CHAPTER: INTRODUCTION

CHAPTER: BACKGROUND: CHASE-LEV DEQUE

CHAPTER: BACKGROUND: IRIS SEPARATION LOGIC

CHAPTER: VERIFICATION OF CHASE-LEV DEQUE WITHOUT RESIZING

CHAPTER: FULL VERIFICATION OF CHASE-LEV DEQUE

CHAPTER: VERIFICATION IN EXTENDED SETTINGS

CHAPTER: CONCLUSION

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# Transport of particles through RO membrane in steady state condition

Rahul Karmakar  J Chakrabarti  Department of Physics of Complex Systems, S. N.
Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake
Kolkata 700106, India.
<EMAIL_ADDRESS>
###### Abstract

Reverse Osmosis(RO) membranes are widespread nowadays for separating the
solvent from a solution. RO membranes are made of polymer matrix. Experiments
show changes in relative interaction of solvent and solute with RO membrane
matrix lead to changes in solvent permeation, solute rejection and fouling.
Here we study microscopically separation of binary mixture through RO
polymeric membrane using Molecular dynamics simulation to rationalize the
experimental observations in terms a simple model. We find better solvent
permeation and solute rejection with increasing interaction between solute and
membrane with increase in fouling of solute inside membrane. We also find that
stronger solvent membrane interaction reduces fouling. However, this leads to
poorer performance of RO in terms of solvent permeation and solute rejection.

##  1 Introduction

Reverse osmosis(RO) is a method where external pressure is applied to overcome
the natural osmotic pressure and enable solvent flow from higher to lower
solute concentration. As a result, RO technique is one of the most advanced
ways to extract solvents from mixtures, and widely used in industries like
water treatment[1, 2], biofuels[3], petrochemical[4], pharmaceuticals
processes[5] and so on. RO is used even in households for purification of
water. RO for separation of smaller molecular size solute from organic
solvents has also been developed in recent times[6]. Such a widespread small
to large scale utility of RO demands thorough understanding of RO membrane
performance from microscopic considerations.

Thin Film Composites (TFC), consisting of polyamide layer around $10-100$ nm
thickness are widely used as RO membrane [7]. TFC membrane is prepared through
interfacial polymerization(IP) process where two monomers in different
phase(one in aqueous and other in organic) reacts only at the interface.
Permeation or solvent recovery ($P$) in RO is estimated by how much solvent is
transported from solution on one side of membrane to other side in presence of
applied pressure overcoming osmotic pressure. The other important aspect is to
reject solute (R) out of the solution allowing only solvent to permeate
through the membrane. Another important criterion of membrane is fouling
($F$). It is the measure of amount of solute accumulated inside the membrane
during permeation. This degrades the future performance of membrane. The
performance of membrane is to increase solvent permeation and solute
rejection, while reducing the membrane fouling.

Current studies on RO revolve around better design of RO membrane[8]. The
performance of membrane is affected by thickness, pore size distribution,
roughness, hydrophilicity, chemical properties[7] etc. Experiments show that
improving hydrophilic tendency by introducing different chemicals improve
water permeation and salt rejection[9]. Studies also suggest that tuning TFC
membrane using nanoparticles[10] as additives for better permeability.
Incorporating ions like copper $Cu^{++}$ [11] in membrane matrix show better
pesticide removal from water and also better antibacterial properties.
Performance in permeation increases introducing polypyrole layer in TFC
membrane[12]. Nonpolar mixtures are also separated using hydrophobic rich TFC
membrane[6, 13]. Selective separation in both organic and polar solvents is
recently developed using trianglamine macro-cycle as a monomer for cross-
linked membranes through inter-facial polymerization(IP) process[14].
Different permeation of solvent is observed in different solutions[15].
Experimental studies show anti-fouling properties by incorporation graphene
oxide (GO) sheets in the polyamide layer[16]. Theoretical studies have been
carried out using full atomistic simulation using non-equilibrium molecular
dynamics(NEMD)[15, 17]. Free energy profiles of water and ions are also
studied inside thin films using the metadynamics technique to estimate barrier
for permeation and rejection process[18]. However, there is yet no attempt
addressing the generic features of solute and solvent interaction with the
membrane which control their transport through the RO membrane.

Here, we investigate a simple microscopic model to study the solvent and
solute motion in the steady state through a RO membrane in presence of an
external force to overcome osmotic pressure. A schematic of the model is shown
in Fig. 1. We place the membrane in contact with two reservoirs: one of the
reservoirs (Mixture reservoir, MR)($z=0$ to $z=z_{A}$) consists of binary
mixture of LJ particles of different sizes, and the other reservoir only the
smaller LJ particles (Pure reservoir, PR)($z=z_{B}$ to $z=z_{C}$). The volume
fraction of solvent in MR reservoir is lower than the PR reservoir, creating
an osmotic pressure which is counterbalanced by an external force, at $z=0$
and falls linearly to zero at $z=z_{A}$.

We study the system using molecular dynamics simulation. We fix size ratio of
solute and solvent, driving force, and ambient temperature. There are several
interaction energy parameters in the system: polymer bead-bead interaction,
polymer bead-wall interactions, solvent-solvent interaction, solute-solute
interaction, solvent-solute interaction which are also held fixed. We tune the
relative strength of the interactions between the solvent and solute particles
with the membrane. We find that the stronger interaction between solute and
the membrane helps in better solvent permeation, solute rejection but with
substantial membrane fouling. On the other hand, stronger solvent membrane
interaction helps to decrease fouling. However, the solvent permeation and
solute rejection also decrease concomitantly.

##  2 Model potential and simulation details

The non-bonded interaction between two monomers i and j of the polymeric
strand with separation $r_{ij}$ is taken as the LJ 12-6 potential:

$V_{\alpha,\beta}(r_{ij})=\begin{cases}4\epsilon_{\alpha,\beta}[(\frac{\sigma}{r_{ij}})^{12}-(\frac{\sigma}{r_{ij}})^{6}],&\text{if
}r_{ij}<2.5\sigma\\\ 0,&\text{otherwise}\end{cases}$

Here $\alpha(=h,p)$ and $\beta(=h,p)$ stand for the bead types and $r_{ij}$ is
separation between two beads. $\epsilon_{h,h}$ is the potential depth for
interaction between h-h beads and $\epsilon_{p,p}$ that for the interaction
between p-p beads. Here the interaction range parameters are $\sigma_{p,p}$
and $\sigma_{h,h}$. The cross-interactions $\epsilon_{p,h}$ and $\sigma_{p,h}$
are taken to follow the Berthelot rule. The bonded interaction corresponding
to stretching between two neighbouring beads at separation $r_{ij}$:

$V_{bond}(r_{ij})=\frac{1}{2}k_{b}(r_{ij}-r_{0})^{2}$ (1)

where $r_{0}=1.5\sigma_{h,h}$ is the equilibrium distance between monomers and
$k_{b}$ the force constant. The change in bond angle also costs energy given
by:

$V_{angle}(\theta)=\frac{1}{2}k_{\theta}(\theta-\theta_{0})^{2}$ (2)

where $k_{\theta}$ the force constant and
$\theta=\cos^{-1}(\frac{\vec{r}_{ij}\cdot{\vec{r}_{jk}}}{|\vec{r}_{ij}||\vec{r}_{jk}|})$
is the angle produced by three consecutive monomers i,j,k and $\theta_{0}$ is
equilibrium angle, set to 114 degrees [19].

Next, we specify the wall interactions with the polymeric beads. The
$\beta(=wh,wp)$-th type wall interacts with a bead of $\alpha$-th type via the
LJ 9-3 potential:

$V_{\alpha,w_{\beta}}(z_{i})=\epsilon_{\alpha,w_{\beta}}[\frac{2}{15}(\frac{\sigma}{z_{i}})^{9}-(\frac{\sigma}{z_{i}})^{3}]$
(3)

Here $z_{i}$ is the z-coordinate of i-th particle. The $\sigma$ values for all
the interactions are taken to be the same but
$\epsilon_{h,wh}>\epsilon_{p,wh}$ and $\epsilon_{p,wp}>\epsilon_{p,wh}$.

The solvent S particles interact with each other via the LJ 12-6 potential
with strength $\epsilon_{S,S}$ and length parameter $\sigma_{S,S}$. The solute
L particles interact with each other with LJ 12-6 parameters $\epsilon_{L,L}$
and $\sigma_{L,L}$. Here the cross interaction parameters $\epsilon_{S,L}$ and
$\sigma_{S,L}$ are taken to follow the Berthelot rule. $\epsilon_{S,h}$ and
$\sigma_{S,h}$ and $\epsilon_{S,p}$ and $\sigma_{S,p}$ interaction parameters
of S with h and p polymeric beads respectively. Similarly $\epsilon_{L,h}$ and
$\sigma_{L,h}$ and $\epsilon_{L,p}$ and $\sigma_{L,p}$ are parameters for
interactions of L with h and p beads respectively.

We take $\epsilon_{h,h}$ as unit of energy. The unit of length is
$\sigma_{S,S}=0.5$ nm. The unit of mass is the mass of water of density 1
$gm/cm^{3}$ in a sphere of diameter 0.5 $nm$. The polymeric bead diameter
$\sigma_{p,p}=\sigma_{h,h}=2\sigma_{S,S}$. The solute particle diameter is
taken $\sigma_{L,L}=1.8\sigma_{S,S}$. For all the wall interaction the length
parameters, $\sigma=\sigma_{S,S}$. The fixed energy parameters are given in
Table 1. The unit mass is taken to be that of S particles $m_{S}$. The masses
of all different particles are given in Table 2. Using the values of mass
$m_{S}$, $\sigma_{S,S}$ and $\epsilon_{h,h}$, we estimate a time scale,
$\tau(=\sqrt{\frac{m_{S}\sigma_{S,S}^{2}}{\epsilon_{h,h}}})\sim 6$
picoseconds. Here we vary: $\bar{\epsilon}_{S}=\epsilon_{S,h}/\epsilon_{S,p}$
and $\bar{\epsilon}_{L}=\epsilon_{L,h}/\epsilon_{L,p}$ at reduced temperature
$T^{*}=\frac{k_{B}T}{\epsilon_{h,h}}=1$.

Table 1: Table for $\epsilon$ Nature of interaction |  dimensionless variable |  dimensionless value
---|---|---
h,h |  $\epsilon_{h,h}$ |  1.0
p,p |  $\epsilon_{p,p}$ |  0.33
h,wh |  $\epsilon_{h,wh}$ |  1.0
p,wh |  $\epsilon_{p,wh}$ |  0.033
p,wp |  $\epsilon_{p,wp}$ |  0.33
h,wp |  $\epsilon_{h,wp}$ |  0.033
S,S |  $\epsilon_{S,S}$ |  1.0
L,L |  $\epsilon_{L,L}$ |  1.0
Table 2: Mass of particles Mass of different particles |  dimensionless value
---|---
$m_{S}$ |  1.0
$m_{L}$ |  5.832
$m_{h}$ |  8.0
$m_{p}$ |  8.0

The model involves several steps: (1) The model RO membrane(M) is prepared
using 9 identical polymeric strands having randomly distributed 50 monomer
beads of two different types in a number ratio 50:50 with overall packing
fraction $\frac{\pi}{6}\frac{N}{V}\sigma_{h,h}^{3}=0.26$. One type of
beads(h-beads) mutually interact more strongly compared to the other class of
beads (p-bead) to mimic non-polar and polar moieties respectively. The network
is confined between two asymmetric walls, each of which is having favourable
interaction with a given type of bead. This mimics the asymmetric interface
between organic and aqueous phases where the membrane is stabilized in
experimental conditions. We consider a rigid network with
$k_{b}=k_{\theta}=k=100$ for simplification. We choose the interaction
parameters such that the polymeric network is uniformly spread throughout the
gap between two walls. (2) We solvate the membrane with the solvent (S)
particles keeping the membrane in contact with two reservoirs of S particles
where the confining walls are removed. (3) Then one of the reservoirs, (MR) is
taken as a mixture of solutes (L) and solvents (S), while the other one, (PR)
with pure S particles to study the reverse osmosis. The z width of membrane
$Z_{M}=12$. The z width of MR and PR are taken $Z_{MR}=Z_{PR}=20$.

We perform all the simulations using the LAMMPS package [20] and the Nose-
Hover thermostat. The time step for integration is taken $0.001\tau$. All the
quantities are averaged over three different independent trajectories each
$1000000\tau$ long. We take $L_{x}=30$ and $L_{y}=20$, in x- and y- directions
respectively with the periodic boundary conditions (PBC) and no PBC in
z-direction. The primary quantity we calculate is the packing fraction for
different species of particles,
$\eta_{\beta}^{\alpha}(t)=\frac{\pi}{6}\rho_{\beta}^{\alpha}(t)\sigma_{\beta,\beta}^{3}$,
here $\rho_{\beta}^{\alpha}(t)$ is number density of particles of the
$\beta$-th (= S,L) species at time $t$ in the $\alpha-th$ (=MR, M and PR)
region. We quantify the membrane performance by the solute rejection rate,
$R=100(1-\frac{\eta_{L,steady}^{PR}}{\eta_{L}^{MR}(0)})$; membrane fouling
$F=100\eta_{L,steady}^{M}$; and solvent permeation or recovery
$P=100(\frac{\eta_{S,steady}^{PR}-\eta_{S}^{PR}(0)}{\eta_{S}^{PR}(0)})$. Here
$\eta_{S}^{PR}(0)$ is initial partial packing fraction of S in PR and
$\eta_{S,steady}^{PR}$ partial packing fraction of S in PR in steady state.
$\eta_{L}^{MR}(0)$ initial partial packing fraction of L in MR and
$\eta_{L,steady}^{PR}$ partial packing fraction of L in PR in steady state.
$\eta_{L,steady}^{M}$ packing fraction of L inside M in steady state.

Figure 1:  Schematic diagram: M is the RO membrane in the middle solvated with
S(grey) particles. MR is on left mixture reservoir containing S(grey) and
L(black) particles. PR is on the right pure reservoir containing S particles
only initially. Two reflecting wall is attached in two extreme end. Force is
applied to MR.

##  3 Results

We characterize interpenetrating network of identical polymeric strands by the
inter-strand correlation function $C(r)$ defined in our previous works [21]
where $r$ is the cross-strand bead-bead distance. The C(r) data in Fig. 2(a)
shows a sharp first peak at $r=2$, equal to the polymeric bead diameter. This
confirms that the monomers belonging to different polymeric chains stay close
to each other which is a signature of an interpenetrating network. The
membrane M is solvated by solvent S particles by placing it in contact with
two identical reservoirs of S particles removing the confining walls. Further,
we keep the z-coordinates of membrane beads fixed and the external force is
set to zero.

Figure 2: (a) Plot of inter-strand correlation function of beads belonging to
two different polymeric strands $C(r)$ versus r. (b) Packing fraction
$\eta_{S}^{MR}(t)$(black line), $\eta_{L}^{MR}(t)$(grey line) as functions of
time t with external force $F_{0}=1$. The parameters are
$\bar{\epsilon}_{S}=1.74$, $\bar{\epsilon}_{L}=3.48$. Inset: Packing fraction
$\eta_{S}^{M}(t)$(black line), $\eta_{L}^{M}(t)$(grey line) over time. (c)
Plot for $\eta_{S}^{PR}(t)$(black line) and $\eta_{L}^{PR}(t)$(grey line)
versus $t$ for the same parameters as panel (b).

The solvated membrane M is placed between a mixture reservoir (MR) consisting
of a binary mixture of 400 L particles ($\eta_{L}^{MR}(0)=0.1)$ and 6000 S
particles ($\eta_{S}^{MR}(0)=0.26)$)and a pure reservoir PR with 6400 S
particles only($\eta_{S}^{PR}(0)=0.28)$. Solvent and solute interaction
parameters with the polymer beads ($\bar{\epsilon}_{S}=1.74$ with
$\epsilon_{S,h}=1$ and $\epsilon_{S,p}=0.574$; and $\bar{\epsilon}_{L}$=3.48
with $\epsilon_{L,h}=2$ and $\epsilon_{L,p}=0.574$). We apply force to
overcome osmotic pressure, $F(z)=F_{0}(1-z/z_{A})$. We use $F_{0}=1$, which
corresponds to experimental range of pressure. We show packing fraction data
$\eta_{S}(t)$ and $\eta_{L}(t)$ with time in different regions MR, M and PR.
We observe in Fig. 2(b) that the $\eta_{S}^{MR}(t)$ decreases in MR with time
from $\eta_{S}^{MR}(0)$ till they saturate in steady state. We also observe
$\eta_{L}^{MR}(t)$ decreases with t and then saturates in the steady state. In
inset of Fig. 2(b) $\eta_{S}^{M}(t),\eta_{L}^{M}(t)$ both saturates at large
time. In Fig. 2(c) both $\eta_{S}^{PR}(t),\eta_{L}^{PR}(t)$ increase and
saturate in steady state. At large time in steady state, we measure packing
fraction of S at PR, $\eta_{S,steady}^{PR}$, packing fraction of L at PR
$\eta_{L,steady}^{PR}$ and packing fraction of L inside M
$\eta_{L,steady}^{M}$. We observe that $\eta_{S,steady}^{PR}=0.36$,
$\eta_{L,steady}^{PR}=0.037$, $\eta_{L,steady}^{M}=0.02$. Using the steady
state values we find for this case $R=62.46\%$, $P=32\%$, $F=2\%$.

We consider the effect of the interaction of the solvent and solute particles
with the polymer beads over the RO performance. We study the case where both
solute and solvent particles favour the h beads compared to the p beads, but
they have different interaction strengths to the h-beads. Since the
interactions with h-bead is stronger for both, we have the restriction
$\bar{\epsilon}_{L}>1$ and $\bar{\epsilon}_{S}>1$. Here we intend to stop L
particles from moving from MR to PR. We study to this end the cases where
$\bar{\epsilon}_{L}$ is varied and $\bar{\epsilon}_{S}$ (=1.74) is fixed. We
observe that $R$ increases with increasing $\bar{\epsilon}_{L}$ in Fig. 3(a)
with, $R\sim\bar{\epsilon}_{L}^{0.4}$ dependence(log-log plot in inset of Fig.
3(a)) and finally comes to saturation for large $\bar{\epsilon}_{L}$. The
physical reason behind the observation is the following: $\bar{\epsilon}_{L}$
gets large, L particles find favourable to stay inside M. This scenario also
suggests that $F$ increases as we observe from Fig. 3(b) that $F$ also
increases rapidly with $\bar{\epsilon}_{L}$, $F\sim\bar{\epsilon}_{L}^{1.82}$
(see log-log plot in the inset). We observe in Fig. 3(c) that $P$ increases as
$P\sim\bar{\epsilon}_{L}^{0.23}$(see the log-log plot in inset) with
$\bar{\epsilon}_{L}$. Since L particles are stuck in MR and M, the S particle
movement is facilitated from MR to PR. Thus, stronger solute and h-bead
interaction, $\epsilon_{L,h}>\epsilon_{L,p}$ is favorable for membrane
performance, but leading to fast growth in fouling.

We also consider increasing solvent-h bead interaction, where we increase
$\bar{\epsilon}_{S}$, fixing $\bar{\epsilon}_{L}$ large (=8.71). In Fig. 3(d)
we plot $F$ with increasing $\bar{\epsilon}_{S}$. We observe $F$ sharply
decreases linearly close to zero with increasing $\bar{\epsilon}_{S}$.
However, Fig. 3(e) shows that $R$ decreases linearly $\bar{\epsilon}_{S}$. On
the other hand, $P$ in Fig. 3(f) decreases linearly as well with increasing
$\bar{\epsilon}_{S}$ beyond $\bar{\epsilon}_{S}>2$. As $\bar{\epsilon}_{S}$
increases S particles are more favourable to stay inside the M. So L particles
do not find favourable to stay inside M and also in MR due to forcing. They
move towards PR which leads to a decrease in $R$, resulting in mixed phase in
PR also. Thus, the increase in $\epsilon_{S,h}$ relative to $\epsilon_{S,p}$
impedes the performance of RO, although the fouling can be reduced to almost
zero. We do not show data larger value for $\bar{\epsilon}_{S}$ as we do not
reach steady state in our calculation.

Figure 3: $R$ vs $\bar{\epsilon}_{L}$ plot with $\bar{\epsilon}_{S}=1.74$. (a)
linear plot and inset, log-log plot. $F$ vs $\bar{\epsilon}_{L}$ plot, (b)
linear plot and inset, log-log plot. $P$ vs $\bar{\epsilon}_{L}$ plot (c)
linear plot and inset, log-log plot. The parameters in (b) and (c) are the
same as in (a). $\bar{\epsilon}_{L}=8.71$ for (d) to (f). (d) $F$ vs
$\bar{\epsilon}_{S}$ and (e)$R$ vs $\bar{\epsilon}_{S}$ plots with the best
fitted lines. $P$ vs $\bar{\epsilon}_{S}$ (f) linear and log-log plots in
inset. The lines in the main panel is guide to the eyes unless otherwise
stated explicitly. Inset: The line is the best fitted line.

Next, we consider the case where solvent and solute particles favour p-beads.
In such cases both $\bar{\epsilon}_{S}$ and $\bar{\epsilon}_{L}<1$. Here we
fix $\bar{\epsilon}_{S}(=0.57)$ and vary $\bar{\epsilon}_{L}$. We observe in
Fig. 4(a) $R$ decreases with increasing $\bar{\epsilon}_{L}$ with
$R\sim\bar{\epsilon}_{L}^{-0.33}$ (inset). Simultaneously, $F$ decreases
rapidly with increasing $\bar{\epsilon}_{L}$,
$F\sim\bar{\epsilon}_{L}^{-1.74}$ (Fig. 4(b)). In Fig. 4(c) we observe that
$P$ also decreases with increasing $\bar{\epsilon}_{L}$
$P\sim\bar{\epsilon}_{L}^{-0.33}$ (inset). Since lower $\bar{\epsilon}_{L}$
denotes a strong L and p bead interaction, data suggests that stronger L and p
interaction leads to better $R$ and $P$ but inferior performance in fouling.
We further vary $\bar{\epsilon}_{S}$ in the regime $\bar{\epsilon}_{S}<1$. We
observe in Fig. 4(d) that $F$ increases with increasing $\bar{\epsilon}_{S}$
with exponent 1.5(inset) and so do both $R$(Fig. 4(e)) and $P$ (Fig. 4(f)).
$R\sim\bar{\epsilon}_{S}^{0.3}$(inset) and $P$ increases linearly till
$\bar{\epsilon}_{S}<0.45$(inset). Less fouling observed for
$\epsilon_{S,h}<\epsilon_{S,p}$. However, this does not provide good RO
performance for $P$ and $R$.

Figure 4:  (a)$R$ vs $\bar{\epsilon}_{L}$ plot with $\bar{\epsilon}_{S}=0.57$.
linear plot and inset, log-log plot. $F$ vs $\bar{\epsilon}_{L}$ plot, (b)
linear plot and inset, log-log plot. $P$ vs $\bar{\epsilon}_{L}$ plot, (c)
linear plot and inset, log-log plot. The parameters in (b) and (c) are the
same as in (a). $\bar{\epsilon}_{L}=0.114$ for (d) to (f). $F$ vs
$\bar{\epsilon}_{S}$ plot, (d) linear and inset, log-log plots. $R$ vs
$\bar{\epsilon}_{S}$ plots, (e) linear and inset, log-log. $P$ vs
$\bar{\epsilon}_{S}$ (f) linear and log-log plots in inset. The line in the
main panels is guide to the eyes. Inset: The line is the best fitted line.

Note that the relevant parameters in the study involve only the energy depth
parameter of LJ interaction. This parameter is sufficiently robust and can be
defined for any inter-molecular interaction having a well-defined minimum.
Hence, it may be interesting to point out the implication of our model for
realistic systems. The solution consisting of S and L particles both
preferring the h-beads can be thought of the case as model for a solution of
non-polar components. Our calculation shows that the interaction of the non-
polar solute should have stronger interaction with the non-polar moieties of
the polymer membrane so that the non-polar solvent can easily flow through the
membrane from mixture to the pure solvent. On the other hand, solution
consisting of S and L particles both preferring the p-beads is a model for
polar solution. Our calculations show that polar solute interaction with polar
moieties of the membrane should be stronger than the solvent interaction for
easy passage of solvent from mixture to pure solvent. Thus, stronger solute
membrane interaction results in larger solvent permeation and solute rejection
but more membrane fouling. Our results at least qualitatively explain
different experimental observations on how tuning the membrane properties
affects RO performance[11].

##  4 Conclusion

To summarize, we study the motion of particles from a mixture of solute and
solvent particles to a pure solvent phase through a RO membrane driven by
force to overcome osmotic pressure. We observe that the solvent recovery and
solute rejection increase with increasing relative interaction of solute and
polymeric membrane. Simultaneously, however, fouling of the RO membrane also
increases. Although stronger solvent and membrane interaction leads to less
fouling, this leads to impede solvent recovery and solute rejection as well.
These results may serve guide for designing RO membrane for targeted
separation applications.

## References

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# Application of S-band for Protection in Multi-band Flexible-Grid Optical
Networks

Varsha Lohani  Anjali Sharma  Yatindra Nath Singh

###### Abstract

The core network is experiencing bandwidth capacity constraints as internet
traffic grows. As a result, the notion of a Multi-band flexible-grid optical
network was established to increase the lifespan of an optical core network.
In this paper, we use the C+L band for working traffic transmission and the
S-band for protection against failure. Furthermore, we compare the proposed
method with the existing ones.

##  I Introduction

According to the Cisco Annual Internet Survey, nearly two-thirds of the
world’s population will have Internet connectivity by 2023. The number of
nodes connected to the IP networks would be more than three times the number
of people worldwide. In 2023, the average of fixed global broadband speeds
will hit 110.4 Mbps, up from 45.9 Mbps in 2018[1]. The evolution of high-speed
technology requires a network that can satisfy bandwidth requirements. Two
networks built upon optical fibre technologies have been proposed and studied
in the past few decades. These are the Fixed-grid optical networks and
Flexible-grid optical networks [2, 3].

Optical networks need to carry enormous amounts of traffic while maintaining
service continuity even in faults. Failure of even a single link111span, link
and edges are used interchangeably will result in the loss of a substantial
amount of data if not protected automatically. Therefore, survivability
against link or path failures is an essential design requirement for high-
speed optical networks. A survivability scheme aims to offer reliable services
for the large volume of traffic even in the presence of failures222Fiber cut,
human-made errors or natural disasters such as earthquakes, hurricanes, etc.
as well as abnormal operating conditions.

Currently, we are working with the C-band of the optical spectrum. Due to
bandwidth capacity crunch at higher traffic loads, many incoming lightpath
requests get blocked. To increase the bandwidth capacity of the fibre, another
dimension of interest is Spatial. Spatial resources can be allocated flexibly
to incoming mixed-rate traffic demands to scale up the network capacity. An
effective and innovative spatial approach uses another band in the
electromagnetic spectrum. The use of bands S and L along with C increases the
network capacity and reduces the blocking probability of the new incoming
connection requests. However, due to the S and L bands, the Generalized
Optical Signal to Noise Ratio (GOSNR) is degraded in the C-band. A model of
GOSNR is formulated in [4].

In this paper, for Routing, Spectrum Assignment [5, 6, 7] and Protection
Provisioning [8] in Multi-band Flexible-grid Optical Networks we consider
C3331530 nm - 1565 nm+L4441565 nm - 1625 nm bands to carry primary traffic due
to their low noise figure [9] whereas S5551460 nm - 1530 nm band is used for
protection against failures.

##  II Proposed Scheme

The Quality of Transmission (QoT) for lightpath provisioning is an essential
parameter for a flexible-grid optical network. In this paper, QoT is modelled
in terms of GOSNR as discussed in [10]. The QoT can be estimated using GNPy,
an open-source optical route planning library [11].

The flowchart of the proposed method for MB flexible-grid optical networks is
shown in figure 1. The terminologies used in the flowchart are defined as:

  * •

$G(V,E,\\{\Delta_{e}\\})$: We represent an optical network as a graph G(V, E)
where G is defined as a set of optical vertices V, and set of optical fiber
edges E. Each edge $\textit{e}\in\textit{E}$ has usable slots $\Delta_{e}$.

  * •

$LR(s,d,|\Delta^{r}|,k)$ is a Lightpath Request where $s$ is the source node,
$s\in V$, and $d$ is the destination node, $d\in V$. In this paper, we use a
grid size of 12.5 GHz and BPSK modulation format, i.e., the modulation level
$m=1$. We are also considering an additional guard band (GB) so that no two
lightpaths interfere if placed next to each other. The maximum number of paths
to be computed by RSA is represented by  k, a positive integer.

  * •

$A_{w}$ is the Availability of the working lightpath. Considering each links
to be independent, therefore, $A_{w}=\prod_{i=1}^{e}A_{i}$, where e are the
edges which are part of the working path w.

  * •

$A_{th}$ is the Threshold Availability.

Figure 1: Flowchart of the proposed scheme.

Whenever a lightpath request arrives, k-shortest path is computed using
Dijkstra’s Algorithm. Next, for spectrum allocation, we will check the
spectrum slots availability first in the C-band and if the required number of
slots are not available in the C-band, then select the L-band. Afterwards,
compute the GOSNR of the lightpath request. If GOSNR is acceptable, allocate
the spectrum slots and set up the lightpath request.

Find the working link-disjoint backup path once the working path is set up.
There is no need to allocate a backup path to every working path. The backup
path is only set up when $A_{w}<A_{th}$ [12]. The process for finding a backup
path is also the same, except for the backup path we consider only S-band.
Here, we are using the Shared Backup Path and Slots Protection Technique.
Therefore, whenever a failure happens, switching from working to backup
happens at the end nodes of the lightpath.

##  III Discussion and Comparison

In this work, according to the grid size of 12.5 GHz [2] for the working path,
we have 868 spectrum slots (L-band has 548 slots and C-band has 320 slots),
whereas, for the backup path, there are 732 slots (S-band). Here, we provide a
comparison of the proposed method with the existing works.

In Standard Single Mode Fibre (SSMF) based Flexible-grid Optical Networks, the
main problem is the bandwidth capacity crunch. To maximize the accommodation
of the connection requests, the Core-based Flexible-grid Optical Networks [13]
and Multi-band (MB) based Flexible-grid Optical Networks are used. The main
problem with core-based flexible-grid optical networks is that as the number
of cores increases, the problem of inter-symbol interference has a greater
impact on the quality of transmission. Therefore, in this paper, we are
considering MB-based Flexible-grid Optical Networks.

Also, in the case of Multi-core Optical for Routing and Spectrum Assignment,
we need to follow spectrum contiguity, continuity, and core continuity
constraints. This additional constraint increases switch complexities within
the network. Additionally, the protection in core-based networks can provide
independent cores than Routing and Spectrum Assignment. However, core
continuity needs to be followed within different cores used for protection.
Whereas there is no additional constraint in Multi-band, only spectrum
contiguity and continuity constraints need to be followed.

Core-based multiband optical networks can be designed to further improve the
capacity of multiband optical networks. However, merging two spatial
components can increase the complexity of designing the optical nodes.

##  IV Conclusion

In this paper, we present a scheme for Routing, Spectrum Assignment and
Protection Provisioning in Multi-band Flexible-grid Optical Networks. Here, we
consider C+L bands to carry direct traffic due to their low noise figure,
whereas S-band is used for survivability against failures. Furthermore, we
also compare the proposed scheme with conventional single-mode fibre and core-
based fibre schemes.

## References

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  * [2] ITU-T G.694.1, ”spectrum grids for WDM applications: DWDM frequency grid”, 2012.
  * [3] O. Gerstel et al., “Elastic optical networking: A new dawn for the optical layer?,” IEEE Commun. Mag., vol. 50, no. 2, pp. 12-20, Feb. 2012.
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  * [9] N. Sambo et al., “Provisioning in multi-band optical networks: A C+L+S-band use case,” 45th European Conference on Optical Communication (ECOC 2019), pp. 1-4, 2019.
  * [10] N. Sambo et al., “Provisioning in Multi-Band Optical Networks,” in Journal of Lightwave Technology, vol. 38, no. 9, pp. 2598-2605, 1 May, 2020.
  * [11] A. Ferrari et al., “GNPy: An open source application for physical layer aware open optical networks,” IEEE/OSA J. Opt. Commun. Netw., vol. 11, no. 10, pp. C94–C108, Jun. 1, 2020.
  * [12] V. Lohani et al., “Availability-Aware Dynamic RSA with Protection using Consecutive Sub-Channels,” In arXiv preprint arXiv:2206.04933, June 2022.
  * [13] A. Sharma et al., “Protected Working Groups-based Resilient Resource Provisioning in MCF-enabled SDM-EONs,” In arXiv preprint arXiv: 2311.11565v1, November 2023.