Department of Mathematics, School of Science, Computing and Engineering
Technologies, Swinburne University of Technology, Melbourne, Victoria 3122,
Australia Aus]Department of Mathematics, School of Science, Computing and
Engineering Technologies, Swinburne University of Technology, Melbourne,
Victoria 3122, Australia India]Department of Applied Mechanics, Indian
Institute of Technology Madras, Chennai 600036, India Aus]Department of
Mathematics, School of Science, Computing and Engineering Technologies,
Swinburne University of Technology, Melbourne, Victoria 3122, Australia
IR,NMR,UV

# Effects of electrostatic interactions on Kapitza resistance in hexagonal
boron nitride-water interfaces

Sobin Alosious  Department of Applied Mechanics, Indian Institute of
Technology Madras, Chennai 600036, India  Sridhar Kumar Kannam  [  Sarith P.
Sathian  [  B.D. Todd  [<EMAIL_ADDRESS>
###### Abstract

Electrostatic interactions in nanoscale systems can influence the heat
transfer mechanism and interfacial properties. This study uses molecular
dynamics simulations to investigate the impact of various electrostatic
interactions on the Kapitza resistance ($R_{k}$) on a hexagonal boron nitride-
water system. The Kapitza resistance at hexagonal boron nitride nanotube
(hBNNT)-water interface reduces with an increase in diameter of the nanotube
due to more aggregation of water molecules per unit surface area. An increase
in the partial charges on boron and nitride caused the reduction in $R_{k}$.
With the increase in partial charge, a better hydrogen bonding between hBNNT
and water was observed, whereas the structure and order of the water molecules
remain the same. Nevertheless, the addition of NaCl salt into water does not
have any influence on interfacial thermal transport. $R_{k}$ remains unchanged
with electrolyte concentration since the cumulative Coulombic interaction
between the ions, and the hBNNT is significantly less when compared with water
molecules. Furthermore, the effect of electric field strength on interfacial
heat transfer is also investigated by providing uniform positive and negative
surface charges on the outermost hBN layers. $R_{k}$ is nearly independent of
the practical range of applied electric fields and decreases with an
increasing electric field for extreme field strengths until the electro-
freezing phenomenon occurs. The ordering of water molecules towards the
charged surface leads to an increase in the layering effect, causing the
reduction in $R_{k}$ in the presence of an electric field.

###### keywords:

American Chemical Society, LaTeX

##  1 Introduction

A profound understanding of the fundamental mechanisms underlying nanoscale
interfacial thermal transport is essential to improve the performance of
various nanoscale devices across a wide range of technologies. 1, 2, 3.
Thermal transport and resistance at interfaces are much more significant in
nanoscale systems than bulk systems since the interface can dramatically
affect the observed properties of various nanoscale materials 4. Carbon
nanotubes (CNTs) and boron nitride nanotubes (BNNTs) have received much
interest as novel two-dimensional materials in the last decade because their
fascinating physical properties make them promising candidates for various
nanofluidic applications. 5, 6, 7, 8, 9 Furthermore, boron nitride materials
have outperformed graphene and carbon nanotubes in some features such as
electrical insulation, chemical inertness, and biological compatibility. 10,
11, 12 Molecular dynamics (MD) simulation techniques are widely used to
compute various properties of boron nitride systems that require a well-
defined force field and water-BN interaction parameters. Multiple studies have
been conducted to develop the BN force field parameters with the help of
different techniques such as diffusion Monte Carlo (DMC), random phase
approximation (RPA) and density functional theory (DFT) 13, 14. With the help
of these force field parameters, researchers have extensively studied
different mechanisms in hexagonal boron nitride systems such as ion
selectivity, wetting behaviour and water transport 15, 16, 17. Nevertheless,
compared to graphene and carbon nanotube (CNT) systems, interfacial heat
transfer and thermal resistance in hexagonal boron nitride-water systems have
received little attention. Uhlig et al. 18 investigated the interfacial water
structure of hydrophobic and hydrophilic surfaces using 3D-AFM imaging with
atomic-scale spatial precision. They showed that the interfacial water
structure on crystalline surfaces has two contrasting configurations at the
atomic scale. Their findings demonstrate the complexities of real water
interfaces, where a small quantity of hydrocarbons may dominate the
interface’s fundamental characteristics.

The properties of confined water significantly change in the presence of
charged surfaces or ionic substrates due to the electrostatic interactions.
The effect of surface charge decoration on the interfacial heat transport in a
graphene water system was studied by Ma et al. 19 with the help of molecular
dynamics simulations. The enhanced Coloumbic force of attraction between water
and diagonally charged graphene layers causes an increase in interfacial
interaction strength, leading to meagre Kapitza resistance. Furthermore, the
diagonal charge decoration caused the formation of ordered water molecules
close to the interface. The spectral analysis reveals that the diagonal charge
decoration enhances spectral thermal conductance for all frequencies. The
diagonal charge decorating on the interfacial graphene sheet improves spectral
thermal conductivity at all frequencies compared to the system without charge
decoration. Furthermore, the characteristic peak in the spectral thermal
conductance shifts to a higher frequency. These results are consistent with
prior research on the solid-liquid interface based on the LJ model. 20. Wang
et al. 21 introduced a promising technique to reduce the interfacial thermal
resistance between copper (Cu) and water by providing surface charges to Cu.
The surfaces charges induced an orientation alignment of water molecules and
reduced the distance between the surface and the water nanolayer. The enhanced
heat transfer is also due to the improved coupling of vibrational density of
states between water and Cu when surface charges are employed. Wei and Luo 22
studied the transport of water and the friction coefficient in carbon
nanotubes and hexagonal boron nitride nanotubes using molecular dynamics
simulations. The friction coefficient in zigzag hBNNT is significantly higher
than zigzag CNT due to the partial charges, which leads to local potential
energy traps and additional electrostatic interaction between hBNNT and water.
Moreover, for an armchair hBNNT, its atomic arrangement does not create local
potential energy traps; thus, the friction coefficient is less than that for a
zigzag hBNNT. Qian et al. 23 reported that the interfacial thermal transport
in ionic liquid (IL)-solid interfaces is dominated by the atomic structure of
EDL rather than the strong solid-liquid interaction using molecular dynamics
simulations. The decrease in ITR with an increase in the surface charge occurs
sharply in the initial stage and slowly in the later stage. They have
calculated the structure factor and geometry state for the quantitative
analysis of the EDL structure near the interface. A very low ITR can be
achieved by the highly ordered interfacial structure of IL, which is justified
by the vibrational spectrum and frequency-dependent heat flow.

Various electrolyte solutions under nano-confinement have been widely studied
due to their significance in biological and nanofluidic applications. Dewan et
al. 24 characterized the fundamental properties of interfacial water by
conducting molecular dynamics simulations of alkali chloride solutions with
two kinds of idealized charged surfaces. They observed a compact layer of
solvents adjacent to the surface beside the diffuse region where water
orientation exhibits no layering. The type of the charge distribution is the
primary deciding factor for the diffuse solvent layer depth. For a realistic
model of negatively charged amorphous silica, the distribution of ions and
water orientation depends on the type of cations and are not well described by
a simple uniform charge distribution model. Hilder et al. 25 showed that a
silicon nitride membrane embedded with a hexagonal boron nitride nanotube
could obtain a 100% (in principle) salt rejection at higher concentrations. A
water flow rate of 10.7 water molecules per nanosecond is possible even for a
concentration as high as 1 M, which has a high energy barrier. Moreover, the
nanotube radius determines whether the nanotube is anion-selective or cation-
selective since the water structure forms a single-file chain for smaller
radius nanotubes. Prakash et al. 26 studied the flow of a symmetric
electrolyte via a charged nanochannel that was exposed to an axial temperature
gradient. They investigated the relative contributions of the Soret effect,
thermoelectric effect, and double-layer potential in the electrical double
layer for different surface charges and temperature gradients. Their findings
show that temperature variations may be used to produce streaming current in a
charged graphene-based nanochannel, depending on the relative impact of the
Soret effect and the double layer potential.

Any structural change of water is sensitive to the electric field due to the
dipole moments in water molecules. Khusnutdinoff and Mokshin 27 studied the
effect of electric field-induced crystalization of supercooled water confined
in a graphene nanochannel. By applying an electric field of 0.5 V/Å to
confined water at a temperature of 268 K and density 0.94 g/cm3, they observed
that the water crystallizes into cubic ice with some defects. When the
electric field is applied, the water dipoles become aligned, and the ordinary
relaxation of the metastable state transforms to an ordered phase leading to
the crystalization of water. Su and Guo 28 investigated the impact of the flow
of single-file water molecules through CNT in the presence of an external
electric field. The applied electric field changes the water molecules’
orientation, disturbs the wavelike density profile, and strongly influences
the water flux. They observed a critical electric field $E_{c}$ beyond which
the electric field strength does not affect the water flux since the frequency
of water dipole flipping vanishes for higher electric field strength. The
electric field-induced water flow can be helpful in various biological
applications and the efficient design of nano pumps. Yenigun and Barisik 29
studied the effect of varying electric fields on interfacial thermal transport
in water confined between two parallel silicon slabs. They observed a fivefold
reduction in the Kapitza length by applying an electric field due to the
change in silicon and water interface energy. Moreover, the thermal
conductivity of the water was slightly reduced due to the alignment of dipoles
in the direction of the electric field. Further increase in the electric field
about 0.53 V/nm leads to electro-freezing since the aligned dipole developed a
crystalline structure. Kapitza length and thermal conductivity remain
unchanged with a further increase in the electric field once the electro-
freezing is formed.

In this paper, we investigate the influence of electrostatic interactions on
Kapitza resistance in a hexagonal boron nitride-water system using equilibrium
molecular dynamics (EMD) simulations. The effect of hBNNT partial charge,
electrolyte concentration and electric field were analyzed in cylindrical and
planar hBN-water systems. The Kapitza resistance/length calculations were
carried out using our previously developed EMD method 30 since the NEMD method
is unsuitable for the cylindrical system used in this study. The Kapitza
resistance is found to be decreasing with an increase in partial charge of
hBNNT and applied electric field, whereas the increase in molarity of the NaCl
electrolyte does not make any impact. A summary of the EMD method,
methodology, detailed analysis of the obtained results, and concluding remarks
are provided in the following sections of the paper.

##  2 Theory

We have recently developed an equilibrium molecular dynamics (EMD) method to
compute the Kapitza resistance in a cylindrical nanoconfinement system (CNT-
water interface). Here we briefly summarise the concept and the final
expressions of the EMD method, which applies to planar and cylindrical
systems. The detailed derivation of the theory is available in the original
paper 30.

The Kapitza resistance or the interfacial thermal resistance, $R_{k}$ is
defined as

$\displaystyle R_{k}=\frac{\Delta T}{J_{q}}.$ (1)

Also, the interfacial thermal conductance, $G_{k}$ can be defined as the
inverse of $R_{k}$,

$\displaystyle G_{k}=\frac{1}{R_{k}}=\frac{J_{q}}{\Delta T}$ (2)

where $J_{q}$ is the heat flux at the fluid-solid interface and $\Delta
T=T_{f}-T_{w}$ is the difference between the temperature of the solid wall,
$T_{w}$ and a fluid slab of some small thickness immediately adjacent to the
solid wall, $T_{f}$. For the graphene-water interface, the average thickness
of the water is found to be about one molecular diameter 31. Here we are
assuming that the time-dependent Kapitza kernel can be expressed as an n-term
Maxwellian memory function and is given by 32,

$\displaystyle
G_{k}\left(t\right)=\sum\limits_{i=1}^{n}{{k_{i}}}{e^{-{\mu_{i}}t}}.$ (3)

Here, $k_{i}$ and $\mu_{i}$ are coefficients in the Maxwellian memory function
and are related by,

$\displaystyle{{\tilde{C}}_{T{J_{q}}}}\left(s\right)=\sum\limits_{i=1}^{n}{\frac{{{k_{i}}}}{{s+{\mu_{i}}}}}{{\tilde{C}}_{{T{T}}}}\left(s\right).$
(4)

${{\tilde{C}}_{T{J_{q}}}}$ and ${{\tilde{C}}_{TT}}$ are the Laplace transforms
of the heat flux-temperature difference cross-correlation function and
temperature difference autocorrelation function, respectively. Thus, for
steady-state conditions ($s=0$) we have,

$\displaystyle
G_{k}\equiv\tilde{G}_{k}\left(0\right)=\sum\limits_{i=1}^{n}{\frac{{{k_{i}}}}{{{\mu_{i}}}}}.$
(5)

Finally, the Kapitza resistance can be computed as,

$\displaystyle R_{k}=\frac{1}{G_{k}}.$ (6)

The Kapitza length can be defined as:

$L_{k}=R_{k}\lambda$ (7)

where $\lambda$ is the thermal conductivity of either the solid or fluid
phase. $L_{k}$ is similar to the slip length in fluid flow and can be defined
as the additional thickness of material required to achieve the same heat
transfer in place of the thermal resistance at the interface. This additional
thickness can be measured either in the direction of solid or fluid from the
interface as per convenience. The thermal conductivity of the solid phase is
used in Eq.(7) to determine the Kapitza length towards the fluid side and vice
versa.

The instantaneous heat flux, $J_{q}(t)$ and the temperature difference $\Delta
T(t)$ can be extracted from the MD simulations with the help of the following
equation 33, 34

$J_{q}(t)=J_{q}^{K}(t)+J_{q}^{\phi}(t),$ (8)

where $J_{q}^{K}(t)$ is the kinetic term and $J_{q}^{\phi}(t)$ is the
potential term of the heat flux. The reader is referred to the original paper
for further details 33, 34.

##  3 Methodology

Figure 1: Schematic depiction of the simulation systems. (a) Water confined in
a hexagonal boron nitride nanotube. (b) Water confined in a planar hexagonal
boron nitride nanochannel.

The simulation models for the planar and cylindrical hexagonal boron nitride
(hBN)-water systems are shown in Figure 1. The planar system consists of a
water block confined between two layers of parallel hBN sheets in which the
outer layers were kept rigid to maintain a constant volume. The dimensions of
the hBN sheets are $L_{x}$=$L_{y}$=4 nm, and the thickness of the water block,
$W$=4 nm. Periodic boundary conditions were given in $x$ and $y$ directions,
and confinement was in the $z$-direction. The hBN sheets were tethered to
their initial position by fixing the centre of mass of the layers to their
initial positions for ensuring a constant channel volume 31. The cylindrical
system consists of water confined inside a hBN nanotube (hBNNT) of diameter,
$D$ and length $L$=5 nm. Periodic boundary conditions were provided in all
three directions. Different hBNNT diameters of 1.38 nm, 2.76 nm, 4.14 nm, 5.52
nm and 6.90 nm were chosen, which correspond to the chiralities of (10-10),
(20-20), (30-30), (40-40) and (50-50) respectively. The diameter of an (m,n)
hBNNT was calculated using the equation $d=\sqrt{3}l/\pi\sqrt{n^{2}+nm+m^{2}}$
where $l$=0.1446 nm is the bond length of boron-nitride. Different parameters
such as partial charge of hBN, salt concentration and external electric field
were studied to understand the effect of electrostatic interactions on the
Kapitza resistance. Boron nitrides (BN) are compounds having covalent–ionic
bonds; hence binding polarity is a crucial factor influencing their physical
characteristics. The dependence of observable parameters on effective static
charges of component atoms is so intricate that it is nearly impossible to
determine experimentally. The theoretically calculated atomic charges in boron
nitrides are characterized by a large dispersion, leaving them practically
unreliable. The absence of a clear distinction of the electron density between
atoms of elements is the general explanation 35. In molecular mechanics force
fields, partial atomic charges are utilized to calculate the electrostatic
interaction energy. They are often utilized to get a qualitative understanding
of a molecule’s structure and reactivity. For BN, different partial charge
values ranging from 0 to 1.05e were reported in various literatures 14, 35,
15, 36 The partial charge of the hBN was varied from 0.0 to $\pm$ 1.50$e$ to
study the effect of partial charge. The effect of the electrolyte
concentration was studied by varying the salt (NaCl) concentration from 0 to
1.0 mol/L. In addition to that, the impact of the external electric field was
studied by varying the electric field strength from 0 to 0.1 V/Å, applied in
the direction perpendicular to the walls. The outer hBN layers on the left and
right sides were provided with equally distributed positive and negative
charges of the same magnitude, respectively. The outer hBN layers act as
electrodes and develop an electric field in the $z$-direction (normal to the
surface). The surface charge density was varied from 0 to 0.1375 $\mu$C/cm2 to
obtain an electric field in the range of 0 to 0.1 V/Å, similar to previous MD
simulations 29, 37, 38. The range of electric fields selected here is higher
than those typically used in experimental studies. However, transient pulse
fields can be used as an alternative to achieve this range of electric field
strengths experimentally. The electro-freezing of water was experimentally
investigated by different researchers using a pulse voltage applied through
two electrodes immersed in water to create large electric fields 39, 40. The
electric field between the two electrodes was reached in the range of 0.1 V/Å,
similar to the field strength used in this study. In addition to that, extreme
field strengths up to 5V/Å were also applied to study the structural changes
of water due to electric fields by assuming that the dielectric breakdown of
water will not occur.

The water molecules were modelled using the simple point charge (SPC/E) 41, 42
water model due to its reliability, precision, and comparatively low
computational cost. The long-range electrostatic forces were calculated using
the particle-particle-particle-mesh (PPPM) 43 solver with an accuracy of
1$\times$10-5 and the water molecules were kept rigid using the SHAKE 44
algorithm. However, to use the PPPM method, periodic boundary conditions in
all directions are required. For the planar system, an Ewald summation
technique with an extended volume ratio of 3.0 was used due to the confinement
in the $z$-direction 45. For the cylindrical system, void spacing of 5 nm in
the $x$ and $y$ directions was provided to eliminate the errors due to
electrostatic interaction between periodic images while implementing the PPPM
solver 46. An optimized Tersoff 47 potential was used to model the hBN
nanosheet/tube interactions. The pairwise interactions between all the atoms
and ions were defined by adding the Lennard-Jones (L-J) and Coloumbic
potentials and which is given by,

$U_{ij}=\sum_{i=1}^{N}\sum_{j=i+1}^{N}\Big{(}4\varepsilon_{ij}\Big{[}\Big{(}\frac{\sigma_{ij}}{r_{ij}}\Big{)}^{12}-\Big{(}\frac{\sigma_{ij}}{r_{ij}}\Big{)}^{6}\Big{]}+\frac{q_{i}q_{j}}{4\pi\varepsilon_{0}r_{ij}}\Big{)}.$
(9)

The values of the L-J interaction parameters are provided in Table 1. The L-J
parameters for the cross-terms were calculated using the Lorentz-Berthelot
mixing rules.

Pair |  $\sigma$ (Å) |  $\epsilon$ (kcal/mol) | $q(e)$ | Reference
---|---|---|---|---
H-H | 0.0 | 0.0 |  $+$0.4238 | Wu et al. 42
O-O | 3.166 | 0.1554 |  $-$0.8476 | Wu et al. 42
B | 3.453 | 0.0949 |  $+$1.05 | Won et al. 15, 14
N | 3.365 | 0.145 |  $-$1.05 | Won et al. 15, 14
Na+ | 2.160 | 0.3526 |  $+$1.0 | Joung et al. 48
Cl- | 4.831 | 0.0128 |  $-$1.0 | Joung et al. 48
Table 1: The L-J parameters used for modeling the hBN-water system.

The cut-off distance values for L-J potential and short-range Coulombic
interactions are set as 1 nm. All the Molecular Dynamics (MD) simulations were
performed by using the Large-scale Atomic/Molecular Massively Parallel
Simulator (LAMMPS) 49 package with a time step of 1.0 fs. Visual Molecular
Dynamics (VMD) 50 was used to visualize the models.

Equilibrium molecular dynamics (EMD) simulations were performed to compute the
Kapitza resistance at the hBN-water interface in planar and cylindrical
geometry. Initially, the system was equilibrated under a canonical (NVT)
ensemble at a reference temperature of 300 K for a time period of 2.0 ns.
Further, the system stability was checked by simulating under a microcanonical
(NVE) ensemble for another 2.0 ns. Finally, the instantaneous heat flux,
$J_{q}$(t) and temperature difference, $\Delta T(t)$, were extracted from
another 5.0 ns production simulation by thermostatting the hBNNT only. The
Kapitza resistance was calculated using the Equations 4-6 with a Maxwellian
one term (n = 1) memory function 51. The fluid slab thickness, $\Delta$ was
taken as 3.165 Å, which is the distance measured from the wall to the first
density peak of water 31, 30. All the calculated parameters were averaged over
five independent simulations with distinct initial configurations.

##  4 Results and discussions

Figure 2: Kapitza resistance/length as a function of the diameter of hBNNT.
The horizontal dashed lines represents the Kapitza resistance/length in a
planar hBN-water system, and the inset shows the change in $F_{NA}$, with the
diameter of the hBNNT. (b) VDOS (arbitrary units) and the overlap, S, of the
hBNNT and the water slab for various hBNNT diameters.

The computation of Kapitza resistance/length in a cylindrical nanoconfinement
system using the NEMD simulation technique is challenging and complex and
limited to a range of diameters. In our previous work, we have developed an
EMD method to compute the Kapitza resistance in a CNT-water interface, and the
results were validated with the NEMD method 30. Results of both the EMD method
and NEMD method are in excellent agreement. In this work, we also use the same
EMD method to compute the Kapitza resistance in the hBN-water interface. Since
the present study also uses the same geometry, this paper does not provide
additional validation of the EMD method for the hBNNT-water interface. Figure
2a shows the effect of diameter of hBNNT on the Kapitza resistance/length. The
dashed lines depict the Kapitza resistance/length of the planar hBN-water
system (water confined in a hBN nanochannel). It is found that the Kapitza
resistance reduces monotonically with an increase in the diameter of hBNNT and
converges to a constant value similar to the CNT-water system. There is a
37.3% reduction in the Kapitza resistance observed when the diameter varied
from 1.38 nm to 6.9 nm, corresponding to the chiralities of (10,10) to
(50,50), respectively. The inset shows the variation of area density factor,
$F_{NA}$ as a function of diameter. $F_{NA}$ is defined as the ratio of the
number of water molecules adjacent to the interface to the surface area of the
hBNNT 30. A slab of water adjacent to the wall is chosen for counting the
number of molecules adjacent to the interface. The water slab thickness is
taken as the distance from the wall to the first density peak of water since
the first hydration shell determines the major part of the interfacial heat
transfer. It is found that $F_{NA}$ increases with an increase in diameter and
reaches a constant value for higher diameters. A higher value of $F_{NA}$
indicates that more water molecules will be present near the interface per
unit surface area leading to a better heat transfer between hBNNT and water,
thereby reducing the Kapitza resistance. This can be replaced by the number of
molecules or the height of the first density peak for planar systems since the
surface area remains the same. Even though the interactions between hBNNT and
water remains the same, for smaller diameters, the slab volume to surface area
is less (curvature effect), leading to a higher Kapitza resistance and this
curvature effect vanishes after a particular diameter. Thus the increase in
area density factor with an increase in diameter is the primary reason for
reducing the Kapitza resistance. Furthermore, we have evaluated the mismatch
of vibrational density of states (VDOS) between hBNNT and water to understand
the interfacial heat transfer. The VDOS can be calculated using the Fourier
transform of the velocity autocorrelation function given by, 52, 53

$P(f)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{C\left(t\right)}e^{ift}dt,$
(10)

where $P(f)$ is the VDOS, $f$ is the phonon frequency, $C(t)$ is the velocity
autocorrelation function given by $C(t)=\langle v(t)v(0)\rangle/\langle
v(0)v(0)\rangle$ and $v(t)$ is the velocity of atom at time $t$.

Also, the VDOS overlap $S$, (generally considered in terms of an arbitrary
unit) of the water and the hBNNT was calculated using the equation 54,

$\displaystyle
S=\frac{\int_{0}^{\infty}P(f)_{hBNNT}P(f)_{water}\,df}{\int_{0}^{\infty}P(f)_{hBNNT}\,df.\int_{0}^{\infty}P(f)_{water}\,df}$
(11)

The energy carriers in water are molecules, and the phonon dynamics in bulk
liquids are not well defined. However, under extreme confinements, surface
charges, electric fields etc., a high ordering of water (solid-like structure)
near the interface is possible. Therefore, here the VDOS overlap is calculated
to show that it doesn’t influence the thermal transport in a solid-fluid
interface, unlike a solid-solid interface. Figure 2b shows the VDOS and its
overlap, $S$ for hBNNT and water. There is no noticeable change in $S$ with an
increase in the diameter, indicating the vibrational coupling between hBNNT
and water are unaltered with a change in diameter. Thus the reduction in the
Kapitza resistance with an increase in diameter can be attributed to the
increased area density factor, similar to the CNT-water system 30.

Figure 3: Kapitza resistance/length as a function of the partial charge of a
(40,40) hBNNT. The inset shows the effect of area density factor $F_{NA}$ and
VDOS overlap, $S$

We further examine how the partial charge of hBNNT can influence the
interfacial heat transfer in the present system. Figure 3 shows the effect of
the partial charge of a (40,40) hBNNT on the Kapitza resistance/length, and
the inset shows the corresponding values of the area density factor $F_{NA}$
and the VDOS overlap, $S$. The partial charge on the hBNNT was varied between
0 to $\pm$ 1.50$e$ to obtain a range of values higher and lower than the
actual value. It is found that the Kapitza resistance decreases with an
increase in partial charge as expected. The Kapitza resistance was reduced to
nearly 71.2% by increasing the partial charge from 0 to $\pm$ 1.50$e$. A
meagre value of Kapitza resistance for the hBNNT-water interface compared to
the CNT-water interface is primarily due to the presence of partial charges on
the hBNNT. To examine how the Kapitza resistance is affected by partial
charge, we have calculated the area density factor, $F_{NA}$, and the VDOS
overlap $S$ for different partial charges. From the inset of Figure 3, we can
see that the area density factor is almost independent of the partial charges.
Alexeev et al. 55 reported that the Kapitza resistance is primarily dependent
on the water density profile near the interface and is inversely proportional
to the height of the first density peak in a water-graphene system. However, a
constant value of the area density factor indicates that the density profile
near the interface is not changing with the addition of partial charge (since
the diameter of hBNNT is the same, $F_{NA}$ and density peaks are
proportional). Similar observations were previously reported in graphene and
hBN nanoconfinement systems such that the partial charge or surface charge
(zero net charge) doesn’t influence the density profile of water 19, 56, 14.
This indicates that the reduction in Kapitza with partial charges are not
related to the change in the structure of the water near the interface. In
addition to that, the VDOS overlap between hBNNT and water, $S$ is also nearly
independent of the partial charges indicating that the reduction in Kapitza
resistance does not originate from the change in vibrational coupling between
water and hBNNT.

Furthermore, hydrogen-bond dynamics were analyzed to understand the physical
mechanism behind the dependency of Kapitza resistance on partial charge. A
hydrogen bond (H-bond) is essentially an electrostatic force of attraction
between a hydrogen (H) atom, which is covalently bonded with a more
electronegative atom or group (donor D), and another electronegative atom
having a lone pair of electrons (acceptor A) 57. The hydrogen bonds in
molecular models are determined by a geometric criterion that includes the
distance between the hydrogen and acceptor atoms (H$\cdots$A) and the donor-
hydrogen-acceptor ($\chemfig[atomsep=2.2em]{D-H}\cdots$A) planar angle value.
In the hBN-water system, two types of H-bonds are possible; one is the bond
between a partially positive hydrogen atom in a water molecule and a partially
negative oxygen atom in a different water molecule
($\chemfig[atomsep=2.2em]{O-H}\cdots$O) and another is between a partially
positive hydrogen atom in a water molecule and partially negative nitride in
the hBN ($\chemfig[atomsep=2.2em]{O-H}\cdots$N). Here, two molecules were
considered H-bonded if (H$\cdots$A) < 3.5 Å and
($\chemfig[atomsep=2.2em]{D-H}\cdots$A) > 140∘ 58, 59. The H-bond lifetime,
$\tau$ can be used to analyze the dynamic behaviour of H-bonds using the time
autocorrelation function given by 60,

$C_{c}(t)=\left\langle\frac{\sum h_{ij}(t_{0})h_{ij}(t_{0}+t)}{\sum
h_{ij}(t_{0})^{2}}\right\rangle$ (12)

Where $h_{ij}$ is a binary measure of whether pair $ij$ forms a H-bond or not
such that, $h_{ij}$ = 1, or $h_{ij}$ = 0. Here we adopted the continuous
lifetime definition, in which once the H-bond is broken, it is always
considered to be broken even if the bond reforms afterwards. The H-bond
lifetime $\tau$ can be defined as 60,

$\tau=\int_{0}^{\infty}C_{c}(t).$ (13)

Figure 4: (a) Number of water-hBNNT hydrogen bonds per molecule and water-
hBNNT hydrogen bond life time as a function of partial charge. (b) Water-hBNNT
hydrogen-bond time autocorrelation function for different partial charges.

We have calculated the number of hydrogen bonds per molecule,
$\tilde{n}_{\mathrm{H-B}}$ and hydrogen bond lifetime, $\tau$ for water-water
($\chemfig[atomsep=2.2em]{O-H}\cdots$O) and water-hBN
($\chemfig[atomsep=2.2em]{O-H}\cdots$N) hydrogen bonds. The number of hydrogen
bonds per water molecule is around 3.18, and the H-bond lifetime is 0.30 ps
for the water-water hydrogen bonds. Both the values remain the same even after
increasing the partial charge of the hBNNT. This indicates that the water-
water hydrogen bond dynamics are not affected by the partial charge of the
hBNNT. In contrast, the H-bond between hBN and water is found to be strongly
dependent on the partial charge. Figure 4a shows the number of hydrogen bonds
per molecule and H-bond lifetime for water-hBN hydrogen bonds. The hydrogen
bond between water and hBN is only possible in a single layer of water near
the interface; therefore, the number of hydrogen bonds per molecule will be
meagre compared to the water-water H-bonds. It is found that the number of
hydrogen bonds per molecule increases with an increase in the partial charge
of the hBNNT. This indicates that the partial charges are favourable for
forming more H-bonds between water and the hBNNT, leading to a better heat
transfer at the interface. Figure 4b shows the H-bond time autocorrelation
function for different partial charges. The correlation function converges
quickly for lower partial charges, and as we increase the partial charge, the
convergence time increases, leading to a higher H-bond lifetime. Thus it is
evident that the H-bond lifetime increases with an increase in partial charges
leading to a stronger H-bond at the interface. From the H-bond analysis, we
can infer that the partial charge of hBNNT strongly influences the hydrogen
bonding dynamics at the interface leading to higher Coloumbic interaction
between water and hBNNT. Thus the reduction in the Kapitza resistance with an
increase in partial charge of the hBNNT can be attributed to the increased
hydrogen bonding between water and hBNNT.

The partial charge does not influence the structuring and ordering of water
near the interface, and the density of water is unaltered with the addition of
partial charge in boron nitride. Similar observations were previously reported
in graphene and BN nanoconfinement systems such that the partial charge or
surface charge (zero net charges) doesn’t influence the density profile of
water 19, 56, 14. However the density profile changes when a net charge
density is provided on the solid. This shows that different charge
arrangements lead to different mechanisms at the interface. Even though the
magnitude of the heat transfer rate across the interface would increase due to
the heat flux contribution from the electrostatic force, the hydrogen analysis
is carried out to understand the physical mechanism underlying interfacial
thermal transport.

Figure 5: Kapitza resistance/length as a function of NaCl concentration for
hBNNT with/without partial charge ($\delta=1.05e$).

Next, we examined whether the addition of salt ions can influence the
interfacial thermal transfer characteristics. Figure 5 shows the effect of the
NaCl concentration on the Kapitza resistance/length for a (40,40) hBNNT
with/without partial charges. The concentration of the NaCl electrolyte varied
from 0 to 1.0 mol/L. For the hBNNT without partial charges, we can see that
the Kapitza resistance is independent of the salt concentrations. Similarly,
for hBNNT with partial charges, the Kapitza resistance reduces due to the
partial charge but remains unchanged with increased concentration. This
indicates that the addition of NaCl does not influence interfacial thermal
transport in both systems. To further understand the underlying mechanism, we
examined the density profiles of the hBNNT-electrolyte system.

Figure 6: Density profile of water and ions for 1.0 mol/L NaCl concentration.
(a),(b) Normalized density profile of water and ions with/without partial
charge. (c),(d) Mass density profile of water and ions with/without partial
charge.

Figure 6 shows the density profiles for different conditions with 1.0 mol/L
NaCl concentration. The normalized density profile for water and ions for the
highest concentration without and with partial charges of the hBNNT is shown
in Figure 6a and 6b, respectively. Similarly, Figure 6c and 6d show the
absolute density profile for both systems. The Kapitza resistance is a strong
function of the height of the first density peak of water 55. We observed that
the height of the density peak is unaltered with the addition of NaCl even at
the highest concentration. From Figure 6a, we can see the normalized density
fluctuation of Na+ and Cl- ions along with the water molecules for the hBNNT
without partial charges. The structure of water governs the density profile of
the ions at the interface in such a way that the ions are concentrated in the
valleys of the water density profile near the interface and equally
distributed in the bulk portion. Since the Kapitza resistance is mostly
dependent on the first layer, the concentration of ions after the first
density peak do not contribute much to the interfacial thermal transport. Thus
the addition of NaCl even at higher concentrations does not influence the
Kapitza resistance. Furthermore, Figure 6c shows the absolute density profile
for the same system. We can see that, compared to the density of water, the
density of both Na+ and Cl- ions are very negligible. Thus the contribution
from NaCl for interfacial heat transfer will be meagre compared to the
contributions from water molecules. So combining both, we can conclude that
the concentration-independent Kapitza resistance is due to the relatively
minor contribution from NaCl ions than water molecules and the position of
ions far from the interfacial region. In addition to that, we have examined
the effect of salt concentration when the hBNNT is applied with partial
charges. When the partial charges are provided, the Coulombic interaction
between water and hBNNT will increase, thereby reducing the Kapitza
resistance. Even though the partial charge of hBNNT and the charge of ions
also enhance the Coulombic interaction, it is significantly less compared to
the water and not enough to make an additional influence on interfacial
thermal transport. Thus the Kapitza resistance is still found to be unaltered
with the addition of NaCl. Figure 6b and 6d show the normalized and absolute
density profile of the electrolyte. Even after applying the partial charges to
the hBNNT, the density profiles nearly match the hBNNT without partial
charges. Thus, adding NaCl to water alone does not affect the interfacial
thermal transport and Kapitza resistance in both systems (with/without partial
charge).

Figure 7: Kapitza resistance/length as a function of low electric field in a
planar hBN with/without partial charge ($\delta=1.05e$).

Applying an electric field to water can influence its properties due to the
polar nature of water. Here we examine the effect of electric field on
interfacial thermal transport in a planar hBN-water nanochannel. An electric
field perpendicular to the interface (along the $z$ direction) is applied to
the system by providing positive and negative surface charges on the outer
layers of hBN as shown in Figure 1. The positive uniform charge density on the
left side acts as an anode, and the negative uniform charge density with the
same magnitude acts as a cathode creating a uniform electric field from left
to right. The electric field varied from 0 to 0.1 V/Å by changing the surface
charge density from 0 to 0.1375 $\mu$C/cm2.

The polarization of water in response to the applied electric field, $E_{0}$,
in the $z$ direction creates an induced electric field given by, 61

$E_{p}(z)=\frac{\int_{-\infty}^{z}\rho_{q}(z^{\prime})dz^{\prime}}{\epsilon_{0}},$
(14)

where $\rho(z^{\prime})$ is the charge density distribution along the $z$
direction and $\epsilon_{0}$ is the dielectric permittivity of vacuum. The
spatial distribution of charge density $\rho(z)$ is defined as

$\rho(z)=\frac{1}{L_{x}L_{y}}\left\langle\sum_{i}^{N}q_{i}\delta(z-z_{i})\right\rangle,$
(15)

where $q_{i}$ is the charge of the $i$th atom. The net electric field is given
by,62

$E(z)=E_{0}+E_{p}(z).$ (16)

At the interface, the induced polarization field, $E_{p}$ is excessively
higher than the applied electric field $E_{0}$ 63. This causes the
overscreening of the applied field by the polarization field. For an applied
field strength of $E_{0}$=0.1 V/Å, the net electric field in the bulk portion
of the channel is obtained as $E$=0.0016 V/Å.

Figure 7 shows the effect of electric field on the Kapitza resistance/length
in the planar hBN-water system with and without partial charges. The Kapitza
resistance is found to be nearly independent of the electric field in the
range of 0 to 0.1 V/Å. Even though the applied fields are higher compared to
experimental values, it is not enough to make any structural changes in the
water. As stated earlier, the density profile and the height of the first peak
is the primary deciding factor controlling the interfacial heat transfer. Here
we found that the density profile coincides with each other for all the
applied external electric fields (not shown here). The electric field can
influence the orientation of water molecules 64, 63; however, it does not make
any impact on the interfacial heat transfer and the Kapitza resistance at this
range of field strengths.

Substantial electric field strength is required to induce any structural
changes in water. Numerous studies using MD simulations have been reported to
understand the structural behaviour of water under extreme magnitudes of
electric fields as high as 5V/Å65, 66, 67, 68, 69. Applying such high electric
fields in experimental setups are nearly impossible due to several constraints
such as difficulty in achieving higher fields and the dielectric breakdown of
water. It has been reported that the dielectric breakdown of water is in the
order of 10-3 V/Å and 10-2 V/Å for millimetre and micrometre scale volumes of
water, respectively 70, 71. This indicates that nanoconfinement of water could
enhance the dielectric strength of water further; however, it is yet to be
experimentally validated. Also, the MD simulations have not reported the
dielectric breakdown of water even in this range of applied electric fields.
However, we cannot completely ignore the possibility that the observations
obtained from MD simulations under these extreme electric field strengths are
simulation artefacts. Nevertheless, to study the structural changes of water
due to electric fields, we applied extensive electric field strengths up to 5
V/Å by assuming that the dielectric breakdown of water will not occur. When
the applied electric field exceeds the dielectric strength of water, the water
will suddenly become an electrical conductor, and electric current flows
through it. Predicting the thermal properties and interfacial thermal
transport during the dielectric breakdown is challenging and is not attempted
in this study.

Figure 8: Evolution of water density profile and snapshots of water molecules
for different electric fields.

Figure 8 shows the evolution of the water density profile and snapshots of
water molecules with increasing electric field strength. While increasing the
electric field, the water molecules start to form an ordered crystalline
structure is known as electro-freezing 72, 73, 74. The density profile shows
that the electro-freezing phenomenon occurs at about 4 V/Å applied field
strength. For zero and lower electric field strengths, we can see from the
snapshots that the water molecules are randomly oriented, and when the field
strength is increased, the ordering starts to occur. A highly ordered
crystalline structure of water molecules starts to be visible from a field
strength of 4 V/Å and above. A similar range of electric field strengths at
which electro-freezing occurs has also been reported in previous MD
simulations of confined water 29, 75. In contrast to that, Yenigun and Barisik
29 reported electro-freezing of confined water at a lower field strength of
0.05 V/Å. However, the electric field they have mentioned is the effective
field inside the water by considering the relative permittivity of water and
not the external applied electric field. From the surface charge density they
have used, it is evident that the effective external field is about 4 V/Å
which is nearly 80 times higher than the effective field (since the relative
permittivity of water, $\epsilon_{r}$ $\approx$ 80). The electro-freezing
phenomenon in subcooled water has been widely reported in experimental studies
76, 77, 78, 79. However, the room temperature electro-freezing has not been
experimentally detected due to different constraints in achieving extreme
electric fields. Experiments are yet to validate the high ordering and
crystallization of water molecules under intense electric fields at room
temperature observed in MD simulations. Therefore the following results and
analysis are strictly based on the assumptions that the dielectric breakdown
of water does not happen and that room temperature electro-freezing is
possible.

Figure 9 shows the effect of electric field on the Kapitza resistance/length
in the planar hBN-water system with partial charges, and the inset shows the
change in order parameter and reduced water density peak for different
electric field strengths. We found that the Kapitza resistance reduced by
about 53.5% when the electric field increased until 4 V/Å. With a further
increase in the field strength, the Kapitza resistance remains almost
constant. Thus with the increase in the electric field, the ordering of water
molecules commences, which reduces the Kapitza resistance, and once electro-
freezing occurs, the Kapitza resistance remains constant with further increase
in the electric field. To quantify the ordering of water molecules with the
electric field, we computed the order parameter 80, which is defined as,

$C=\frac{1}{2}\langle(3cos^{2}\theta-1)\rangle$ (17)

where C = 0 for random alignment, C = 1 for parallel alignment and C = -1/2
for perpendicular alignment of molecular dipoles. The angle $\theta$ is
defined by cos $\theta$ = $\hat{\textbf{{n}}}.\hat{\textbf{{E}}}$, where
$\hat{\textbf{{E}}}$ is the direction of applied field and
$\hat{\textbf{{n}}}$ is the direction of dipole moment.

Figure 9: Kapitza resistance/length as a function of high electric fields in a
planar hBN-water system. The inset shows the reduced water density peak and
order parameter as a function of electric field.

From the inset of Figure 9 we can observe that when the electric field
increases from 0 to 4 V/Å, the order parameter increases monotonically and
converges to a steady value once the electro-freezing started to appear.
Similarly, the reduced first water density peak
($\rho_{r}$=$\rho_{max}/\rho_{bulk}$) also increases with an increase in the
electric field. When the electric field is applied, due to the positive
surface charge on the left hBN layer, the oxygen atoms with negative partial
charge will orient towards the left wall, and the hydrogen atoms with positive
partial charge will orient towards the negatively charged left hBN layer. This
orientation will increase with an increase in electric fields and be reflected
in the value of the order parameter. Moreover, the ordering of water molecules
towards the charged surface intensifies the layering effect, which increases
the height of the first water density peak with an increase in the electric
field. Combining the above observations, we can conclude that the reduction in
Kapitza resistance with electric field strength is mainly due to the higher
first water density peak and the ordering of water molecules leading to more
aggregation of water molecules at the interface.

##  5 Conclusions

We have investigated the effect of different types of electrostatic
interactions on the Kapitza resistance in cylindrical and planar hexagonal
boron nitride-water systems using MD simulations. The cylindrical system
consists of water confined in a hexagonal boron nitride nanotube (hBNNT), and
the planar system consists of water confined between parallel hexagonal boron
nitride (hBN) nanochannel. The calculation of Kapitza resistance was carried
out using our previously developed EMD method due to the limitations of the
NEMD method in cylindrical geometries. Our EMD method utilises the correlation
functions of instantaneous heat flux and temperature difference data at the
interface and provides correlation time-independent results. We examined the
effect of different factors affecting the Kapitza resistance, such as the
diameter of hBNNT, partial charge, electrolyte concentration, and electric
field strength. The Kapitza resistance monotonically decreases with an
increase in diameter of the hBNNT and nearly converges to the value of the
planar system, precisely similar to the trend in a CNT-water system. The
Kapitza resistance was reduced to almost 37.37% when the diameter increased
from 1.38 nm to 6.9 nm primarily due to the increase in area density factor,
$F_{NA}$ and the overlap of VDOS between water and hBNNT. The primary reason
for a lower value of Kapitza resistance for hBNNT-water than the CNT-water
interface is due to the presence of partial charges on hBNNT. A reduction in
71.2% in Kapitza resistance was found when the partial charge increased from 0
to $\pm$ 1.50$e$. The partial charge does not influence the structuring and
ordering of water near the interface. Whereas the hydrogen bonding between
water and hBN wall increases with higher partial charges leading to a better
heat transfer, thereby reducing the Kapitza resistance. Furthermore, we found
that the addition of NaCl to water doesn’t impact interfacial heat transfer.
The Kapitza resistance is nearly independent of the electrolyte concentration
since the contribution of electrostatic interaction of both Na+ and Cl- ions
are significantly less compared to the water molecules. Finally, the effect of
the electric field on Kapitza resistance was studied by providing a uniform
surface charge to the left and right hBN layers. The Kapitza resistance is
found to be nearly independent of the electric field within the experimentally
achieved range of 0 to 0.1 V/Å. To study the structural changes of water due
to electric fields, extensive electric fields up to 5 V/Å were applied by
assuming that the dielectric breakdown of water will not occur. The Kapitza
resistance reduced by about 53.5% when the electric field increased until
electro-freezing occurs at approximately 4 V/Å. When the electric field is
applied, the ordering of water molecules towards the charged surface
intensifies the layering effect, which leads to an increase in the height of
the first water density peak, thereby increasing the interfacial thermal
transport. These findings provide an insight into the contributions of
electrostatic interactions on Kapitza resistance, which helps to manage the
interfacial thermal transport in various nanoscale systems.

## Acknowledgements

The authors are grateful for the computational resources provided by the P. G.
Senapathy Center for Computing Resources at IIT Madras (AQUA supercluster) and
the Swinburne supercomputing OzSTAR facility. Authors S.P.S and S.A gratefully
acknowledge the financial support received from the Ministry of Education,
Government of India, under MHRD-STARS (Project Number: STARS1/336) to carry
out a part of this work. This research is also supported by the Indian
Institute of Technology Madras to the Micro Nano-Bio Fluidics Group under the
funding for the Institutions of Eminence scheme of Ministry of Education,
Government of India [Sanction. No: 11/9/2019-U.3(A)].

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