# Understanding the electrochemical performance and diffusion kinetics of
HC$||$Na3V2(PO4)3/C full cell battery for energy storage applications

Madhav Sharma  Department of Physics, Indian Institute of Technology Delhi,
Hauz Khas, New Delhi-110016, India  Rajendra S. Dhaka
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Delhi, Hauz Khas, New Delhi-110016, India

###### Abstract

The efficient energy storage devices are crucial to meet the soaring global
energy demand for sustainable future. Recently, the sodium-ion batteries
(SIBs) have emerged as one of the excellent cost effective solution due to the
uniform geographical distribution and abundance of sodium. Here, we use hard
carbon (HC) as an anode and Na3V2(PO4)3/C (NVP/C) as a cathode to fabricate a
HC$||$NVP/C full cell battery and understand its electrochemical performance
and diffusion kinetics. These materials are characterized through the analysis
of x-ray diffraction and Raman spectroscopy to confirm their single phase and
structure. The full cell demonstrates a high operating voltage of $\sim$3.3 V,
with minimal polarization of 0.05 V, attributed to the lower working voltage
of the HC. Interestingly, for the full cell battery we find the specific
capacity of around 70 mAh/g at 0.1 C and even around 35 mAh/g at high current
rate of 5 C along with high rate capability up to 55 cycles. The diffusion
kinetics of the full cell battery is investigated through detailed analysis of
CV curves at various scan rates, and the diffusion coefficient is found to be
5–8$\times$10-11 cm2/s for the anodic as well as cathodic peaks.

##  I  Introduction

Sodium-ion batteries (SIBs) have emerged as potential candidates for cost
effective energy storage devices to meet the requirements of future generation
[1, 2]. Although sodium’s larger size and higher standard electrode potential
limit the energy density of SIBs compared to lithium-ion batteries (LIBs), the
focus on cathode materials with similar intercalation chemistry has
facilitated the exploration of viable compounds for both systems [1, 2].
Despite many challenges, ongoing research seeks to optimize SIB performance,
making them a compelling candidate for large-scale energy storage applications
[3]. A vast category of anodes and cathodes have been discovered, showing
potential for their utilization in SIBs, but each has its own merits and
demerits. Anodes pose a challenge, with graphite, common in LIBs, showing
unsatisfactory sodium ion intercalation [4]. However, non-graphitic anodes,
particularly hard carbon (HC) derived from various precursors, have emerged as
promising alternatives for SIBs due to its remarkable sodium retention ability
[5, 6, 7]. Also, HC is an eco-friendly anode option derived largely from
biomass and the disordered nature of HCs, enriched with defects as well as
hetero-atoms, offers more avenues for Na+ ion storage and diffusion. Moreover,
compared to graphite, its expanded interlayer space ensures a stable
intercalation/de-intercalation mechanism for sodium ions during charging-
discharging process [5, 8].

For the cathode side, the polyanionic-based materials have emerged as a
prominent category for SIBs [9] with Na3V2(PO4)3 (NVP) standing out as a
leading candidate due to its noteworthy energy storage capacity and remarkable
structural resilience [10, 11]. This material boasts a NASICON-type crystal
framework characterized by its three-dimensional open lattice that ensures
high ionic conductivity [12]. Nevertheless, a significant challenge
encountered with NVP lies in its low electron conductivity; therefore, various
strategies have been used to surmount this hurdle, notably incorporating
carbon coatings [13]. There have been many studies on these electrode
materials in half-cell configuration; however, detailed investigation of the
electrochemical performance in the full-cell configuration are still limited.
Here, the selection of both anode and cathode materials plays a critical role
in determining the working voltage and energy density of the assembled full
cell. A symmetric full cell using NVP has been investigated due to its dual
functionality as both a cathode (V3+/V4+ at $\sim$3.4 V) and an anode (V2+/V3+
at $\sim$1.7 V) electrode [14]. However, the relatively high potential on the
anodic side and a low capacity of around 59 mAh/g lead to a moderate working
potential of 1.7 V and low gravimetric energy density in the NVP symmetric
full cell [15]. While efforts have been made to boost the gravimetric energy
density by incorporating high-capacity anode materials in sodium-ion
batteries, but giving a nominal voltage of around 2 V only [16, 17], hence
there is still room for improvement in achieving an overall higher working
potential. To tackle these challenges, the HC is advantageous due to its lower
operating potential and higher Na+ ion storage capacity [18].

Therefore, we present a comprehensive analysis of the electrochemical
properties of full cell sodium-ion batteries utilizing NVP/C as a cathode and
HC as an anode. Using the x-ray diffraction (XRD) and Raman analysis, we
confirm the NASICON-type crystal structure of the synthesized cathode material
NVP/C. Also, the anode material (hard carbon) displayed characteristics
typical of disordered graphite domains. Our electrochemical testing involve
cyclic voltammetry (CV), electrochemical impedance spectroscopy (EIS), and
galvanostatic charge-discharge (GCD) measurements, which provided insightful
outcomes for both NVP/C and HC half cells, as well as the assembled full cell
(HC$||$NVP/C). The half cells demonstrated a decent performance, revealing a
reversible discharge capacity of 85 mAh/g (operating voltage: $\sim$3.4 V) at
0.1 C for the NVP/C cathode and a charge capacity of 252 mAh/g (operating
voltage: $\sim$0.19 V) at 50 mA/g for the HC anode. Interestingly, we get good
performance of HC$||$NVP/C full cell having the capacity of around 70 mAh/g at
0.1 C and around 35 mAh/g at 5 C. Additionally, owing to the lower working
voltage of HC, the full cell exhibits a notable high operating voltage of
$\sim$3.3 V, coupled with minimal polarization of 0.05 V. We perform detailed
analysis of CV data at different scan rates to understand the diffusion
kinetics of the full cell where the extracted values of the diffusion
coefficient are in the range of 5–8$\times$10-11 cm2/s for the anodic/cathodic
peaks.

##  II  Experimental

Synthesis of cathode material:

The NVP/C composite sample is synthesized using the sol-gel method where
NaH2PO4, V2O5, Na2CO3, and oxalic acid were taken as precursors in their
stoichiometric ratio. First, 11 mmol of oxalic acid and 2.2 mmol of V2O5 were
dissolved in 30 mL of DI water taken in a Duran bottle and magnetically
stirred at 70oC for 30 mins. Then, 6.6 mmol of NaH2PO4 and 0.11 mmol of Na2CO3
are dissolved in 10 mL of DI water, magnetically stirred separately, and later
drop wise added to the above solution. The Duran bottle was closed, heated,
and magnetically stirred overnight at 80oC. After that, the Duran bottle was
opened, heated at 80oC, and stirred until a gel was formed, which was later
vacuum-heated overnight to remove any excess water. The obtained dried
precursors were grinded for 2 hrs and heated at 750oC for 12 hrs in a high-
temperature tube furnace under an Ar+H2 (9:1) environment [19].

Physical characterization:

We conducted x-ray diffraction measurements using a Panalytical Xpert 3
instrument at room temperature, employing CuKα radiation with a wavelength of
1.5406 $\AA$. The Raman spectrum was collected using a Renishaw inVia confocal
microscope equipped with a 532 nm wavelength laser and 2400 lines per mm
grating. The elemental composition/mapping was checked through energy
dispersive x-ray spectroscopy (EDX) using an Oxford-made LN2 free SDD X-max 80
EDS detector. The surface morphology was examined through Field Emission
Scanning Electron Microscopy (FE-SEM) using a JEOL JSM-7800F Prime system.

Half-cell fabrication:

To conduct the electrochemical test, initially, half cells were prepared with
NVP/C as the working electrode, sodium metal anode as the counter/reference
electrode, and 1M NaPF6 dissolved in ethylene carbonate (EC) and diethyl
carbonate (DEC) in ratio of 1:1 as electrolyte. The electrode preparation
process involved creating slurries of the active materials. Specifically, for
NVP/C, a mixture of the active material, carbon (super P), and polyvinylidene
fluoride (PVDF) in a ratio of 70:20:10 was utilized. N-methylpyrrolidone (NMP)
was used as the solvent for the slurry. Following this, the slurry was applied
onto a battery-grade aluminum foil using the doctor’s blade casting technique.
The coated foil was dried overnight at 80oC to ensure complete solvent
evaporation. After calendaring, 12 mm diameter electrodes were precisely cut
out for further utilization. The assembly of 2032 coin cells were carried out
inside a glove box (UniLab Pro SP from MBraun) filled with argon while
maintaining a controlled atmosphere with O2 and H2O levels below 0.1 ppm.

Full-cell fabrication:

To construct the full cells, we use NVP/C as cathode and HC (purchased from
Shaldong Gelon LIB Co., Ltd., having a median particle size (D50) of 9 $\mu$m
and a specific surface area (SSA) of $\leq$ 5 m2/g) as an anode. The
fabrication involves two steps: First, the HC anode was subjected to three
cycles of sodiation in the half-cell configuration have sodium-metal as the
counter/reference electrode. The slurry for the HC anode comprised a mixture
of hard carbon, super-P, and PVDF in a mixture of 8:1:1 ratio. This slurry was
coated onto a battery-grade copper current collector. These electrodes were
then used to make half cells utilizing 1M NaPF6 dissolved in EC:DEC as the
electrolyte. The specific capacity of each cell was recorded, and the actual
capacity of each cell was determined by multiplying it with the respective
electrode weight of the cathode. The cells were disassembled in a fully
charged state inside the glove box with an open-circuit voltage (OCV) of
$\approx$1.1 V.

We use this sodiated hard carbon electrode material for the fabrication of
full cell batteries. The weight of the electrodes was carefully selected to
match the actual capacity of both the anode and cathode materials,
respectively. These cells were assembled using the same electrolyte employed
in the initial cycling of HC. The OCV of the NVP/C half cells was
approximately 2.5 V, and the Hard-carbon half cells were disassembled at an
OCV of roughly 1.1 V. Therefore, it is anticipated that the successfully
fabricated full-cell will have an expected OCV in the vicinity of 1.4 V.

Electrochemical characterization:

The electrochemical performance is tested through detailed analysis of
electrochemical impedance spectroscopy (EIS) and cyclic voltammetry (CV),
which were performed using the Biologic VMP-3 model. The EIS covered a
frequency range from 10 mHz to 100 kHz, with an ac voltage amplitude of 10 mV
applied at the cell’s OCV state. The galvanostatic charge-discharge (GCD)
profiles were obtained using a Neware battery analyzer (model no.: BTS400).

Figure 1: (a) The Rietveld refined XRD pattern, (b) the crystal structure
generated via VESTA software using refined parameters, and (c) the Raman
spectrum of the prepared NVP/C sample. (d, e) The XRD pattern and the Raman
spectrum of the HC. The FE-SEM images of (f) NVP/C and (g) HC samples. (h) The
EDX layered image of NVP/C and the corresponding elemental mapping of (i)
sodium (cyan), (j) vanadium (yellow), (k) phosphorus (pink), (l) oxygen
(green), and (m) carbon (red).

##  III  Results and discussion

Figure 2: First three cyclic voltammetry profiles at a scan rate of 0.05 mV/s
in a potential range of (a1) 2.5–4.0 V for the NVP/C half cell, (a2) 0.01–3.0
V for the HC half cell, and (a3) 1.5–4.0 V for the HC$||$NVP/C full cell. The
electrochemical impedance spectroscopy of the (b1) fabricated NVP/C half cell,
(b2) the HC half cell, and (b3) the HC$||$NVP/C full cell with the
corresponding equivalent circuit. The zoomed view is shown in the inset of
(b1) and (b2). The galvanostatic charge-discharge profiles of (c1) NVP/C half
cell in a potential range of 2.5–4.0 V at 0.1 C for 5 cycles, (c2) the HC half
cell in a potential range of 0.01–2.5 V at 50 mA/g for 3 cycles, and (c3) the
full cell in a potential window of 1.5–4.0 V at 0.1 C for 5 cycles.

Figure 1(a) presents the XRD pattern along with the Rietveld refinement which
confirms the reflections corresponding to a rhombohedral lattice and
dimensions $a$($b$) = 8.72 Å and $c$ = 21.78 Å, classified under the space
group R-3c, consistent with ref. [11]. The atomic occupancies and Wyckoff
positions are provided in Table I.

Table 1: Detailed structural information of NVP/C extracted from Rietveld refinement Atom |  x |  y |  z |  Occ. |  Site
---|---|---|---|---|---
Na1 |  0.00000 |  0.00000 |  0.00000 |  1.000 |  6b
Na2 |  0.63625 |  0.00000 |  0.25000 |  0.626 |  18e
V |  0.00000 |  0.00000 |  0.14728 |  1.000 |  12e
P |  0.28492 |  0.0000 |  0.25000 |  1.000 |  18e
O1 |  0.19422 |  0.16724 |  0.09202 |  1.000 |  36f
O2 |  0.03414 |  0.21398 |  0.19213 |  1.000 |  36f

As depicted in Fig. 1(b), the sodium ions (Na+) are situated at the 6b (Na1)
and 18e (Na2) positions, V3+ ions fill the 12c octahedral sites, while P5+ at
the 18e tetrahedral sites [20]. The Raman spectroscopy was employed to
investigate the characteristics of the synthesized NVP/C, as illustrated in
Fig. 1(c). The pronounced D band observed at 1341.5 cm-1 indicates the
defects, vacancies and edge structures of carbon. Concurrently, the G band,
detected at 1589.5 cm-1 is representative of the graphitic carbon. The
intensity ratio between the G and D bands, IG/ID is found to be 0.85, which
underscores the disordered configuration of the carbon layer enveloping the
NVP particles [21]. Additionally, the three distinct signals detected at
around 805 cm-1 and 960 cm-1 can be attributed to the bridging mode, where the
1035.5 cm-1 belong to the stretching mode of the [PO4] tetrahedra [22, 23].
The XRD profile of the HC exhibits two broad peaks at the 2$\theta$ value of
approximately 23.5°and 44°, which are corresponding to the (002) and (100)
crystal planes of disordered graphite domains. These features are indicative
of the distinctive traits commonly observed in HC [24]. The Raman spectrum
reveals distinctive features marked as D1, D3, D4, and G bands, with peak
positions at 1389 cm-1, 1548.5 cm-1, 1339 cm-1, and 1602.5 cm-1, respectively.
The D1 band, attributed to disorder-induced modes, signifies the disordered
carbon structure in HC. The D3 and D4 bands are the indicators of sp3
hybridized carbon, implying the existence of minute crystalline domains within
the disorderly matrix. Additionally, the G band, aligned with the E2g
vibrational mode in graphite, emphasizes the graphitic quality inherent in the
carbon material [25]. The microstructure of NVP/C and HC is examined through
FE-SEM, as shown in Figs. 1(f, g), respectively. The image reveals aggregated
nanoparticles of NVP/C with a particle size of less than 1 $\mu$m. On the
other hand, an irregular morphology having a particle size within the range of
5-10 $\mu$m is observed for HC. Additionally, Figs. 1(h–m) present the layered
EDX image and elemental mapping of Na, V, P, O, and C, providing evidence of
the uniform distribution of these constituent elements in the sample.

Figure 3: The first cycle of the GCD profiles at each current rate from 0.1 C
to 5.0 C for (a) the NVP/C half cell in the potential range of 2.5–4.0 V, and
(b) the HC$||$NVP/C full cell in the potential range of 1.5–4.0 V. The
corresponding rate capability, along with the Coulombic efficiency, are
presented in (c) and (d) for both the cases, respectively.

Now, we first probe the electrochemical behavior of NVP/C and HC half cells.
The cyclic voltammetry (CV) curves of NVP/C half cell are displayed in Fig.
2(a1) up to three cycles at a scan rate of 0.05 mV/s within the potential
window of 2.5-4.0 V against Na+/Na. Notably, the oxidation peak at 3.44 V and
the reduction peak at 3.31 V revealed the presence of the V3+/V4+ redox
couple. While a slight decrease in peak current values post the first cycle
indicated the formation of the solid electrolyte interface (SEI) layer, the
peak voltage values remain consistent, with minimal polarization of around 0.1
V [9]. Likewise, the CV curves of the HC half cell are depicted in Fig. 2(a2)
covered the potential range of 0.01-3.0 V with a scan rate of 0.05 mV/s up to
three cycles. In the first cathodic region, distinct peaks appeared at
potential values of 0.01, 0.65, and 0.99 V. The anodic region displayed a
sharp peak at 0.19 V and a broad region extending up to approximately 1.5 V,
observed reversibly in both the anodic and cathodic regions (sharp peak at
0.01 V and a broad region extending to 1.5 V). The irreversibility of the
peaks at 0.65 and 0.99 V could be attributed to initial Coulombic losses due
to the SEI formation [26]. The CV analysis of the full cell is conducted
within the potential window of 1.5 to 4.0 V at a scan rate of 0.05 mV/s, as
illustrated in Fig. 2(a3). These CV profiles reveal a combination of features
from the HC and NVP/C electrodes. In the anodic region, a distinct peak
emerges at 3.37 V, accompanied by a preceding hump characteristic of the HC.
Interestingly, the cathodic region displays two peaks at 3.32 and 3.23 V. This
splitting of the cathode peak corresponds to the rearrangements in the local
redox environment when two sodium ions sequentially insert into the Na(2)
sites [27, 28]. Notably, the polarization observed between the anodic and
cathodic peaks is minimal around 0.05 V only. This value is lower than the
potential gap observed in NVP/C half cell, indicating the superior
electrochemical compatibility between the HC anode and the NVP/C cathode in
full cell battery system.

Further, the electrochemical impedance spectroscopy (EIS) measurements are
conducted on freshly prepared cells (following a 9-hour rest) at open-circuit
voltage (OCV). The obtained data are represented in the form of Nyquist plots,
illustrated in Figs. 2(b1–b3) for the NVP/C half cell, HC half cell, and
HC$||$NVP/C full cell, respectively. These plots were fitted with an
equivalent circuit displayed in the inset of Fig. 2(b3). We find that the EIS
behavior of NVP/C and HC half cells, in Figs. 2(b1, b2), exhibit similar
features: two depressed semicircles in the high and mid-frequency regions,
indicative of solvation layer resistance (Rsl) and charge transfer resistance
(Rct), along with a straight line representing Na+ ion diffusion in the bulk
electrode at low frequencies. For the HC half cell, the Rsl and Rct are
measured to be 169.4 $\ohm$ and 261.3 $\ohm$, respectively, while for the
NVP/C half cell, these values are found to be 165.4 $\ohm$ and 276.5 $\ohm$,
respectively [29, 30]. In contrast, the full cell exhibits a higher Rct value
of 2241 $\ohm$, which is notably greater than the Rct values for both the
NVP/C and HC half cells. This difference may be attributed to the blocking
effect of the solid electrolyte interface (SEI) [31] on the HC anode,
particularly because it was disassembled at a charged state and the same
extracted material was used in the fabrication of the full cell battery.

Moreover, the galvanostatic charge-discharge (GCD) measurements are performed
for the NVP/C half cell at a current rate of 0.1 C (1 C = 117.6 mAh/g) within
a potential window of 2.5–4.0 V, where the initial five cycles are depicted in
Fig. 2(c1). In these GCD profiles we find a flat plateau at around 3.4 V,
which is attributed to the V3+/V4+ redox couple [32]. The initial discharge
capacity is observed around 85 mAh/g with the very good capacity retention in
five cycles, reflecting an impressive stability rate of over 97% with respect
to the initial value. Also, the GCD profiles of the HC half cell, taken in a
potential window of 0.01-2.5 V at a current density of 50 mA/g, revealed two
distinct regions: sloping and flat, as shown in Fig. 2(c2). The sloping region
is indicative of Na+ ion storage in defected carbon sites, while the flat
region corresponds to intercalation of Na+ ion between graphene layers and the
filling of nano-pores [6, 33]. The initial discharge capacity reached around
351 mAh/g, but upon charging, only 252 mAh/g was achieved, which is 71.2% of
the first discharge capacity due to the initial loss from irreversible surface
reactions [34]. However, in the second cycle, the charging capacity almost
matches the discharge capacity, having 96% of the Coulombic efficiency. The
GCD profiles of the full cell measured between 1.5 V and 4.0 V at 0.1 C
current rate are presented in Fig. 2(c3). Here, it is crucial to establish the
voltage window with reference to the anode; therefore, the effective voltage
values are obtained by subtracting the anode voltage from the cathode voltage,
and we observe a combined effect of both the anode and cathode profiles. The
GCD curves of the full cell featuring an ascending region starting around 2.7
V and stabilizing at 3.4 V, providing a working voltage of approximately 3.3
V, which found to be relatively higher than reported 2.7 V in ref. [22].
During charging, the Na+ ions move from the cathode to the anode, and the
ascending part reflects the filling of defect carbon sites, while the flat
region signifies reaching the plateau for the HC anode. Interestingly, we find
the stable charge-discharge capacity of 65–70 mAh/g (slight variation based on
the mass loading of the cathode material) for the HC$||$NVP/C full cell up to
five cycles, see Fig. 2(c3). These values are found to be consistent with
reported iparameters, as summarized in Table II.

Table 2: Performance comparison of the sodium ion full cell with reported parameters. Anode$||$Cathode |  working voltage |  specific capacity |  ref.
---|---|---|---
NVP/C-T$||$NVP/C-T |  1.7 V |  50.9 mAh/g at 0.4 C w.r.t. anode |  [15]
MoTe2$||$NVP |  2 V |  230 mAh/g at 0.2 A/g w.r.t. anode |  [16]
HC$||$NVP/C |  2.75 V |  unsatisfactory w.r.t. cathode |  [22]
HCC$||$NVP |  - |  46 mAh/g at 100 mA/g w.r.t cathode |  [35]
GDHC$||$NVP |  2.4 V |  80 mAh/g at 0.05 C w.r.t cathode |  [36]
NC@MoSe2@rGO $||$NVOPF |  2.1 V |  286.7 mAh/g at 25 mA/g w.r.t. anode |  [37]
HC$||$NVP/C |  3.3 V |  70 mAh/g at 0.1 C w.r.t. cathode |  present work.

To test the rate capability, we further tested the electrochemical performance
of the NVP/C half-cell and the HC$||$NVP/C full-cell battery. The first cycle
galvanostatic charge-discharge (GCD) profiles are shown in Figs. 3(a, b) for
the NVP/C half-cell and HC$||$NVP/C full cell, respectively, at various
current densities from 0.1 C to 5 C. The corresponding rate capability and the
Coulombic efficiency at each current rate are presented in Figs. 3(c, d). In
the NVP/C half cell, we got a specific capacity of around 83 mAh/g at 0.1 C
with remarkable capacity retention of 97.2% after testing up to 55 cycles at
various current rates. However, the specific capacity of around 70 mAh/g is
observed in the full-cell at 0.1 C, which decreased to around 60 mAh/g having
the retention of 77.5% after 55 cycles at various current rates [38].
Nevertheless, in terms of high current rate performance, the full-cell
outperformed the NVP/C half-cell at 5 C. The full cell retained 47.5% of its
initial capacity, while the NVP/C half-cell could hold only 21.7% when the
current rate increases from 0.1 C to 5 C. This suggests a complementary
interaction between the HC anode and NVP/C cathode, consistent with the CV
results.

Figure 4: The plot of the real part of impedance versus the inverse square
root of the angular frequency in the Warburg region obtained from EIS data
shown in Figs. 2(b1–b3)

In order to understand the kinetics mechanism in these cells, we have done
further analysis of the EIS data and estimated the diffusion coefficient using
the following equations [39]:-

$D_{Na^{+}}=\frac{R^{2}T^{2}}{2A^{2}\eta^{4}F^{4}C^{2}\sigma^{2}}$ (1)
$Z^{\prime}=R_{S}+R_{CT}+\sigma\omega^{-0.5}$ (2)

where R, T, A, $\eta$, F, C, and $\sigma$ are gas constant, temperature, the
geometrical surface area of the electrode, number of sodium ions participated
in the reaction, the Faraday’s constant, Na concentration in the electrode
material (mol/cm3) and the Warburg coefficient, respectively. To find the
Warburg coefficient ($\sigma$), we plot the real part of impedance (Z′) as a
function of the inverse square root of the angular frequency
($\omega^{-0.5}$). We then fitted these data points using equation 2, see Fig.
4, where the estimated values of $\sigma$ are found to be 134.2, 1006.7 and
280.4 $\ohm/s^{1/2}$ for the NVP/C half cell, HC half cell and the full cell,
respectively. The significantly higher value of $\sigma$ for the HC half cell
suggests that the electrochemical process in the case of HC is primarily
limited by diffusion, with mass transport occurring at a comparatively slower
rate when compared with the NVP/C half cell as well as the full cell [40]. By
considering all the values in equation 1, the estimated values of the
diffusion coefficient are: 0.28$\times$10-12 cm2/s for the NVP/C half cell,
0.27$\times$10-13 cm2/s for the HC half cell, and 0.54$\times$10-13 cm2/s for
the full cell battery.

The cyclic voltammetry (CV) measurements were done on the fabricated full cell
(having NVP/C as cathode and HC as anode) at different scan rates ranging from
0.05 to 1 mV/s in a voltage window of 1.5 to 4.0 V. Interestingly, the CV
curve displayed prominent peaks in the cathodic as well as in the anodic
region as marked by C and A in Fig. 5(a), respectively. We observe that the
peak height increases from 0.1 mA to 1.3 mA when the scan rate vary from 0.05
mV/s to 1 mV/s. To probe the kinetic behavior of the sodium-ion battery, we
calculate the diffusion coefficient values for both cathodic and anodic peaks
using the Randles-Sevcik equation [41].

$i_{p}=(2.69\times
10^{5})AD^{\frac{1}{2}}C\eta^{\frac{3}{2}}\upsilon^{\frac{1}{2}}$ (3)

where $i_{p}$ (mA) is the peak current, A (cm2) is the electrode surface area,
D (cm2/s) is the diffusion coefficient, C (mol/cm3) is the concentration of Na
ions in bulk, $\eta$ is the number of transferred electrons in the redox
reaction, and $\upsilon$ (mV/s) is the applied scan rate. The peak current
obtained from the oxidation/reduction peaks is plotted as a function of the
square root of the scan rate $\upsilon$, shown in Fig. 5(b), and the values of
$D$ are calculated using the slope we got from the linear fitting of the
$i_{p}$ versus $\upsilon^{\frac{1}{2}}$. The diffusion coefficient values are
found to be 8$\times$10-11 and 4.7$\times$10-11 cm2/s for A, and C peaks,
respectively. These values match well with Na+ ion diffusion in both NVP/C and
HC half cells, justifying the coordinated performance of the HC$||$NVP/C full-
cell [42, 43, 44].

Figure 5: (a) The CV curves of the HC$||$NVP/C full cell at different scan
rates in the voltage range of 1.5–4.0 V, (b) the linear fit of the peak
current ip of A and C as a function of the square root of the scan rate, (c)
the linear fit between ip/$\upsilon$1/2 as a function of $\upsilon$1/2, (d)
the linear fit between log(ip) as a function of log($\upsilon$), the
capacitive and diffusive contributions for the peaks (e) A and (f) C at
different scan rates.

Finally, to attribute the contribution of diffusion and surface-controlled
reactions, we further analyzed the data using the Dunn method [45, 46].

$\frac{i}{\upsilon^{\frac{1}{2}}}=k_{1}\upsilon^{\frac{1}{2}}+k_{2}$ (4)

where k1 and k2 are the coefficients for the respective capacitive and
diffusion currents, which are calculated by linear fitting of the
$i$/$\upsilon^{1\textfractionsolidus 2}$ versus
$\upsilon^{1\textfractionsolidus 2}$, as shown in Fig. 5(c), the values for k1
and k2 are found to be 0.91 and 0.43 for anodic peak and 0.48 and 0.61 for the
cathodic peak, respectively. These values indicate that the diffusion part is
dominant in cathodic and has a high capacitive contribution in the case of
anodic reaction. In order to determine the contribution quantitatively at
different scan rates, we show the log($i$) versus log($\upsilon$) plot in Fig.
5(d) and performed the linear fit using the power law [47]:

$i=a\upsilon^{b}$ (5)

here $i$ is the current, $\upsilon$ is the scan rate, and $a$ and $b$ are the
fitting parameters. The value of $b$ determines the nature of the sodium ion
charge/discharge reaction mechanism. If the $b=$ 0.5, the storage process is
diffusion controlled, and if the $b=$ 1, the process is capacitive, and the
intermediate value of $b$ marks the contribution from both. The obtained
values of $b$ are 0.75 and 0.65 for anodic and cathodic peaks, respectively.
These contributions are shown at various scan rates in Figs. 5(e, f) for the
anodic and cathodic peaks, respectively. These values confirm the dominating
diffusive controlled process at low scan rates and the capacitive contribution
found to increase at higher scan rates. For example, the diffusive
contribution in the total current at 0.05 mV/s is around 68 and 85% for anodic
and cathodic peaks, respectively, which drops to 32 and 56% at a high scan
rate of 1 mV/s. For both the anodic and cathodic reactions, the general trend
follows the lowering of diffusion-controlled contribution with increasing the
scan rates.

##  IV  Conclusions

In summary, we synthesized NVP/C cathode material and investigated its
electrochemical performance in a full cell configuration with HC as an anode.
The phase of the material was confirmed using Rietveld refinement of the XRD
pattern and Raman spectroscopy measurement. In the half-cell configuration,
both the NVP/C and HC demonstrated good electrochemical performance against
the sodium metal anode, yielding a reversible discharge capacity of 85 mAh/g
at 0.1 C and a charge capacity of 252 mAh/g at 50 mA/g, respectively. The
HC$||$NVP/C full cell demonstrated a decent specific capacity of approximately
70 mAh/g at 0.1 C and exhibited a high-rate capability, with a capacity of
around 35 mAh/g at 5 C. The diffusion coefficient for the HC$||$NVP/C full
cell is found to be 5–8$\times$10-11 cm2/s for both the anodic and the
cathodic peaks using thorough cyclic voltammetry (CV) analysis. The
collaborative effect of HC and NVP/C enabled a high operational voltage of
approximately 3.3 V, with a minimal polarization of only 0.05 V. This
investigation into the HC$||$NVP/C full cell contributes valuable insights
into the development of cost-effective high energy density sodium-ion battery
system for energy storage applications, potentially advancing the
affordability of these devices for the common public.

##  V  Acknowledgments

Madhav thanks CSIR-HRDG for the fellowship support and Central Research
Facility (CRF) of IIT Delhi for the FE-SEM measurements. We acknowledge
Department of Science and Technology (DST), Govt. of India for financially
supporting the research facilities for sodium-ion battery project through
“DST-IIT Delhi Energy Storage Platform on Batteries” (project no.
DST/TMD/MECSP/2K17/07), and from SERB-DST through a core research grant (file
no.: CRG/2020/003436). We thank IIT Delhi for providing research facilities
for sample characterization (XRD and Raman at the physics department).

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# SPECT Angle Interpolation Based on Deep Learning Methodologies

Charalambos Chrysostomou, Loizos Koutsantonis, Christos Lemesios and Costas N.
Papanicolas, C. Chrysostomou, L. Koutsantonis, C. Lemesios and C.N.
Papanicolas are with the computation-based Science and Technology Research
Center, The Cyprus Institute, 20 Konstantinou Kavafi Street, 2121, Aglantzia,
Nicosia, Cyprus

###### Abstract

A novel method for SPECT angle interpolation based on deep learning
methodologies is presented. Projection data from software phantoms were used
to train the proposed model. For evaluation of the efficacy of the method,
phantoms based on Shepp Logan, with various noise levels added were used, and
the resulting interpolated sinograms are reconstructed using Ordered Subset
Expectation Maximization (OSEM) and compared to the reconstructions of the
original sinograms. The proposed method can quadruple the projections, and
denoise the original sinogram, in the same process. As the results show, the
proposed model significantly improves the reconstruction accuracy. Finally, to
demonstrate the efficacy and capability of the proposed method results from
real-world DAT-SPECT sinograms are presented.

###### Index Terms:

Convolutional Neural Networks (CNN), Ordered Subset Expectation Maximization
(OSEM), Single Photon Emission Computerized Tomography (SPECT), SPECT angle
interpolation

††Manuscript received December 20, 2020

##  I Introduction

Emission tomography is leading the medical imaging field, with Positron
Emission Tomography (PET) [1, 2] and Single Photon Emission Computerized
Tomography (SPECT) [3, 4, 5] as the primary techniques by detecting the trace
concentrations of radioactivity. In all emission tomography methods, and more
particularly in SPECT, the quality of the reconstruction is restricted by the
background radiation, rate of absorption and re-scattering effects [6, 7, 8].
One essential solution is to increase the radiopharmaceuticals doses given to
the patients. However, the use of these radiopharmaceuticals can have
unfavourable consequences on the health of the patients, and it is sought to
conserve the doses to a minimum and keep the quality of the image
reconstruction. Unfortunately, by reducing the radiopharmaceuticals dose on
the patients, the quality of the image reconstructions is limited. In order to
address the constraints as mentioned earlier, a novel method has been proposed
and developed to perform SPECT angle interpolation based on deep learning
methodologies.

The paper is organised as follows: Section II-A presents the data generated
and used for training the proposed model, Section II-B, presents the proposed
model. Section III, presents the results and discussions and finally Section
IV is conclusions.

##  II Methods and Materials

###  II-A Training Data

In order to adequately train the proposed model, 200,000 software phantoms
were randomly generated of $128\times 128$ pixels. Moreover, sets of
vectorised projections (sinograms) were generated from the software phantoms
by simulating 32 and 128 projections, evenly spaced in 360 degrees. The
generated 32 projections obtained from the phantom images were further
randomised with a Poisson probability distribution to provide the noisy sets
of projections. Samples of the randomly generated phantoms are shown in figure
1.

Figure 1: Sample software phantoms randomly generated in order to train the
proposed model

###  II-B Proposed Model

Deep learning methodologies have been successfully applied in image analysis,
and classification [9, 10]. The proposed novel methodology is based on
convolutional neural network architecture and more specifically, a U-Net
architucture[11]. Initially, the U-Net architecture was developed for
biomedical image segmentation and be very powerful for tasks like creating
segmentation masks and for image processing and generation such as super-
resolution or colourisation. We propose a method that can quadruple the
projections and denoise the original sinogram in the same process.

Figure 2: Proposed model based on U-Net architecture for SPECT Angle
interpolation.

The proposed method is based on standard U-Net architecture. U-Net
architecture is based on fully convolutional neural networks (FCN) [12] and
was initially developed for biomedical image segmentation [11]. The advantage
of using a U-Net architecture is that it can be successfully trained and
operate with less training samples and generate more precise results.

In a U-Net, architectures based on FCN, two key features are observed: (1)
U-net architectures are symmetric, and (2) U-Net architectures have skip
connections between the down-sampling and up-sampling layers by applying a
concatenation operator. The purpose of the skip connections is to carry local
information from the down-sampling stage of the model and combine it with the
global information in the up-sampling stage, in order to improve the
reconstruction accuracy. Figure 2 shows the proposed model. The proposed
architecture is separated into three parts, the contracting or down-sampling
path, the bottleneck and the expanding or up-sampling path.

The contracting or down-sampling path consists of four blocks, of two
convolutional layers [13] each, with 3x3 kernel and rectified linear
activation function (ReLU) [14], followed by an 2x2 average pooling layer
[13]. For each block, the number of kernels increases, beginning with 32
kernels for the first block, 64, 128 and 256 for the second, third and fourth
blocks respectively. The main objective of the contracting path is to obtain
the core context of the input sinogram in order to perform the denoising. The
common contextual information of the sinograms will be carried to the
upsampling path using skip connections. The bottleneck part of the proposed
architecture consists of two convolutional layers, with 3x3 kernel and ReLU as
the activation function.

The expanding or up-sampling path consists of seven blocks. The first six
blocks consist of a transposed convolutional layer and two convolutional
layers with 3x3 kernels and ReLU as the activation function. The last block is
a single convolutional layer of 1x1 kernel. The first four blocks of the
expanding path are to allow accurate translation of the core context of the
sinograms and to connect the contextual information carried using skip
connections from the contracting path. The last three blocks are used to
combine the core context and contextual information accurately into a denoised
sinogram with quadruple the projections.

The advantage of using the proposed model is the ability to connect contextual
information from the up-sampling path with the location information from the
down-sampling path, which is essential to perform SPECT angle interpolation.
An additional benefit of the proposed model is that no dense layers are used,
which allows variable sizes of sinograms. The only parameters that need to be
optimised are the kernel in the convolutional layers which are independent of
input size.

##  III Results and Discussions

The efficiency of the proposed method and comparison with existing methods was
performed using, the Mean Absolute Percentage Error (MAPE), Mean Square Error
(MSE), Structural Similarity (SSIM) Index [15], and the Peak signal-to-noise
ratio (PSNR) were used. The Shepp Logan phantom [16], as shown in Fig. 3 was
used to demonstrate and evaluate the capabilities of the proposed method.
Table I shows the results of the denoised method for the three different
levels of noise compared to the original, noise-free 128 projections sinogram.
Table II and Figure 4 shows the reconstruction results based on the standard
OSEM method versus the proposed method. As the results show, the proposed
method significantly denoises the sinograms and improves the reconstructions,
in comparison with the original 32 projections sinogram. Finally, in order to
test the efficacy of the proposed model, we used real-world DAT-SPECT
sinograms and produced reconstructions based on the standard OSEM and proposed
method. The results are presented in Fig. 5. As the results show, the proposed
model is capable of been applied and improve real-world problems, where the
collection of training data is limited or challenging, by using
computationally inexpensive software phantoms.

TABLE I: Sinogram DeNoising Results Noise | MAPE | MSE | SSIM | PSNR
---|---|---|---|---
Low | 2.64% | 0.0006 | 0.977 | 32.51
Medium | 4.13% | 0.0009 | 0.972 | 30.64
High | 6.34% | 0.0018 | 0.948 | 27.51
TABLE II: Reconstruction Results based on OSEM | Standard Method | Proposed Method
---|---|---
Noise Level | MSE | SSIM | PSNR | MSE | SSIM | PSNR
Low | 0.0071 | 0.84 | 21.49 | 0.0039 | 0.90 | 24.07
Medium | 0.0076 | 0.82 | 21.20 | 0.0036 | 0.90 | 24.46
High | 0.0093 | 0.76 | 20.33 | 0.0051 | 0.86 | 22.94
Figure 3: Shepp Logan Phantom used to evaluate and demonstrate the
capabilities of the proposed method Figure 4: Results of the sinogram
reconstructions by using OSEM for the proposed method versus the standard
method Figure 5: Results of the DAT-SPECT sinogram reconstructions by using
OSEM for the proposed method versus the standard method

##  IV Conclusions

In this study, we demonstrate the capability of the convolutional neural
networks to perform SPECT angle interpolation in SPECT imaging. The proposed
method was trained using software phantoms, and the capabilities of the system
were tested using software-generated sinograms as well as real-world DAT-SPECT
sinograms. As the results show, the proposed method significantly improves the
results and outperforms the standard OSEM method. Although the phantom and the
results presented in this paper is appropriate for illustrating the
capabilities of the proposed method, further experimentation is needed for
evaluating the potential application of the method in clinical studies.

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