# Influence of solvent polarization and non-uniform ion size on electrostatic
properties between charged surfaces in an electrolyte solution

Jun-Sik Sin<EMAIL_ADDRESS>Department of Physics, Kim Il Sung
University, Taesong District, Pyongyang, Democratic People’s Republic of Korea

###### Abstract

In this paper, we study electrostatic properties between two similar or
oppositely charged surfaces immersed in an electrolyte solution by using mean-
field approach accounting for solvent polarization and non-uniform size
effect. Applying a free energy formalism accounting for unequal ion sizes and
orientational ordering of water dipoles, we derive coupled and self-consistent
equations to calculate electrostatic properties between charged surfaces.
Electrostatic properties for similarly charged surfaces depend on counterion
size but not coion size. Moreover, electrostatic potential and osmotic
pressure between similarly charged surfaces are found to be increased with
increasing counterion size. On the other hand, the corresponding ones between
oppositely charged surfaces are related to both sizes of positive and negative
ions. For oppositely charged surfaces, the electrostatic potential, number
density of solvent molecules and relative permittivity of an electrolyte
having unequal ion sizes are not symmetric about the centerline between the
charged surfaces. For either case, the consideration of solvent polarization
results in an decrease in the electrostatic potential and the osmotic pressure
compared to the case without the effect.

Electrolyte, Osmotic Pressure, Solvent Polarization, ion size Effect

###### pacs:

82.45.Gj

††preprint: Preprint

##  I Introduction

The study of electrostatic properties between charged surfaces in an
electolyte is of great significance in material science and biology. It is
well known that the interaction between two charged surfaces is attributed to
two kinds of physical natures. (i.e., Van der Waals and Electrostatic
interaction).Israel_book_1985 ; Lyklema_book_2005 ; Hunter_book_1981 ;
Hansen_ARPC_2000 ; Kampf_PRL_2009

Although classical Poisson-Boltzmann (PB) theory has been a fundamental tool
describing electric double layer electrostatic potential and osmotic-pressure
between two charged surfaces for a long time Gouy_JPhysF_1910 ;
Chapman_PhilosMag_1913 , many researchers have devoted a great deal of effort
to amending the PB theory which cannot be applicable to the case being short
the distance between two charged surfaces. To properly describe the electric
double layer properties, there exist attempts to develop mean-field theories
describing finite sizes of ions and water molecules and/or water
polarizationBikerman_PhilosMag_1942 ; Wicke_ZEC_1952 ; Iglic_JPhysF_1996 ;
Andelman_PRL_1997 . In the last decade, the consideration of non-uniform ion
sizes has attracted massive attention Chu_BJ_2007 ; Kornyshev_JPCB_2007 ;
Biesheuvel_JCIS_2007 ; Li_PRE_2011 ; Li_PRE_2012 ; Boschitsch_JCC_2012 ;
Siber_PRE_2013 ; Gongadze_EA_2015 ; Mohajernia_EA_2017 . In particular, the
authors of Iglic_Bioelechem_2010 ; Iglic_Bioelechem_2012  simultaneously
accounted for orientational ordering of water dipoles and finite sizes of ions
and water molecules.

Recently, it was demonstrated in the papers Sin_EA_2015 ; Gongadze_EA_2015 ;
Sin_PCCP_2016_1 ; Sin_EA_2016 ; Sin_PCCP_2016_2 ; Sin_CSA_2017 ;
Mohajernia_EA_2017  that the simultaneous considerations of unequal ion size
and solvent polarization are very important for describing electrostatic
properties of an electrolyte near a single charged surface.

On the other hand, the electrostatic properties between charged surfaces were
studied by using mean-field theories taking into account finite ion size,
solvent polarization and composition of solvent mixture. Using a mean-field
theory accounting only for ion size effect, the authors of Das_PRE_2011
demonstrated the valuable fact that nontrivial interactions between ion size
effect and electric double layer overlap phenomena may augment the effective
extent of electric double layer overlap in narrow fluidic confinements.
Considering ions as point-like charge, the authors of Das_PRE_2012
demonstrated the effect of solvent polarization on the electric double layer
electrostatic potential distribution and the effective EDL thickness in narrow
nanofluidic confinements. Although the authors of Abrashkin_PRL_2007
addressed osmotic pressure between charged surfaces by accounting for dipole
moment of water molecules and ion size, some of their results were contrary to
common sense such as behavior of spatial distribution of permittivity.
Teschke_CPL_2000 ; Teschke_JMCA_2001  Namely, in Abrashkin_PRL_2007 , the
increase of relative permittivity near the charged surface was predicted for
point-like ions. This is a consequence of predicted accumulation of water
dipoles near the charged surface due to an assumed Boltzmann distribution for
water molecules, which prevails over the saturation effect in polarizability
as shown in Gongadze_GPB_2011 . The authors of Andelman_JPCB_2009  derived a
general expression for osmotic pressure for the case when the free energy of
the system does not depend explicitly on the coordinate. Even though studies
Urbanija_JCP_2008 ; Andelman_PRE_2013 ; Bohinc_SM_2016 ; Andelman_PRE_2016
have been investigated electric double layer forces between charged surfaces,
they did not considered the effect of solvent polarization.

Recently, the authors of Das_JCP_2013 ; Gongadze_EA_2014  presented a more
satisfactory answer on the osmotic pressure by using Langevin-Bikerman. Their
model well represents electrostatic properties by simultaneous considerations
of size effects and polarization of water molecules, barring the difference in
size between positive and negative ions.

Although the authors of Gongadze_EA_2015 ; Mohajernia_EA_2017  described
differential capacitance and permeation through charged nanotube membranes by
accounting for both solvent polarization and disparity of ion sizes, there
does not exist study to describe osmotic pressure between charged surfaces
with consideration of the effects. On the other hand, Monte Carlo simulation
Zelko_JCP_2010 ; Guerrero_JCP_2011 ; Wang_JCP_2012 was extensively used to
correct the theoretical predictions of classical PB method. In fact,
discreteness-of-charge and the image effects, considered by using Monte Carlo
method, are of important roles in the compact part of the electric double
layer and may substantially affect the zeta potential of the surface. In the
present paper, these effects are beyond our scope and we provide results only
for constant zeta potential because compared to such effects, ion size
asymmetry and solvent polarization crucially affect electrostatic interaction
between charged surfaces for high electrolyte concentration.

In this paper, we study the effect of the difference in size between positive
and negative ions as well as solvent polarization on electrostatic potential,
number density of ions and water molecules, permittivity and osmotic pressure
by using a mean field approach. Iglic_Bioelechem_2010 ; Iglic_Bioelechem_2012
; Sin_EA_2016 ; Gongadze_EA_2014 ; Gongadze_EA_2015  The first result is that
evaluating electrostatic properties between oppositely charged surfaces
requires considering difference in size between positive and negative ions,
while these between similarly charged surfaces can depend only on counterion
size. Next, it is shown that for a constant surface potential, solvent
polarization diminishes ion size effects on electrostatic properties between
similar and oppositely charged surfaces. Finally, we emphasize that our method
can consistently explain the experimental results of interaction force between
similar or oppositely charged surfaces, by correcting the large under-
prediction made by the corresponding PB model or the over-prediction made by
considering only the ion size.

##  II Theory

We consider two parallel plates (similar or oppositely charged) separated by a
distance $H$ in an electrolyte. The transverse direction is noticed by x; the
left plate is placed at $x=-H/2$ and the right plate $x=-H/2$. The resulting
electrostatic properties at the interfaces between the plates and the
electrolyte solution should be addressed by setting the free energy of total
thermodynamic system as follows

$\displaystyle F=\int
d{\bf{r}}\left[{-\frac{{\varepsilon_{0}E}}{2}^{2}+e_{0}z\psi\left({\bf{r}}\right)\left({n_{+}-n_{-}}\right)+\left\langle{p_{0}E\cos\omega\rho\left(\omega\right)}\right\rangle_{\omega}-\mu_{+}n_{+}-\mu_{-}n_{-}-\left\langle{\mu_{\omega}\left(\omega\right)\rho\left(\omega\right)}\right\rangle_{\omega}-Ts}\right]..$
(1)

While the local electrostatic potential is denoted by
$\Psi\left({\bf{r}}\right)$, the number density of different ionic species and
the number density of water molecules are expressed as
$n_{i}\left({\bf{r}}\right),i=+,-$ and
$n_{w}\left({\bf{r}}\right)=\left\langle{\rho\left({\omega,{\bf{r}}}\right)}\right\rangle_{\omega}$,
respectively.

Here
$\left\langle{f\left(\omega\right)}\right\rangle_{\omega}=\int{f\left(\omega\right)2\pi\sin\left(\omega\right)d\omega}$
and $\omega$ stands for the angle between the dipole moment vector p and the
normal to the charged surface. ${\bf{p}}$ and ${\bf{E}}$ notice the dipole
moment of water molecules and electric field strength, respectively, where
$p_{0}=\left|{\bf{p}}\right|$ and $E=\left|{\bf{E}}\right|$.

In Eq. (1), the first term describes the self energy of the electrostatic
field, where stands for the vacuum permittivity. The second term means the
electrostatic energy of the ions. It is noticeable that unlike the case in
Sin_EA_2016 , the the third term i.e., electrostatic energy of water dipoles,
is equal to one of Iglic_Bioelechem_2010 ; Gongadze_EA_2014  where the formula
for osmotic pressure was derived. Coupling the system to a bulk reservoir
necessitates the next three terms, where $\mu_{+}$ and $\mu_{-}$ mean the
chemical potentials of positive and negative ions, respectively, and
$\mu_{w}\left(\omega\right)$ corresponds to the chemical potential of water
dipoles with orientational angle $\omega$ . $T$ and $s$ are the temperature
and the entropy density, respectively. (Please refer Iglic_JPhysF_1996 ;
Iglic_Bioelechem_2010 ; Iglic_Bioelechem_2012 ; Sin_EA_2016 .) From this free
energy functional, we derive the self-consistent equations determining
electrostatic properties by performing minimization of the corresponding free
energy describing the electric double layer and subsequently find the formula
for the osmotic pressure. The Lagrangian of the total system can be expressed
so that the volume conservation is satisfied

$\displaystyle
L=F-\int{\lambda\left({\bf{r}}\right)}\left({1-n_{+}V_{+}-n_{-}V_{-}-n_{w}V_{w}}\right)d{\bf{r}},$
(2)

where $\lambda$ is a local Lagrange parameter. When the origin of the electric
potential is located at $x=\infty$, $\psi\left({x=\infty}\right)=0$. At the
origin, $n_{i}\left({x=\infty}\right)=n_{ib}$ and
$\lambda\left({x=\infty}\right)=\lambda_{b}$, where $n_{ib}$ and $\lambda_{b}$
represent the bulk ionic concentration and the Lagrange parameter at
$x=\infty$, respectively. The number densities of ions and water molecules can
be obtained by applying the boundary conditions and by writing Euler-Lagrange
equations of Eq. (2) in terms of the number densities of particles.

$n_{+}={{n_{+b}\exp\left({-V_{+}h-e_{0}z\beta\psi}\right)}}/{D},$ (3a)
$\displaystyle
n_{-}={{n_{-b}\exp\left({-V_{-}h+e_{0}z\beta\psi}\right)}}/{D},$ (3b)
$\displaystyle
n_{w}={{n_{wb}\exp\left({-V_{w}h}\right)\frac{{\sinh\left({p_{0}\beta
E}\right)}}{{p_{0}\beta E}}}}/{D},$ (3c) $\displaystyle
n_{+b}\left({e^{-V_{+}h-\beta ze\phi}-1}\right)+n_{-b}\left({e^{-V_{-}h+\beta
ze\phi}-1}\right)+n_{wb}\left({e^{-V_{w}h}\frac{{\sinh(p_{0}\beta
E)}}{{p_{0}\beta E}}-1}\right)=0,$ (3d)

where
$D=n_{+b}V_{+}\exp\left({-V_{+}h-e_{0}z\beta\psi}\right)+n_{-b}V_{-}\exp\left({-V_{-}h+e_{0}z\beta\psi}\right)+n_{w}V_{w}\exp\left({-V_{w}h}\right)\frac{{\sinh\left({p_{0}\beta
E}\right)}}{{p_{0}\beta E}}$, $h=\lambda-\lambda_{b}$ and
$\left\langle{e^{-p_{0}E\beta\cos\left(\omega\right)}}\right\rangle_{\omega}=\frac{{2\pi\int_{\pi}^{0}{d\left({\cos\omega}\right)e^{-p_{0}E\beta\cos\left(\omega\right)}}}}{{4\pi}}=\frac{{\sinh\left({p_{0}E\beta}\right)}}{{p_{0}E\beta}}$.
Iglic_Bioelechem_2010 ; Iglic_Bioelechem_2012

In addition to the above equations, we have the following equations for the
chemical potentials of ions and water molecules

$\mu_{+}=k_{B}T\ln\left({n_{+b}/N_{b}}\right)+V_{+}\lambda_{b},$ (4a)
$\mu_{-}=k_{B}T\ln\left({n_{-b}/N_{b}}\right)+V_{-}\lambda_{b},$ (4b)
$\mu_{w}\left(\omega_{i}\right)=k_{B}T\ln\left({\rho\left(\omega_{i}\right)\Delta\Omega/N_{b}}\right)+V_{w}\lambda_{b},$
(4c)

where $\rho\left(\omega_{i}\right)$ stands for the number density of water
molecules with orientational angle $\omega_{i}$ and
$n_{w}=\left\langle{\rho\left(\omega\right)}\right\rangle_{\omega}$.

The Euler-Lagrangian equation for $\psi\left(r\right)$ yields the Poisson
equation

$\nabla\left({\varepsilon_{0}\varepsilon_{r}\nabla\psi}\right)=-e_{0}z\left({n_{+}-n_{-}}\right),$
(5)

where $\varepsilon_{r}\equiv
1+\frac{{\left|{\bf{P}}\right|}}{{\varepsilon_{0}E}}$. Due to the symmetry of
the present study, $\bf{P}$, the polarization vector due to a total
orientation of point-like water dipoles, is parallel to the normal to charged
surfaces, as in Iglic_Bioelechem_2010 ; Das_JCP_2013 ; Gongadze_GPB_2011

${\bf{P}}\left(x\right)=n_{w}\left(x\right)p_{0}L\left({p_{0}E\beta}\right),$
(6)

where $\beta=1/\left({k_{B}T}\right)$, $p_{0}=4.8D$ and the function
$L\left(u\right)=\coth\left(u\right)-1/u$ means the Langevin function.

For now, let us derive a new formula for osmotic pressure accounting for
unequal ion sizes and solvent polarization. In fact, the authors of
Andelman_JPCB_2009  proved the fact that when the free energy density of the
total system does not depend on the coordinate, osmotic pressure between two
charged surfaces can be derived from the following expression

$f-\left({\partial f/\partial\psi^{\prime}}\right)\psi^{\prime}=consant=-P,$
(7)

where $\psi^{\prime}$ is the derivative of $\psi$ with respect to $x$ and the
constant is the negative of the local pressure $P$ that is defined to be the
sum of the osmotic pressure and the bulk pressure, i.e.,

$P=P_{osm}+P_{bulk}.$ (8)

The free energy density for the present study does not depend explicitly on
the coordinate x, as can be seen in Eq.(1). Using Eqs. (1) and (7), we can
therefore get

$\left({\partial
f/\partial\psi^{\prime}}\right)\psi^{\prime}=-\varepsilon_{0}\psi^{\prime
2}+\left\langle{\rho\left(\omega\right)p_{0}E\cos\omega}\right\rangle_{\omega}$
(9)
$P=-\frac{{\varepsilon_{0}E}}{2}^{2}-e_{0}z\psi\left({n_{+}-n_{-}}\right)+\mu_{+}n_{+}+\mu_{-}n_{-}+\left\langle{\mu_{\omega}\left(\omega\right)\rho\left(\omega\right)}\right\rangle_{\omega}+Ts$
(10)

Substituting Eq. (4) in the above equation, we get the following equation

$\begin{array}[]{l}P=-\frac{{\varepsilon_{0}E}}{2}^{2}-e_{0}z\psi\left({n_{+}-n_{-}}\right)+\left({k_{B}T\ln\left({n_{+b}/N_{b}}\right)+V_{+}\lambda_{b}}\right)n_{+}+\left({k_{B}T\ln\left({n_{-b}/N_{b}}\right)+V_{+}\lambda_{b}}\right)n_{-}\\\
+\left\langle{\left({k_{B}T\ln\left({\rho\left(\omega\right)\Delta\Omega_{i}/N_{b}}\right)+V_{w}\lambda_{b}}\right)\rho_{w}\left(\omega\right)}\right\rangle_{\omega}\\\
+k_{B}T\left[{N\ln N-\sum\limits_{i=\\{+,-\\}}{n_{i}\ln
n_{i}}-\mathop{\lim}\limits_{k\to\infty}\sum\limits_{i=1}^{k}{\left\\{{\left[{\rho\left({\omega_{i}}\right)\Delta\Omega_{i}}\right]\ln\left[{\rho\left({\omega_{i}}\right)\Delta\Omega_{i}}\right]}\right\\}}}\right]\\\
=-\frac{{\varepsilon_{0}E}}{2}^{2}-e_{0}z\psi\left({n_{+}-n_{-}}\right)+k_{B}Tn_{+}\ln\left({\frac{{n_{+b}}}{{N_{b}}}\frac{N}{{n_{+}}}}\right)+k_{B}Tn_{-}\ln\left({\frac{{n_{-b}}}{{N_{b}}}\frac{N}{{n_{-}}}}\right)\\\
+k_{B}T\left\langle{\rho\left(\omega\right)\ln\left({\frac{{\rho_{b}\left(\omega\right)\Delta\Omega_{i}}}{{N_{b}}}\frac{{N}}{{\rho\left(\omega\right)\Delta\Omega_{i}}}}\right)}\right\rangle_{\omega}+\lambda_{b}.\end{array}$
(11)

As the distance between the charged surfaces approaches the positive infinity,
$P=P_{bulk}$. In consequence, we obtain $\lambda_{b}=P_{bulk}$ from Eq. (11).

We can find the formula for osmotic pressure by comparing the above fact, Eq.
(11) and Eq. (8)

$\begin{array}[]{l}P_{osm}=-\frac{{\varepsilon_{0}E}}{2}^{2}-e_{0}z\psi\left({n_{+}-n_{-}}\right)+k_{B}Tn_{+}\ln\left({\frac{{n_{+b}}}{{N_{b}}}\frac{N}{{n_{+}}}}\right)+k_{B}Tn_{-}\ln\left({\frac{{n_{-b}}}{{N_{b}}}\frac{N}{{n_{-}}}}\right)\\\
+k_{B}T\left\langle{\rho\left(\omega\right)\ln\left({\frac{{\rho_{b}\left(\omega\right)\Delta\Omega_{i}}}{{N_{b}}}\frac{{N}}{{\rho\left(\omega\right)\Delta\Omega_{i}}}}\right)}\right\rangle_{\omega}.\end{array}$
(12)

On the other hand, Eq. (3d) can be rewritten as follows.

$n_{+b}e^{-V_{+}h-\beta ze\phi}+n_{-b}e^{-V_{-}h+\beta
ze\phi}+n_{wb}e^{-V_{w}h}\frac{{\sinh(p_{0}\beta E)}}{{p_{0}\beta
E}}=n_{+b}+n_{-b}+n_{wb}=N_{b}$ (13)

Rearranging Eq. (13) also results in a new relation

$N=n_{+}+n_{-}+n_{w}=\frac{{n_{+b}\exp\left({-V_{+}h-e_{0}z\beta\psi}\right)+n_{-b}\exp\left({-V_{-}h+e_{0}z\beta\psi}\right)+n_{wb}e^{-V_{w}h}\frac{{\sinh(p_{0}\beta
E)}}{{p_{0}\beta E}}}}{D}=\frac{{N_{b}}}{D}$ (14)

Substituting the above relation in Eq. (12) and using the condition of volume
conservation, we eventually obtain the following relation

$\begin{array}[]{l}P_{osm}=-\frac{{\varepsilon_{0}E}}{2}^{2}-e_{0}z\psi\left({n_{+}-n_{-}}\right)+k_{B}Tn_{+}\ln\left({\frac{1}{{\exp\left({-V_{+}h-e_{0}z\beta\psi}\right)}}}\right)+k_{B}Tn_{-}\ln\left({\frac{1}{{\exp\left({-V_{-}h+e_{0}z\beta\psi}\right)}}}\right)+\\\
+k_{B}T\left\langle{\rho\left(\omega\right)\ln\left({\frac{{\rho_{b}\left(\omega\right)\Delta\Omega_{i}}}{{N_{b}}}\frac{N}{{\rho\left(\omega\right)\Delta\Omega_{i}}}}\right)}\right\rangle_{\omega}\\\
=-\frac{{\varepsilon_{0}E}}{2}^{2}+k_{B}T(n_{+}V_{+}+n_{-}V_{-})h+\\\
+k_{B}T\left({n_{wb}V_{w}h\exp\left({-V_{w}h}\right)\frac{{\sinh\left({p_{0}\beta
E}\right)}}{{p_{0}\beta
E}}/D+\exp\left({-V_{w}h}\right)\left\langle{\exp(p_{0}\beta
E\cos\omega)(p_{0}\beta E\cos\omega)}\right\rangle}\right)=\\\
=-\frac{{\varepsilon_{0}E}}{2}^{2}+k_{B}T(n_{+}V_{+}+n_{-}V_{-}+n_{w}V_{w})h+k_{B}T\left({\exp\left({-V_{w}h}\right)\left\langle{\exp(p_{0}\beta
E\cos\omega)\left({p_{0}\beta
E\cos\omega}\right)}\right\rangle_{\omega}}\right)\\\
=-\frac{{\varepsilon_{0}E}}{2}^{2}+k_{B}Th+k_{B}Tn_{wb}\exp\left({-V_{w}h}\right)/D\left\langle{\exp(p_{0}\beta
E\cos\omega)\left({p_{0}\beta E\cos\omega}\right)}\right\rangle_{\omega}=\\\
=-\frac{{\varepsilon_{0}E}}{2}^{2}+k_{B}Th-
k_{B}Tn_{wb}\exp\left({-V_{w}h}\right)/D\left({\cosh\left({p_{0}\beta
E}\right)-\frac{{\sinh\left({p_{0}\beta E}\right)}}{{\left({p_{0}\beta
E}\right)}}}\right)=\\\ =-\frac{{\varepsilon_{0}E}}{2}^{2}+k_{B}Th-
k_{B}Tn_{w}\left({\cosh\left({p_{0}\beta E}\right)\frac{{\left({p_{0}\beta
E}\right)}}{{\sinh\left({p_{0}\beta
E}\right)}}-1}\right)=-\frac{{\varepsilon_{0}E}}{2}^{2}+k_{B}Th-
k_{B}Tn_{w}\left({p_{0}\beta E}\right)L\left({p_{0}\beta
E}\right).\end{array}$ (15)

Eq. (15) corresponds to the osmotic pressure between two charged surfaces.
This osmotic pressure is constant across the channel, and therefore, can be
estimated at any point within the channel. We can easily identify that the
formula Eq. (15) is for a more general situation containing ones of Poisson-
Boltzmann, Bikerman, Langevin-Poisson-Boltzmann, Modified Langevin-Poisson-
Boltzmann, Langevin-Bikerman approach Chapman_PhilosMag_1913 ;
Bikerman_PhilosMag_1942 ; Gongadze_GPB_2011 ; Velikonja_IJMS_2013 ;
Das_JCP_2013 ; Gongadze_EA_2014  and the approach with ion size effect and
without solvent polarization Boschitsch_JCC_2012 . For the simplicity of
calculation, we can express all the quantities in dimensionless forms as

$\begin{array}[]{l}\bar{x}=x/\lambda,\bar{\psi}=e_{0}z\beta\psi,\bar{d}=d/\lambda,\bar{\varepsilon}_{r}=\varepsilon_{r}/\varepsilon_{p},\lambda=\sqrt{\frac{{\varepsilon_{0}\varepsilon_{p}k_{B}T}}{{2n_{b}e_{0}^{2}z^{2}}}},\bar{h}=hV_{w},\\\
\bar{V}=V/V_{w},\eta=\frac{1}{{n_{wb}V_{w}}},\bar{n}_{b}=\frac{{n_{b}}}{{n_{wb}}},p_{0}\beta
E=p_{0}\sqrt{\frac{{2n_{b}}}{{\varepsilon_{0}\varepsilon_{p}k_{B}T}}}\frac{{d\bar{\psi}}}{{d\bar{r}}}=\chi\bar{E},\end{array}$
(16)

Based on the dimensionless parameters, Eq. 4 and Eq. 5 can be rewritten as
follows

$\bar{n}_{+b}\left({e^{-\bar{\psi}-\bar{V}_{+}\bar{h}}-1}\right)+\bar{n}_{-b}\left({e^{\bar{\psi}-\bar{V}_{-}\bar{h}}-1}\right)+\left({e^{-\bar{h}}\frac{{\sinh(\chi\bar{E})}}{{\chi\bar{E}}}-1}\right)=0,$
(17)

$\frac{d}{{d\bar{x}}}\left({\bar{\varepsilon}_{r}\frac{{d\bar{\psi}}}{{d\bar{x}}}}\right)=\frac{\eta}{2}\frac{{\exp\left({\bar{\psi}-\bar{V}_{-}\bar{h}}\right)-\exp\left({-\bar{\psi}-\bar{V}_{+}\bar{h}}\right)}}{{\frac{{\sinh\left({\chi\bar{E}}\right)}}{{\chi\bar{E}}}\exp\left({-\bar{h}}\right)+\bar{n}_{b}\bar{V}_{-}\exp\left({\bar{\psi}-\bar{V}_{-}\bar{h}}\right)+\bar{n}_{b}\bar{V}_{+}\exp\left({-\bar{\psi}-\bar{V}_{+}\bar{h}}\right)}}.$
(18)

##  III Results and Discussion

All the calculations in the present study are performed by using the fourth
order Runge-Kutta method combined with shooting method. For clarity, we choose
0.01M for the ionic concentration in the buk electrolyte solution and 298K for
the temperature.

###  III.1 Similarly Charged Surfaces

Without loss of generality, we assume that the surfaces are positively
charged.

Figure 1: (Color online)For similarly charged surfaces, variation of
electrostatic potential (a), the number density of counterions (b), the number
density and water molecules(c) and the permittivity (d) with the position for
different sets of ion sizes. The separation distance between charged surfaces
is $H=5$nm and the surface potential is
$\psi\left(x=H/2)\right)=\psi\left(x=-H/2)\right)=+0.5$V.  Figure 2: (Color
online) For similarly charged surfaces, (a) Variation of the centerline
potential with the separation distance between the charged surfaces for
$\psi\left(x=H/2)\right)=\psi\left(x=-H/2)\right)=+0.5$V. (b) Variation of the
centerline potential with the surface potential for different sets of ion
sizes. The separation distance between charged surfaces is $H=5$nm. Figure 3:
(Color online) For similarly charged surfaces, (a) Variation of osmotic
pressure with the separation distance between the charged surfaces for
$\psi\left(x=H/2)\right)=\psi\left(x=-H/2)\right)=+0.5$V. (b) Variation of the
osmotic pressure as a function of the surface potential for different sets of
ion sizes and the separation distance between charged surfaces, $H=5$nm.

Fig. 1(a) shows the electrostatic potential profile between similarly charged
surfaces for the case where the distance between charged surfaces is $H=5$nm
and the surface potential is $\psi\left(x=h/2\right)=0.5$V. The cases for
$\left(V_{-}=V_{+}=V_{w}=0.03nm^{3}\right)$,
$\left(V_{-}=V_{+}=0.3nm^{3},V_{w}=0.03nm^{3}\right)$ and
$\left(V_{-}=0.3nm^{3},V_{+}=V_{w}=0.03nm^{3}\right)$ are represented by
squares, circles and solid line, respectively. In Fig. 1(a), it is illustrated
that the electrostatic potential has a symmetric distribution attributed to
the geometry of this system and the sign of the surface potentials. We can
also find that an increase in counterion size makes the electrostatic
potential increase in the region between two charged surfaces. This is
understood by the fact that the screening property of a counterion with a
large size is weaker than corresponding one of a counterion with a smaller
size. Importantly, we should emphasize that the electrostatic potential
profile is not related to coion size, as in Sin_EA_2016 ; Sin_CSA_2017 .

Fig. 1(b) demonstrates the spatial dependence of the number density of
counterions. For the cases when a counterion has the small size
($V_{-}=0.03nm^{3}$), the number density of counterions is larger than
corresponding one for a counterion of the larger size ($V_{-}=0.3nm^{3}$)in
the vicinity of a charged surface. Such a phenomenon is attributed to the
excluded volume effect, as explained in Iglic_Bioelechem_2010 .

Fig. 1(c) shows the transverse variation of the number density of water
molecules. It is clearly seen that the density for a counterion of the large
size is smaller than corresponding one for a counterion of the smaller size.
This is explained by combining excluded volume effect of counterions and the
fact that due to overlap of the electric double layers of similarly charged
surfaces, the number density of counterions gets larger than the bulk value.

Fig. 1(d) shows the variation of the permittivity with the position between
two charged surfaces. According to the permittivity formula of Eq. (6), the
permittivity is strongly affected by the number density of water molecules and
the electric field strength. Near the centerline between the charged surfaces,
the permittivity is proportional to the number density of water molecules,
since due to the geometrical symmetry of the present system the electric field
strength is zero at the centerline between two charged surfaces. However, in
the vicinity of a charged surface, in spite of the fact that the number
density of water molecules for the case when a counterion has the large size
is smaller than corresponding one for a counterion of the smaller size, the
permittivity for the former case is larger than one for the latter case. That
is why a high electric field strength lowers the permittivity Booth_JCP_1955
.

Fig. 2(a) shows the electrostatic potential at the centerline between two
charged surfaces as a function of the separation distance between the
surfaces. In Figs. (2, 3, 4, 5), we use an identical convention for the
studied cases. Case 1, Case 3 and Case 5 mean
$\left(V_{-}=V_{+}=V_{w}=0.03nm^{3}\right)$,
$\left(V_{-}=V_{+}=0.3nm^{3},V_{w}=0.03nm^{3}\right)$ and
$\left(V_{-}=0.3nm^{3},V_{+}=V_{w}=0.03nm^{3}\right)$ for the case when both
solvent polarization and ion sizes are considered, respectively. Case 2 , Case
4 and Case 6 represent $\left(V_{-}=V_{+}=V_{w}=0.03nm^{3}\right)$,
$\left(V_{-}=V_{+}=0.3nm^{3},V_{w}=0.03nm^{3}\right)$ and
$\left(V_{-}=0.3nm^{3},V_{+}=V_{w}=0.03nm^{3}\right)$ for the case accounting
only for ion sizes, respectively.

For similarly charged surfaces, the electrostatic potential at the centerline
between them is a main characteristics that illustrates overlap of electric
double layers. It is noticeable that under the boundary condition of a given
potential, solvent polarization lowers the potential at the centerline. In
fact, as pointed out in Das_PRE_2011 , ion size effect enhances electrostatic
potential at the centerline due to lowering of screening property of ions.
However, solvent polarization lowers the permittivity as shown in Fig. 1(d).
From the viewpoint of physics, lowering permittivity yields enhancement of
magnitude of electric field strength in an electrolyte. Since at any position
between two charged surfaces the permittivity value is not higher than the
bulk value of the permittivity, the electric field strength is not lower than
corresponding one for the case with only ion size effects. Finally for the
case with solvent polarization, the centerline potential is lower than one for
the case without the effect. As can be expected, for either case, an increase
in the separation distance between two charged surfaces decreases the
centerline potential. This is deduced by using the fact that the longer the
distance between two charged surfaces, the weaker the overlap of electric
double layers of the charged surfaces. The inset in Fig. 2(a) shows that
$\Delta\psi$, the difference in the centerline potential between the cases
with and without solvent polarization, decreases with the distance between
charged surfaces. It can be explained by the fact that as the distance between
charged surfaces gets longer, the centerline potential for either case tends
to zero.

Fig. 2(b) shows the dependence of the centerline potential on surface
potential at charged surfaces. It is clearly demonstrated that an increase in
the surface potential allows the centerline potential to increase. This is
attributed to the fact that for a given distance between two charged surfaces,
an increase in the surface potential enhances the overlap of electric double
layers of two charged surface and therefore induces the increase in the
centerline potential.The inset in Fig. 2(b) shows that the difference in the
centerline potential between the cases with and without solvent polarization,
increases with increasing the surfaces potential. It can be explained by the
fact that an increase in the surface potential involves enhancement of solvent
polarization .

Fig. 3(a) shows the variation of osmotic pressure with the distance separation
between two charged surfaces. As one can see, a large ion size makes repulsive
osmotic pressure higher than for the smaller one and solvent polarization
decreases the osmotic pressure. For all the cases, an increase in the distance
between charged surfaces yields diminished osmotic pressure. The above facts
are attributed to the following fact. As shown in Sin_EA_2016 , the value of h
increases with increasing the electric potential. On the other hand, as
mentioned above, ion size effects increases the centerline potential. Namely,
h increases with ion size. At the centerline, the electric field strength is
zero due to the geometrical symmetry of this problem. As a result, the formula
for osmotic pressure between the similarly charged surfaces is rewritten as
follows:

$P_{osm}=k_{B}Th(x=0).$ (19)

Consequently, the osmotic pressure increases with counterion size, decreases
when we consider solvent polarization. As the distance between charged
surfaces approaches infinity, it also tends to zero. The inset of Fig. 3(a)
shows that an increase in the distance between the surfaces allows $\Delta
P_{osm}$, being the difference in osmotic pressure between the cases with and
without solvent polarization, to be decreased. Combining the Eq. (19) and the
explanation for Fig. 2(a) provides the reason for the phenomenon.

Fig. 3(b) shows the surface potential dependence of osmotic pressure between
two charged surfaces. Fig. 3(b) demonstrates that an increase in the surface
potential induces an increase of osmotic pressure. This phenomenon is
elucidated by combining Eq. (19) and the fact that the centerline potential
increases with increasing the surface potential. Although we have treated the
cases of positively charged surfaces, the same is true for the cases of
negatively charged surfaces.

Figure 4: (Color online) For similarly charged surfaces, Variation of osmotic
pressure with the separation distance between the charged surfaces for
$\psi\left(x=H/2)\right)=\psi\left(x=-H/2)\right)=+0.5$V for different types
of electric double layer model. Circles, crosses, plus signs, asterisks,
squares and diamonds stand for Poisson-Boltzmann(PB), Langevin-Poisson-
Boltzmann(LPB), Modified Langevin-Poisson-Boltzmann(MLPB), Bikerman, Langevin-
Bikerman(LB) and present approach($V_{-}=V_{+}=0.1nm^{3}$), respectively.

Fig.4 shows the dependence of osmotic pressure on the distance separation
between the surfaces for similarly charged surfaces
($\psi\left(x=H/2)\right)=\psi\left(x=-H/2)\right)=+0.5$V) for Poisson-
Boltzmann(PB), Langevin-Poisson-Boltzmann(LPB), Modified Langevin-Poisson-
Boltzmann(MLPB), Bikerman, Langevin-Bikerman(LB) and the present
approach($V_{-}=V_{+}=0.1nm^{3}$). Here circles, crosses, plus signs,
asterisks, squares and diamonds stand for Poisson-Boltzmann, Langevin-Poisson-
Boltzmann, Modified Langevin-Poisson-Boltzmann, Bikerman, Langevin-Bikerman
and present approach($V_{-}=V_{+}=0.1nm^{3}$), respectively. Comparison of the
osmotic pressure calculated using different types of electric double layer
model shows the following facts.

First, we confirm again that the models with solvent polarization(Langevin-
Poisson-Boltzmann and Langevin-Bikerman) predict lower values of osmotic
pressure than for corresponding models (Poisson-Boltzmann and Bikerman)
without solvent polarization, respectively.

It is also identified that the models with larger volumes of ions result in
larger values of osmotic pressure. In other words, Poisson-Boltzmann
approach(point-like ion) predicts lower values of osmotic pressure than for
Bikerman approach($V_{-}=V_{+}=V_{w}=0.03nm^{3}$). And Langevin-Poisson-
Boltzmann approach(point-like ions) yields lower values of osmotic pressure
than for Langevin-Bikerman approach($V_{-}=V_{+}=V_{w}=0.03nm^{3}$) which
provides lower osmotic pressures than for present
approach($V_{-}=V_{+}=0.1nm^{3}$).

Finally, it should be pointed out that modified Langevin-Poisson-Boltzmann
approach predicts slightly lower values of osmotic pressure than for Langevin-
Poisson-Boltzmann approach. The difference is attributed to a stronger
decrease in permittivity due to consideration of cavity field as pointed out
in Iglic_Bioelechem_2012 . However, since the difference in osmotic pressure
for the two approaches is small, we confirm that the present approach without
consideration of cavity field is quite reliable for computing osmotic
pressure.

###  III.2 Oppositely Charged Surfaces

Fig. 5(a) shows the electrostatic potential profile for oppositely charged
surfaces. The symbols in Fig. (5) have the same meanings as in Fig. (1). As
can be seen, for the cases when negative and positive ions are the same size,
the profile has point symmetry in the position. This is attributed to the
geometrical and electrical symmetry of the system. For the case of unequal
sizes of ions($V_{-}=0.3nm^{3},V_{+}=V_{w}=0.03nm^{3}$), the profile has not
point symmetry. For the case of unequal sizes, near the surface with negative
potential the profile is equal to one for the case of
$V_{-}=V_{+}=V_{w}=0.03nm^{3}$, near the surface with positive potential the
profile is equivalent to one for the case of
$V_{-}=V_{+}=0.3nm^{3},V_{w}=0.03nm^{3}$. As mentioned-above, this fact is
easily understood by the fact that electric double layer near a charged
surface is determined mainly by counterions.

Fig. 5(b) and Fig. 5(c) show the number density of water molecules and the
permittivity between the oppositely charged surfaces, respectively. Those
quantities exhibits such a behavior as in Fig. 5(a) due to the same reason.

Fig. 6(a) show the centerline potential according to the distance between
charged surfaces for the case of $V_{-}=0.3nm^{3},V_{+}=V_{w}=0.03nm^{3}$.

Figure 5: (Color online)For oppositely charged surfaces, variation of
electrostatic potential (a), the number density and water molecules(b) and the
permittivity (c) with the position for different sets of ion sizes. The
separation distance between charged surfaces is $H=5$nm and the surface
potential is $\psi\left(x=H/2)\right)=-\psi\left(x=-H/2)\right)=+0.5$V Figure
6: (Color online) For oppositely charged surfaces, (a) Variation of the
centerline potential with the separation distance between the charged surfaces
for $\psi\left(x=H/2)\right)=-\psi\left(x=-H/2)\right)=+0.5$V. (b) Variation
of the centerline potential with the surface potential for the case of unequal
ion sizes($V_{-}=0.3nm^{3},V_{+}=V_{w}=0.03nm^{3}$) and the separation
distance between charged surfaces, $H=5$nm. Figure 7: (Color online) For
oppositely charged surfaces, (a) Variation of osmotic pressure with the
separation distance between the charged surfaces for
$\psi\left(x=H/2)\right)=-\psi\left(x=-H/2)\right)=+0.5$V. (b) Variation of
the osmotic pressure with the surface potential for different sets of ion
sizes and the separation distance between charged surfaces, $H=5$nm.

Due to the symmetry, for different cases the centerline potential is zero at
any distance and any magnitude of surface potential. It is clearly seen that
for the cases with and without solvent polarization, the centerline potential
decreases with the distance between two charged surfaces. The value for the
case with solvent polarization is lower than that for the case without solvent
polarization. The fact is explained by the fact that the low permittivity
induces high magnitude of electric field strength.

Fig. 6(b) shows the centerline potential with the surface potential. Due to
the same reason in Fig. 5(a), the centerline potential increases with
increasing magnitude of the surface potential. It is noticeable that the
difference in the centerline potential between the two charged surfaces is
enhanced with the surface potential.

Fig. 7(a) is a graph of the attractive osmotic pressure as a function of the
distance between charged surfaces. As one can see, an increase in ion size
involves the enhancement of the attractive osmotic pressure due to the same
reason in the Fig. 3(a). Unlike the case of similarly charged surfaces, both
sizes of a positive and negative ion are important for determining the
pressure. This is understood by considering the fact that counter-ions of the
two surfaces have the different sign of electric charge. The inset of Fig.
7(a) shows that as the distance between the oppositely charged surfaces
increases, the attractive osmotic pressure vanishes due to the same reason as
in the inset of Fig. 3(a)

Fig. 7(b) shows the variation of osmotic pressure with the surface voltage.
Fig. 7(b) demonstrates that as the surface potentials increases, the
attractive osmotic pressure increases due to the same reason as in the inset
of Fig. 3(b). The inset of Fig. 7(b) shows that as the surface potentials
increases, the difference in the attractive osmotic pressures between charged
surfaces for the cases with and without solvent polarization, increases due to
the same reason as in the inset of Fig. 3(b).

Summarizing Figs. (2, 3, 7), it is elucidated that for the case of constant
surface potential, solvent polarization reduces ion size effect. Namely, the
centerline potential and osmotic pressure between two charged surfaces are
diminished for the case with solvent polarization than for the case without
the effect. This result distinguishes our theory from one of Das_JCP_2013 . In
fact, they considered uniform size effect as well as solvent polarization,
while they asserted that experiment results are understood by the increase due
to only the consideration of solvent polarization. We believe that the reason
is not so simple as their ones. On one hand, ion size effect, as shown in
Figs. (2, 3, 6, 7), enhances the centerline potential and osmotic pressure for
similar or oppositely charged surfaces. On the other hand, the consideration
of solvent polarization involves lowering the properties. As a result, solvent
polarization reasonably lowers excessive increases in centerline potential and
osmotic pressure due to ion size effect. We conclude that simultaneous
considerations of solvent polarization and ion size effect is mandatory for
elucidating experimental results.

Also, the present theory can take into account the difference in size not only
between ions but also between ions and a water molecule. This fact
demonstrates the clear advantage that unlike in the previous theory
Das_JCP_2013  where ions and water molecules have the same size, the present
method can treat more realistic situations where sizes of ions and water
molecules are not equal to each other.

Our results can be compared with Monte Carlo simulation describing
orientational ordering of solvent dipoles and ion size effects. The simulation
will require vast computational cost due to a large number of degrees of
freedom for the present study.

##  IV Conclusions

Using a mean-field theory accounting for solvent polarization and unequal size
effect, we have studied electrostatic properties between two charged surfaces.
We have shown that the electrostatic properties are unconditionally
symmetrical about the centerline between similarly charged surfaces but not
between oppositely charged surfaces.

We have demonstrated that for the case of similarly charged surfaces,
electrostatic properties are determined mainly by counterions but not by
coions. In contrast to the case, the properties for oppositely charged
surfaces are determined by both positive and negative ions. Moreover, the
centerline potential, being a quantity representing overlap of electric double
layers of similarly charged surfaces, and the osmotic pressure between the
surfaces increase with the counterion size. Most importantly, we have found
that under the condition of constant surface potential, the consideration of
solvent polarization reduces the centerline potential and osmotic pressure
augmented by ion size effect.

##  V Reference

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# Trust and Reputation Management for Blockchain-enabled IoT

Guntur Dharma Putra14, Sidra Malik12, Volkan Dedeoglu2, Salil S. Kanhere14 and
Raja Jurdak3  1UNSW Sydney 2CSIRO’s Data61 3QUT, Brisbane 4CSCRC, Australia
{gdputra<EMAIL_ADDRESS>{volkan.dedeoglu,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In recent years, there has been an increasing interest in incorporating
blockchain for the Internet of Things (IoT) to address the inherent issues of
IoT, such as single point of failure and data silos. However, blockchain alone
cannot ascertain the authenticity and veracity of the data coming from IoT
devices. The append-only nature of blockchain exacerbates this issue, as it
would not be possible to alter the data once recorded on-chain. Trust and
Reputation Management (TRM) is an effective approach to overcome the
aforementioned trust issues. However, designing TRM frameworks for blockchain-
enabled IoT applications is a non-trivial task, as each application has its
unique trust challenges with their unique features and requirements. In this
paper, we present our experiences in designing TRM framework for various
blockchain-enabled IoT applications to provide insights and highlight open
research challenges for future opportunities.

###### Index Terms:

trust, reputation, blockchain, IoT, CPS, TRM

Figure 1: The summary of our research experiences in designing TRM framework
for various IoT applications. While there is convergence in the employed TRM
techniques and methods, each application has its own trust challenges with
their unique features and requirements, which significantly influence the
final TRM framework. We explain our experience in more detail in Section 3.

##  1 Introduction

Blockchain, the underpinning technology of cryptocurrency, has shown its
potential to revolutionise the Internet of Things (IoT) [1]. The salient
features of blockchain, such as decentralisation and auditability, have solved
the inherent challenges of IoT, such as centralisation, data silos and
interoperability [2]. For example, in the context of supply chain management,
blockchain integration with IoT has increased the transparency of how the
traded commodities are handled from production to the retail shelves, thus
increasing traceability [3].

Although distributed consensus algorithms remove the need to trust a
centralised entity, blockchain by itself cannot ascertain the veracity of the
collected observation data [4], thus raising concerns about the integrity and
authenticity of the logged data. Moreover, blockchain mechanisms cannot
guarantee the trustworthiness of data at the origin, as the data is collected
from the physical world by an entity that may be malicious or prone to errors
[4]. In fact, invalid observational data cannot be corrected once stored on-
chain, which would overload the blockchain with immutable bad data.

Blockchain-enabled IoT networks utilise authentication and authorisation
schemes to limit access only to only authorised nodes. However, these schemes
can not ascertain in situ reliability and trustworthiness of authorised nodes,
as these schemes do not monitor the behaviour of nodes over the operational
period. The case of Mirai Botnet exemplifies how IoT nodes can be compromised
post-authentication and become malicious [5], which could severely impede the
security and resiliency of the network. As critical infrastructures are being
increasingly connected to IoT networks [6], the presence of malicious
adversaries could eventually cause severe detrimental effects, as exemplified
in the recent Colonial Pipeline attack, where a ransomware attack halted a
major gas pipeline in the US [7].

Trust and Reputation Management (TRM) is an effective approach to overcome the
aforementioned trust issues, wherein the trustworthiness of each node is
continuously evaluated. TRM performs trust evaluation by processing feedback
and ratings from other nodes, after which TRM quantifies the nodes’
trustworthiness [8]. In TRM, each node is assigned trust or reputation scores
that represent its trustworthiness level, using which other nodes in the
network may conveniently infer their trustworthiness level and the data they
generate. TRM can benefit from the incorporation of blockchain with its
salient features, such as transparency, tamper-resilience, verifiability, and
pseudonymity. These features improve TRM resiliency and address the
aforementioned issues of conventional TRM [9]. For instance, the
decentralisation inherent in blockchain eliminates the reliance on a Trusted
Third Party. Trust related data can also be stored on the shared immutable
ledger, maintaining integrity and high availability. Smart contracts can be
employed to enforce trustworthy collection of collaboration evidence and trust
calculation.

However, designing a blockchain-based TRM to provide strong assurance of
trustworthiness in data and participants behaviour is a non-trivial task.
There exist many challenges in incorporating TRM for blockchain-enabled IoT,
which specifically depend on the IoT applications, as each application has its
trust challenges with their unique features and requirements.

In this paper, we present our research experiences in designing TRM framework
for various applications of blockchain-enabled IoT, namely access control,
intrusion detection systems, supply chain management and network resource
sharing. First, we describe the definition of trust and reputation with its
subtle differences and outline the fundamental stages in TRM. Subsequently, we
discuss the salient features of blockchain that would improve the design of
TRM. We highlight specific trust issues in each application domain that
necessitates the incorporation of TRM for addressing these specific
challenges. We summarise the lessons we learned in the process, which can
guide future research in this area. Lastly, we highlight several open research
challenges in blockchain-based TRM for future work. We present a summary of
our research experiences in Fig. 1.

##  2 Overview of Trust and Reputation Management

Trust represents a subjective belief towards an entity (trustee) regarding its
behaviour, which is gradually built from repetitive interaction [10]. The
higher the trust to a trustee, the more it is expected to do things that would
benefit the trustor, i.e., an entity that trusts the trustee. On the other
hand, reputation can be seen as a collection of trust opinions towards an
entity, which represents a global view of an entity’s trustworthiness [11].

TRM is designed to quantitatively evaluate the accountability of each node in
an IoT network by assigning numerical scores to each node which represents its
trustworthiness level [9]. TRM works by collecting direct experience or
feedback from the peer nodes to quantify the trust level. In practice, these
scores are useful for managing the risks in communicating with other nodes in
a dynamic and hostile IoT network. For instance, in a mobile crowdsourcing
scenario, wherein each node individually collects data and contributes to the
system, TRM is essential to provide a first layer safeguard to determine the
trustworthiness of the collected data [11].

In the following subsections, we describe the inherent building blocks of a
TRM with its salient properties. We also discuss how blockchain would enhance
TRM in the context of IoT applications.

Figure 2: The five fundamental stages of TRM. While each IoT application has
its own trust issues and unique challenges, the TRM follows these generic
stages from modelling until the scores are applied to address the trust
issues.

###  2.1 TRM Fundamentals

In general, the evolution of trust and reputation follows similar pattern as
in real world social interactions, wherein, these properties are gradually
developed as more interactions happen. Naturally, these interactions may
increase or decrease the trust level, depending on the perceived interaction
experience. In the following, we describe the fundamental properties of trust
and also the stages in TRM from trust modelling to the application of trust
scores.

####  2.1.1 Trust Dimensions

Depending on specific applications, TRM can model the trust using a simple
single approach where trust is derived from single attribute. TRM can also use
multi-dimensional approach where more detailed considerations are taken into
account [9].

  * •

Single-dimension. Single-dimension trust model could be an option where there
is a need to focus on a specific metric to build trust. In addition, this type
of trust model may be the best option if the network consists of mainly
resource-constrained nodes.

  * •

Multi-dimension. To capture multiple aspects of trust, multi-dimension trust
model can be applied, where trust is derived from multiple attributes with
certain weightings to put emphasise on several attributes over another. In
practice, this model is preferred as it provides a richer representation of
trust.

####  2.1.2 Types of Trust

In general, IoT applications rely on the collection of sensor observations and
the interactions between nodes. As such, trust can be generally classified
into three main categories as follows.

  * •

Data-based. In this category, trust is derived and calculated from the quality
of the observation data, which is prone to noise, sensor drift, or bias. Here,
the trust is directly correlated with the veracity and the authenticity of the
data. Therefore, data-based trust relies heavily on data validation, which may
include data correlation with other nearby sensor observations to validate if
there is any degradation of the data quality.

  * •

Behaviour-based. On the other hand, behaviour-based trust puts more emphasis
on trusting the behaviour of the nodes. For instance, positive interaction is
defined as a behaviour that conforms to the expected behaviour, which would
increase the trust level. Conversely, a negative experience would degrade the
trust level. Here, policy enforcement plays a significant role as a means to
validate node’s behaviour.

  * •

Hybrid. In certain scenarios, utilising one trust category may not be adequate
to derive trustworthiness. For example, supply chain applications necessitate
that trusted traders are seen as the parties who supply reliable data and
conform to the pre-agreed trade rules. In this scenario, trust and reputation
could be evaluated with a consideration of both the data-based and behaviour-
based approach, where weightings can be used to balance these two trust
approaches.

####  2.1.3 Fundamental Stages in TRM

TRM empirically evaluates trust in five sequential stages, shown in Fig. 2. We
discuss the stages in more details as follows.

  1. 1.

Trust modelling. The initial stage in TRM deals with how the trust and
reputation should be quantified, which defines the attributes and parameters
for trust quantification. The trust model oftentimes is application specific
and cannot be generalised. Some attributes include the veracity of the sensor
observation correlated with adjacent sensor data and the adherence of a node
to the pre-determined policies.

  2. 2.

Information gathering. Subsequently, TRM gathers necessary information as per
the trust model, using which the trust and reputation for each node are later
evaluated. In general, TRM collects the information from two sources: 1)
direct experience with a particular node; and 2) in the absence of prior
interaction, TRM collects recommendation or feedback from other nodes.

  3. 3.

Score calculation. TRM could utilise multiple techniques to calculate the
trust and reputation scores, depending on the pre-defined trust model and
collected trust information. In blockchain-enabled TRM, smart contracts may be
used to calculate the trust and reputation score in a transparent manner,
which we explain in more detail in Section 2.2.

  4. 4.

Score propagation and update. The trust and reputation scores should be
propagated to all participating nodes in the network, such that everyone is
notified of the new scores. In addition, TRM may follow 1) time-driven update,
in which the scores are continuously updated on regular basis, or 2) event-
driven approach, where the scores are updated on certain events. The
propagation may follow a centralised or distributed model, depending on the
application architecture. In blockchain-based TRM, the score propagation
occurs in parallel with block propagation, as the updated scores are stored in
each new block.

  5. 5.

Score application. Finally, the trust and reputation scores could be applied
to certain IoT scenario. For instance, the scores can be used as the basis to
apply incentives and penalty mechanisms. We discuss various applications of
TRM in more detail, based on our experience, in Section 3.

###  2.2 Blockchain Properties for TRM

As an integral component in blockchain-enabled IoT, TRM can benefit from the
incorporation of blockchain, such as the transparent and deterministic nature
of smart contracts, which we discuss in the following.

  * •

Decentralisation. In blockchain-enabled TRM, there is no centralised authority
that manages the collection of trust evidence and evaluation of trust scores.
That way, the risks of having a single point of failure or malicious authority
can be effectively eliminated. Blockchain is managed in decentralised manner,
wherein all nodes are participating in the process, with consensus algorithm
as the safeguard to any fraudulent actions. Blockchain introduces a
distributed ledger where the trust data is stored in an immutable and
transparent fashion.

  * •

Immutability. The inherent data structure of the distributed ledger, which
consists of inter-linked blocks with hash cryptography, makes it difficult if
not almost impractical to tamper with the data. TRM can benefit for this
immutability property as the trust evidence could be securely stored without
any risk of data alteration.

  * •

Transparency. The transparent nature of blockchain would allow for reliable
auditability, as the distributed ledger could act as a traceable source for
auditing the trust calculation with the corresponding trust evidence. In
addition, the smart contract provides a transparent trust calculation.

  * •

Smart contract. Smart contracts allow for a deterministic and trusted
execution of business logic in a decentralised manner. In blockchain-enabled
TRM, smart contracts play a significant role of enforcing the TRM business
logic in a transparent manner. For example, an IoT node may submit feedback
about its recent experience with a service provider, after which the smart
contract updates the provider’s trust and reputation score.

  * •

Pseudonymity. Blockchain utilises public-key cryptography instead of real
identities for its authentication mechanism. An IoT node is thus recognisable
by its public key, which to some extent protects its privacy, as the real
identities are not exposed to public.

##  3 Applications of TRM for IoT

In this section, we discuss our research experiences in TRM for blockchain-
enabled IoT. We first discuss specific trust issues in different application
domains and present TRM frameworks to address them. We then identify the
unique challenges of incorporating TRM to these application domains, which
influence the design of the TRM framework. We present a summary of our
experiences in Fig. 1.

Figure 3: Blockchain-based TRM for dynamic access control schemes in IoT [12].
Blockchain is an instrumental building block where smart contract is utilised
to enforce transparent and auditable trust calculation and access control
enforcements. Figure 4: The design of our trustworthy CIDS framework [17],
where TRM plays an important role in validating contributed detection rules
and assigning score to each of the validated rules.

###  3.1 IoT Access Control

Access control aims to enforce restrictions on the actions that authenticated
users can execute on a particular IoT system [13]. With access control, or
authorisation, any illegitimate access to protected resources or security
breaches can be avoided. For instance, resource owners can define an access
policy to a particular resources in IoT networks, wherein other nodes need to
authorise themselves to access the resource.

Challenges. There have been several proposals for blockchain-based IoT access
control schemes [14, 15], where the schemes work by enforcing static access
policies, based on an assumption that the authenticated nodes would always
show benign behaviour. However, authenticated nodes may be compromised post-
authentication and become malicious, which may significantly disrupt the
network. A prominent example is the extensive damage caused by the Mirai
botnet [5]. This motivates the need to design dynamic access control
mechanisms, without overlooking the fact that authorisation demands strict
consideration of certain attributes to access a resource.

Contributions. To address the aforementioned challenges, we proposed a trust-
based blockchain authorisation for IoT [12] (see Fig. 3), where we utilise TRM
to supply additional attributes to a decentralised Attribute-Based Access
Control (ABAC) mechanism. Our TRM framework progressively quantifies the
trustworthiness of both Service Consumers (SC) and Providers (SP) into trust
and reputation scores, using which we define more attributes to enrich the
ABAC mechanism, hence achieving dynamic access policies. In general, the trust
and reputation scores are calculated based on the nodes’ adherence to the
access control policies. In our definition, trust is a subjective belief on
SC’s past behaviour, which may aid in forecasting SC’s future behaviour. On
the other hand, reputation is seen as the public view of SC’s past behaviour
from multiple trust sources. We define three smart contract to manage the
access control mechanism in a transparent and auditable manner, namely policy,
attribute and TRM contracts, as seen in Fig 3. Policy contract stores and
enforces the access policies based on the required attributes, while attribute
contract is the point of contact when policy contract needs to verify some
attributes; and TRM contract calculates and updates the trust and reputation
scores. In our framework, we note that continuous evaluation of the nodes
behaviour can be a potential way to detect and eliminate compromised nodes in
the IoT network [16]. Our solution is implemented on a private Ethereum
network, using which we benchmark our TRM framework. The experimental results,
which include the latency, score evolution and gas consumption of the smart
contracts, indicate the applicability of our solution in the IoT context.

In [18], we proposed an improvement of our dynamic access control framework,
where we emphasise privacy preservation and efficient trust computation. To
conceal users’ personal information from public access, where we utilise
multiple private side-chains to privately store sensitive users’ attributes.
We conceive a lightweight TRM model that reduces the trust computation load by
using a simple recursion. Our improved framework is implemented on a public
Rinkeby Ethereum test-network interconnected with a lab-scale testbed of
Raspberry Pi computers. The evaluations consider various performance metrics
to highlight the applicability of the proposed solution for IoT contexts.

Figure 5: The layered architecture of TrustChain [22], which constitutes three
interconnected layers. Data layer supplies multi-faceted observation from
which the trust and reputation scores are derived. Figure 6: An architecture
overview of DeTRM [24], which consists of four layers. The modular design of
DeTRM allows for more applications as other application can be built on top of
the blockchain layer.

###  3.2 Intrusion Detection System

Intrusion Detection Systems (IDS) have been widely used in the industry to
alert the network administrator of any possible threats in the IoT networks
[17]. The proliferation of IoT devices in diverse application domain has
consequently enlarge the attack surface, which has raised the urgency to
defend the network against emerging attacks [19]. In recent years, blockchain
has prompted the new notion of decentralised Collaborative IDS (CIDS), wherein
blockchain enables IDS nodes to collaboratively contribute relevant
information, such as intrusion detection rules, which helps increase the
overall capability of the IDS nodes to detect unknown threats [20].

Challenges. However, the existing work of blockchain-based CIDS generally
presume that the contributor nodes are continually honest and overlooks the
necessity to regularly evaluate the trustworthiness of the participating IDS
nodes. Without proper trust evaluation in CIDS, compromised nodes may share
deceptive detection rules on purpose, which would jeopardise the network to
detrimental attacks [21].

Contributions. To address the trustworthiness issues in CIDS, we proposed a
trustworthy blockchain-based CIDS framework [17]. In our proposed framework,
the TRM progressively evaluates the trust level of each CIDS node by assessing
the quality of the contributed detection rules. In Fig. 4, we group the CIDS
nodes into three categories, namely contributor, validator and regular nodes.
We introduce the notion of validator nodes, who perform off-chain assessment
of the submitted detection rules. Subsequently, validator nodes assign
numerical scores to each contributed rule, using which the trustworthiness
level of CIDS node is determined. The TRM framework employs smart contracts to
transparently update and calculate the scores. As such, each CIDS node can
conveniently update its local detection database with the validated detection
rules. To achieve decentralisation in the storage of detection rules, we
utilise a peer-to-peer storage system, such as The InterPlanetary File System
(IPFS) [23]. Our framework is designed to be blockchain-agnostic, such that it
can be implemented on any blockchain platforms with support in smart
contracts. Signature-based CIDS is used as an illustrative example, but our
TRM framework can be also generalised to other CIDS types, such as Machine
Learning (ML) or anomaly-based. We tested our proposed framework on a lab-
scale Ethereum private network, where results indicate that our proof-of-
concept performs within the expected benchmarks of Ethereum, highlighting its
feasibility.

###  3.3 Supply Chain

Supply Chain Management Systems (SCMS) provide an end-to-end tracking of the
traded commodities, which spans from sourcing the raw materials to the final
retail shops [25]. SCMS create digital records of the traded goods, referred
to as digital assets, using which SCMS tracks the transfers of commodities and
ownership changes. IoT has been implemented to improve SCMS [26] by
information gathering at various stages of supply chain, for instance sensor
networks that provide automated monitoring of commodity storage. In recent
years, blockchain has shown its potential to enhance conventional SCMS due to
its inherent data structure of immutable time-stamped records [22]. Supply
chain activities such as sourcing goods, trading, manufacturing, ownership
transfer etc., can thus be logged on blockchain in an immutable way. This
increases transparency of products as they move through their life cycle. In
addition, blockchain’s smart contracts are able to automate certain
operations, such as, provenance, compliance checking and trust computation,
based on supply chain events and activities in a secure and verifiable manner.

Challenges. However, blockchain alone cannot ascertain the authenticity and
veracity of the IoT observations provided by supply chain participants. In
fact, IoT nodes may turn faulty from time to time and thus record and send
inaccurate observation. As such, the authenticity and the integrity of the
supply chain data become questionable. The fact that the recorded data becomes
immutable once stored on blockchain exacerbates this trust issue on supply
chain data, as the ledger would be contaminated with bad data [27].

Contributions. To address the trust issues in blockchain-enabled supply chain,
we argue that TRM would be an effective solution. In [22], we proposed
TrustChain, where we derive and evaluate trust from multiple sources, namely
sensor observations, regulatory inspections and trade events, as seen in Fig.
5. TrustChain provides flexibility in calculating the trust scores at
different perspective, for instance, at the commodity level or a particular
supply chain participant to evaluate its role in the supply chain. Thus we
assign trust scores to the traded commodities and the traders both. We use two
types of smart contracts, namely quality and rating contracts, to enlist
certain quality criteria for the corresponding commodity, e.g., temperature
boundary and contract identifier, and to compute the scores after a trade
event, respectively. The use of smart contracts enables transparent automation
of trust score calculation, using which we apply reward and penalty mechanisms
to encourage truthful contribution of data to the blockchain. Our proof-of-
concept implementation of TrustChain in Hyperledger Fabric reveals that our
proposed trust mechanisms for supply chain only incur minimal overheads in
throughput and latency.

While TrustChain was among the first proposals in incorporating TRM for IoT-
enabled supply chains, we note that TrustChain assumes that there are no
alterations of commodities as they make their way from the primary producers
to the retail shelves. However, this assumption is unlikely in real world
supply chain scenarios, as it is quite common to have production processes
somewhere in the supply chain, such as production of cheese products from raw
milk. In addition, TrustChain fully assigns blockchain operation and
maintenance to a business network administrator, which makes the use of
blockchain questionable as it is essentially a centralised TTP.

To address the issues, we proposed DeTRM [24], where we design a tailored TRM
framework for SCMS by considering inherent supply chain operations into our
trust model, for instance, production of new products from existing
commodities in the SCMS. Therefore, DeTRM tracks all trust information within
the entire supply chain life cycle, i.e., from sourcing the raw material to
the retail shelf. To derive trustworthiness, DeTRM correlates sensor
observations to evaluate the data quality into numerical measures and assesses
the behaviour of supply chain entities against the pre-agreed trade contracts.
To establish a solid decentralised framework, we require all supply chain
entities, such as producers, traders and distributors, to run blockchain peer
nodes for maintaining a network of consortium blockchain. In our framework, we
exclude the end product customers from the blockchain network, but they could
track the trust and provenance information of the products by scanning the
imprinted QR code, see Fig. 6. The proof-of-concept implementation of DeTRM in
a lab-scale Hyperledger Fabric network highlights that our framework incurs
minimal overheads, with regards to resource utilisation, latency and
throughput.

Figure 7: The overview of our resource sharing mechanism [28], where the
resource users offload computation tasks to resource owners. Here, the side-
chain hosts the TRM contract, while the main-chain records the sharing
agreements via broker and sharing contracts.

###  3.4 IoT Resource Sharing

Resource sharing in IoT network is intended to achieve effective utilisation
of scarce network resources, such as radio spectrum and computation
infrastructures [29]. Resource sharing can also alleviate the computation
workload on constrained IoT devices, such as edge computing [30]. Blockchain
has received significant attention from research and industry communities as
an integral building block to realise the Dynamic Resource Sharing (DRS) [31].
Due to the verifiable automation enabled by blockchain, DRS can significantly
improve the resource utilisation, compared to the conventional resource
allocation schemes, which tend to be static and inflexible. Blockchain can
effectively act as a trustless intermediary to facilitate communications
between parties in resource sharing, making the coordination and cooperation
more effective and efficient.

Challenges. However, blockchain-based resource sharing for IoT still suffers
from several challenges. While blockchain provides consistency and trust in
the stored data within a resource sharing environment, blockchain alone cannot
ascertain the trust in the participating nodes when performing resource
sharing. The lack of trust would discourage nodes in the network to share or
use other resources, as some malicious nodes may present and can potentially
launch adverse attacks. These trust issues would undermine the initial goal of
resource sharing, as its effectiveness is reduced.

Contributions. To address the above mentioned trust issues, we proposed a TRM
framework to support resource sharing, which aims to encourage cooperation in
resource sharing by evaluating participants’ reliability after each sharing
instance [28]. We consider the use case of computation resource sharing in
edge computing, where the resource users offload computation tasks to resource
owners (see Fig.7), after which the trustworthiness of the resource owners are
determined. Our framework also allows for both resource owners and users to
employ changeable keys to obfuscate their transaction traces in the network,
avoiding the de-anonymisation attacks that may reveal sensitive information of
critical network infrastructures. Our framework exploits smart contracts to
provide auditable trust calculation, where a recursive trust computation is
proposed to meet the efficiency requirements of trust calculation in 6G
networks. Our framework demonstrates how blockchain, with the aid of a TRM,
could provide reliable assurance in the trust between participating nodes in
the network, thus reducing the inherent risk of resource sharing. We developed
a proof-of-concept implementation of our proposed framework on lab-scale
private networks. The experimental results signify the feasibility of our
framework, as it only incurs minimal overheads with regards to gas consumption
and overall latency.

##  4 Discussion

In this section, we summarise the lessons learned from our journey in
incorporating blockchain-based TRM in various IoT applications. We also
discuss several future research directions for TRM in blockchain-enabled IoT.

###  4.1 Lessons Learned

We note that there are four lessons from our research in blockchain-based TRM,
which are discussed in the following.

  * •

Trust model. Various IoT applications, discussed in Section 3, point out that
each application requires a specific trust model that is influenced by its
unique challenges. While one can borrow a trust model conception from a
particular domain, the model still needs to be adjusted to match with the way
trust is evaluated in the adopted domain.

  * •

Smart contracts. With the incorporation of blockchain for TRM, smart contracts
become an instrumental building block, where the main TRM logic is
implemented. All IoT applications in Section 3 utilise smart contracts to
transparently collect trust evidence and calculate the corresponding trust and
reputation scores. Smart contract also allows for better auditability in the
TRM process, which bring more confidence in the TRM framework as all TRM
operations can be transparently tracked on-chain.

  * •

Off-chain component. While blockchain-based TRM benefits from transparent on-
chain computation, some underlying TRM processes cannot be implemented on-
chain due to their computation complexity. In certain conditions, off-chain
computation is required to complement the smart contract, as demonstrated in
our TRM framework for CIDS. In addition, blockchain may not be an ideal
storage medium for large amount of data, as in the case of supply chains,
which highlights the need to incorporate off-chain storage, such as cloud or
peer-to-peer storage.

  * •

Changeable keys. Blockchain-based TRM benefits from the utilisation of
pseudonyms, which conceal the node’s real identity. However, it possible to
de-anonymise the pseudonyms [32], especially if the public keys are used
repeatedly. IoT nodes may opt to use changeable keys in different
transactions. However, the TRM needs to be aware that multiple keys correspond
to the same node, when calculating the trust and reputation scores.

###  4.2 Future Research Directions

While there have been several works that have developed TRM for blockchain-
enabled IoT, we note that there are still several open challenges, which
include security, privacy, interoperability and scalability.

Security. It is known that TRM is susceptible to attacks, wherein the
adversaries attempt to illegitimately increase their reputation score and ruin
others. Incorporation of blockchain to TRM may render the conventional defence
schemes unusable, which highlights the need to design a robust blockchain-
based TRM against these trust-related attacks. Some common attacks include: 1)
Sybil attack, where adversaries create fake identities to increase or ruin
trust scores to their benefit; 2) ballot-stuffing, where a node illegitimately
increases its score, for instance by colluding with other nodes; and 3) bad-
mouthing, where the adversaries attempt to ruin honest users’ reputation
scores, for instance by submitting fake negative feedback. Further research
should also study the possible impact on the network reliability if these
security measures fail.

Privacy. As discussed in Section 4.1, there is the need to design privacy-
preserving TRM using changeable keys. Zero-Knowledge Proof (ZKP) is a
potential technique for concealing real identities, which works by validating
a given mathematical proof without revealing any information that could be
traced back to the user’s real identity. In ML-based TRM [28], federated
learning could be an approach to achieve privacy-preservation as some trust
related data may reveal sensitive information. In addition, future research
also needs to study the mechanisms to conceal feedback information, which
would encourage IoT nodes to submit truthful feedback.

Interoperability. Future IoT networks, for instance 6G-enabled IoT, will offer
integration of autonomous networks, which may run an independent TRM [28]. To
allow collaboration between these autonomous networks with their own TRM,
future work should aim to devise a reliable and trustworthy trust and
reputation scores transfer between independent TRMs. In addition, TRM often
experience the lack of prior interactions which may hinder the calculation of
trust and reputation scores. With interoperable TRMs, the absence of prior
interactions can be addressed by transferring trust or scores from another TRM
that has already collected trust evidence about a particular entity. To this
end, future research should also aim to provide cross-chain data transfer for
transferring trust evidence between independent blockchains.

Scalability. Blockchain-based TRM inherently suffers from limited blockchain
throughput, which makes it difficult to scale. As the throughput issues
actually stem from the resource consumptive consensus algorithm, future
research should aim to provide fast consensus algorithm, such as Proof-of-
Stake, which has recently become the default consensus algorithm for Ethereum
[33]. The use of fast and efficient consensus can help TRM scale, especially
in large-scale IoT deployments. In addition, future research is required to
enhance the scalability of the append-only shared ledger, which can grow
exponentially as old trust and reputation data would remain in the ledger as
the data for new interactions continues to be recorded on-chain.

##  5 Conclusion

In this paper, we presented our research journey over the past few years,
where we address trust issues in various IoT applications, including access
control, intrusion detection systems, resource sharing and supply chains. We
noted that each application has its own trust issues and unique challenges
which influence the design of the corresponding TRM. We also discussed the
lessons we learned in the process and highlighted several open research
challenges in blockchain-based TRM research for future work, which include
security, privacy, interoperability and scalability.

## Acknowledgement

The work has been partially supported by the Cyber Security Research Centre
Limited whose activities are partially funded by the Australian Government’s
Cooperative Research Centres Programme.

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