# Surfactant-laden Liquid Thread Breakup Driven by Thermal Fluctuations

Luís H. Carnevale<EMAIL_ADDRESS>Institute of Physics, Polish Academy
of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland  Piotr Deuar  Institute
of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw,
Poland  Zhizhao Che  State Key Laboratory of Engines, Tianjin University,
300350 Tianjin, China  Panagiotis E. Theodorakis<EMAIL_ADDRESS>Institute
of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw,
Poland

###### Abstract

The breakup of liquid threads into droplets is crucial in various applications
such as nanoprinting, nanomanufacturing, and inkjet printing, where a detailed
understanding of the thinning neck dynamics allows for a precise droplet
control. Here, the role of surfactant in the breakup process is studied by
many-body dissipative particle dynamics, in particular the various regime
transitions and thread profiles, shedding light on molecular-level intricacies
of this process hitherto inaccessible to continuum theory and experiments.
Moreover, the role of surfactant in the most unstable perturbation, the formed
droplet size, and surfactant distributions have been unraveled. As surfactant
concentration rises, both the wavelength and time to breakup steadily increase
due to the lowering of surface tension below the critical micelle
concentration (CMC) and viscous effects introduced by micelles above the CMC.
These changes prior to the breakup lead to larger droplets being formed in
cases with higher surfactant concentration. We also compared the thinning
dynamics to existing theoretical predictions, revealing that the surfactant-
laden breakup starts at the inertial regime and transitions into the thermal
fluctuation regime when the concentration is increased. Thus, we illuminate
the hitherto poorly investigated and intricate breakup process of surfactant-
laden liquid threads driven by thermal fluctuations, contributing to a deeper
understanding of this process at molecular scales.

##  I Introduction

The breakup of liquid threads into droplets is a ubiquitous natural phenomenon
with diverse applications,Moseler and Landman (2000) such as
nanoprinting,Basaran, Gao, and Bhat (2013) nanoscale manufacturing and
chemical processing,Ye  _et al._ (2003) spraying,Wu, Zhang, and Zhang (2021)
and inkjet printing.Hoath (2016) Precise control over droplet size and a
comprehensive understanding of the dynamics surrounding the thinning neck,
also known as the thinning bridge, near the pinch-off point, is crucial in
many of these and other applications.Mousavi  _et al._ (2023); Mousavi and Lee
(2022); Mousavi  _et al._ (2022); Heydarpoor and Famili (2024); Zhao  _et al._
(2024) The pinch-off process has been extensively examined in the literature,
beginning with the foundational works of Plateau and Rayleigh,Plateau (1857);
Rayleigh (1878) who delineated the perturbations leading to the
destabilization and subsequent breakup of liquid threads or jets. Emphasizing
surface tension’s significance and assuming surface free energy minimization,
Plateau established the stability condition, i.e. $1>2\pi R_{0}/\lambda=\chi$,
where $\lambda$ represents the perturbation wavelength, $\chi$ signifies the
nondimensional wavenumber, and $R_{0}$ denotes the initial radius of the
undisturbed fluid. Instabilities only arise from perturbations with
wavelengths longer than the initial circumference of the liquid thread.
Rayleigh, via linear stability analysis of inviscid liquids, derived the
growth rate of such perturbations, identifying the wavenumber $\chi=0.697$ as
the point of maximum growth (Rayleigh mode). Expanding on Rayleigh’s work,
Weber conducted a similar analysis for viscous fluids, establishing that the
characteristic wavenumber, determining the highest growth rate, relies on the
ratio between viscous forces, inertia, and surface tension.Weber (1931) This
dependence can be expressed through the non-dimensional Ohnesorge number
Oh$=\mu/\sqrt{\rho\gamma R_{0}}$, where $\mu$ is the viscosity, $\rho$
density, and $\gamma$ surface tension.

Given surface tension’s pivotal role in liquid thread breakup, it is
unsurprising that numerous industrial applications utilize surfactants to
modify fluid properties and enhance process control, particularly in droplet
stabilization within emulsions.Goodarzi and Zendehboudi (2019) Surfactants, as
amphiphilic molecules, have a hydrophobic and a hydrophilic part. This
characteristic leads them to favorably adsorb at the surface of liquids
reducing surface tension. However, there exists a maximum surface
concentration, $\Gamma_{\infty}$, beyond which the fluid surface cannot
accommodate more surfactant. After this point, further increase in the
surfactant amount would increase the number of surfactant monomers in the
bulk. As a result, these will come together and form aggregates of micellar or
other morphologies, depending on the structural characteristics of the
surfactant. The minimum bulk surfactant concentration that allows for the
formation of these aggregates is known as the critical aggregation
concentration (CAC). The more specific term critical micelle concentration
(CMC) is also widely used, but this term rather indicates that these
aggregates are actually micelles. Importantly, a fluid with surfactants above
the CAC will not experience any more lowering of its surface tension due to
the saturation of surfactant concentration at the interface. Hence, any
further changes in properties above the CAC can be attributed to a higher
concentration and the presence of aggregates in the bulk.

During the thinning process of a liquid thread, surfactants are advected away
from the pinching point and, in this case, there is a competition between the
advection and the adsorption of surfactants from the bulk toward the
interface. If adsorption is much slower than advection, a surface
concentration gradient is established that leads to Marangoni stresses that
slow down the bridge thinning.Kovalchuk  _et al._ (2018) Various thinning
regimes can occur depending on the balance of forces near the pinch-off. These
regimes are typically characterized by the time before breakup $\tau=t_{b}-t$,
where $t_{b}$ is the time that breakup occurs. When inertial forces dominate
(I regimeEggers (1993)), the minimum thread radius at the pinch-off region
varies as $h_{min}\sim\tau^{2/3}$. This result was provided by Eggers, who
used self-similar theory and the lubrication approximation to simplify the
Navier–Stokes equation for an inviscid liquid. He has also shown that a
universal regime exists when considering a balance between viscous and
inertial forces (VI regime), where $h_{min}\sim\tau$.Eggers (1993) There is
also an intermediate regime where viscous forces are dominant (V regime),
which also scales as $h_{min}\sim\tau$ as found by Papageorgiou when analyzing
the pinch-off for the Stokes flow.Papageorgiou (1995) However, in the V regime
only viscous and capillary forces are relevant with Re depending on time and
reaching infinity as pinch-off is approached. In the VI regime, the inertial
forces balance the other forces with Re number going to unity. For this
reason, the V regime may be considered as only an intermediate regime, while
VI is rather a universal regime from a continuum perspective. Transitions
between these regimes may also take place during the thinning process. In
particular, Castrejón-Pita et al. have shown the wide range of possible
transitions through simulation and experiments.Castrejón-Pita  _et al._ (2015)
Moreover, when the adsorption of surfactants is much slower than advection,
Wee et al. have demonstrated that minimum radius also changes as
$h_{min}\sim\tau$, following Papageorgiou’s solution with some corrections to
accommodate surface rheological effects.Wee  _et al._ (2020) In the opposite
limit, when adsorption is much faster, the interface remains at a constant
concentration and, instead of a power-law, the thinning dynamics follows an
exponential regime where $h_{min}\sim e^{\tau}$. Martínez-Calvo and Sevilla
(2020) An exponential regime has also been found in the breakup of
viscoelastic fluids.Chang, Demekhin, and Kalaidin (1999)

The different regimes discussed so far are only valid until the minimum radius
reaches a small enough length scale in which thermal fluctuations become
relevant, due to the molecular motion. This thermocapillary length scale can
be defined as $l_{T}=\sqrt{k_{B}T/\gamma}$ and depends on the thermal energy
$k_{B}T$, where $T$ is temperature, $k_{B}$ Boltzmann’s constant, and $\gamma$
surface tension. Moseler and LandmanMoseler and Landman (2000) have studied
their influence on the breakup process by applying the lubrication
approximation to the Landau–Lifshitz–Navier–Stokes equation, which contains a
stochastic stress tensor to model the thermal fluctuations and this model is
referred to as the stochastic lubrication equation (SLE). They have found that
the surface profile near the pinching point becomes a symmetrical double cone
which suppresses the formation of satellite droplets and the model yields
predictions comparable to molecular dynamics (MD) simulations. Eggers has
demonstrated that in the thermal-fluctuation thinning regime (TF)
$h_{min}\sim\tau^{0.418}$ by arguing that surface tension becomes less
important in driving the breakup and looking for symmetric self-similar
solutions to the SLE.Eggers (2002) This behavior has been confirmed, by
Hennequin et al., from experiments using ultralow interfacial tension phases
of a colloid-polymer solution.Hennequin  _et al._ (2006) Petit et al.
experimentally observed the transition from the VI regime to the TF
regime.Petit  _et al._ (2012)

Despite the above studies, much less is known about what is happening when
surfactant is present, especially at molecular-level scales. To investigate
the behavior of surfactant-laden liquid threads at this small length scale, we
use the many-body dissipative particle dynamics method (MDPD).Pagonabarraga
and Frenkel (2000, 2001); Warren (2003) At its core, MDPD employs a coarse-
grained representation, grouping particles into clusters to capture the
collective behavior of molecular systems through soft-core potential
interactions, while at the same time, it can be used to simulate molecular
chains, such as surfactants. Combined with the ability to model thermal
fluctuations, MDPD allows for a computationally efficient way to simulate
systems at scales that are inaccessible to purely macroscopic or atomistic
models. Mostly, MDPD and its predecessor dissipative particle dynamics (DPD)
have been successful in describing the breakup process for surfactant-free
systems. For example, Tiwari et al. Tiwari and Abraham (2008); Tiwari  _et
al._ (2008) have used DPD and observed the symmetric double cone profile shape
during pinch-off and were also able to recover the TF regime minimum radius
scaling. Mo et al. Mo  _et al._ (2015) have done a more detailed analysis of
the thinning process using DPD and were able to observe the I regime and the
TF regime by changing the viscosity and surface tension of the system. They
have also shown that the viscosity of a DPD fluid is non-Newtonian and this
might affect the power-law exponent of the V regime. Arienti et al. Arienti
_et al._ (2011) have revealed that MDPD is also capable of describing the
breakup process at different coarse-grinning levels and they have also
demonstrated the transitions from the I to V regime and then to the TF regime.
Zhao et al. Zhao, Lockerby, and Sprittles (2020) used MDPD to simulate fluids
with different properties and observed different regime transitions when
increasing the Oh number and also have proposed a new final regime that
precedes pinch-off, although no explanation on the nature of this regime was
given. Moreover, Zhao et al. Zhao  _et al._ (2021) used both MDPD and SLE to
study liquid thread breakup and obtained the same results, however, their TF
regime scaling differed from Eggers’ solution and this discrepancy was
attributed to the neglected influence of surface tension when deriving the
power-law. In our previous work,Carnevale  _et al._ (2023) we have
investigated different systems with MDPD and found that the formation of
satellite droplets followed a power-law that depended on Oh and a thermo-
capillary number Th$=l_{T}/R_{0}$. Despite these studies, we have not been
able to find molecular simulations of the breakup process for a surfactant-
laden liquid thread. However, different models to simulate systems with
surfactant have been used within the MDPD framework. For example, Ghoufi et
al.Ghoufi, Emile, and Malfreyt (2013) proposed one of the first models of a
sodium dodecyl sulfate (SDS) molecule via a coarse-grained hydrophilic and
three hydrophobic particles, where parametrization of intermolecular
interactions is based on the Flory–Huggins theory. In another study, Zhou et
al. Zhou  _et al._ (2019) modeled SDS and dodecyltrimethylammonium bromide
(DTAB) surfactants by first matching the surface tension of each particle to
experimental values. Then, the intermolecular interactions were tuned by
constructing different multi-component systems. Recently, Hendrikse et al.
Hendrikse, Amador, and Wilson (2023) proposed a different parametrization
scheme to model alkyl ethoxylate surfactants. In this case, they first match
the surface tension and density of each particle to experimental values at a
given coarse-grained level and the cross-interactions were obtained from the
activity coefficients at infinite dilution computed from the excess chemical
potential.

In view of the insufficient fundamental understanding of the breakup process
in the presence of surfactant at a molecular scale, we have undertaken the
task of exploring this phenomenon in depth by investigating various key
properties. Specifically, we have carried out MDPD simulations of threads that
are long enough to examine how the surfactant concentration changes the
characteristic wavenumber that identifies the perturbation with the highest
growth rate leading to breakup. In addition, we check the difference in sizes
of main droplets and satellite droplets formed after the breakup for each case
considered, and also the minimum radius thinning dynamics was investigated and
compared with scaling laws of the aforementioned regimes. Lastly, we present
the different thread profiles near pinch-off along with the surfactant
distribution on the surface and in the bulk phase, elucidating the surfactant
transport mechanism as the system evolves towards breakup. Through this
thorough investigation, we aim to shed light on the nuanced molecular
mechanisms underpinning this phenomenon, thereby contributing to a more
comprehensive understanding of surfactant-laden breakup processes at the
molecular scale. This will offer further possibilities in tailored designs for
the relevant applications.

##  II Model and Methodology

###  II.1 Many-Body Dissipative Particle Dynamics

The MDPD model is a mesoscale, particle-based model that evolved from its
predecessor DPD Hoogerbrugge and Koelman (1992); Lavagnini  _et al._ (2021) by
adding an attractive and a repulsive contribution between the particles that
depends on their local density. This change enables the simulation of systems
with liquid–vapor coexistence.Pagonabarraga and Frenkel (2000); Warren (2003)
MDPD consists in integrating the equation of motion Eq. 1 for every particle
$i$ that interacts with the other particles $j$ through a conservative force,
$\bm{F}^{C}$, a random force, $\bm{F}^{R}$, and a dissipative force,
$\bm{F}^{D}$. The integration of the equation of motion for each particle $i$
is done by using the modified velocity-Verlet algorithm,Groot and Warren
(1997) where the equation reads

$\displaystyle m\frac{d\bm{v}_{i}}{dt}=\sum_{j\neq
i}\bm{F}_{ij}^{C}+\bm{F}_{ij}^{R}+\bm{F}_{ij}^{D}.$ (1)

The conservative force has a repulsive and an attractive term that act at
different lengths. Its most common form is

$\displaystyle\bm{F}^{C}_{ij}=A\omega^{C}(r_{ij})\bm{e}_{ij}+B\left(\bar{\rho_{i}}+\bar{\rho_{j}}\right)\omega^{d}(r_{ij})\bm{e}_{ij},$
(2)

where $A<0$ and $B>0$ are the attractive and repulsive parameters,
respectively, $r_{ij}$ is the distance between particles, $\bm{e}_{ij}$ is the
direction vector from particle i to particle j, $\omega^{C}(r_{ij})$ and
$\omega^{d}(r_{ij})$ are linear weight functions defined as follows

$\displaystyle\omega^{C}(r_{ij})=\begin{cases}&1-\frac{r_{ij}}{r_{c}},\ \
r_{ij}\leq r_{c}\\\ &0,\ \ r_{ij}>r_{c},\end{cases}$ (3)
$\displaystyle\omega^{d}(r_{ij})=\begin{cases}&1-\frac{r_{ij}}{r_{d}},\ \
r_{ij}\leq r_{d}\\\ &0,\ \ r_{ij}>r_{d},\end{cases}$ (4)

with $r_{c}$ being a cutoff distance for the attractive interaction, usually
set to unity, while the repulsive interaction cutoff is usually
$r_{d}=0.75r_{c}$. Although not necessary for our purposes, changing $r_{d}$
affects the parameter space $(A,B)$ where liquid, vapor, and solid phases are
present.Vanya  _et al._ (2018)

The many-body contributions in the repulsive force that come from the
dependence on local densities $\bar{\rho_{i}}$ and $\bar{\rho_{j}}$ are
calculated as

$\displaystyle\bar{\rho_{i}}=\sum_{j\neq i}\frac{15}{2\pi
r_{d}^{3}}\left(1-\frac{r_{ij}}{r_{d}}\right)^{2}.$ (5)

Other functions can be used to compute the local densities, such as the kernel
functions commonly used in the smoothed particle hydrodynamics (SPH)
method.Wang  _et al._ (2016)

To incorporate thermal fluctuations in MDPD, random and dissipative forces are
introduced and act as a thermostat, keeping the temperature in the simulation
constant and equal to unity. Both can be expressed as

$\displaystyle\bm{F}^{D}_{ij}=-\sigma\omega^{D}(r_{ij})(\bm{e}_{ij}\cdot\bm{v}_{ij})\bm{e}_{ij},$
(6)
$\displaystyle\bm{F}^{R}_{ij}=\xi\omega^{R}(r_{ij})\theta_{ij}{\color[rgb]{0,0,0}{\Delta
t^{-1/2}}}\bm{e}_{ij},$ (7)

where $\sigma$ is the dissipative strength, $\xi$ is the strength of the
random force, $\bm{v}_{ij}$ is the relative velocity between particles,
$\theta_{ij}$ is a random variable from a Gaussian distribution with zero mean
and unit variance.  $\Delta t$ is the timestep, taken to be equal to $0.01$.
According to the fluctuation–dissipation theorem,Español and Warren (1995)
$\sigma$ and $\xi$ are related to each other by

$\displaystyle\sigma=\frac{\xi^{2}}{2k_{B}T},$ (8)

and the weight functions for the forces are

$\displaystyle\omega^{D}(r_{ij})=\left[\omega^{R}(r_{ij})\right]^{2}=\left(1-\frac{r_{ij}}{r_{c}}\right)^{2}.$
(9)

For multi-component systems, $A=A_{ij}$ is the attraction parameter between
particles of type $i$ and of type $j$. The repulsive parameter $B$ has to be
the same for all interactions due to the no-go theorem otherwise the force
wouldn’t be conservative.Warren (2013) Other types of local density functions
might be used to circumvent this restriction, however, they have to be
carefully defined to avoid unphysical behavior in the simulations.Vanya and
Elliott (2020)

###  II.2 Parametrization

Figure 1:  Schematic representation of the coarse-grained $HT_{3}$ surfactant
model (see text for details).Zhou  _et al._ (2019)

To simulate a physical system with different components in MDPD, it is
necessary to define the coarse-graining level, i.e., how many atoms or
molecules are represented by one MDPD particle. A typical choice is to define
one particle as three water molecules and match the model reduced units to
real units by measuring some key properties such as density and surface
tension for a given set of simulation parameters. In this paper, we have
adopted the commonly used values $A_{ww}=-40$ and $B=25$ (MDPD units), where
the subscript ‘w’ denotes the self interaction between water beads, and the
dissipative coefficient $\sigma=4.5$.Ghoufi and Malfreyt (2011); Arienti  _et
al._ (2011) The conversion between reduced and real units is detailed in Table
1. With the interaction between our liquid particles being set, we follow the
parametrization of sodium dodecyl sulfate (SDS) surfactant molecules in the
same manner as Zhou et al.,Zhou  _et al._ (2019) which was chosen due to its
ample use in industrial processes. At the coarse-grained level, the molecule
is represented by one ‘H’-type particle, which is the hydrophilic head group
of the surfactant and by three ‘T’-type particles, which model the alkane
hydrophobic tail of the molecule. Figure 1 shows, schematically, the coarse-
grained SDS model.

Table 1:  Conversion between MDPD units and real units. The scaling is done by matching surface tension and density of water to values measured from MDPD simulations using $A=-40$ and $B=25$. The coarse-graining level is defined so that one MDPD particle represents three water molecules. Parameter | MDPD value | Real value
---|---|---
Particle | 1 | 3 H2O
$r_{c}$ | 1 | 8.17 Å
$\rho$ | 6.05 | 997 kg/m3
$\gamma$ | 7.62 | 72 mN/m

The self interaction between T particles $A_{TT}$ is tuned to reproduce the
surface tension of hexadecane and the cross interaction between $w$ and T
particles $A_{wT}$ was adjusted with the interfacial tension between water and
hexadecane (T-type particles). The self and cross interactions with the H-type
particle were fitted by also measuring the interfacial tension between
hexadecane and water with SDS molecules at the interface. All interactions are
summarized in Table 2. Moreover, we used harmonic potentials to model the
bonds and angles between particles in the surfactant molecule in order to
preserve the structure of the molecule. We found that the bond length and
strength proposed by Ref. (36) were not stable in our simulations, so we
adopted the values $r_{0}=0.45$, $k=400$ of Ref. (47), while the same angle
potential parametersZhou  _et al._ (2019) were kept, namely,
$\theta_{0}=160^{\circ}$, $k=100$.

Table 2:  MDPD interaction parameters for the HT3 surfactant model and water. W is the water particle, H the hydrophilic particle, and T the hydrophobic particle. The MDPD repulsive parameter is the same for all interactions and is $B=25$. $A_{ij}$ | W | H | T
---|---|---|---
W | -40 |  |
H | -32.18 | -19 |
T | -27 | -5.98 | -22

##  III Results and Discussion

###  III.1 Surfactant properties

Figure 2:  Example of a simulation box used to measure the surface tension of
a liquid slab in the presence of surfactants. The liquid particles are
represented by the shaded blue volume for easier visualization. In this case,
the concentration of surfactants is below CMC and the surfactant molecules are
not aggregated in the bulk phase.

We have conducted a series of simulations to comprehensively characterize the
pertinent attributes of our system when surfactants are introduced. The
primary focal point of our validation investigation pertains to the surface
tension, specifically its change by varying surfactant concentration. Figure 2
shows the standard setup used in simulation to measure the surface tension
based on the Kirkwood–Buff method.Kirkwood and Buff (1949) Since the CMC is
reached when the interface is saturated, the initial surfactant concentration
can be defined as the total number of surfactant molecules $N_{t}$ divided by
the initial surface area of the system $A_{s}$, namely, $C=N_{t}/A_{s}$, while
the surface excess concentration is obtained by counting only the number of
molecules on the surface divided by the surface area, $\Gamma=N_{s}/A_{s}$.
Lastly, bulk concentrations are also presented in terms of the surface area
$C_{b}=N_{b}/A_{s}$, and, therefore, $N_{t}=N_{b}+N_{s}$.

Figure 3:  Dependence of surface tension, $\gamma$ (circles) and surface
excess concentration, $\Gamma$ (triangles), on the number of surfactant
molecules per surface area, $C$. Both values reach a plateau after crossing
the CMC (dashed line) indicating that the interface is fully saturated.

Our simulations for determining the surface tension for the water–SDS-
surfactant system start with the initialization of a water slab with a
thickness $h=20$ and particle number density $\rho=6.05$ at the midpoint of
the periodic simulation box. This box was defined by dimensions of
$L_{x}=L_{y}=20$ and $L_{z}=70$ (Figure 2). Subsequently, surfactant molecules
were positioned directly at the interfaces of this water slab, and we allowed
the system to run $10^{5}$ time steps to attain thermodynamic equilibrium.
Once equilibrium was reached, we performed an additional $10^{6}$ time steps
of simulation to acquire data for calculating the properties of the system. A
representative configuration of an equilibrated system is visually depicted in
Figure 2. To determine the surface tension, we employed the Kirkwood–Buff
method,Kirkwood and Buff (1949) i.e.,

$\gamma=\frac{L_{z}}{2}\left(P_{zz}-\frac{P_{xx}+P_{yy}}{2}\right)$ (10)

where $P_{zz}$ is the normal component of the pressure tensor, while $P_{xx}$
and $P_{yy}$ are the components of the tangential directions to the
liquid–vapor surfaces. An equally significant parameter is the surface-excess
concentration of surfactants. These compounds have a natural propensity to
adsorb at the liquid interface until reaching a point of saturation, where the
surface-excess concentration attains its maximum, while surface tension
$\gamma$ reaches its minimum. Beyond this saturation threshold, the
introduction of additional surfactants leads to their aggregation within the
bulk liquid, forming the micelles. In Figure 3, we present the findings that
illustrate the variation of both the surface tension and the surface excess
concentration in response to differing surfactant concentrations. In addition,
the CMC point is clearly discernible in our results and shown in the plot.

Figure 4:  Change in the interfacial thickness with the increase in the number
of surfactant molecules per surface area ($\alpha$ gives the slope.). It
remains at a constant value upon reaching the CMC. Inset: example of a density
distribution near the interface for a simulation below CMC. The vertical lines
indicate the position of the interface and its thickness from a hyperbolic
tangent fit (Eq. 11). W, H, T are the liquid, head, and tail beads,
respectively.

The interfacial thickness is another important quantity in describing the
region affected by surfactants along a fluid’s interface. This measurement can
be obtained by fitting the density distribution of $w$ particles across the
interface using the hyperbolic tangent function

$\rho(z)=\frac{\rho_{l}}{2}\left[1-\tanh{\left(\frac{2(r-R_{0})}{\delta}\right)}\right]$
(11)

where $\rho_{l}$ represents the bulk density of the liquid phase, $R_{0}$ is
the Gibbs dividing surface position and $\delta$ the thickness of the
interface. An example of such a distribution is depicted in the inset plot of
Fig. 4 for a simulation with $C=1.5$. We can see how the density of $w$
particles changes from the liquid phase where $\rho=6.05$ to the vapor phase
with $\rho\approx 0$ and the vertical lines indicate the Gibbs dividing
surface (dash-dot) and the thickness of the interface (dashed). The density of
each surfactant particle is also notable and it shows that the head group H
tends to stay closer to the liquid phase while the tail particles are much
closer to the vapor phase. The primary plot in Fig.4 showcases the variation
in interfacial thickness. As surfactant concentration rises, $\delta$
increases until it reaches a plateau value when $C=1.8$, indicating the CMC
saturation point. For concentrations below $0.6$, the thickness increases
linearly with slope $\alpha\approx 0.5$ and between $C=0.6$ and the CMC with
slope $\alpha\approx 0.9$. These two linear regimes can be explained by the
orientation of the surfactant molecules with respect to the interface. At
lower concentrations, the surfactant tails stay in contact with the liquid
surface, while at a higher concentration they become more packed orienting in
the normal direction to the surface due to the decrease in available surface
area per molecule.Hendrikse, Amador, and Wilson (2023) In this sense, the
cross-over at $C=0.6$ reflects the range of the interactions between
individual surfactant molecules at the fluid surface.

###  III.2 Wavenumber and droplet sizes

Figure 5:  Simulation snapshots of the breakup process of a small liquid
thread. The system develops perturbations that grow with time and lead to the
pinch-off and subsequent formation of a droplet. The concentration of
surfactants is below CMC in this example. The threads used for statistical
analysis were four times longer in the axial direction than the simulation
shown here, which was shortened for the sake of clarity.

Following our previous work with pure liquid threads,Carnevale  _et al._
(2023) we have determined the characteristic wavelength of the breakup
process, expressed by the reduced wavenumber $\chi=2\pi R_{0}/\lambda$. This
is done by computing the density correlation function along the $z$
direction,Theodorakis, Paul, and Binder (2009) that is the direction along the
thread. To construct our initial configuration, we first make a water cylinder
with unperturbed radius $R_{0}=8$ and length $L_{z}=24$, which is stable
($L_{z}<2\pi R_{0}$) and does not break. In turn, we add the surfactant
molecules on its surface, according to the desired concentration and we run
the system for $2\times 10^{5}$ time steps to reach equilibrium. After
reaching equilibrium, we replicate the system in the $z$ direction $48$ times
to obtain a long thread and let it evolve in time until it breaks up and all
droplets are formed. We have already verified that possible finite-size
effects in the direction along the thread quickly disappear, but long threads
enable a more accurate calculation of the characteristic length scale of the
breakup process when realizing the Fourier transform of the density–density
correlation, as has already been shown for pure liquid droplets.Carnevale  _et
al._ (2023) Moreover, long threads naturally lead to the formation of a larger
number of droplets, which in turn allows for better statistics on the
properties of the main and satellite droplets. We realized 20 such simulations
for each surfactant concentration and the averaged results are discussed in
this study. A time sequence of snapshots that leads to the formation of a
single droplet is presented in Figure 5. In this example, a shorter thread
simulation is being shown just for clarity as the longer threads would be
poorly represented on a plot due to its dimensions. To form this thread, we
replicate the equilibrated system $12$ times in the axial direction, giving
thread length $L=12L_{z}$. It is clearly visible that a perturbation starts to
develop with $\lambda\approx L/3$, but, due to asynchronous breakup, only one
final droplet is formed, a mechanism that has been explained in detail in our
previous work with pure liquids.Carnevale  _et al._ (2023)

As indicated by our results, increasing surfactant concentration leads to a
notable reduction in the surface tension. Since surface tension acts as the
primary force of the instability, its decrease subsequently decelerates the
overall breakup process. This is visually evident in Figure 6, showcasing a
discernible growth in the time required for the thread to break ($t_{break-
up}$), which shows a linear growth with surfactant concentration, also beyond
the CMC. Moreover, the characteristic wavenumber, $\chi$ (see Ref. (34) for
details on calculating $\chi$), decreases as more surfactant is added to the
system, which means that perturbations with longer wavelengths become more
effective in destabilizing the liquid thread. However, this decrease is not
only due to the change in surface tension caused by the surfactants as
$\gamma$ remains constant for concentrations above the CMC. Thus, the
formation of micelles also adds viscous effects in the liquid bulk which
explains the further slow-down of the breakup process above CMC. We also
observe an increase in the standard deviation in our measurements, which can
be attributed to the increase of the thermocapillary length consequently
intensifying the significance of fluctuations within the system.

Figure 6:  Change in reduced characteristic wavenumber and time to breakup vs
total surfactant concentration (total number of surfactant molecules divided
by the initial surface area of the system).

After the breakup, we track the number of main and satellite droplets over
time by doing a cluster analysis with the software OVITO.Stukowski (2010) The
clusters were defined as a group of particles that are close to each other
within a threshold value and the droplet sizes were obtained from the radius
of gyration of spherical clusters following the protocols of our previous
work.Carnevale  _et al._ (2023) Figure 7 shows the increase in size of the
main droplets formed as surfactant concentration increases. This change in
size is expected as the wavelengths that lead to the breakup also increase,
subsequently forming fewer pinching points along the thread axis. There is a
slight difference between systems below and above CMC in how the main droplet
sizes increase. This can be explained by the slowing down of the breakup
process, which favors the suppression of pinching in a few regions of the
thread, similar to what is depicted in Figure 5. Furthermore, we could not
observe any statistically significant variation in the satellite droplet size
with concentration, it having values in the range $R_{s}/R_{0}=0.24\pm 0.03$
in all the simulated cases. However, there is a clear reduction in the
proportion in which they are formed as $C$ increases that could also be
attributed to the larger thermocapillary length scale, since thermal
fluctuations are believed to inhibit their formation.Moseler and Landman
(2000) This proportion approximately attains an asymptotic value above CAC as
shown in the inset of Figure 7.

Figure 7:  Size of main droplets $R_{m}$ normalized with the initial thread
radius vs total surfactant concentration, $C$. The radius of satellite
droplets did not present any statistically significant variation with respect
to the change in concentration, and therefore data are now plotted here (see
main text for further details). Inset shows the change in the proportion
between the number of satellite droplets and the total number of droplets
formed.

###  III.3 Surfactant distribution at the liquid–vapor interface

Since the breakup is influenced by surface tension, describing how the
surfactant is distributed along the interface is of great importance in
understanding the pinch-off process and the formation of droplets. To capture
these distributions, we investigated two systems below and two above the CMC,
conducting 20 simulations for each system. These simulations were done with
short liquid threads where $L_{z}=\lambda_{c}=2\pi R_{0}/\chi$, ensuring
breakup driven by the most unstable perturbation. Utilizing OVITO
software,Stukowski (2010) we separated the particles into axial bins and
tracked interface particles. Circles were fitted to these particles’ positions
in each bin, with their radii defining the interface profile $h$.
Additionally, surfactant molecules within each bin were quantified, and
assigned to the surface if within an interface thickness $\delta$ from $h$ or
otherwise to the bulk phase. The number of molecules on the surface $N_{s}$
and in the bulk $N_{b}$ are divided by the surface area $A$ on each bin to
obtain the surface excess $\Gamma$ and the bulk concentration $C_{b}$.

Figure 8 shows the breakup profile near the pinch-off of all the studied cases
and the surfactant concentration distribution on the surface and in the bulk
phases. Because of the stochastic nature of the breakup, the averages were
taken by shifting the pinching point to $z=0$ and by flipping the profiles
whenever $\int_{-L/2}^{0}hdz$ $>$ $\int^{L/2}_{0}hdz$ to preserve large anti-
symmetrical effects.Zhao, Lockerby, and Sprittles (2020) The lines show the
ensemble-averaged profile and the shaded area is within one standard
deviation. In the instance of the lowest surfactant concentration, a prolonged
neck emerges between droplets with an asymmetrical pinching point, deviating
from the symmetrical double-cone profile predicted for thermal fluctuation
breakup. As surfactant concentration increases, an asymmetrical double cone
profile begins to manifest, becoming almost symmetrical for the highest
concentration.

Figure 8:  Profile shapes, $h/R_{0}$, and surfactant concentration along the
$z$ axis at the surface and in the bulk of the fluid. Lines represent the
ensemble average and shaded area is one standard deviation. The averages were
obtained at the moment of pinch-off and for two surfactant concentrations
below CMC, a) $C=0.5$ and b) $C=1.0$, and two above CMC, c) $C=2.3$ and d)
$C=2.9$. $\Gamma_{\infty}$ is the highest surfactant excess concentration at
surface saturation. $\Gamma$ is the surface excess concentration and $C_{b}$
is the bulk concentration (number of molecules per surface area).

The different panels of Fig. 8 illustrate the distribution patterns of
surfactant molecules within both the surface and bulk regions along the axis
of the liquid thread. In the scenario where the initial concentration is at
its lowest (panel a), an expected absence of surfactant molecules in the bulk
phase is observed. This absence is attributed to the distance from the CMC.
Furthermore, a discernible advection of surfactants is witnessed,
characterized by a reduction in surface concentration around the pinching
point and a subsequent increase beyond its initial value of $\Gamma=0.5$
within the region occupied by the primary droplet. Contrary to this case, in
all other instances, the presence of molecules within the bulk phase is
evident. Both surface and bulk surfactants exhibit an advective motion away
from the pinching point. In addition, it is noteworthy that the surface
concentration consistently remains below the saturation concentration
$\Gamma_{\infty}$. An analysis of the temporal evolution, specifically
examining the case with the highest initial concentration as illustrated in
Fig. 9, indicates a trajectory wherein the advected surface surfactants
migrate toward the main droplet region and subsequently infiltrate the bulk
phase. This observation suggests a directional flow pattern, wherein
surfactant molecules, once advected from the surface around the pinching
point, are transported towards the primary droplet region and eventually
integrate into the bulk phase.

Figure 9:  Time evolution of the surfactant distribution for a case above CMC
($C=2.9$) showing the advection of surfactants from the pinching point and
increase in bulk surfactant in the main droplet region. $\Gamma$ is the
surface excess concentration and $C_{b}$ is the bulk concentration (number of
molecules per surface area). Each panel corresponds to a different point in
time before the breakup moment ($t=t_{b}$), i.e. a) $t_{b}-t=400$, b)
$t_{b}-t=100$, c) $t_{b}-t=50$, and d) $t_{b}-t=0$, as indicated.

###  III.4 Neck-radius scaling

Thinning dynamics of liquid threads under varying surfactant concentrations
were systematically examined by tracking the minimum neck radius $h_{min}$
until the pinch-off event. Four distinct cases, encompassing two surfactant
concentrations below the CMC and two above, were investigated. For each case,
20 simulations were conducted, and the averaged neck radius values were
obtained with respect to their relative time from the breakup ($t_{b}$)
$\tau=(t_{b}-t)$ as presented in Fig. 10. According to a theoretical analysis
of surfactant-laden breakup at the limit when surfactants are advected away
from the pinching point, the minimum radius should follow the viscous regime
with linear scaling $h_{min}\sim\tau$.Wee  _et al._ (2020) However, we did not
observe this scaling and at the lowest surfactant concentration, the thinning
process appears to be close to the inertial regime, indicated by the scaling
$h_{min}\sim\tau^{2/3}$. Interestingly, for low concentration we did not
observe a clear transition or the presence of the thermal fluctuation regime
$h_{min}\sim\tau^{0.418}$.Eggers (2002) MDPD simulations of a liquid without
surfactants also have shown a deviation from the thermal fluctuation
scaling.Zhao  _et al._ (2021)

As we increase surfactant concentration, consequently reducing surface
tension, the breakup dynamics still exhibits a power-law behavior. However,
the observed exponent deviates from the inertial regime and approaches the
thermal fluctuation one. These results substantiate the conclusions drawn by
Zhao et al.,Zhao, Lockerby, and Sprittles (2020) who demonstrated that the
scaling exponent converges towards $0.418$ when the surface tension
$\gamma\rightarrow 0$ from simulations using the 1D SLE model. In all of our
simulations cases, we can observe a transition between the power-law to a
different regime immediately preceding the pinch-off event. This last regime
has been proposed by Zhao et al. Zhao  _et al._ (2020) as the “breakup”
regime, following a power-law with exponent 0. However, no further explanation
was given as to the physical meaning and origin of this behavior. Moreover,
they only analyzed the evolution of a single simulation trajectory instead of
an ensemble average. From our results, the transition actually occurs when
$h_{min}$ reaches below the cutoff radius of the particle interactions and it
could be an artifact from the MDPD method. When the thread reaches such
scales, the local density $\bar{\rho}$ (see Eq. 5) starts to decrease with
time and this, in turn, reduces the strength of the repulsive term in the
conservative force, leading to a stronger relative attraction between the
particles that slows down the thinning process. This possible regime scales
not as power-law but seemingly as $h_{min}\sim e^{\tau}$. Exponential thinning
is also found in the breakup of viscoelastic fluidsChang, Demekhin, and
Kalaidin (1999) and when surface rheological effects are taken into
account.Martínez-Calvo and Sevilla (2020); Wee, Wagoner, and Basaran (2022)

Figure 10:  Radius scaling of the thinning process for threads with four
different surfactant concentrations and comparison with known breakup regimes.
a) $C=0.5$; b) $C=1.6$; c) $C=2.3$; d) $C=2.9$. Each case is an ensemble
average of 20 simulations. $r_{c}$ is the cutoff distance for the attractive
interaction. Dashed lines were added just as reference to the theoretical
regimes and have the same values in all plots.

##  IV Conclusions

In this study, we investigated the impact of surfactants on the breakup of a
liquid thread under thermal fluctuations by employing simulations via the
particle-based mesoscale method MDPD. We characterized the HT3 coarse-grained
surfactant model in terms of its relevant interfacial properties such as
surface tension and interface thickness. Furthermore, the model also features
the spontaneous formation of micelles above the CMC. The results indicate that
the model is adequate in reproducing the behavior of real surfactant
molecules. Then, a detailed study of the breakup of liquid threads in the
presence of surfactant was carried out. We find that increased surfactant
concentration would naturally lead to the increase in the most unstable
wavelength that characterizes the breakup, and, also, in the time for it to
occur. At first, this could be explained by the decrease in surface tension
which, in turn, increases the Oh number. However, both the wavelength and time
to breakup keep increasing for concentrations above the CMC, where surface
tension remains constant. This might therefore be due to bulk viscous effects
from the increase in surfactant micelle concentration. We also discussed the
minimum radius scaling of the thinning process and showed that the observed
power-law behavior matches the I regime and approaches the TF regime,
depending on the surfactant concentration — low and high, respectively. At low
concentrations, we found that the breakup is in the I regime instead of the
universal VI or V regimes. Nonetheless, this result is in agreement with
experimental measurements.Kovalchuk, Nowak, and Simmons (2016) At higher
concentrations, the thinning dynamics transition into the thermal fluctuation
regime. Moreover, we argue that the “breakup” thinning regime proposed by Zhao
et al. Zhao, Lockerby, and Sprittles (2020) might be an artifact of the MDPD
model as the transition into it occurs when $h_{min}$ approaches the cutoff of
the interactions. When the minimum radius is below the cutoff, the attraction
between particles becomes increasingly more dominant than the repulsion
leading to the slow-down of the thinning dynamics. Finally, we have examined
the breakup profiles and surfactant distribution of the threads providing
molecular-level insights into the mechanisms of this phenomenon. Thus, we
anticipate that our study sheds light on the thus far poorly investigated
breakup of liquid threads laden with surfactant.

###### Acknowledgements.

This research has been supported by the National Science Centre, Poland, under
grant No. 2019/34/E/ST3/00232. We gratefully acknowledge Polish high-
performance computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH)
for providing computer facilities and support within computational grant no.
PLG/2022/015747.

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# Toward Efficient Transfer Learning in 6G

Saeedeh Parsaeefard and Alberto Leon-Garcia
Electrical and Computer Engineering Department, University of Toronto
saeideh.fard and alberto.leongarcia @utoronto.ca

###### Abstract

6G networks will greatly expand the support for data-oriented, autonomous
applications for over the top (OTT) and networking use cases. The success of
these use cases will depend on the availability of big data sets which is not
practical in many real scenarios due to the highly dynamic behavior of systems
and the cost of data collection procedures. Transfer learning (TL) is a
promising approach to deal with these challenges through the sharing of
knowledge among diverse learning algorithms. with TL, the learning rate and
learning accuracy can be considerably improved. However, there are
implementation challenges to efficiently deploy and utilize TL in 6G. In this
paper, we initiate this discussion by providing some performance metrics to
measure the TL success. Then, we show how infrastructure, application,
management, and training planes of 6G can be adapted to handle TL. We provide
examples of TL in 6G and highlight the spatio-temporal features of data in 6G
that can lead to efficient TL. By simulation results, we demonstrate how
transferring the quantized neural network weights between two use cases can
make a trade-off between overheads and performance and attain more efficient
TL in 6G. We also provide a list of future research directions in TL for 6G.

###### Index Terms:

Six generation wireless networks (6G), transfer learning, efficient learning
algorithm

##  I Introduction

6G will intensify the handling of AI and ML for use cases from various corners
of industry and life. 6G will be a network of ”integrated intelligence” [1, 2]
in which AI/ML algorithms are applied for autonomous networking and where the
network will be redesigned to support the AI/ML based applications [3, 4, 5,
6].

The success of AI/ML algorithms requires large volume, updated data sets with
similar statistical conditions in the training and testing phases [7].
However, this is not the case for many real scenarios for two main reasons.
First, collecting data and providing labeling are not always accessible (e.g.,
cyber attacks) and may involve high cost. Second, dynamic environments in real
scenarios call for updating data sets regularly (e.g., fast and dynamic nature
of the wireless channel and uncertainty in the behavior/mobility patterns of
end users in intelligent transportation systems (ITS)). Re-collecting data in
these scenarios and rebuilding models from scratch impose huge burdens for
these use cases.

Transfer learning (TL) or knowledge transfer (KT) is a promising approach to
handle both of the above-mentioned challenges [8, 9, 7, 10]. In this context,
it is possible to transfer lesson-learned parameters (knowledge) among
learning algorithms that includes data sets, models, hyper parameters, and
optimization approaches. In particular, each learning algorithm can be divided
into two parts: 1) a domain containing feature data and its statistics; 2) a
task involving models and labels. Typically in TL there are at least one
source and one target AI/ML algorithms in TL where the knowledge of domain
and/or task of the source is transferred to a target.

Experience with TL shows that sharing domains and tasks can improve the speed
of convergence and learning accuracy. this case will be refereed to ”a
positive TL”. However, when there is not enough similarity and correlation
among domains and tasks of the source and the target, TL might have negative
effect on the performance of the target. Applications of TL in 6G and its
positive potential in future networking has recently drawn a lot of attention,
e.g., [11, 12]. In this paper, our goal is to study how we can attain
efficient TL in 6G.

We begin by looking at the TL categories. Then, we explain how 6G architecture
planes can be modified to provide the efficient TL. We discuss the deployment
of TL pipe-lines in the infrastructure, novel procedures in management plane
to handle TL among sources and targets, new repositories and knowledge
extraction modules in a training plane, and modifications in application
plane. Based on the network layers, we provide examples about data sets and
models that may be saved in repositories of the training plane in 6G.

Fortunately, in wireless networks, there exist spatio-temporal and
hierarchical correlations among AI/ML algorithms running in different nodes
that are conducive to positive and efficient TL. Despite its great potential,
TL in 6G introduces various implementation challenges. For example, use cases
can impose diverse granularity requirements of TL in 6G involving real time,
online/offline, and on-demand TL. And there are concerns about robustness,
reliability, security, and privacy of pipe-lines for TL. We will discuss these
implementation challenges in 6G and highlight the overheads of TL in 6G. In an
evaluation section, we show how quantization approaches can be used with TL to
reduce the extra overhead inside of 6G.

The organization of this paper is as follows. In Section II, we briefly review
TL. In Section III, we show how TL changes the 6G architecture. Section IV
contains future research directions. In Section V, a deployment scenario for
TL in 6G will be discussed; followed by conclusions in Section VI.

##  II Transfer Learning Preliminaries

Consider an AI/ML algorithm $m$ that has [8]:

  * •

Domain: consisting of a feature space $\mathcal{X}_{m}$ for algorithm $m$ and
a marginal probability distribution $P_{m}(\textbf{X}_{m})$, where
$\textbf{X}_{m}=\\{x^{1}_{m},\cdots,x^{n}_{m}\\}\in\mathcal{X}_{m}$, which
constitute the domain
$\mathcal{D}_{m}=\\{\mathcal{X}_{m},P_{m}(\textbf{X}_{m})\\}$

  * •

Task: involving a label space $\mathcal{Y}_{m}$ and a predictive function
$f_{m}(\dot{)}$ which shows the relation between $\mathcal{X}_{m}$ and
$\mathcal{Y}_{m}$, and the task is denoted by
$\mathcal{T}_{m}=\\{\mathcal{Y}_{m},f_{m}(.)\\}$.

In supervised learning mode, the data set $m$ consists of
$\mathfrak{D}_{m}=\\{(x_{m}^{1},y_{m}^{1}),\cdots(x_{m}^{n},y_{m}^{n})\\}$ and
$f_{m}(.)$ is found by minimizing the distance between the feature and the
label spaces. In unsupervised learning mode, the task changes to
$\mathcal{T}_{m}=\\{f_{m}()\\}$ since the labels are not available and the
best predictive function is matched to the feature space. Traditionally, the
domain $\mathcal{D}_{m}$ and the task $\mathcal{T}_{m}$ are fully disjoint
from any other $\mathcal{D}_{n}$ and $\mathcal{T}_{n}$ for $m\neq n$. However,
in TL, parameters among domains and tasks $m$ and $n$ for all $m$ and $n$ can
be shared, see Fig. 1. Any shared part of the domains or the tasks can be
referred as ”knowledge”, e.g., all or the portion of the feature and the label
spaces (data sets), or $f(.)$ and its hyper-parameters (models). TL aims to
improve the performance of $\mathcal{T}_{n}$ with $\mathcal{D}_{m}$ and
$\mathcal{T}_{m}$. We consider the following two key performance measures for
target $n$:

  * •

Accuracy, consistency and generalization of learning in the target domain:
$\eta^{TL}_{n}=\frac{P^{\text{TL}}_{n}}{P^{\text{Traditional}}_{n}}$ where
$P^{\text{TL}}_{n}$ and $P^{\text{Traditional}}_{n}$ are the performance
measures of the AI algorithm $n$ with and without TL, respectively.

  * •

Training time in the target domain:
$\tau^{TL}_{n}=\frac{T^{\text{Traditional}}_{n}}{T^{\text{TL}}_{n}}$ where
$T^{\text{TL}}_{n}$ and $T^{\text{Traditional}}_{n}$ are the training times of
the AI algorithm $n$ with and without TL, respectively.

Now we can defined positive TL as a case that $\eta^{TL}_{n}>1$ and
$\tau^{TL}_{n}>1$. The above definitions are for one source and one target and
it can be extended to any set of sources and targets.

From 6G’s preservatives, to have successful TL, required bandwidth, delay, the
level of security and privacy of passed knowledge between sources and targets
should be determined. Also, the granularity of TL algorithm and the iteration
number to pass knowledge between source and target should be determined. For
instance, we can have following classes of interaction among source and
targets:

  * •

Real-time TL where the knowledge is passed between source and target as soon
as a new update is available;

  * •

Non real-time or semi-real time TL where the knowledge between the sources and
the targets are passed according to a schedule;

  * •

On-demand TL where the knowledge is passed from the source to the target when
a request from the target is initiated.

These types of interactions together with the end-to-end (E2E) delay determine
the required bandwidth between each source-target pair. These parameters can
be used to deploy the pipe-line for TL, demonstrated in Fig. 1. An important
point is that the TL pipe-line causes additional overhead and cost for passing
knowledge. Below we identify different types of overheads and costs are
incurred in TL. One approach to obtaining a single measure for overhead and
cost is to use a weighted sum of the components:

$\displaystyle\Theta^{\text{TL}}_{mn}=\alpha_{1}f_{1}(W_{mn})+\alpha_{2}f_{2}(D_{mn})$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,+\alpha_{3}g_{mn}(\text{
interaction})+\alpha_{4}h_{mn}(\text{security}),$

where $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ are regularization weights
and functions are defined as:

  * •

$f_{1}(W_{mn})$ and $f_{2}(D_{mn})$ are increasing and decreasing functions of
required E2E bandwidth $W_{mn}$ and delay $D_{mn}$, respectively;

  * •

$g_{mn}(\text{ interaction})$ shows the cost per interaction class, defined as

$g_{mn}(\text{
interaction})=\left\\{\begin{array}[]{lr}M_{1},&\text{if}\,\,\text{Real-
time},\\\ M_{2},&\text{if }\,\,\text{Non-Real time},\\\
M_{3}&\,\,\text{Otherwise,}\end{array}\right.$

where $M_{1}$, $M_{2}$, $M_{3}$ are positive values demonstrating the costs
related to each class and they can be set as $M_{1}\geq M_{2}\geq M_{3}$ by
wireless network infrastructure providers. The good policy also can be
designed to reduce the cost of overhead, for instance by sending the TL
knowledge among the source and the target in non-rush hour time windows.

  * •

$h_{mn}(\text{security})$ is the security related cost function and it is an
increasing function with the level of requested by source $m$ and target $n$.

Clearly, decreasing the overhead while attaining positive performance of the
learning AI are major indicators of effectiveness of TL in 6G.

Figure 1: Transfer learning and its pipe-line inside of 6G

There are several categories of TL based on what can be transferred among
sources and targets and the similarities and differences between their domains
and tasks. For example, based on the availability of labels, there are three
categories of TL:

  * •

Transductive TL where the label data is available only from the source not the
target and $\mathcal{D}_{m}\neq\mathcal{D}_{n}$ while
$\mathcal{T}_{m}=\mathcal{T}_{n}$

  * •

Inductive TL where label data are available for the source and targets but the
tasks are different, i.e., $\mathcal{T}_{m}\neq\mathcal{T}_{n}$

  * •

Unsupervised TL where the label data is not available in neither source or
target

Based on domains, we have

  * •

Heterogeneous TL where $\mathcal{D}_{m}\neq\mathcal{D}_{n}$

  * •

Homogeneous TL where $\mathcal{D}_{m}=\mathcal{D}_{n}$

and based on the solution approach, we have

  * •

Instance-based TL where samples of data sets are shared between the source and
target

  * •

Feature-based TL where original features of the source are matched to target
ones, or where for both source and target features, latent features or common
features are discovered.

  * •

Parameter-based TL which involves transferring the parameters of models

  * •

Relational-based TL which transfer the learned rules from source to the target

In the following, we will discuss how TL should be processed in 6G and give
examples of the above categories of TL in 6G.

##  III Aspects of Efficient TL in 6G

In this section, we study effects of TL in 6G and show how 6G should be
adapted to serve AI/ML use cases. Many AI/ML use cases in 6G involve
learning/control loops containing monitoring, analysis, policy, execution plus
knowledge steps, denoted as (MAPE-K) loops. A MAPE-K loop provides a unified
way to represent these AI/ML based applications [6, 2]. TL can be deployed
between ML algorithms that are normally present in the analysis step of MAPE-K
loops associated diverse services in OTT, and/or networking applications
(NAL). In this paper, for simplicity, we assume there is one MAPE-K loop per
each AI/ML algorithm which can be a target or a source of TL. Let’s start our
discussions with some examples to show how TL can be used in 6G.

###  III-A Examples of TL in 6G

In 6G, many use cases in networking share the spatio-temporal features or
similar information in performing various tasks. Let’s focus on resource
allocation (RA), energy efficiency (EE), and admission control (AC) of radio
access networks (RANs) for one specific area in 6G. There are several access
nodes, e.g., ORANs, in a region that provide coverage (see Fig. 2).

All of these tasks utilize the mobility pattern and traffic pattern of end-
users. Therefore, the domains are similar but the tasks are different. Any
ORAN can share its own domain among its own different tasks. Also, ORAN 1 can
share its own task and domain for the resource allocation (RA) with the
resource allocation (RA) in ORAN 2; which is an example of to transduction TL
where the source is RA in ORAN 1 and the target is RA in ORAN 2. If the
resource allocation (RA) in RAN 2 also shares its domain or tasks with the
resource allocation (RA) in RAN 1, we have inductive TL. If both RANs do not
have labeled data, there is unsupervised learning among them. The resource
allocation (RA) in ORAN 1 and ORAN 2 can share their knowledge in real-time,
non-real time and on-demand cases. For the last two cases, depending on the
amount of available bandwidth, TL among ORAN 1 and ORAN 2 can be scheduled to
reduce $\Theta^{\text{TL}}_{\text{ORAN1},\,\text{ORAN2}}$. The same is true
for the energy efficiency (EE) in both ORAN 1 and ORAN 2. However, there is a
chance that the energy efficiency (EE) decision is in contrast with an RA
decision. For instance, the resource allocation (RA) requires more resource to
guarantee the QoS while the energy efficiency (EE) asks for less power
consumption and releasing some resources. These examples demonstrate the
importance of interaction and TL management in 6G.

These AI/ML algorithms can share the domains and tasks with another part of
the RAN in different cities or different locations. In this case, we can have
parameter or instance or feature based TL. The resource allocation, admission
control, and the energy efficiency in ORANs of one area can share their models
and data sets with the core elements to derive more large scale mobility
patterns and network user behaviors. In this case, one probable scenario is to
share the domain of RANs with the core to increase the size of data sets of
core elements without additional cost. The knowledge can be fully or partially
shared among AI/ML algorithms in 6G.

Figure 2: One scheme of the TL examples in zone 1 and zone 2 of 6G

The mobility pattern of 6G nodes, which arises from NAL applications, can be
transferred to ITS, which is an OTT application, or vice versa. Various types
of TL can be applied here. The similarity between these two use cases is
obvious. However, 6G can encounter more complex scenarios. For example, for
cyber-security, 6G may require a behaviour and traffic model of legitimate
users of OTT use cases. In this case, the domains and the tasks of both cases
are not correlated. Therefore, we have heterogeneous TL here, and any solution
based categories of TL can be applied.

###  III-B Hierarchy and Spatio-temporal Features of 6G and TL

Wireless networks have hierarchical structures. Also, our examples in the
previous section show that there is an inherent spatio-temporal feature among
different location zones, RANs, and core entities. These features help create
positive TL in 6G for clustering and classifications AI/ML algorithms.
Therefore, hierarchy and Spatio-temporal features of traffic of users in 6G
can be used as the criteria for the clustering AI/ML algorithms for TL. In
Figs. 3 (a) and (b), we plot some models of interaction of AI/ML algorithms
for TL in 6G which can basically derived from both hierarchy or spatio-
temporal feature of 6G. In Fig. 3, we present three models:

  * •

Cascade AI/ML algorithms in (i) which can show the neighboring ORANs
transferring their models and data sets for mobility or traffic patterns of
users

  * •

Hierarchical AI/ML algorithms in (ii) which can represent the interaction
between access and core elements in TL

  * •

Parallel AI/ML algorithms (iii) belong to the case that the similar nodes
(base stations, switches, storage devices) collaborate to learn the similar
models based on their local data sets through the concept of federated
learning (FL). Also, they can share their public (not very sensitive data sets
in terms of privacy) with their peers.

This hierarchical structure inside of 6G can help to find the correlation,
similarities, and relation among sources and targets.

Figure 3: The hierarchy and interaction among AI/ML use cases in 6G for three
models of interaction (i) cascade model (ii) Hierarchical model, (iii)
parallel model.

###  III-C 6G Architecture Modifications

We see the 6G architecture as including four planes to serve these use cases
and to handle their needs associated with TL [13, 2]. We define
responsibilities of each plane as follows:

  * •

Application Plane which represents the MAPE-K loops as chains of
functions/services that implement the data collection processing and action
pipe-lines that implement the loops, along with their required QoS and QoE.
The ML algorithm(s) in each MAPE-K loop can be the source or target for
algorithms in other MAPE-K loops. Application plane determines following items
for TL:

    * –

A list of trustful targets;

    * –

A list of trustful sources;

    * –

Parts of models and tasks that are allowed to be shared;

    * –

Required levels of security and privacy;

    * –

TL angularity with any specific source and target;

    * –

Privacy and security level;

    * –

Required E2E delay for TL with any specific source and target;

    * –

Required bandwidth for any allowable TL.

The above parameters help to determine the service level agreement (SLA) for
any pair of the source and target in TL. These SLAs will be used in 6G to
establish appropriate pipe-lines per each MAPE-K loop.

  * •

Infrastructure Plane which is an E2E cloud-native, multi-tier, programmable
structure with virtualized communication and computation resources. This plane
inherits the software defined and virtualized structures of 5G and it is
responsible to provide TL pipe-lines from sources and targets for both OTT and
NAL applications. The request for setting up a TL pipe-line is sent by a
management plane to the infrastructure plane. Other pipe-line parameters are
derived from SLAs in application plane.

  * •

Management Plane which is in charge of monitoring all running services and
Required admission for new requests of services in fully autonomous manner. In
6G, this plane should include procedures to handle the conflicts that may
arise among MAPE-K loops, pertaining to the consistency and coherency of loops
[2], and TL management. These new procedures along with orchestrator and
infrastructure management are shown in boxes in Fig. 4. TL manager has
following tasks:

    * –

Determining who can initiate TL among MAPE-K loops and services in 6G. For
instance, among networking applications, TL request can be issued by network
elements (decentralized manner) or can be handled by the orchestrator
(centralized manner). For the OTT applications, the request can be originate
with the OTT applications;

    * –

Determining authorized TLs among MAPE-K loops;

    * –

Initiating and handling TL for one or multiple service/infrastructure
providers and OTT slices for different organizations.

The request for setting up TL pipe-line among MAPE-K loops can be initiated
from use cases. For example in an OTT, an Intelligent Transportation System
(ITS) may send a request for TL to get data about the mobility pattern of
users. In a NAL setting, a request for TL can be started by network management
plane. In both cases, the orchestartor will handle the setup procedure for
providing the TL pipe-line in the infrastructure by monitoring security,
reliability, and privacy. In general, all the coordination among sources and
targets are handled by this plane. This task is more complex when dealing with
multi-source multi-task TL and when the sources and targets are in different
networks or OTT applications.

  * •

Training Plane which is a new plane in 6G to integrate AI and ML into the
network [2, 4] and to provide sandboxes that mimic the real environment with
the goal of training, retraining and examining any AI based use cases in
offline, near-online and online manner [4]. To support TL, this plane provides
an environment to measure the TL positive effects on the targets’ learning
procedures. It should learn the similarity and correlations among sources and
targets to attain positive TL for both NAL and OTT use cases. It also needs to
be equipped with the means to extract the knowledge from domains and tasks of
each AI based application. We represent these tasks in four boxes in Fig. 4,
defined as:
1) Sandboxes which are responsible to provide simulation/ emulation
environments for TL, clustering and finding the beneficial correlation among
sources and targets. These reports should be sent to the second box of the
management plane.
2) Knowledge extraction  which can provide parameters from domains and tasks
of sources and targets for TL (knowledge). This knowledge is ready to transfer
between source-target pairs depending on TL categories.
3) Repositories of data and knowledge which are storage units to store and
retrieve knowledge for TL among sources and targets.

In 6G, the training and management plane have tight interaction to attain
efficient TL. All actions in boxes in the training plane are sent to the
management plane to execute the interaction and knowledge transfer among the
MAPE-K loops. In Fig. 4, we show arrows between boxes to highlight the
importance of these interfaces.

Figure 4: Training and management aspects of 6G to provide TL.

###  III-D Repository of Knowledge of TL in 6G

In the repository, the models and data sets of each target-source pair can be
stored and used for TL among similar/correlated OTT and networking use cases
when permitted. For instance, the following information can be stored in the
repository:

  * •

End-Users which are smart phones, IoT devices (e.g., smart watches),
autonomous vehicles, and sensors. These devices can share the following:

    * –

Models and data sets related to UE/Session related parameters, e.g., mobility
pattern, spatio-temporal traffic model, wireless channel and coverage model,
QoS, and QoE.

  * •

Radio-access level devices including base stations, radio remote heads (RRHs),
cloud radio access networks (CRAN) or open RANs (ORANs) where they provide
coverage for specific area. These entities can share the following:

    * –

Models and data sets per each region for spatio-temporal traffic and mobility
models of users in that area, network measurements and logs, RAN-related
parameters (e.g., channel quality), load, interference, user density. resource
condition parameters (e.g., availability of front-haul), reliability and
robustness of services.

  * •

Core-level devices including switches, firewalls, packet gateways, computing
devices, storage devices of the users’ information, and clouds inside of the
networks, which can share the following:

    * –

Models and data sets for processing load and availability of core functions
(both virtual and physical), load and availability of computing resources,
large scale spatio-temporal traffic and mobility per zone for each type of
services and slices, core network measurements and logs, and procedure level
task information.

  * •

OTT and service applications including traditional data and voice services,
smart services for different sectors of industry, and slices which can provide

    * –

Models and data sets for slice QoS and QoE, slice isolation, spatio-temporal
traffic and mobility per slice, measurements, and logs per slices,

  * •

Management level which includes all the OSS and BSS parameters and they can
share the following parameters inside of the network or with another network
provider:

    * –

Models and data sets for fault, configuration, accounting, performance, and
security (FCAPS) and the management reports

Repository of knowledge in TL can be handled by training plane where seasonal,
daily, and weekly patterns of any knowledge, models and data sets can be
provided for different MAPE-K loops. However, the management plane takes care
of initiating and allowing the TL procedures between any source and target in
private and secure manner.

##  IV Challenges and Future Works

TL will play an important role in simplifying and streamlining the use of AI
and ML in 6G networks. One of the major challenges of deploying TL is to deal
with the vast number of possible approaches and formulations, so choosing the
best TL approach is a major implementation challenge in 6G. Also, TL mechanism
should be tuned based on the use case which poses the challenge of finding new
procedures to handle TL in more general manner.

There exist overlaps between TL and other concepts such as federated learning,
semi-supervised learning, multi-view learning, and most importantly, multi-
task learning. Having a unified view of TL and highlighting its overlaps with
other context can unlock the potential of TL in 6G.

Adaptation of TL for reinforcement learning (RL) in dynamic environment is
another challenge [14, 15]. Here, to have positive TL, transformation of
different parts of the information from the source (e.g., state-space, action-
space, reward, and transition dynamics) to the target should be studied. Since
there is more spatio-temporal dynamic behavior in the system, TL needs more
adaptation of the parameters. Most of the recent works are well-tuned
approaches for specific applications of TL in RL. Framework-agnostic TL for RL
is of desirable in 6G.

Positive TL requires appropriate clustering, correlation, and similarity
detection among sources and targets. This can be efficiently handled for
homogeneous TL among sources and tasks. However, in heterogeneous scenarios,
adapting the source and target domains and tasks are not trivial. Deriving
latent variables in data sets of different AI/ML algorithms can also lead to
positive TL and attain the better clustering and similarities among tasks.

In the most works, TL is used for a pair of one source and one target.
However, as mentioned in the examples above, in 6G, there is a set of sources
aiming to share their information with a set of targets. Expanding all the
existing formulations of one source-target pair TL to multi-source multi-
target TL in 6G can be considered as another future research direction.

Preserving the required security and privacy and preventing any threat from
attackers during the knowledge transfer among AI/ML algorithms are also
another important issue for efficient TL in 6G. TL in 6G should be immunized
against eavesdropping, GAN attacks (for models and data sets), poisoning, and
Sybil attacks.

##  V Deployment Scenario:
Quantization to Reduce TL Overhead

To overcome TL extra overhead in 6G, in this section, we study how
quantization algorithms can compensate this practical challenge. Quantization
in this context is a class of techniques for performing computations and
storing tensors at less number of bits than floating point case. Consequently,
a quantized model executes some or all of the operations on tensors with
integers rather than floating point values, leading to a more compact model
representation for DNNs.

We implement TL pipe-line between two neural networks in which one of them is
trained on the CIFAR-10 data set and the neural network model is MobileNetV2
[16] and we apply [17]. Fig. 5 shows the performance of quantized algorithms
with the case that the exact values of weights are passed among neural
networks, i.e., Floating point method which requires 32 bits of data per
weights of neural networks. In Fig. 5, the accuracy of training phase shows
the performance of floating point approach versus following quantization
algorithms for 8 bits per each weight [17]:

  * •

Default quantization approach which applies uniform quantization levels
between minimum and maximum values of weights

  * •

FBGEMM where by considering the distribution of the weights, quantization
levels is adjusted which leads to better accuracy compared to Default
quantization approach

  * •

Quantization aware training (QAT) In the above approaches, quantization is
applied at the end of the model training. In this approach, after that, the
model is fine-tuned on the quantized weights by training a couple of more
iterations to improve the accuracy. In our experiments, this method leads to
1% better accuracy compared to FBGEMM from Fig. 5.

Fig. 5 highlights that by 75% decreasing the load of the network (8 bits/32
bits) ($f_{mn}(\text{interaction})$), still the accuracy of training phase is
in the acceptable range for QAT. To demonstrate it figuratively, in Fig. 6,
the size of the quantized model is compared to that of the original model. As
shown in this figure, the size of the model is reduced to 75%. This simulation
shows that the quantization is an appropriate method to make a trade-off
between performance and amount of messages in the AI-aaS based networks.

After passing the quantized weights, we can re-tune the final layer of the
model in the target to increase the accuracy. We use a Pytorch tutorial of
”Quantized Transfer Learning for Computer Vision” and its colab version [18].
By re-training the model at the target, the accuracy level will be increased
up to the case without quantization. The re-training phase in the target takes
$3$ m and $16$ sec versus while full training requires $10$ m and $16$ sec for
convergence. On the other hand, we have $\tau^{\text{TL}}=3.142$, meaning that
TL can improve the energy efficiency with the factor $\tau^{\text{TL}}$, and
consequently, improving sustainability in 6G.

Figure 5: Percentage of accuracy of target versus quantization approaches
Figure 6: The amount of overhead of quantized versus non-quantized TL in 6G

##  VI Conclusion

In this paper, we address how efficient transfer learning (TL) in 6G can be
attained. We introduce metrics to assess efficiency TL from both learning
algorithm and networking aspects. We discuss possible modifications in 6G
architecture to handle TL and we define new elements in the network management
plane. We focus on new activities in training plane and its repositories for
TL in 6G. With some examples and looking at spatio-temporal features of
wireless networks, we show how classifications and clustering of algorithms in
6G can be attained to reach efficient TL. For positive TL, we require more
unified view of TL approaches with security and privacy guarantees discussed
in this paper.

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