-\phi_{PZC}(x)}Q(x,\phi)dV$ (12)

The final term $\mu_{H^{+}}$ (proton’s chemical potential) is given by,

$\mu_{H^{+}}=\frac{1}{2}G[H_{2}]+e\Phi_{SHE}-0.059\times pH$ (13)

where $G[H_{2}]$ is

$G[H_{2}]=E[H_{2}]+ZPE[H_{2}]+\frac{7}{2}k_{B}T-TS_{H_{2}}$ (14)

All the physical quantites in Equation (14) can be obtained by DFT
calculations and standard thermodynamic database. To this end, we have used
the JDFT method with an implicit solvation method as implemented in the
simulation package JDFTxSundararaman  _et al._ (2017) to obtain the electronic
structure and the potential at the PZC of each H coverage for calculation of
the Free energy function. The implicit electrolyte is described by the charge-
asymmetric non-locally determined local-electric (CANDLE) model. $\Phi_{SHE}$
is the computational standard hydrogen electrode. It has been determined to be
4.66 V from PZC calibration of the CANDLE solvation in JDFTxSundararaman and
Goddard III (2015).$E(x)$ and PZC is determined form the JDFTx simulation for
different $x$ coverages and configurations. We fit the data to obtain
relationships of $E(x)$ and PZC with $x$ and feed them to Equations (2) and
(3) to calculate relative free energy with H-coverage at any given electrode
potential. Once the relative free energy is obtained,H-coverage $x$ and net
surface charge $Q$ are calculated by taking ensemble averages. As there are
different H coverage configurations for any given value of coverage ,
degenerate statistical mechanics is used to calculate the ensemble
averages.The charge associated with the H-coverage leads to the redox
capacitance and the one associated with surface charge $Q$ yields the EDL
capacitance. The slope of the charge with electrode potential gives total
electric capacitance ($C_{E}$) at any given electrode potential. The
Generalised Gradient Approximation with the Perdew-Burke-Enzerhof functional
(GGA-PBE)Perdew  _et al._ (1996) was considered as the approximate electron
exchange-correlation for the DFT calculations. Ultrasoft
pseudopotentialsGarrity  _et al._ (2014) are used to describe the ion-electron
interaction. Kinetic energy cut-off of 20 Hartree and 30 Hartree were used for
structure optimization and single point energy calculation at the optimized
geometry with convergence criteria of $10^{-6}$ Hartree. For all calculations,
$3\times 3\times 1$ supercells of the unit cells were considered.

##  III Results And Discussions

###  III.1 Structural models and properties

Ti3CNO2 and Mo2TiC2O2 are modelled by replacing 50$\%$ of carbon layer (of
Ti3C2O2) by nitrogen and by replacing the outer Ti layers (of Ti3C2 O2)by Mo,
respectively. Modelling the nitrogen doped Ti3C2O2 is, however, different. It
is because the nitrogen atom can occupy three possible positions when doped
into Ti3C2O2 MXene. Accordingly, three structural models can be constructed:
Lattice-Site(LS) model, Functional-Site(FS) model, and Surface-Site(SS) model.
The doped nitrogen replace carbon and Oxygen atoms in LS and FS models,
respectively. In the SS model, they do not replace any atoms, but position
themselves on the surface of the MXene. They can occupy either the positions
on top of the functional atoms or the positions between the functional and Ti
atom in the outer layers. The structural models are shown in Figure 1.

Figure 1: Structural models of Nitrogen doped $Ti_{3}C_{2}T_{x}$: (a)LS model
, (b)FS model , and (c)SS model, where N is located above the O atom , (d)SS
model, where N is located between O and Ti atoms.Blue , brown, red and sky-
blue balls represent Ti, C, O and N atom respectively.

In this work we have considered 10% nitrogen doping in Ti3C2O2. Total energy
calculations are done to obtain optimised structures for the FS and LS doping
models. The SS model was not considered as calculations in Reference Chengjie
_et al._ , 2020 showed it to be dynamically unstable. The structural
parameters of pristine and doped Ti3C2O2 are given in Table-1. The results are
in excellent agreement with the experimental ones Chengjie  _et al._ (2020).

Table 1: Optimised lattice parameters and bond lengths of pristine and nitrogen doped $Ti_{3}C_{2}T_{x}$ in Å System | Lattice parameter | Bond lengths
---|---|---
|  | Ti-C | Ti-O | Ti-N
Un-doped | 3.03 | 2.19 | 1.97 | -
Lattice-Site | 3.01 | 2.12 | 1.99 | 2.20
Functional-Site | 3.04 | 2.26 | 1.96 | 1.91

We find that the lattice contracts upon doping at the lattice site and expands
upon doping at the functional site. This trend in the lattice parameters is
consistent with the size of the atoms. The atomic radius of Oxygen, Carbon and
Nitrogen are 60pm,70pm and 65pm respectively. The changes in the lattice
parameters upon doping are consistent with this. The length of the Ti-C bond
expands(contracts) in the FS(LS) model, in comparison to that in the pristine
system. A reverse trend is shown in the case of Ti-O bond lengths, although
the changes in this case are almost negligible. The Ti-N bond length changes
substantially between the two models. This is expected since different atoms
are replaced in different models. It suggests that the charge sharing between
Ti and N will be more in case of FS model than the LS one. Overall, the trends
in the lattice parameters follow the trends in the Ti-C bond lengths.

###  III.2 Electronic Structure and Quantum Capacitance

The quantum capacitance of an electrode material depends on its electronic
structure near the Fermi level. In Figure 2 we present the electronic
structure of pristine and nitrogen doped (both FS and LS) $Ti_{3}C_{2}O_{2}$.
The nitrogen atom has one more valence electron than carbon and one less
electron than the oxygen. Therefore, when a nitrogen atom replaces a
carbon(oxygen) atom, total number of electrons in the system increases
(reduces).As a consequence, the electronic bands around the Fermi-levels are
shifted. We find that in case of doping at the functional site (middle pnel of
Figure 2), the electronic band shifts towards higher energies. This happens
due to the reduction in the number of electrons in the system. In case of
doping at the lattice site, the shift of the electronic bands is towards lower
energies(lower panel of Figure 2). This is due to increase in total number of
electrons in the system.

Figure 2: Electronic Structure of pristine (upper panel),FS doped (middle
panel) and LS doped $Ti_{3}C_{2}O_{2}$(lower panel)

Close inspection of the densities of states reveals the following features:
(a) the electronic structure near the Fermi level is most substantially
affected in case of doping at the functional site. This is mostly due to the
nitrogen $p$ states occupying higher energy levels in the valence band (b) in
case of doping at the lattice sites, the nitrogen $p$ states are deep within
the valence band unable to make an impact on the features near the Fermi
level. The features near the Fermi level, in this case, are mostly due to the
Ti $d$ states (c) the unoccupied part of the spectra are overwhelmingly due to
Ti $d$-bands which implies serious consequences on the charge transfer. From
these we can predict that it is more likely that the largest contribution to
$C_{Q}$ will come from the functional site doped system.

In Figure 3, we present the variation of Integrated Quantum
capacitance($C_{Q}^{int}$) with absolute voltage. The voltage range is set
between $\pm$1 V. This range is chosen as because at room temperature, the
electrochemical stability window of the electrolyte solvent is about 1.25 eV
Kuhnel  _et al._ (2020). In the negative part of the voltage window, maximum
$C_{Q}^{int}$ of FS doped Ti3C2O2 is four times larger than that of the
pristine MXene. This comparative behaviour correlates to the changes in the
electronic structure near the Fermi level. As expected from the features in
the electronic structure, $C_{Q}^{int}$ of LS doped system, though higher than
the pristine system throughout the voltage window, is not as amplified as the
FS doped system.

Figure 3: Integrated Quantum Capacitance of pristine and doped
$Ti_{3}C_{2}O_{2}$

Figure 4 shows electronic structures of $Ti_{3}CNO_{2}$ and
$Mo_{2}TiC_{2}O_{2}$. A comparison between LS doped Ti3C2O2 and Ti3CNO2 can be
instructive. One of the prominent features is that with introduction of more
nitrogen replacing carbon, the bands move towards lower energy, in general. A
consistent trend emerges as one makes a comparison between pristine, LS doped
and carbon substituted Ti3C2O2. In Ti3CNO2, states near Fermi level are
populated due to more contributions from nitrogen. The shoulder at around 3 eV
in LS doped Ti3C2O2 transforms to a peak in Ti3CNO2. The small peak closest to
the Fermi level, in the unoccupied part in LS doped system transforms into a
prominent one with nitrogen substitution. Upon substitution of Mo at Ti
positions, however, the electronic structure changes considerably. The
densities of states in Mo2TiC2O2 is dominated by contributions from Mo and O
near the Fermi level in the occupied part of the spectrum. In the unoccupied
part, states near the Fermi level have dominant contributions from Ti and Mo.
The bands are more extended in Mo2TiC2O2 in comparison to Ti3C2O2 as Mo being
from the $4d$ series have wider bands. These features reflect in variations in
$C_{Q}^{int}$. For $Ti_{3}CNO_{2}$, the higher density of states in the
conduction band around the Fermi-level leads to significantly larger
$C_{Q}^{int}$ in the positive side of the voltage window in comparison with
that in the negative side. As expected from the features of the densities of
states, $Mo_{2}TiC_{2}O_{2}$ have the lowest integrated quantum capacitance
throughout the voltage window.

Figure 4: Electronic Structure of $Ti_{3}CNO_{2}$(Top) and
$Mo_{2}TiC_{2}O_{2}$(Bottom) Figure 5: Integrated Quantum Capacitance of
$Ti_{3}CNO_{2}$ and $Mo_{2}TiC_{2}O_{2}$

Equation (1) implies that larger value of quantum capacitance will lead to
higher value of the total capacitance where the electrical part of the total
capacitance will be the sole contributor. If both electrical and quantum parts
of the capacitance are comparable, the total capacitance will be lesser than
the least. Our results suggest that in comparison to others, FS doped Ti3C2O2
(Ti3CNO2) may lead to higher total capacitances when operated in the
negative(positive) voltage windows. A final call on this will be possible only
when the electrical part of the capacitances are calculated.In the next sub-
sections we explore this point.

###  III.3 Electrical capacitance

As discussed in Section II, we have used the model of MXene electrode in
contact with an acidic electrolyte H2SO4 for calculation of the EDLC and
Pseudocapacitances. We specifically looked at the effects of doping and
substitution at different sites on the evolution of the electron and proton
transfers between the electrolyte and the electrode, giving rise to
simultaneous contributions of EDL and redox capacitances. The literature on
nitrogen doped functionalised Ti3C2 has proposed EDL as a primary mechanism
behind elevated value of the total capacitance due to nitrogen doping Wen  _et
al._ (2017); Chengjie  _et al._ (2020). On the other hand, analysis based
solely on the electronic structures of functionalisedTi3CN monolayers inferred
that the primary mechanism for O functionalised system is the surface redox
reaction while for F and (OH) functionalisation it is EDL Zhang  _et al._
(2016b). One of our motivations, therefore, is to investigate this aspect.

####  III.3.1 Nitrogen Doped Ti3C2O2

Acoording to Equation (8), the surface redox behaviour is controlled by
adsorption of proton on the electrode surface and subsequent electron transfer
to the electrode at a given electrode potential. To determine the extent of
the surface redox phenomenon we calculated $G(x,\phi)$ (Equation 9) for each
H-coverage, $x$, at different fixed electrode potentials ($\Phi$); $\Phi$
varying from -1 V to +1 V with respect to SHE. In Figure-6 (a)-(c), we show
$G(x,\phi)$ as a function of $x$ for pristine, nitrogen doped FS and LS
$Ti_{3}C_{2}O_{2}$ , respectively. For all systems, and for all $\Phi$,
$G(x,\Phi)$ is parabolic in nature. From the minima of $G(x,\Phi)$, we find
that the ensemble averaged coverage $x_{avg}=0.63,0.93.0.73$ for pristine,
functional-site and lattice-site doped $Ti_{3}C_{2}O_{2}$, respectively. The
result for pristine compound agrees very well with the existing one Zhan  _et
al._ (2018). The results imply that the electrode surfaces are not fully
protonated even at -1 V for all systems.However, nitrogen doping enhances
protonation with the FS system having more than 90% covered, the maximum among
the three systems. Another common feature in all systems is that the $x_{avg}$
continuously decreases with increasing $\Phi$ leading to $x_{avg}=0$ at
$\Phi=1$ V.

Figure 6: Variations in the Relative Free Energy with H-coverage at different
fixed potentials for pristine(a), nitrogen doped FS (b) nitrogen doped LS (c )
$Ti_{3}C_{2}O_{2}$. The charge storage and electrical capacitances as a
function of electrode potential are shown for pristine (d,g), nitrogen doped
FS (e,h) nitrogen doped LS (f,i) systems. The negative values of the EDL
capacitances imply that the EDL mechanism is acting against the surface redox
mechanism.

The electrochemical behaviour of a capacitor in acidic electrolyte like H2SO4
can be understood in terms of the net charge on the surface that is
responsible for EDL, and the proton transfer from the electrolyte ions to the
electrode that is responsible for the redox reaction. The net number of
electron transfer or the total charge storage in the electrode is decided by
the cumulative effect of these two. The electron transfer, surface charge, and
proton transfer numbers as a function of $\Phi$ are shown in the Figure-6
(d)-(f) for pristine, nitrogen doped FS and LS systems. The proton transfer
number is given by $x_{avg}$ while the surface charge is the ensemble average
of $Q$ in Equation (11). The electron transfer happening across the charge
layer accumulated over the electrode surface is obtained by adding these two.
We find that for all three systems accumulated surface charge prohibits the
proton transfer to the surface. For the LS doped system, the variations in
surface charge, proton transfer and consequently the electron transfer numbers
are linear with $\Phi$. The surface, in all three cases, are slightly
positively charged. This is due to the fact that part of the positive charge
from proton transfer becomes the net surface charge. In case of the pristine
and the FS doped systems, the variations are piecewise linear with changes in
slopes at intermediate voltages, the slopes being larger for FS doped system.
The pseudo-capacitance, the EDLC and the total electrical capacitance are
obtained from the slopes of proton transfer number, surface charge, and
electron transfer number curves (Fig 6(d-f)) respectively.Figure 6 (g)-(i)
shows variations in the capacitances for the three systems under
consideration.

Figure 7: Variations in the Relative Free Energy with H-coverage at different
fixed potentials for $Ti_{3}CNO_{2}$(a) and $Mo_{2}TiC_{2}O_{2}$(b).The charge
storage and electrical capacitances as a function of electrode potential are
shown for $Ti_{3}CNO_{2}$(c,e) and $Mo_{2}TiC_{2}O_{2}$(d,f)

In case of the pristine system, total electric capacitance is nearly constant
at 250 F/g between -1 V and 0 V. It decreases linearly to 200 F/g for higher
potentials. Both qualitative and quantitative agreements with the existing
result Zhan  _et al._ (2018) on the variation in the total capacitance with
$\Phi$ are excellent. For the LS doped system, the total capacitance is near
constant at a value of 250 F/g throughout the range of the voltage. This is
because of the near constant contributions from both EDLC and
pseudocapacitance which in turn is due to the linear variations in proton
transfer number and the surface charge. Therefore, nitrogen doping at the
lattice site does not lead to any improvement in the electrical capacitance of
$Ti_{3}C_{2}O_{2}$. The doping at the functional site, on the other hand,
improves the total capacitance. This is due to higher value of $C_{redox}$ and
lower value of $C_{EDL}$ for this system. Since in these cases, EDLC acts
against pseudocapacitance, a lower value of $C_{EDL}$ implies that the surface
electrochemistry is overwhelmingly determined by the redox mechanism. In case
of the pristine and LS doped $Ti_{3}C_{2}O_{2}$, the EDLC, both qualitatively
and quantitatively, contribute substantially to the total electrical
capacitance. In these two cases, though redox appears to be the dominating
mechanism, the effect of EDL is recognizable. The dominance of redox reaction
as found in our calculations, however, is in contrast with the predictions
made from the experimental results Wen  _et al._ (2017); Chengjie  _et al._
(2020).

A qualitative understanding of the larger value of redox capacitance for FS
doped system as compared to the LS doped one can be understood from Bader
charge analysis Bader (1990). With full H-coverage, the average Bader charge
transfer from the electrode to the $H^{+}$ ion is 0.463$e$, 0.501$e$ and
0.455$e$ for pristine, FS doped, and LS doped systems, respectively. The
largest charge transfer for FS doped system can explain the largest
pseudocapacitance. The reason FS doped system has a larger charge transfer
than the LS doped one is the following: from Table 1, we find that the Ti-N
bond length for FS doped system is 15% less than that in the LS doped system.
In fact Ti-N and Ti-O bond lengths are comparable for FS doped systems as
doped nitrogen occupies the functional sites. Due to significantly smaller
Ti-N bond length, the charge sharing in Ti-N bond is greater in FS doped
system than that in LS doped one. This, in turn, is responsible for larger
charge transfer to the $H^{+}$ ions as can be seen quantitatively from
calculation of Bader charges on each atom.

####  III.3.2 Substituted $Ti_{3}C_{2}O_{2}$

The electrochemical behaviours of the substituted systems Ti3CNO2 and
Mo2TiC2O2 throw up quite an interesting picture. Figure 7(a)-(b) show the
$G(x,\phi)$ as a function of H-coverage for $Ti_{3}CNO_{2}$ and
$Mo_{2}TiC_{2}O_{2}$, respectively. Unlike the doped systems, $Ti_{3}CNO_{2}$
and $Mo_{2}TiC_{2}O_{2}$ have significantly low coverages, the highest
$x_{avg}$ being 0.35 and 0.31 respectively at -1 V. Behaviour of charge
storages (Figure 7 (c)-(d)) too, are very different from the doped systems.The
magnitude of the charges are less for substituted systems in comparison to the
pristine and doped systems.However, the surface charge and protons do not act
against each other. Lower amount of charge storage (in comparison with doped
systems) and the negative sign of the surface charge (unlike the doped
systems) can be correlated with the substantially low H-coverage. The
qualitative and quantitative variations of various components in charge
storage are reflected in the behaviour of EDLC and pseudocapacitances. The
non-linear variations of the proton and surface charges in Ti3CNO2 determine
the variations of the capacitances. While pseudocapacitance was the dominant
contributor for negative voltages, EDLC starts to play a significant role with
increasing voltage, finally becoming the dominant one for positive voltages.
Thus, in nitrogen substituted Ti3C2O2, the mechanism of charge storage is
dependent on the voltage window. Our quantitative calculations are in
agreement with the quantitative analysis that Ti3CNO2 cathode has surface
redox reaction as the dominating mechanism Zhang  _et al._ (2016b). Behaviour
of the components in electrical capacitance of Mo2TiC2O2 is found to be
different from all other systems. In here, the pseudocapacitance and EDLC do
not oppose each other but collaborate. While pseudocapacitance turns out to be
dominant, and both components remain near constant for most of the voltage
window, beyond +0.5 V EDLC starts to assume more significant role. However,
the absolute values are the least among the systems considered. This can be
understood from the magnitudes of the charge storage which in turn is
connected to the lowest H-coverage and the fact that Mo being 60% more
electronegative than Ti, is unable to transfer as much charge as Ti in outer
layers of the MXene.

###  III.4 Total Capacitance and comparison with experiments

In Figure 8 we show the total capacitance (calculated by Equation (1)of
pristine and doped Ti3C2O2(left) ,$Ti_{3}CNO_{2}$ and
$Mo_{2}TiC_{2}O_{2}$(right). The variations in the capacitances are plotted
with respect to SHE and Ag/AgCl electrode potentials. We find that the
capacitance profiles for pristine and doped systems follow that of their
quantum capacitances. This was to be expected as the orders of magnitude of
$C_{E}$ and $C_{Q}$ are same for all systems except FS doped one at negative
electrode potentials. As a consequence, we obtain quite high capacitance for
FS doped system in the negative potential range. For the positive part of the
SHE potential window the capacitances of pristine, FS doped and LS doped
systems are comparable. An one to one comparison with the experimental results
is somewhat difficult because of various factors. The capacitance crucially
depends on the sample preparation method, the exact composition, the
functional groups and the electrolyte. Accordingly, our results are
contradictory to the ones obtained in Reference Chengjie  _et al._ , 2020
where the electrolyte is basic. The capacitances obtained in that work are
significantly lower than our results. Also, they obtained the highest value of
capacitance for LS doped system and identified EDL as the primary mechanism
which is aided by the surface redox reactions due to the surface functional
groups. On the other hand, some correspondences can be made with the
experimental results of Reference Wen  _et al._ , 2017 as the electrolyte used
was 1 M H2SO4, exactly the one used in the model implemented in our
calculations. Although the maximum capacitance reported by them is only 32 F/g
for the pristine system as opposed to 176 F/g in our calculations, there is
good agreement for the maximum capacitance of nitrogen doped system. The
reported value of 192 F/g in the Ag/AgCl electrode potential window of
-0.35-0.35 eV agrees well with the calculated values for FS (230 F/g ) and LS
(192 F/g) doped systems in the same window. However, there are certain
qualitative differences between the results from the experiment and our
calculations. From the high resolution XPS study it was suggested that the
doped nitrogens occupy the carbon sites. Therefore the capacitance reported in
their work is that of a LS doped system. It is yet to be observed whether
further gain in capacitance can be obtained in a FS doped system in agreement
with our results. The experiment also suggested that this gain in capacitance
is due to larger inter-layer distance due to doping leading to an increase in
EDLC and therefore the primary mechanism in the doped system is EDL. Our
calculations could not confirm it. The EDL part of the capacitance, in our
calculations, rather went the opposite way. However it is worth mentioning
that our calculations are on a monolayer implying infinite inter-layer
distances. Moreover no other functional group except -O was used in our model
while the experimental sample had -F functional groups in significant
proportions. The presence of -F groups that are less redox active than -O in
the experimental sample may explain the discrepancy between the experiment and
theory in identifying the primary mechanism for charge storage. The total
capacitance of the substituted systems are less than the doped systems. While
the variation in the total capacitance of Mo2TiC2O2 follows that of its
quantum capacitance, the situation is different for Ti3CNO2. Though it had
larger quantum capacitance for higher voltages, the drop in the electrical
capacitance with increasing voltage brings the maximum of total capacitance in
the negative part of SHE voltage window. These results show that the quantum
capacitance is an important component having substantial influence on the
quantitative estimation of capacitances of supercapacitors as observed earlier
Das and Ghosh (2022).

Figure 8: Total Capacitance as a function of electrode potential for nitrogen-
doped (left) and substituted (right) $Ti_{3}C_{2}O_{2}$. The vertical dotted
lines in the left panel indicate the potential window used in the experiment
on nitrogen doped $Ti_{3}C_{2}O_{2}$ Wen  _et al._ (2017).

##  IV Conclusions

Using an implicit solvation model in conjunction with DFT we have modelled the
electrolyte electrode interaction at the electrode surface for computation of
the electrochemical parameters of doped and substituted oxygen functionalised
Ti3C2 MXene. This model addresses the surface EDL and redox effects in H2SO4
electrolyte solution. We have computed the contributions to stored charges and
the capacitances due to both effects when nitrogen is doped at different sites
of Ti3C2O2 as well as in substituted systems Ti3CNO2 (nitrogen substitution at
Carbon site) and Mo2TiC2O2 (molybdenum substitution at outer Ti layers). We
find that upon inclusion of the quantum capacitance, the maximum gain in the
capacitance is obtained for nitrogen doped at functional sites of Ti3C2O2 at
the negative SHE potentials. This two-fold gain, in comparison to the pristine
compound is due to larger storage of charge dominated by the redox activity at
the surface. The presence of nitrogen dopants at the surface along with the
largest coverage of H+ ions from the electrolyte are responsible for this. Our
quantitative results suggest that for the pristine and doped systems the
surface redox activity is primary responsible for the electrochemical
parameters although EDL mechanism competes with it. The two mechanisms act
against each other and influence the overall charge storage and electrical
part of the capacitance. Quantitatively our results are in excellent agreement
with the experiment Wen  _et al._ (2017). However, the qualitative explanation
of increased capacitance value differs with ours. With increase in nitrogen
content such that it substitutes 50% carbon from the lattice site, the
capacitance value degrades. It further degrades upon substituting the surface
Ti atoms with more electronegative Mo.These are yet to be verified through
electrochemical measurements although both compounds have been synthesised
Naguib  _et al._ (2012b); Maughan  _et al._ (2021). In cases of substitution,
though EDL and surface redox phenomena co-operate each other, poorer charge
transfer hinders growth of capacitances. In case of nitrogen substitution we
found evolution of EDL mechanism to become the dominant one upon increase in
voltage. The reverse trend in comparison to doped systems can be correlated
with remarkably low ion coverage from the electrolyte and subdued charge
transfer in the surface due to presence of more electronegative substituents.
Our results suggest that nitrogen doping is a better strategy to get better
electrochemical performances in Ti3C2O2 MXene electrodes.We also suggest that
much better values of capacitances in nitrogen doped Ti3C2O2 can be obtained
if samples are prepared with doping at the functional site and electrochemical
measurements are done in a larger negative voltage window with Ag/AgCl as
reference electrodes.

##  V Acknowledgement

The authors gratefully acknowledge the Department of Science and Technology,
India, for the computational facilities under Grant No. SR/FST/P-II/020/2009
and IIT Guwahati for the PARAM supercomputing facility.

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