# Configurational Entropy and braneworlds in $f(T,B)$ gravity

A. R. P. Moreira111Email<EMAIL_ADDRESS>F. C. E. Lima222Email:
<EMAIL_ADDRESS>and C. A. S. Almeida333Email:
<EMAIL_ADDRESS>Universidade Federal do Ceará (UFC), Departamento de
Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.

###### Abstract

Abstract

The thick brane scenario built on the $f(T,B)$ teleparallel gravity theory was
considered for the study of phase transitions, internal structures and new
classes of solutions in a model. In this theory, $T$ denotes the torsion
scalar, and $B$ is a boundary term. An interesting result was observed when
brane splitting occurs, i. e., internal structures in the model arise as a
consequence of the appearance of new domain walls in the theory. In fact, this
preliminary result influences the profile of the matter field (from kink to
multi-kink) so that for appropriate values of the parameters $k_{1,2}$
multiple phase transitions are identified. To perform this analysis, the
Differential Configurational Entropy (DCE) that has the ability to predict the
existence of phase transitions through critical points was used. Furthermore,
the DCE is able to select the most stable solutions since it gives us details
about the informational content to the field settings.

##  I Introduction

Investigation of the anisotropy of the cosmic microwave background WMAP ,
WMAP1 , supernovae Supernova , Supernova1 , and baryonic acoustic oscillations
SDSS  indicate an accelerated expansion of the universe. This expansion occurs
due to the existence of a repulsive gravitational energy, known as dark energy
Sharif . Currently, there are two lines of investigation that seek to
understand the acceleration of cosmic expansion. Several researchers propose
the study of exotic energy as an explanation for the problem Sahni , Sahni1 ,
Padmana , Caldwell , Nojiri , Feng , Kamenshchik . A new attempt at explaining
this question arises when Einstein’s Lagrangian is modified. Therefore,
modified gravity theories have caught the attention of several researchers
Sharif , Sotiriou , Sotiriou1 .

In 1920, the work of Kaluza Kaluza  and Klein Klein  brought the discussion of
extra dimensions as a narrative to unify Einstein’s general relativity and
Wang  electromagnetism. In 1998, Arkani-Hamed et. al. Arkani  propose a new
structure to solve the hierarchy problem that does not depend on supersymmetry
or technicolor. In this context, gravitational and gauge interactions become
joined at the weak scale. Soon later, L. Randall and R. Sundrum propose a new
high-dimensional mechanism to solve the hierarchy problem Randall&Sundrum ,
Randall&Sundrum1 .

The braneworlds scenario has been intensively studied in different theories of
modified gravity, e. g., the scalar-tensor theory of gravity Bogdanos ,
Bogdanos1 , Yang , Farakos , Liu , Guo , the theory of metric $f(R)$ gravity
Parry , Petrov , Petrov1 , and the $f(R)$ Palatini theory Bazeia , Gu , Olmo ,
Olmo1 . The theory of $f(T)$ gravity appears as an alternative to dark energy
to explain the acceleration of the universe Wang , Cai . These theories have
been intensively studied, as seen in Refs. Behdoodi , Nozari , Moreira2021b ,
Moreira2021a .

Recently, new teleparallel gravity models have emerged, namely: the
$f(T,T_{G})$ gravity with $T_{G}$ being the Gauss-Bonnet torsion scalar
Kofinas , Kofinas1 , Chatto , $f(T,\mathcal{T})$ gravity with $\mathcal{T}$
the stress-energy tensor trace Gomez , Harko , and the $f(T,B)$ gravity model
with $B$ the term of boundary Bahamonde , Franco , Rivera . The $f(T,B)$ model
has become an interesting model due to its agreement of theoretical results
with experimental data describing the accelerated expansion of the universe
Franco , Rivera , Said . As a matter of fact, the combination of the scalar
torsion and boundary term can allow Lorentz invariant theories BLi2010 ,
Sotiriou2010 , Capozziello2019 . Studies of cosmological perturbations
Bahamonde1 , thermodynamic properties Pourbagher , Pourbagher1 , and study of
gravitational waves Abedi  have also been developed considering $f(T,B)$
gravity Bhattacharjee .

The Mathematical Theory of communication of C. E. Shannon made it possible to
quantify the information and understand how the information can be transferred
efficiently from source to the receiver GleiserSowinki . Motivated by
Shannon’s theory, Configuration Information Measures (CIMs) present the
proposal to quantify the content and complexity of information contained in a
localized physical structure. These structures arise in topological field
theories, as can be seen in Refs. Ranada , Dva , LPA . Formulated in the
reciprocal space, Configurational Entropy (CE) and Configurational Complexity
(CC), and their differential variants, DCE and DCC, are the bases of CIMs. CE
plays an important role in theory, i. e., quantifies the number of bits needed
to build a stable field configuration out of wave modes GleiserSowinki , GS ,
Correa2015a .

In 2012, Gleiser and Stamatopoulos (GS) Gleiser2011  show that CE brings
information about some parameters of a given model for which the energy
density is located. In Ref. Gleiser2011  it was shown that the higher the
energy that approximates the actual solution, the higher its relative CE,
which is defined as the absolute difference between the actual function and
the test function of the CE Correa2015a . CE applications have been shown to
be a promising path for the physical understanding of some systems, e. g., the
non-equilibrium dynamics of spontaneous symmetry breaking G1 , the stability
bound for compact objects G2 , and for investigate the emergence of objects
located during the inflationary preheating G3 .

In braneworld scenario, DCE can be used to measure the degree of informational
organization in the structures of the system. In fact, DCE plays an important
role to decide the most suitable intrinsic parameters of the models
Correa2015c , Correa2016a , Correa2016b , Correa2016p . The approach of the
calculate of DCE can be applied to study the formation of internal structures
in the brane Cruz2017 , Cruz2018 . Additionally, DCE was studied in braneworld
scenarios in a context of modified gravity of the type $f(R)$ Correa2015a ,
and $f(R,\mathcal{T})$ Correa2015b . Inspired by these results, our purpose
here is make a more accurate analysis of the conditions that lead to the phase
transition in a braneworld scenario in a modified teleparallel $f(T,B)$
gravity. For this the DCE is calculated in our model.

Our work is organized as follows: In Sec. II, the fundamental concepts of
$f(T,B)$ gravity in a braneworld scenario is presented. In addition, the
energy densities of the brane and the matter field solutions of the models
$f_{1}(T,B)=T+k_{1}B^{n_{1}}$, and $f_{2}(T,B)=T+k_{2}(-T+B)^{n_{2}}$ were
investigated. In Sec. III, a brief review of CE and DCE concepts is performed.
Then, we study the DCE in the mentioned models. Finally, in Sec. IV we discuss
our findings.

##  II $f(T,B)$ brane models

Motivated by recent works on modified teleparallel $f(T,B)$ gravity
Moreira2021a , in this section we present the fundamental concepts of the
$f(T,B)$ gravity in a braneworld scenario.

###  II.1 $f(T,B)$ gravity

In teleparallel gravity, the dynamic variable for $f(T,B)$ gravity are the
vielbein, which are related to metric as follows,

$\displaystyle g_{MN}=\eta_{ab}h^{a}\ _{M}h^{b}\ _{N},$ (1)

where the metric signature is $\eta_{ab}=diag(-1,1,1,1,1)$, the uppercase
Latin letters $M,N=0,1,2,3,4$ represent the indices of the bulk coordinates
and the lowercase Latin letters $a,b=0,1,2,3,4$ represent the indices of the
tangent space coordinates.

The interesting thing about teleparallel gravity is that it assumes an non-
curvature connection, which is known as Weitzenböck connection
$\widetilde{\Gamma}^{P}\ _{NM}=h_{a}\ ^{P}\partial_{M}h^{a}\ _{N}$ Aldrovandi
. In this context, the torsion is not null, and is represented in terms of the
Weitzenböck connection as Aldrovandi

$\displaystyle T^{P}\ _{MN}=\widetilde{\Gamma}^{P}\
_{NM}-\widetilde{\Gamma}^{P}\ _{MN}.$ (2)

Through torsion, the contorsion tensor Aldrovandi  is define as

$\displaystyle K^{P}\ _{NM}=\frac{1}{2}\Big{(}T_{N}\ ^{P}\ {}_{M}+T_{M}\ ^{P}\
{}_{N}-T^{P}\ _{NM}\Big{)}.$ (3)

On the other hand, the dual torsion tensor Aldrovandi  is

$\displaystyle S_{P}\ ^{MN}=\frac{1}{2}\Big{(}K^{MN}\
_{P}-\delta^{N}_{P}T^{QM}\ _{Q}+\delta^{M}_{P}T^{QN}\ _{Q}\Big{)},$ (4)

and the torsion scalar $T=T_{PMN}S^{PMN}$. The curvature scalar and the
torsion scalar are related as follows,

$\displaystyle R=-T+B,$ (5)

where

$\displaystyle B\equiv-2\nabla^{M}T^{N}\
_{MN}=\frac{2}{h}\partial_{M}(hT^{M})$ (6)

is the boundary term, being that $T_{M}=T^{N}\ _{MN}$.

The gravitational Lagrangian is written in terms of $T$ and $B$, i. e.,
$\mathcal{L}=-hf(T,B)/4\kappa_{g}$, where $h=\sqrt{g}$, and $\kappa_{g}=4\pi
G/c^{4}$ is the gravitational constant Abedi2017 . So, the gravitational
action can be written as

$\displaystyle\mathcal{S}=-\frac{1}{4\kappa_{g}}\int
h\Big{[}f(T,B)+4\kappa_{g}\mathcal{L}_{m}\Big{]}d^{5}x,$ (7)

where $\mathcal{L}_{m}$ is the matter Lagrangian.

By varying the action (7) in terms of the dynamic variable (vielbein), we
obtain the field equations, namely,

$\displaystyle\frac{1}{h}f_{T}\Big{[}\partial_{Q}(hS_{N}\
^{MQ})-h\widetilde{\Gamma}^{R}\ _{SN}S_{R}\
^{MS}\Big{]}+\frac{1}{4}\Big{[}f-Bf_{B}\Big{]}\delta_{N}^{M}$
$\displaystyle+\Big{[}(\partial_{Q}f_{T})+(\partial_{Q}f_{B})\Big{]}S_{N}\
^{MQ}+\frac{1}{2}\Big{[}\nabla^{M}\nabla_{N}f_{B}-\delta^{M}_{N}\Box
f_{B}\Big{]}$ $\displaystyle=$ $\displaystyle\kappa_{g}\mathcal{T}_{N}\ ^{M},$
(8)

where $f\equiv f(T,B)$, $f_{T}\equiv\partial f(T,B)/\partial T$,
$f_{B}\equiv\partial f(T,B)/\partial B$, $\Box\equiv\nabla^{M}\nabla_{M}$ and
$\mathcal{T}_{N}\ ^{M}$ is the stress-energy tensor, that arises from the
Lagrangian variation of matter in terms of vielbein.

###  II.2 $f(T,B)$ braneworld

To analyze the braneworld scenario in a modified teleparallel $f(T,B)$
gravity, the following metric ansatz is considered Yang2012 ,

$ds^{2}=e^{2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$ (9)

where $e^{A(y)}$ is the warp factor and $\eta_{\mu\nu}=(-1,1,1,1)$ is the
Minkowski metric.

For the metric (9), we choose vielbein in the form $h^{a}\
_{M}=diag(e^{A},e^{A},e^{A},e^{A},1)$, which represents a good choice, since
the equations of the gravitational field do not add any additional
restrictions to the functions $f(T,B)$ and neither to the torsion scalar and
the boundary term Moreira2021a , Moreira2021b . Namely,

$\displaystyle T=-12A^{\prime 2}\ \ \ \text{and}\ \ \
B=-8(A^{\prime\prime}+4A^{\prime 2}),$ (10)

where the prime $(\ ^{\prime}\ )$ represents differentiation from $y$. Since
the curvature scalar can be written in terms of the torsion scalar and the
boundary term (5), we have

$\displaystyle R=-4(5A^{\prime 2}+2A^{\prime\prime}).$ (11)

Finally, we add a matter Lagrangian with a real scalar field
$\phi\equiv\phi(y)$, given in the form

$\mathcal{L}_{m}=\frac{1}{2}\partial^{M}\phi\partial_{M}\phi+V(\phi).$ (12)

The energy density is obtained as follows,

$\rho(y)=-e^{2A}\mathcal{L}_{m}.$ (13)

With Eq. (II.1), the gravitational field equations can be written as,

$\displaystyle\phi^{\prime\prime}+4A^{\prime}\phi^{\prime}=\frac{\partial
V}{\partial\phi},$ (14)
$\displaystyle\frac{1}{4}\Big{[}f+8(A^{\prime\prime}+4A^{\prime
2})f_{B}\Big{]}+6A^{\prime 2}f_{T}=\kappa_{g}\Big{(}\frac{1}{2}\phi^{\prime
2}-V\Big{)},$ (15) $\displaystyle
12\Big{[}A^{\prime}(A^{\prime\prime\prime}+8A^{\prime}A^{\prime\prime})(f_{BB}+f_{TB})+3A^{\prime\prime}A^{\prime
2}(f_{TT}+f_{BT})\Big{]}-\frac{1}{2}(A^{\prime\prime}+4A^{\prime 2})\times$
$\displaystyle(4f_{B}+3f_{T})-\frac{1}{4}f=\kappa_{g}\Big{(}\frac{1}{2}\phi^{\prime
2}+V\Big{)}.$ (16)

Although Eqs. (14), (15) and (16) look simple, it is difficult to find an
analytical solution for the most general cases. For simplicity, let us propose
an ansatz warp factor of the form Gremm1999 ,

$\displaystyle e^{2A(y)}=\cosh^{-2p}(\lambda y),$ (17)

where $p$ and $\lambda$ are the parameters that determine the width and width
of the source, respectively.

Now we propose two models of $f(T,B)$, i. e., $f_{1}(T,B)=T+k_{1}B^{n_{1}}$,
that is a model already analyzed in the braneworld scenario Moreira2021a ,
Moreira2021b , and $f_{2}(T,B)=T+k_{2}(-T+B)^{n_{2}}$ which gives a very
interesting model. This choice of $f(T,B)$ leads to results very similar to
that obtained in Ref. Correa2015b , where is considered a $f(T,R)$ gravity
model with $f(T,R)=T+kR^{n}$. The $k_{1,2}$ and $n_{1,2}$ parameters control
the modification of the usual teleparallel theory. To simplify our analysis,
the gravitational constant $\kappa_{g}=1$ is considered.

####  II.2.1 $f_{1}(T,B)=T+k_{1}B^{n_{1}}$

As shown in Refs. Moreira2021a , Moreira2021b , for $f_{1}(T,B)$ the equations
of the metric components and the scalar field are

$\displaystyle\phi^{\prime 2}(y)=$
$\displaystyle\frac{3}{2}p\lambda^{2}\mathrm{sech}^{2}(\lambda
y)-\frac{\alpha^{n_{1}}}{(2\beta-1)^{2}}\Big{[}8^{n_{1}-1}k_{1}n_{1}(n_{1}-1)(1+4p)\sinh^{2}(\lambda
y)\Big{]},$ $\displaystyle V(\phi(y))=$
$\displaystyle\frac{3}{4}\alpha+2^{3n_{1}-4}k_{1}(n_{1}-1)(p\lambda^{2})^{2}\alpha^{n_{1}-2}\Big{\\{}4\mathrm{sech}^{4}(\lambda
y)+64p^{2}\tanh^{4}(\lambda y)$ (19)
$\displaystyle-[3n_{1}+4(8+3n_{1})]\mathrm{sech}^{2}(\lambda
y)\tanh^{2}(\lambda y)\Big{\\}},$

where $\alpha\equiv p\lambda^{2}[\mathrm{sech}^{2}(\lambda
y)-4p\tanh^{2}(\lambda y)]$ and $\beta\equiv 1+p-p\cosh(2\lambda y)$. With
equation (13), we can find the energy density for $f_{1}(T,B)$, which is

$\displaystyle\rho(y)$ $\displaystyle=$
$\displaystyle\Bigg{\\{}2^{3n_{1}-2}\alpha(n_{1}-1)k_{1}-\frac{8^{n_{1}-1}}{(2\beta-1)^{2}}\Big{[}\alpha(n_{1}-1)3n_{1}k_{1}(1+4p)\sinh^{2}(\lambda
y)\Big{]}$ (20)
$\displaystyle-3\Big{[}(p\lambda)^{2}-\frac{1}{2}(1+2p)p\lambda^{2}\mathrm{sech}^{2}(\lambda
y)\Big{]}\Bigg{\\}}\cosh^{-2p}(\lambda y).$

The behavior of the energy density of the brane is shown in Fig. 1. For the
case $n_{1}=2$ (Fig.1$a$), varying the value of the parameter $k_{1}$ is
observed the emergence of internal structures in the brane. Further, for
$n_{1}=3$ (Fig.1$b$), varying the parameter $k_{1}$ the internal structures of
the model are intensified (this is seen with the increase amount of maximal
points). In fact, the appearance of such structures is related to the division
of the brane Moreira2021b , Moreira2021a .

---
(a) (b)
Figure 1: The energy density for $p=\lambda=1$. (a) $n_{1}=2$ . (b) $n_{1}=3$.

Through Eq. (II.2.1) we obtain the scalar field. As the general expression for
the scalar field is difficult to obtain, the numerical solutions for two
particular cases $n_{1}=2$ and $n_{1}=3$ are investigated. Figs. 2 and 3
describe the behavior of the scalar field (Fig. 2 for $n_{1}=2$ and Fig. 3 for
$n_{1}=3$). It seems to us that the model supports multiple phase transitions
transforming kink-like solutions into more complex configurations as the brane
splitting occurs.

A particularly interesting result is the existence of branes with internal
structure for certain values of $k_{1}$. As seen in Ref. D , the existence of
such structures in low-dimensional models coupled by gauge fields lead to the
existence of structures with superconducting properties. In the study of
cosmic strings the internal structures are related to the presence of
superconducting strings (see for example Hind , Hart , Cao ). At this point,
let us to give a possible interpretation of the phenomenon in our model. Note
that for the value of $k_{1}=-0.01$ and $n_{1}=2$ the topological scalar field
configuration have a kink-like behavior. In this case, the expected behavior
for energy density when $y\to 0$ is $\rho(y)\approx$ sech${}^{2}(y)$. However,
in the case that $n_{1}$ as well as the parameter $k_{1}$ decrease, the
appearance of internal structures on the brane it becomes evident. This is
seen as a direct consequence of a new domain wall that starts to form in the
vicinity of $y=0$. The appearance of these new domain walls makes the
topological configuration of the scalar field (kink-like) to have a profile
similar to a double-kink or even multi-kink configuration. Here it is
interesting to mention that similar behavior occurs for other values of
$n_{i}$, so this discussion can be generalized to all cases showed in this
work. We return to this discussion in the next section when we introduce the
DCE formalism to identify the location of domain walls and model phase
transitions.

---
Figure 2:  The scalar field solution for $n_{1}=2$ with $p=\lambda=1$.
---
Figure 3: The scalar field solution for $n_{1}=3$ with $p=\lambda=1$.

####  II.2.2 $f_{2}(T,B)=T+k_{2}(-T+B)^{n_{2}}$

Using Eqs. (15) and (16), we obtain the equations that are related to the
metric components and the scalar field as

$\displaystyle\phi^{\prime
2}(y)=\frac{3}{2}\Big{\\{}1-4^{n_{2}-1}k_{2}n_{2}[(2+5p)\mathrm{sech}^{2}(\lambda
y)-5p]^{n_{2}-1}\Big{\\}}p\lambda^{2}\mathrm{sech}^{2}(\lambda y),$ (21)
$\displaystyle V(\phi(y))$ $\displaystyle=$
$\displaystyle\frac{1}{8}\Bigg{\\{}6p\lambda^{2}(\beta-1)\mathrm{sech}^{2}(\lambda
y)-2{n_{2}}k_{2}(5\beta-1)^{n_{2}-1}[p\lambda^{2}\mathrm{sech}^{2}(\lambda
y)]^{n_{2}}$ (22)
$\displaystyle\times\Big{[}8-5n_{2}+4p(2n_{2}-5)\sinh^{2}(\lambda
y)\Big{]}\Bigg{\\}},$

remembering that $\beta\equiv 1+p-p\cosh(2\lambda y)$. The energy density is
obtained for $f_{2}(T,B)$ using Eq.(13), which here has the form

$\displaystyle\rho(y)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{\\{}(4p\lambda^{2})^{n_{2}}k_{2}[(2+5p)\mathrm{sech}^{2}(\lambda
y)-5p]^{n_{2}}\Big{[}\frac{5\beta-1-n_{2}(2\beta-1)}{1-5\beta}\Big{]}$ (23)
$\displaystyle+6p\lambda^{2}\beta\sinh^{2}(\lambda
y)\Big{\\}}\cosh^{-2p}(\lambda y),$

The energy densities of the brane are shown in Fig. 4. For $n_{2}=2$ (Fig.
4$a$), varying the value of the parameter $k_{2}$, arise the internal
structures in the model, i. e., two symmetric (maximum) critical points appear
near the origin. Meanwhile, the wells around the structures increase in depth.
For $n_{2}=3$ (Fig. 4$b$), varying the value of the parameter $k_{2}$, we have
the appearance of structures with three peaks. This energy density profile
shows the existence of localized structures.

Now, let us draw attention to the region around $y=0$ of the energy density.
In this region, it is evident for this case that there is multiple phase
transition making the energy density more intense and located around the
origin. In fact, analyzing the profile of the scalar field is observed that
this is a consequence of the emergence of settings similar to multi-kink. Here
it is interesting to mention that these results are related to brane
splitting.

---
(a) (b)
Figure 4: The energy density for $p=\lambda=1$. (a) $n_{2}=2$ . (b) $n_{2}=3$.

Using Eq. (21) the scalar field is obtained. For simplicity, we solve
numerically the equation since the general expression for the scalar field is
difficult to obtain. The behavior of the solutions of the field for two
particular cases, i. e., $n_{2}=2$, and $n_{2}=3$ are presented respectively
in Figs. 5 and 6. In fact, it is evident that the scalar field assumes a
behavior kink or multi-kink type (in some case). It seems to us that for some
values of $k_{2}$ we have the appearance of a new domain wall, and
consequently, multiple phase transitions when the brane is splitting. This
will be detailed in the next section.

---
Figure 5: The scalar field solution for $n_{2}=2$ with $p=\lambda=1$.
---
Figure 6: The scalar field solution for $n_{2}=3$ with $p=\lambda=1$.

##  III Configurational Entropy in brane Models

In 2012, Gleiser and Stamatopoulos (GS) GS  suggest a representation of the
detailed measure of the complexity of fields , i. e., the so-called
configurational entropy (CE). As discussed in Ref. GleiserSowinki  the
configurational information measures (CIMs) arises from the desire to quantify
the informational content and complexity contained in the form of localized
structures that arise in field theories. Constructed in reciprocal space,
configurational entropy and its differential variant, e. g., differential
configurational entropy (DCE), is a concept that is at the core of CIMs. In
this section, a brief discussion of the property of DCE measurements in a
braneworld scenario is presented. Using the concept of DCE, the possible phase
transitions that the model supports are investigated.

To calculate the DCE, let us define the Fourier transform as discussed in
Refs. Gleiser2011 , G2 , namely,

$\displaystyle\mathcal{F}[\omega]=\frac{1}{\sqrt{2\pi}}\int e^{i\omega
y}\rho(y)dy.$ (24)

In the braneworld $\mathcal{L}_{m}$ is the Lagrangian density of matter and
$e^{2A}$ is the warp factor. As defined in Eq. (13), the energy density is
$\rho(y)=-e^{2A}\mathcal{L}_{m}$ Correa2015c , Correa2016a , so Eq. (24) takes
the form

$\displaystyle\mathcal{F}[\omega]=-\frac{1}{\sqrt{2\pi}}\int e^{2A(y)+i\omega
y}\mathcal{L}_{m}dy.$ (25)

We define now the modal fraction $f(\omega)$, which measures the relative
weight of each mode $\omega$ as Gleiser2011 , G2 , Correa2015a , Correa2015b ,
Correa2015c , Correa2016a , Correa2016b , Cruz2017

$\displaystyle
f(\omega)=\frac{|\mathcal{F}[\omega]|^{2}}{\int|\mathcal{F}[\omega]|^{2}d\omega}.$
(26)

The CE emerged inspired by information theory, in particular Shannon’s theory
Shannon , so the CE can be described as Correa2015a , Correa2016a ,
Correa2016b

$\displaystyle S_{C}[f]=-\sum f_{n}\ln(f_{n}),$ (27)

which is able to provide us with informational content of settings that are
compatible with specific restrictions of a given physical system. So we have
some properties that are important to mention, as: $f_{n}=1/N$ when all $N$
modes carry the same weight. On the other hand, $S_{C}=0$ if only one mode is
present, and we have a discrete DCE maximum when $S_{C}=\ln N$ Correa2015a ,
Correa2016a , Correa2016b .

Finally, for the case of an arbitrary continuous function $f(\omega)$ on an
open interval, the DCE has the form Gleiser2011 , G2 , Correa2015a ,
Correa2015b , Correa2015c , Correa2016a , Correa2016b

$\displaystyle S_{C}[f]=-\int\bar{f}(\omega)\ln[\bar{f}(\omega)]d\omega,$ (28)

where $\bar{f}(\omega)=f(\omega)/f_{max}(\omega)$ is the normalized modal
fraction, and $f_{max}(\omega)$ denotes the maximum fraction.

###  III.1 $f_{1}(T,B)=T+k_{1}B^{n_{1}}$

For the model $f_{1}(T,B)$, we use the energy density (II.2.1) to find the
modal fraction $f(\omega)$. In the case of $n_{1}=2$, we have that the modal
fraction is

$\displaystyle
f(\omega)=\frac{385\pi\omega^{2}[9\omega^{2}+32k_{1}(4-15\omega^{2}+3\omega^{4})]^{2}}{1152[165+256k_{1}(600k_{1}-11)]}\mathrm{csch}^{2}\Big{(}\frac{\pi\omega}{2}\Big{)}.$
(29)

For $n_{1}=3$ the modal fraction is

$\displaystyle
f(\omega)=\frac{715\pi\omega^{2}[189\omega^{2}+32k_{1}(16420\omega^{2}-6912-3122\omega^{4}+95\omega^{6})]^{2}}{8064[19305+14336k_{1}(1454080k_{1}+663)]}\mathrm{csch}^{2}\Big{(}\frac{\pi\omega}{2}\Big{)}.$

In Fig. 7$a$ for $n_{1}=2$ and Fig. 7$b$ for $n_{1}=3$, the plots of modal
fractions are shown. The profile of the modal fraction is undergoing
modifications as we vary the $n_{1}$ and $k_{1}$ parameters. For $n_{1}=2$,
there are two maximal points that are symmetric in relation to origin, and
when we increase the value of $k_{1}$, new symmetric peaks arise. A similar
behavior occurs when $n_{1}=3$. Nonetheless, when $n_{1}=3$ the modal fraction
near the origin becomes more significant.

Let us now connect the discussion in the previous section to the results
obtained of the modal fraction. Note that when $k_{1}$ is positive the modal
fraction tends to have several oscillations. However, when $k_{1}$ takes on
negative values as $k=-0.03$ the oscillations tend to decrease so that two
symmetric critical points (maximum) and one local maximum in $y=0$ arise. In
fact, when comparing the modal fraction with our previous results, we notice
that the two peaks of greatest intensity are located near to the points where
the scalar field reaches its stability. As a matter of fact, it seems to us
that the peak (at $y=0$) indicates the region where the kink starts to
transform into a configuration similar to a multi-kink (in this case, similar
to double-kink). A consequence of this behavior is the appearance of internal
structures in the model indicating the existence of phase transitions and
brane splitting. Further, when $n_{1}=3$ a similar interpretation can be done.
Indeed, in this case, we have that the energy density is more intense
generating more significant internal structure with a small localization. This
leads to smoother one-kink to double-kink phase transitions. Due to the
existence of more internal structures close to $y=0$, the modal fraction
presents a more intense peak in this region.

We stress here that the calculation of DCE is performed by a numerical
approach. This occurs because of great complexity to calculate the integral
(28). The behavior of DCE is shown in Fig. 8. Note that if $n_{1}=2$ (Fig.
8$a$) the DCE has two minimum points and one maximum point, and the same
happens for the case $n_{1}=3$ (Fig. 8$b$).

In the case $n_{1}=2$, it is observed that the “absolute minimum point” is
located in the range $-0.02<k_{1}<-0.01$. Here it is important to mention that
in this range of values occur the emergence of internal structures that are a
consequence of the brane split. From topological field theory when
$-0.02<k_{1}<-0.01$ the phase transition, i. e., the transition from kink-like
structure to structures that look like double-kink occur. In this range of
values the emergence of new domain walls generating the possibility of new
topological structures is observed. In turn, for $n_{1}=3$ with
$0<k_{1}<0.001$, the brane splitting and the appearance of internal structure
in the model are also observed. However, in the case $n_{1}=3$ for the range
of values of $k_{1}$ due to the DCE profile, it is possible to verify the
existence of a structure similar to a triple-kink. This result is interesting
and promising as it suggests that more exotic topological structures may
appear in the model.

---
(a) (b)
Figure 7: Plots of modal fraction for $p=\lambda=1$. (a) $n_{1}=2$ . (b)
$n_{1}=3$.
---
(a) (b)
Figure 8: Plots of DCE for $p=\lambda=1$. (a) $n_{1}=2$ . (b) $n_{1}=3$.

###  III.2 $f_{2}(T,B)=T+k_{2}(-T+B)^{n_{2}}$

For the model $f_{2}(T,B)$, we use the energy density (21) to find the modal
fraction $f(\omega)$. Initially, when $n_{2}=2$ the modal fraction is

$\displaystyle
f(\omega)=\frac{77\pi\omega^{2}[45\omega^{2}-4k_{2}(68-30\omega^{2}+7\omega^{4})]^{2}}{5760[165+8k_{2}(422k_{2}-121)]}\mathrm{csch}^{2}\Big{(}\frac{\pi\omega}{2}\Big{)}.$
(31)

On the other hand, if $n_{2}=3$ the modal fraction is

$\displaystyle
f(\omega)=\frac{1001\pi\omega^{2}[135\omega^{2}+2k_{2}(5128\omega^{2}-5184-616\omega^{4}+7\omega^{6})]^{2}}{5760(19305+126048k_{2}+32905472k_{2}^{2})}\mathrm{csch}^{2}\Big{(}\frac{\pi\omega}{2}\Big{)}.$

Similar to the $f_{1}(T,B)$ model, the modal fraction profile in the
$f_{2}(T,B)$ model changes as the parameters $n_{2}$ and $k_{2}$ are modified.
For $n_{2}=2$ when the parameter $k_{2}$ increases the appearance of several
oscillations occur, and arise an absolute maximal point at $y=0$. When $k_{2}$
is near the range that identifies the appearance of internal structures the
modal fraction assumes a new behavior, i. e., the modal fraction has a local
minimum that seems to identify the region of phase transition. Further, it is
observed the appearance of two symmetrical and absolute maximum points. These
points seem to identify the values at which the scalar field reaches
stability. It is important to mention that in this transition we have the
fields going from kink-like structures to structures that seem to behave like
the double-kink when occur the brane splittting. Identical behavior occur when
$n_{2}=3$ and $k_{2}$ is changed. However, it is interesting to mention that
in the second case, due to the existence of multi-kink (or three-kink)
solutions, the modal fraction has a more relevant contribution around the
origin, as discussed in the previous model.

As in the previous case, to obtain the DCE result of the model, the
calculations were numerically performed. The behavior of DCE is shown in Fig.
10. Note that for $n_{2}=2$ (Fig. 9$a$) the DCE presents something quite
different, where the minimum two points are very close. Further, in the case
$n_{2}=3$ (Fig. 9$b$), the DCE have an absolute maximal point followed by an
almost exponential decay.

In the case $n_{2}=2$, it is observed that the first minimal is located in the
range $0<k_{2}<0.1$. For this range of values also occurs the emergence of
internal structures that are a consequence of the brane splitting. For
$0<k_{2}<0.1$, the phase transition occurs, i. e., the transition from kink-
like structure to structures that look like double-kink. Something unusual
happens for $n_{2}=3$. The DCE maximal point located at $-0.01<k_{2}<0.12$,
marks the division of the brane (Fig. 4$b$) and the phase transition of the
scalar field (Fig. 6). In other words, the critical point in DCE seems to
identify the location of multiple walls that generate multi-kink
configurations, i. e., it seems to us that this region registers multiple
phase transitions in theory.

---
(a) (b)
Figure 9: Plots of modal fraction for $p=\lambda=1$. (a) $n_{2}=2$ . (b)
$n_{2}=3$.
---
(a) (b)
Figure 10: Plots of DCE for $p=\lambda=1$. (a) $n_{2}=2$ . (b) $n_{2}=3$.

##  IV Final remarks

In this work, we studied teleparallel gravity models of the type
$f_{1}(T,B)=T+k_{1}B^{n_{1}}$ and $f_{2}(T,B)=-T+k_{2}(-T+B)^{n_{2}}$. In both
cases, it was observed that the parameters $k_{1,2}$ and $n_{1,2}$ are
responsible for the division of the brane that generates internal structures
that are related to the appearance of new domain walls. In fact, with the
appearance of these walls, new classes of topological structures for the
matter field were obtained.

Taking into account the criteria of the approach presented by Gleiser and
Stamatopoulos Gleiser2011 , the DCE is directly related to the energy of the
system, where the smaller (larger) the DCE, the smaller (greater) the energy
of the solutions. DCE is able to provide criteria to control the stability of
settings based on the informative content of your profiles G2 . Thus, with the
help of DCE it was possible to select the most prominent solutions for our
$f_{1,2}(T,B)$ models, analyzing some specific values of the $n_{1,2}$ and
$k_{1,2}$ parameters.

In the model $f_{1}(T,B)$ with $n_{1}=2$, the DCE minimum point located around
$-0.02<k_{1}<-0.01$ marks the phase transition of the brane, which tends to
splitting. The same is true for $n_{1}=3$, with the DCE minimal point located
around $0<k_{1}<0.001$. On the other hand, for $f_{2}(T,B)$ with $n_{2}=2$,
the DCE minimum point located around $0<k_{2}<0.1$ marks the phase transition
of the brane, which tends to splitting. When $n_{2}=3$, the brane phase
transition is marked by the maximum point of the DCE which is located in the
interval $-0.01<k_{2}<0.12$. Here it is important to mention that the
topological configurations most likely to be found in the system are described
by the range of values of the parameters $k_{1,2}$ mentioned above.

Finally, it is observed that DCE provides a complementary perspective to
investigate the phase transition, splitting of the brane, and the arising of
new class of structures in braneworld in the context of modified teleparallel
gravity theories, or other theories of gravitation.

## Acknowledgments

The authors thank the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), no 309553/2021-0 (CASA), and Coordenaçao de
Aperfeiçoamento do Pessoal de Nível Superior (CAPES), for financial support.

## References

  * [1] WMAP collaboration, C. Bennett et al., Astrophys. J. Suppl. 148, 1 (2003) [astro-ph/0302207];
  * [2] WMAP collaboration, D.N. Spergel et al., Astrophys. J. Suppl. 148, 175 (2003) [astro-ph/0302209].
  * [3] Supernova Cosmology Project collaboration, S. Perlmutter et al., Nature 391, 51 (1998).
  * [4] Supernova Cosmology Project collaboration, S. Perlmutter et al., Astrophys. J. 517, 565 (1999).
  * [5] SDSS collaboration, D.J. Eisenstein et al., Astrophys. J. 633, 560 (2005).
  * [6] M. Sharif, and M. Zubair, J. Cosm. Astrophys. 2012, 28 (2012).
  * [7] V. Sahni, and A. A. Starobinsky, Int. J. Mod. Phys. D 9 373 (2000).
  * [8] V. Sahni, Lect. Notes Phys. 653, 141 (2004).
  * [9] T. Padmanabhan, Gen. Relativ. Gravit. 40, 529 (2008).
  * [10] R. Caldwell, Phys. Lett. B 545, 23 (2002).
  * [11] S. Nojiri, and S.D. Odintsov, Phys. Lett. B 562, 147 (2003).
  * [12] B. Feng, X.-L. Wang and X.-M. Zhang, Phys. Lett. B 607, 35 (2005).
  * [13] A. Y. Kamenshchik, U. Moschella, and V. Pasquier, Phys. Lett. B 511, 265 (2001).
  * [14] T.P. Sotiriou and S. Liberati, Ann. Phys. 322, 935 (2007).
  * [15] T.P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010).
  * [16] T. Kaluza, Math. Phys. 1921, 966 (1921).
  * [17] O. Klein, Nature 118, 516 (1926).
  * [18] J. Wang, Wen-Di Guo, Zi-Chao Lin, and Yu-Xiao Liu, Phys. Rev. D 98, 084046 (2018).
  * [19] N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B 429, 263 (1998).
  * [20] L. Randall, and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999).
  * [21] L. Randall, and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999).
  * [22] C. Bogdanos, A. Dimitriadis, and K. Tamvakis, Phys. Rev. D 74, 045003 (2006).
  * [23] C. Bogdanos, J. Phys. Conf. Ser. 68, 012045 (2007).
  * [24] K. Yang, Y.-X. Liu, Y. Zhong, X.-L. Du, and S.-W. Wei, Phys. Rev. D 86, 127502 (2012).
  * [25] K. Farakos, G. Koutsoumbas, and P. Pasipoularides, Phys. Rev. D 76, 064025 (2007).
  * [26] Y.-X.Liu, F.-W.Chen, H.Guo,and X.-N.Zhou, J. High Energy Phys. 05, (2012) 108.
  * [27] H. Guo, Y.-X. Liu, Z.-H. Zhao, and F.-W. Chen, Phys. Rev. D 85, 124033 (2012).
  * [28] M. Parry, S. Pichler, and D. Deeg, J. Cosmol. Astropart. Phys. 04 (2005) 014.
  * [29] D. Bazeia, A. S. Lobao, R. Menezes, A. Yu. Petrov, and A. J. Silva, Phys, Lett. B 729, 127 (2014).
  * [30] V. I. Afonso, D. Bazeia, R. Menezes, and A. Yu. Petrov, Phys. Lett. B 658, 71 (2007).
  * [31] D. Bazeia, L. Losano, R. Menezes, G. J. Olmo, and D. Rubiera-Garcia, Eur. Phys. J. C 75, 569 (2015).
  * [32] B.-M. Gu, B. Guo, H. Yu, and Y.-X. Liu, Phys. Rev. D 92, 024011 (2015).
  * [33] C. Barragán, and G. J. Olmo. Phys. Rev. D 82, 084015 (2010).
  * [34] G. J. Olmo, Intern. J. Mod. Phys. D 20, 413 (2011).
  * [35] Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, Rep. Prog. Phys. 79, 106901 (2016).
  * [36] A. Behboodi, S. Akhshabi, and K. Nozari, Phys. Lett. B 723, 201 (2013).
  * [37] K. Nozari, and N. Sadeghnezhad, Intern. J. Geom. Meth. Mod. Phys. 16, 1950042 (2019).
  * [38] A. R. P. Moreira, J. E. G. Silva, and C. A. S. Almeida, Europ. Phys. J. C 81, 1-9 (2021).
  * [39] A. R. P. Moreira, J. E. G. Silva, F. C. E. Lima, and C. A. S. Almeida, Phys. Rev. D 103, 064046 (2021).
  * [40] G. Kofinas, and E. N. Saridakis, Phys. Rev. D 90, 084044 (2014).
  * [41] G. Kofinas, and E. N. Saridakis, Phys. Rev. D 90, 084045 (2014).
  * [42] S. Chattopadhyay, A. Jawad, D. Momeni, and R. Myrzakulov, Astrophys. Space Sci. 353, 279 (2014).
  * [43] D. Saez-Gomez, C. S. Carvalho, F. S. N. Lobo and I. Tereno, Phys. Rev. D 94, 024034 (2016).
  * [44] T. Harko, F. S. N. Lobo, G. Otalora, and E. N. Saridakis, J. Cosm. Astrop. Phys. 2014, 21 (2014).
  * [45] S. Bahamonde, and S. Capozziello, Eur. Phys. J. C 77, 107 (2017).
  * [46] G. A. R. Franco, C. Escamilla-Rivera and J. Levi Said, Eur. Phys. J. C 80, 677 (2020).
  * [47] C. Escamilla-Rivera, and J. Levi Said, Class. Quant. Grav. 37, 165002 (2020).
  * [48] S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. L. Said, J. Mifsud, and E. Di Valentino, Teleparallel Gravity: From Theory to Cosmology. arXiv 2021, arXiv:2106.13793v3.
  * [49] B. Li, T. P. Sotiriou and J. D. Barrow, Phys. Rev. D 83 (2011), 064035.
  * [50] T. P. Sotiriou, B. Li and J. D. Barrow, Phys. Rev. D 83 (2011), 104030.
  * [51] S. Capozziello, M. Capriolo and L. Caso, Eur. Phys. J. C 80 (2020) no.2, 156
  * [52] S. Bahamonde, V. Gakis, S. Kiorpelidi, T. Koivisto, J. L. Said, and E. N. Saridakis, Eur. Phys. J. C 81, 53 (2021).
  * [53] A. Pourbagher, and A. Amani, Astrophysics Space and Science, 364, 1-8 (2018).
  * [54] A. Pourbagher, and A. Amani, Modern Phys. Lett. A 35, 2050166 (2020).
  * [55] H. Abedi and S. Capozziello, Eur. Phys. J. C 78, 474 (2018).
  * [56] S. Bhattacharjee, Phys. Dark Universe 30, 100612 (2020).
  * [57] M. Gleiser, and D. Sowinski, Phys. Rev. D 98, (2018) 056026.
  * [58] A. F. Rañada, Lett. Math. Phys. 18, 97 (1989).
  * [59] G. Dvali, I. I. Kogan, and M. Shifman. Phys. Rev. D 62, 106001 (2000).
  * [60] F. C. E. Lima, A. Yu. Petrov, and C. A. S. Almeida, Phys. Rev. D 103, 096019 (2021).
  * [61] M. Gleiser, and N. Stamatopoulos, Phys. Lett. B 713 (2012) 304.
  * [62] R. A. C. Correa, P. H. R. S. Moraes, A. de Souza Dutra and R. da Rocha, Phys. Rev. D 92 (2015) no.12, 126005.
  * [63] M. Gleiser and N. Stamatopoulos, Phys. Lett. B 713 (2012), 304-307.
  * [64] M. Gleiser and N. Stamatopoulos, Phys. Rev. D 86, 045004 (2012).
  * [65] M. Gleiser and D. Sowinski, Phys. Lett. B 727, 272 (2013).
  * [66] M. Gleiser and N. Graham, Phys. Rev. D 89, 083502 (2014).
  * [67] R. A. C. Correa and R. da Rocha, Eur. Phys. J. C 75 (2015) no.11, 522.
  * [68] R. A. C. Correa, D. Monteiro Dantas, P. H. R. da Silva Moraes, A. de Souza Dutra and C. A. S. de Almeida, Annalen Phys. 530 (2018) no.7, 1700188.
  * [69] R. A. C. Correa, P. H. R. S. Moraes, A. de Souza Dutra, W. de Paula and T. Frederico, Phys. Rev. D 94 (2016) no.8, 083509.
  * [70] R. A. C. Correa, D. M. Dantas, C. A. S. Almeida and R. da Rocha, Phys. Lett. B 755 (2016), 358-362.
  * [71] W. T. Cruz, D. M. Dantas, R. A. C. Correa and C. A. S. Almeida, Phys. Lett. B 772 (2017), 592-598.
  * [72] W. T. Cruz, D. M. Dantas, R. V. Maluf and C. A. S. Almeida, Annalen Phys. 531 (2019) no.10, 1900178.
  * [73] R. A. C. Correa and P. H. R. S. Moraes, Eur. Phys. J. C 76 (2016) no.2, 100.
  * [74] R. Aldrovandi and J. G. Pereira, “Teleparallel Gravity: An Introduction,” (Springer,Berlin, 2013).
  * [75] H. Abedi and S. Capozziello, Eur. Phys. J. C 78, 474 (2018).
  * [76] J.Yang, Y.-L. Li, Y. Zhong and Y. Li, Phys. Rev. D 85, 084033 (2012).
  * [77] M. Gremm, Phys. Lett. B 478, 434 (2000).
  * [78] D. Bazeia, D. A. Ferreira, and M. A. Marques, Europ. Phys. J. C 81, 619 (2021).
  * [79] M. Hindmarsh, K. Rummukainen, and D.J. Weir, Phys. Rev. Lett. 117, 251601 (2016).
  * [80] B. Hartmann, F. Michel, and P. Peter, Phys. Rev. D 96, 123531 (2017).
  * [81] X.-F. Cao and Y.-W. Yu, Phys. Rev. D 97, 023022 (2018).
  * [82] C. E. Shannon, The Bell system technical journal 27, 379 (1948).


# Cluster Formation and Relaxation in Particle Rafts under Uniform Radial
Expansion

Khá-Î Tô  The Department of Physics and the James Franck and Enrico Fermi
Institute
The University of Chicago IL 60637

###### Abstract

Particle rafts floating at an air-liquid interface exhibit a variety of
behaviors when interfacial flow is introduced. Motivated by previous
experimental observations, the failure pattern of particle rafts under uniform
radial expansion is reported in this paper. The expansion process is
specifically designed to expand the system affinely in the radial direction
and to keep the velocity gradient constant throughout. A strong resemblance to
the results of particle rafts under uniform uniaxial expansion [1] is found.
The size of the cluster emerging as the rafts are pulled apart scales
inversely with the pulling velocity. This is a result of two competing
velocities: the inter-particle separation speed provided by the flow and a
size-dependent relaxation speed for clustering. A model, generalized from a
one-dimensional linear (in)stability calculation, is in agreement with the
failure morphology found for this radially expanded system. Nonlinear
relaxation and particle rearrangement is observed after the initial clustering
occurs. This is a feature unique to a two-dimensional system. With its easily
accessible particle dynamics at the microscopic level, this system provides
insights into the morphology controlled by two competing mechanisms in two or
higher dimensions and across different scales.

††preprint: APS/123-QED

##  I Introduction

Sub-millimeter particles floating at an air-liquid interface can aggregate
into particle rafts. The inter-particle attraction arises from the coupling
between the particles and the fluid interface [2, 3, 4]. The resulting
particle rafts can be treated like two-dimensional solids [5]. For example,
compressing the raft from two of its opposing ends at small strain can induce
an instability in which the particles buckle out of the plane [6, 7, 8].
Likewise, pulling the raft from its opposing ends quasi-statically can induce
ductile deformation [9]. The underlying fluid plays a crucial role. Not only,
does it produce the capillary forces that create the inter-particle
attractions, it also provides substantial expansion or compression forces when
the fluid surface area is varied sufficiently rapidly. Thus the fluid
interface is a valuable tool with which to control particle-raft dynamics when
the flow is relevant.

Different regimes of particle-raft behavior have been investigated. A previous
study [1] explored the failure of particle rafts under expansion with a
uniformly expanding metric in one dimension. This produced cracks throughout
the material. By keeping the velocity gradient uniform along the extensional
axis, the failure was found to be homogeneous with no single crack dominating
the resultant morphology. A smooth change in the distance between neighboring
cracks was observed as the pulling speed was varied; the failure pattern
evolved from an almost unperturbed, densely-packed raft at low shear rates to
a uniformly distributed set of rifts separating small clusters of particles at
high shear rates. In these experiments, the orientations of the particle
clusters were found to be nearly isotropic; despite having a specific pulling
direction, the clusters were oriented with no strong variation with respect to
the extension axis. This suggests that the direction of the applied shear
plays no significant role in determining the failure pattern of particle
rafts.

In other studies, the rafts were pulled apart radially so that the extension
was in all directions simultaneously. In one case, this expansion was applied
by adding surfactant near its center in order to impose a surface-tension
gradient across the raft [10, 11]. In another study, advection was generated
by pumping water into a funnel so that the raft expanded as the water level
rose [12]. In both cases, as the systems evolved under the applied forces,
wide cracks appeared preferentially near the center of the rafts. The
morphology in these cases is thus significantly different from what was found
in the one-dimensional experiments.

This raises a puzzle as to what aspects of the expansion gives rise to the
differences observed between the radial and linear pulling geometries. To
address this puzzle, this paper examines the failure morphology of particle
rafts pulled apart under radial expansion using a protocol that more closely
aligns with that used in the one-dimensional experiments. In particular, the
experiments are designed to produce a nearly uniform velocity gradient
everywhere on the air-water interface. I verify the velocity field using
Particle Image Velocimetry (PIV). The results of these experiments are more
similar to the one-dimensional pulling studies suggesting the importance of
minimizing velocity gradients in the expansion dynamics.

In order to analyze these results, I generalize two dimensions to an
(in)stability calculation used to determine the cluster sizes, i.e., the
distance between rifts, in the one-dimensional (1D) situation [1]. With no
further assumptions on the elastic properties of the aggregate, the
calculation showed that the morphology was determined by a cross-over
phenomenon due to the competition between the rate of separation (caused by
the pulling velocity) and the rate of relaxation (caused by the inter-particle
interactions) at different cluster sizes. In this model, the average cluster
length scales inversely with the velocity, in good agreement with the
experimental results.

In addition to exploring the failure pattern of particle rafts under a
different expansion metric, the present study is also inspired by the
recognition that the study of structure formation under isotropic competing
mechanisms is relevant in many systems across different scales [13]. At the
largest scale, the competition between gravity and cosmological expansion
leads to the cluster formation in the universe [14]. In particle rafts, the
capillary attraction is asymptotically inversely proportional to the distance
between particles, which has the same form as two-dimensional gravity.
Therefore, working with an radial-expansion metric provides a two-dimensional
table-top experiment for studying this type of phenomenon. At much smaller
scales, the clustering of colloids [15, 16] and the microbial growth on liquid
substrate [17] both have isotropic competing interaction/flow. Two-dimensional
particle aggregates under radial expansion can offer a macroscopic platform
(with access to the microscopic entities involved) to study the morphology
controlled by these competing processes.

Figure 1: Failure morphology at different pulling velocity $V$. (A) A
schematic of the experimental apparatus. Left: The side view of the whole
apparatus. A radial expander inserted into water creates an nearly-radial flow
to stretch the particle raft enclosed inside. Middle: The side view of the
boundary of the Mylar strip boundary. Right: The top view of the expander at
early and late time. The gray arms are connected and move coherently to make
the twelve orange inner nodes move outward. The inner nodes are connected by
Mylar strips painted in blue, which set the boundaries of the system. The
dotted black lines show the weight tied to the Mylar strips to keep them tout
during the motion. (B, C) Snapshots and zoomed-in images of experiments at a
fixed stretch ratio $\lambda=1.5$ for different pulling velocities $V$.

##  II Experimental Details

The particle rafts consist of polyethylene spheres floating at an air-water
interface. The polyethlene particles have density $1080$ $kgm^{-3}$ and are a
mixture of two diameter ranges: $d=550\pm 50\mu m$ (small) and $d=655\pm 55\mu
m$ (large). Polydisperse packings are made by mixing roughly equal volumes of
these two sizes. The particles naturally aggregate into a raft due to the
lateral capillary attraction, also known as the “Cheerios” effect [2, 3, 4]. A
monolayer of densely-packed particles is prepared by gathering the particles
manually into the center of the apparatus as shown in the left panel of Fig.
1(A). The rafts are compacted by slight vibrations of the interface and have
an initial packing fraction of $0.72\pm 0.01$. The particle raft floats on
deionized water which has density $\rho_{w}=998$ $kgm^{-3}$; dynamic viscosity
$\eta_{w}=0.95$ $mPas$ and surface tension, $\sigma_{w}=0.073$ $Nm^{-1}$ at
$22^{\circ}C$.

To create an isotropic flow, a two-dimensional radial expander is custom-built
to create a coordinated movement of twelve nodes going radially outward
simultaneously. A schematic of the experimental apparatus is shown in Fig. 1A.
The nodes are connected by twelve moving Mylar strips to form an expanding
twelve-sided polygon. The Mylar strips are inserted vertically into the water
as shown. Because Mylar is slightly hydrophilic, the particles rafts are
enclosed inside the boundaries but do not make direct contact with the
boundaries. This is shown in the middle panel of Fig. 1A. The rafts are thus
pulled only by the flow of the fluid which rises incrementally and uniformly
everywhere inside the boundaries as the Mylar boundaries are extended outward
radially. The fluid interface can appropriately be considered as an expanding
metric on which the particles float. As the nodes simultaneously move outward,
as shown in “Top View” in Fig. 1A, an approximately isotropic flow is
generated by the moving Mylar strips. (See Appendix A about how Mylar strips
are attached and how the expander is driven.)

The distance between the midpoint of each side to the center of the dodecagon
is $L_{0}/2$ and $L_{0}\approx 61$ $mm$. Each side moves outwards away from
the center at the same pulling speed $V/2$. The Particle Image Velocimetry
(PIV) measurement at various $V$ is shown in Appendix B. This allows the flow
field $\bm{u}(x,y)$ to be described as an affine expansion:

$\bm{u}(x,y)=\frac{V}{L}(x\bm{\hat{x}}+y\bm{\hat{y}})$ (1)

where the origin is the center of the raft and $x$ and $y$ are the two
orthogonal coordinates in the plane. The flow is radially outward away from
the center.

The pulling speed, $V$, is varied from
$2.8\text{\,}\mathrm{mm}\text{/}\mathrm{s}260\text{\,}\mathrm{mm}\text{/}\mathrm{s}$.
The raft diameter increases as $L=L_{0}+Vt$ and the stretch ratio is defined
as $\lambda=L/L_{0}$. The initial rate of stretch,
$\dot{\lambda}_{0}=V/L_{0}$, is varied from
$5\text{\times}{10}^{-2}\text{\,}\mathrm{s}^{-1}4.6\text{\,}\mathrm{s}^{-1}$.
The experiments are stopped at a maximum stretch ratio $\lambda_{max}\sim
1.75$.

I use the method described in [1] to measure the cluster length, $\ell$,
inside each raft. This requires counting the number of pixels between rifts on
each line of the raster-scanned image. The only complication is due to the
difference in pixelization at angles different from the rectangular
coordinates of the images produced by the camera.

##  III Experimental Results

As the twelve-sided boundary moves outward so that the diameter increases at
velocity $V$, the raft of particles is stretched by the underlying flow so
that micro-cracks, or rifts, begin to form as shown in Fig. 1B. At
intermediate to high $V$, the rifts emerge soon after the onset of expansion.
They are distributed diffusely throughout the entire raft and grow larger with
time. The experiments are stopped at a stretch ratio ($\lambda_{max}\sim
1.75$). After their formation and initial growth, the same rifts remain up to
a larger stretch ratio; once a rift is formed, it does not collapse at later
times. This can be observed in the movies included in the Supplementary
Information (SI). Likewise, the particles in between the initially-formed
rifts remain connected. These clusters of particles maintain their
configurations and are distributed homogeneously throughout the entire system
with a characteristic size. The average size of these clusters of particles,
$\ell$, is strongly dependent on the pulling speed $V$, as shown in Fig. 1B,
C.

When $V$ is small enough, the raft shape is only very slightly perturbed at
the initiation of expansion and remains unchanged afterwards. There is a small
drift in the position due to secondary flows but the internal structure is
undisturbed. In this low-velocity regime, no significant rifts can be observed
and the cluster width, $\ell$, remains close to the initial system size,
$L_{0}$.

With increasing $V$, the number of rifts increases while the cluster size,
$\ell$, decreases. The cluster size, $\ell$ decreases until it reaches the
size of a single particle at large velocities, as shown in the right column of
Fig. 1B. Figure 1C shows how the internal features change with increasing $V$:
a single relatively closely-packed structure at low $V$, evolves upon
increasing $V$ into separate clusters consisting of only a few (sometimes only
one) particles.

Figure 2: Angle-averaged distribution of cluster lengths, $\ell$, for
different velocities at fixed stretch ratio, $\lambda=1.5$. The most dominant
length, that is the probability of cluster length $\ell$ multiplied by $\ell$
normalized by $\langle\ell\rangle=\sum\ell P(\ell)$, is plotted versus $\ell$
with three pulling speeds, $V$: $2.8$ $mm/s$ (blue), $26$ $mm/s$ (green) and
$260$ $mm/s$ (purple).

The average cluster length, $\langle\ell\rangle$, is obtained by averaging the
measurement at all different angles. This is because there is no specific
tensile axis in this radial expansion system. The percentage standard
deviation of this average quantity is smaller than $3\%$ for all velocities,
which is comparable to the fluctuation due to rotating pixelated images. This
indicates that the variation in cluster orientations is very low and below the
uncertainty level of detection.

Figure 2 shows the angular-averaged distribution of cluster sizes at a liquid
stretch ratio $\lambda=1.5$ for different pulling velocities. The ordinate is
the most dominant cluster length, $\ell P(\ell)/\langle\ell\rangle$, where
$P(\ell)$ is the probability of finding a cluster of length $\ell$ and the
average cluster length $\langle\ell\rangle=\sum\ell P(\ell)$. Since the
statistics with small cluster lengths always outnumber those with larger
$\ell$, I plot the most dominant length, that is how much material has cluster
length $\ell$. This can help better identify the cluster distribution at each
velocity, $V$. At small $V$, a large portion of the distribution remains close
to the initial size of the raft, $L_{0}$. The distribution shifts to smaller
$\ell$ as $V$ increases. At the highest pulling speed, $V$, most clusters have
lengths between $d$ and $2d$.

Figure 3: Averaged cluster length, $\langle\ell\rangle=\sum\ell P(\ell)$,
versus pulling speed, $V$ at fixed stretch ratio $\lambda=1.5$. (A) The blue
points are measurements of $\langle\ell\rangle$ versus $V$. The blue curve is
a fit of Eq. 2. The red solid line indicates $L_{0}/\sqrt{2}$, which is the
size of the measuring box. The red dashed line shows the largest particle
diameter $d_{max}$. The blue curve is a fit to Eq. 2. (B) $\langle\ell\rangle$
is plotted versus $V$ for both radial (blue dots) and uniaxial (green squares)
experiments at similar expansion condition ($\lambda=1.5$ and
$\varepsilon=0.5$). The black line shows the theoretical prediction for water
from Ref. [1].

Figure 3A shows the average cluster length $\langle\ell\rangle$ versus pulling
speed $V$ at a fixed value of $\lambda=1.5$, for all the experiments. One can
see that $\langle\ell\rangle$ decreases monotonically with increasing $V$ and
levels off at high velocity. Because a cluster cannot be significantly larger
than the initial system size, $L_{0}$, or smaller than a particle diameter,
$d$. The data can be interpolated between these two extremes using a similar
form as was used in the one-dimensional pulling experiments [1]:

$\langle\ell\rangle=\frac{1}{aV^{b}+1/(L_{0}/\sqrt{2}-d_{max})}+d_{max}.$ (2)

This assumes a power-law dependence of the average cluster length away from
the two extremes. The difference from the one-dimensional fitting form is that
initial raft size $L_{0}$ is scaled by $\sqrt{2}$ because the cluster lengths
are measured in a square box of size $L_{0}/\sqrt{2}$ located in the center of
the raft to avoid complexity caused by the round shape of the raft.

The fit gives us $a=(9\pm 2)\times 10^{3}$ and $b=1.20\pm 0.05$. This result
is similar to the results in uniaxial expansion. The average cluster length,
$\langle\ell\rangle$, scales inversely with the pulling velocity $V$.

##  IV Discussion

Figure 3B compares the results from these radial expansion experiments with
those found in the 1D experiments. Both data sets are shown at a similar time
in the expansion: the liquid strain $\varepsilon=0.5$ for 1D experiment and
$\lambda=1.5$ for radial experiment both occur when the system size is $1.5$
times larger than the original size of the raft. The experiments under radial,
isotropic expansion give very similar results to the 1D experiments. This
suggests that the experiments with isotropic expansion can in fact be
generalized from the analysis used in one-dimension.

To understand this similarity, let us reexamine the flow field described in
Eq. 1. The velocity field can be expanded around any arbitrary point,
$(x_{0},y_{0})$, within the boundaries:

$\bm{u}(x,y)=\bm{u_{0}}+\frac{V}{L}((x-x_{0})\bm{\hat{x}}+(y-y_{0})\bm{\hat{y}})$

where $\bm{u_{0}}$ is the velocity at $(x_{0},y_{0})$. Independent of the
position, $(x_{0},y_{0})$, the velocity field expands radially outward from
that point; all points are equivalent and every particle feels the surrounding
fluid retreating at the same constant velocity in all directions. Thus, the
local motion can be reduced to multiple one-dimensional chains as treated in
the case of uniaxial expansion.

The large cracks that were observed near the raft centers in previous
experiments where the rafts were pulled radially in two-dimensions [10, 11,
12], are not present in the data presented here. The difference in the way the
pulling of the raft is accomplished is likely the reason for this difference.
The present experiment was designed to minimize higher-order gradients in the
flow. When using the funnel method [12], I also observed the formation of
large cracks which I attributed to gradients in the velocity field produced by
the influx of liquid near the center of the raft.

Figure 4: Time evolution of cluster sizes in both radial and uniaxial
experiments. (A) Average cluster length, $\langle\ell\rangle$, versus liquid
stretch ratio $\lambda$ or strain $\varepsilon$ at all $V$. The results from
radial and uniaxial expansion experiments are shown in crosses and dots
respectively. The color from dark purple to light orange represent results
from fast to slow $V$, and span the entire pulling velocity range in both
experiments.) (B) The ratio of particles occupied in the cluster versus strain
$\varepsilon$ for uniaxial experiments. The results from four fastest pulling
speeds, $V=20$ $mm/s$ (orange) to $200$ $mm/s$ (purple), are presented. (C)
The change in $\langle\ell\rangle$ , divided by $\langle\ell\rangle$ at
$\varepsilon=1.0$ (solid circles), and the inverse of normalized number of
clusters, $n_{c}$, scaled by $n_{c}$ at $\varepsilon=1.0$ (empty circles), are
plotted versus $\varepsilon$.

To understand the two-dimensional effects intrinsic to this systems, I analyze
both the uniaxial and radial expansion datasets. As shown in Fig. 4A, the time
evolution of the cluster sizes shows a similar trend in both cases. Here I
focus on the time evolution in the uniaxial expansion experiments because they
have the greater range. Only the behavior for liquid strain $\varepsilon\geq
1.0$ are examined because the cluster sizes at early times is complicated by
limitations of image processing. A square observation window of size $L_{x0}$
located at the center of the rafts is also chosen to avoid dealing with the
irregular edges of the rafts.

At low $V$, the cluster size is nearly independent of $\varepsilon$. At higher
pulling speed, however, the average cluster sizes consistently decrease with
increasing $\varepsilon$. This suggests non-linear, late-time relaxation of
the raft.

One possibility for this is that, after the initial expansion, the particles
which are further apart experience a smaller velocity gradient. However, the
particles that are initially more compact continue to experience strong
attraction from their neighbors; they rearrange so the particle clusters
become more compact. Another possibility is that the clusters simply continue
to break up as time goes on.

To understand the cause of the decrease in cluster sizes, I calculate the
particle ratio inside the clusters that have been identified. The particle
ratio is defined as the area of particles inside the clusters divided by the
total area of the clusters. This quantity indicates if the voids inside the
clusters are shrinking in time. As shown in Fig. 4B, for four higher pulling
velocities, the particle ratios all increase in time. This suggests that
rearrangement of neighboring particles inside a cluster is indeed taking
place.

On the other hand, Fig. 4C examines whether the clusters are breaking up with
time. The solid dots represent the cluster lengths $\langle\ell\rangle$
normalized by $\langle\ell_{\varepsilon=1.0}\rangle$, the measurement at
$\varepsilon=1.0$ versus strain. I also plot the inverse of the normalized
number of clusters, $n_{c}$ (scaled by $n_{c,\varepsilon=1.0}$) as empty
circles. $n_{c}$ is the number of clusters in the observation region
normalized by the total area of particles. This quantity shows how many
clusters are formed per unit area of materials. If the inverse of $n_{c}$
matches the change in cluster length, this implies that the change in the
number of clusters is the main contributor to the change in cluster sizes,
which is the case at $V=20$ $mm/s$ Moreover, the overall change in
$\langle\ell\rangle$ is small because the particles are already at a lower-
energy configuration to start with. The slight increase in $n_{c}^{-1}$ after
$\varepsilon=1.7$, that is the decrease in the number of clusters, suggests
that a larger-scale coalescence has occurred as the global shear became very
weak.

At $V=200$ $mm/s$, a significant difference between $n_{c}^{-1}$ and
$\langle\ell\rangle$ can be observed. The average cluster length decreases by
more than $20\%$ while the inverse of $n_{c}$ only changed by $10\%$. A
substantial rearrangement between neighboring particles is therefore
responsible for this change, as shown in Fig. 4B as well.

##  V Conclusions

In this paper, the morphology of particle rafts floating at an air-water
interface is studied as the underlying fluid is forced to flow radially
outward at a uniform constant velocity gradient. The experiments are designed
to minimize any high-order velocity gradients in the system. By examining the
flow field of the radial expansion with particle image velocimetry (PIV), the
local expansion metric is found to be uniform inside the apparatus. As was
found in the case of uniaxial expansion described in Ref. [1], the failure
morphology due to the outward flow varies smoothly as a function of the fluid
expansion velocity. When the velocity is low, the rafts remain intact
throughout the expansion without the formation of visible internal cracks. For
larger expansion velocities, rifts open up inside the rafts. The rifts are
distributed diffusely with a spacing between them that decreases with
increasing velocity. At the largest velocities used in the experiment, the
separation decreases until the cluster size, that is the length between rifts,
becomes only slightly larger than the size of a single particle. The measured
average cluster sizes are very close to that of the one-dimensional expansion
experiment and follow the same scaling behavior despite the fact that the
geometry of the expansion metric is different. This indicates that the 1D
(in)stability model that was used to describe the relaxation and cluster
formation in [1] can be generalized to two-dimensions and is sufficient to
explain the radial experiment.

By measuring the time evolution of the clusters, a non-linear relaxation is
found at a later time; after the initial formation of clusters, the clusters
start to shrink in sizes. The decrease in particle ratio inside the clusters
shows that the particles continue to rearrange and become more compact even
while the overall outward expansion continues. This can be understood because,
once the initial expansion rate sets the dominant cluster length, the velocity
gradient between clusters is no longer that used initially. Because the
particles are farther apart (in the rifts), the rifts will increase in size
faster than the spaces between the more densely packed particles. Moreover,
the global velocity gradient is also decreasing monotonically as the system
size becomes larger. The particles inside an individual cluster will rearrange
according to the competition between the local relaxation rate and the local
shear rate set by neighboring particles. This later-time aspect of the
relaxation is not captured by the one-dimensional linear (in)stability
analysis.

This experiment provides a flexible platform for studying how morphology can
form by the competition between two isotropic mechanisms. There are many other
systems that share this common attribute. As pointed out in [1], an obvious
analog is cosmological expansion [14]. In this table top experiment, the
interaction between particles has the same form as gravity in two dimensions.
Other examples include gel formation[15] and structure formation by colloids
in polymer solutions[16]. This two dimensional system gives us a direct access
to the clustering morphology with minimal imaging complexity which would be
troublesome in three dimensional systems. In addition, without tuning the
details of the potentials chemically or at a small scale, one of the competing
processes can be easily manipulated by changing the mechanical driving
protocol, while the other can be controlled by changing specific particle-
liquid or particle-particle interactions.

##  VI Acknowledgement

I thank T. A. Witten, V. Vitelli and M. M. Bandi for insightful discussion. I
am deeply grateful to S. R. Nagel for his support and mentorship. This work
was supported by the National Science Foundation (MRSEC program NSF-DMR
2011854), by the Simons Foundation for the collaboration Cracking the Glass
Problem Award #348125 and by Government Scholarship to Study Abroad by the
Ministry of Education in Taiwan.

##  Appendix A Details of the Radial Expander

Figure 5: Details of the radial expander design. (A) Schematics of the top
view of the radial expander. The inner nodes are shown as orange and the outer
nodes are green. Two of the inner (orange) nodes are connected to the opposite
sides of a belt which is driven by a motor. The movement of the belt drives
those two nodes in opposite directions. This opens (closes) the entire
structure when the belt is rotated clockwise (counter-clockwise). Because of
the geometry connecting all the nodes, pulling on just two of the inner nodes
is sufficient for the entire structure to open smoothly. (B) A side view of
the green and orange nodes. The green nodes are placed to ensure that the
entire structure is stable and does not wobble vertically during the opening
of the expander. The bottom of a green node is attached to a ball bearing so
that it slides smoothly on a clear acrylic platform. Each orange node is a
screw with a 3D-printed attachment at its tip. This attachment is composed of
two closely-spaced columns that are threaded by two adjacent Mylar ribbons.
These ribbons slide smoothly around the two columns as the expander is moving.
Each ribbon is held in place by the attachments at two adjacent orange nodes.
To keep the Mylar ribbon taut during movement of the expander, a weight is
attached that pulls the ribbon outwards on the opposite side from where the
particles are located. The Mylar ribbons are partly submerged below the liquid
surface. It is these ribbons which create the boundary for the particles as
shown in Fig. 1A. (C) Top: A photograph of the angled top view of the
expander. Bottom: The side view of the chips and Mylar strips before inserting
into water.

The design of the radial expander used to create the flow with minimal
velocity gradients is shown in Fig. 5. The twenty-four arms are connected by
binding posts (green nodes), which allow them to rotate freely, as shown in
the gray pieces in the Fig. 5A. Half of them are on the top (light gray) and
half of them are at the bottom (dark gray). Two of the opposite inner nodes
are each connected to the opposite side of a belt. As the belt is driven by a
motor, a motion of the arms to move the twelve inner (orange) nodes outward is
created. Ball bearings are glued underneath the binding posts (shown in Fig.
5B) to hold the expander and make it glide smoothly on the platform. The inner
nodes are long screws, that go below the platform and a 3D-printed chip is
attached to the tip. This chip, as shown in Fig. 5B, has two threaded
cylinders with a $1.22$ $mm$ slit in between them. Each of the cylinder serves
as a column for one side of the Mylar strip. Each Mylar strip forms a triangle
and is attached to a weight at the corner against the side connecting adjacent
nodes. This provides sufficient tension to the strip during the motion to
create an outward-going boundary.

##  Appendix B Particle Image Velocimetry

Figure 6: The Particle Image Velocimetry results of the radial expander across
whole velocity range. The magnitude of the velocity $\bm{u}$ normalized by the
pulling speed $V$ is plotted versus the position relative to the center,
normalized by the radius of the raft, $L_{0}/2$. A linear relation can be
observed in all $V$, indicating an radially affine flow field can be created
by the expander. The measurements near the boundaries are excluded due to the
difficulty of image processing.

To measure the expansion of the liquid surface due to the extension of the
pullers, the Particle Image Velocimetry (PIV) measurement on water at $V=2.8$,
$26$ and $260$ $mm/s$. The result is shown in Fig. 6. A layer of light, non-
interacting floating particles are spread sparsely on the water surface prior
to expansion. By tracking the motion of these particles and computing the
correlation between adjacent frames, the underlying fluid flow is shown to
lead to an uniform radial expansion of the surface, as described in 1; the
spacing of the particles increases proportional to their relative position.
Fig. 6 shows the gradient of the velocity is constant radially.

## References

  * To and Nagel [2023] K.-I. To and S. R. Nagel, Rifts in rafts, Soft Matter , (2023).
  * Paunov  _et al._ [1993] V. Paunov, P. Kralchevsky, N. Denkov, and K. Nagayama, Lateral capillary forces between floating submillimeter particles, Journal of colloid and interface science 157, 100 (1993).
  * Vella and Mahadevan [2005] D. Vella and L. Mahadevan, The “cheerios effect”, American journal of physics 73, 817 (2005).
  * Dalbe  _et al._ [2011] M.-J. Dalbe, D. Cosic, M. Berhanu, and A. Kudrolli, Aggregation of frictional particles due to capillary attraction, Phys. Rev. E 83, 051403 (2011).
  * Cicuta and Vella [2009] P. Cicuta and D. Vella, Granular character of particle rafts, Phys. Rev. Lett. 102, 138302 (2009).
  * Varshney  _et al._ [2012] A. Varshney, A. Sane, S. Ghosh, and S. Bhattacharya, Amorphous to amorphous transition in particle rafts, Phys. Rev. E 86, 031402 (2012).
  * Jambon-Puillet  _et al._ [2017] E. Jambon-Puillet, C. Josserand, and S. Protière, Wrinkles, folds, and plasticity in granular rafts, Phys. Rev. Materials 1, 042601 (2017).
  * Razavi  _et al._ [2015] S. Razavi, K. D. Cao, B. Lin, K. Y. C. Lee, R. S. Tu, and I. Kretzschmar, Collapse of particle-laden interfaces under compression: Buckling vs particle expulsion, Langmuir 31, 7764 (2015), pMID: 26099031, https://doi.org/10.1021/acs.langmuir.5b01652 .
  * Xiao  _et al._ [2020] H. Xiao, R. J. S. Ivancic, and D. J. Durian, Strain localization and failure of disordered particle rafts with tunable ductility during tensile deformation, Soft Matter 16, 8226 (2020).
  * Vella  _et al._ [2004] D. Vella, P. Aussillous, and L. Mahadevan, Elasticity of an interfacial particle raft, Europhysics Letters (EPL) 68, 212 (2004).
  * Bandi  _et al._ [2011] M. M. Bandi, T. Tallinen, and L. Mahadevan, Shock-driven jamming and periodic fracture of particulate rafts, EPL (Europhysics Letters) 96, 36008 (2011).
  * Kim  _et al._ [2019] B. L. Kim, A. Rendos, P. Ganesh, and K. A. Brown, Failure of particle-laden interfaces studied using the funnel method, Colloid and Interface Science Communications 28, 54 (2019).
  * Das  _et al._ [2016] T. Das, T. Lookman, and M. M. Bandi, Morphology dictated heterogeneous dynamics in two-dimensional aggregates, Soft Matter 12, 9674 (2016).
  * Kravtsov and Borgani [2012] A. V. Kravtsov and S. Borgani, Formation of galaxy clusters, Annual Review of Astronomy and Astrophysics 50, 353 (2012), https://doi.org/10.1146/annurev-astro-081811-125502 .
  * Lu  _et al._ [2008] P. J. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield, F. Sciortino, and D. A. Weitz, Gelation of particles with short-range attraction, Nature 453, 499 (2008).
  * Tanaka [2009] H. Tanaka, Formation of network and cellular structures by viscoelastic phase separation, Advanced Materials 21, 1872 (2009), https://onlinelibrary.wiley.com/doi/pdf/10.1002/adma.200802763 .
  * Atis  _et al._ [2019] S. Atis, B. T. Weinstein, A. W. Murray, and D. R. Nelson, Microbial range expansions on liquid substrates, Phys. Rev. X 9, 021058 (2019).