# Energy pathways in large- and small-scale convection-driven dynamos.

Souvik Naskar1  Anikesh Pal1<EMAIL_ADDRESS>1Department of Mechanical
Engineering, Indian Institute of Technology, Kanpur 208016, India

###### Abstract

We investigate the energy pathways between the velocity and the magnetic
fields in a rotating plane layer dynamo driven by Rayleigh-Bénard convection
using direct numerical simulations. The kinetic and magnetic energies are
divided into mean and turbulent components to study the production, transport,
and dissipation associated with large and small-scale dynamos. This energy
balance-based characterization reveals distinct mechanisms for large- and
small-scale magnetic field generation in dynamos, depending on the nature of
the velocity field and the conditions imposed at the boundaries.

###### keywords:

##  1 Introduction

Hydromagnetic dynamo action is the commonly accepted source of magnetic fields
in planets and stars. We explore the energy balance in a plane layer
convection-driven Childress-Soward (CS) dynamo (Childress & Soward, 1972). In
this simple Cartesian dynamo model, driven by rotating Rayleigh-Bénard
convection (RBC), the velocity field provides energy to the magnetic field to
sustain the dynamo action against Joule dissipation. Our analysis
distinguishes between the part of the kinetic energy that produces the
horizontally-averaged large-scale mean magnetic field and the small-scale
turbulent magnetic field. Based on this decomposition, we derive the kinetic
and magnetic energy budgets and characterize a dynamo with both large- and
small-scale magnetic fields in terms of their energy conversion mechanisms.

Dynamos are generally classified based on the scale of the magnetic field
generated by them Brandenburg & Subramanian (2005). The magnetic field for a
large-scale dynamo has a length scale larger than the velocity field, whereas
the length scale of the magnetic field is smaller than the velocity field for
a small-scale dynamo. For a plane layer dynamo, this classification is
apparent from the fraction of energy in the mean magnetic field to the total
magnetic energy (Tilgner, 2012). The magnetic Reynolds number at the
convective scale $\widetilde{Rm}$, representing the dominance of
electromagnetic induction over ohmic diffusion, is one of the important
diagnostic quantities that decides this mean energy fraction. Large-scale
dynamos with high mean energy fraction, $O(0.1)$ has been found to operate at
low magnetic Reynolds numbers $\widetilde{Rm}\lesssim 13$ (Tilgner, 2012),
where kinetic helicity has been proposed as a driving mechanism. Conversely,
higher values of $\widetilde{Rm}$ lead to small-scale magnetic field
generation driven by the stretching of magnetic field lines by the velocity
field, with comparatively lower mean energy fraction. However, a recent study
Yan & Calkins (2022) demonstrates the existence of large-scale dynamos,
despite the presence of a strongly turbulent velocity field with low helicity.
In the present investigation, we use the kinetic and magnetic energy budgets
to understand the various mechanisms associated with large-and small-scale
magnetic field generation.

##  2 Method

###  2.1 Governing Equations

In this study, the CS dynamo is driven by a classical Rayleigh-Bénard
convection setup with a plane layer of incompressible, electrically
conducting, Boussinesq fluid kept between two parallel plates with a distance
$d$ and temperature difference $\Delta T$, where the lower plate is hotter
than the upper plate. The system rotates with a constant angular velocity
$\Omega$ about the vertical axis, anti-parallel to the gravity $g$. This fluid
has the kinematic viscosity $\nu$, thermal diffusivity $\kappa$, adiabatic
volume expansion coefficient $\alpha$, and magnetic diffusivity $\eta$. The
Navier-Stokes equations, coupled with the energy equation, the Maxwell
equation, and the solenoidal field conditions, govern the velocity, pressure,
temperature, and the magnetic field $\\{u_{i},p,\theta,B_{i}\\}$ (Naskar &
Pal, 2022a, b) as presented below.

$\frac{\partial u_{j}}{\partial x_{j}}=\frac{\partial B_{j}}{\partial
x_{j}}=0,$ (1) $\begin{split}\frac{\partial u_{i}}{\partial
t}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}=-\frac{\partial p}{\partial
x_{i}}+\frac{1}{E}\sqrt{\frac{Pr}{Ra}}\epsilon_{ij3}u_{j}\hat{e_{3}}+B_{j}\frac{\partial
B_{i}}{\partial
x_{j}}+\theta\delta_{i3}+\sqrt{\frac{Pr}{Ra}}\frac{\partial^{2}u_{i}}{\partial
x_{j}\partial x_{j}},\end{split}$ (2) $\frac{\partial\theta}{\partial
t}+u_{j}\frac{\partial\theta}{\partial
x_{j}}=\frac{1}{\sqrt{RaPr}}\frac{\partial^{2}\theta}{\partial x_{j}\partial
x_{j}},$ (3) $\frac{\partial B_{i}}{\partial t}+u_{j}\frac{\partial
B_{i}}{\partial x_{j}}=B_{j}\frac{\partial u_{i}}{\partial
x_{j}}+\sqrt{\frac{Pr}{Ra}}\frac{1}{Pr_{m}}\frac{\partial^{2}B_{i}}{\partial
x_{j}\partial x_{j}}.$ (4)

The governing non-dimensional parameters are the Rayleigh number
($Ra=g\alpha\Delta Td^{3}/\kappa\nu$) and Ekman number ($E=\nu/2\Omega d^{2}$)
representing the thermal forcing and rotation rates, respectively, whereas,
the thermal and magnetic Prandtl numbers ($Pr=\nu/\kappa$ and $Pm=\nu/\eta$)
are the properties of the fluid. The current setup is a local approximation to
the astrophysically more relevant spherical shell dynamo models (Tilgner,
2012). In the horizontal directions ($x_{1},x_{2}$) periodic boundary
conditions are applied. Both no-slip and free-slip boundary conditions have
been used in the vertical direction ($x_{3}$).

$\begin{split}u_{1}=u_{2}=u_{3}=0\;\textrm{ at }\;x_{3}=\pm
1/2\qquad\text{(no-slip)}\\\ \frac{\partial u_{1}}{\partial
x_{3}}=\frac{\partial u_{2}}{\partial x_{3}}=0,\ u_{3}=0\ \textrm{ at }\
x_{3}=\pm 1/2\qquad\text{(free-slip)}.\end{split}$ (5)

Thermal boundary conditions are isothermal with unstable temperature
gradients.

$\begin{split}\quad\theta=1/2\;\textrm{ at
}\;x_{3}=-1/2,\quad\theta=-1/2\;\textrm{ at }\;x_{3}=1/2\end{split}$ (6)

Perfectly conducting and pseudo-vacuum boundary conditions have been used for
the magnetic field.

$\begin{split}\frac{\partial B_{1}}{\partial x_{3}}=\frac{\partial
B_{2}}{\partial x_{3}}=B_{3}=0\ \textrm{ at }\ x_{3}=\pm
1/2\qquad\text{(perfectly conducting)}\\\ B_{1}=B_{2}=\ \frac{\partial
B_{3}}{\partial x_{3}}=0\ \textrm{ at }\ x_{3}=\pm 1/2\qquad\text{(pseudo-
vacuum)}\end{split}$ (7)

###  2.2 Energy Budget

We perform a Reynolds decomposition of the variables into mean and fluctuating
parts such that $\phi(x,y,z,t)=\bar{\phi}(z,t)+\phi^{\prime}(x,y,z,t)$ where
$\phi=\\{u_{i},p,\theta,B_{i}\\}$. Here, the over-bar denotes an average over
the horizontal directions(Tilgner, 2012; Naskar & Pal, 2022b). The kinetic
energy($K=1/2u_{i}u_{i}$) and magnetic energy ($M=1/2B_{i}B_{i}$) are also
divided into the mean ($\mathfrak{K}$ and $\mathfrak{M}$) and turbulent
($\mathcal{K}$ and $\mathcal{M}$) components, and are presented in equations
8-11. These equations can be derived in three steps: (1) horizontally
averaging equations 2 and 4 to get the mean momentum and mean Maxwell
equations respectively, and then subtracting these mean equations from 2 and 4
to get the fluctuating parts of momentum and Maxwell equations (2) multiplying
the mean velocity and mean magnetic field with mean momentum and mean Maxwell
equations respectively, to get evolution equations for $\mathfrak{K}$ and
$\mathfrak{M}$, (3) multiplying the fluctuating velocity and magnetic fields
with fluctuating parts of momentum and Maxwell equations respectively, and
averaging them to get evolution equations for $\mathcal{K}$ and $\mathcal{M}$.
This decomposition into mean and turbulent parts of the energies is the
primary distinctive feature of the present study from an earlier budget
analysis of a dynamo (Brandenburg et al., 1996). The turbulent kinetic energy
(TKE) (Naskar & Pal, 2022b) evolves as:

$d\mathcal{K}/dt=\mathcal{S}+\mathcal{B}-\mathcal{D}-\partial\mathcal{T}_{j}/\partial
x_{j}+\mathcal{P},$ (8)

where $\mathcal{K}=1/2\overline{u^{\prime}_{i}{}u^{\prime}_{i}{}}$ is the TKE,
$\mathcal{S}=-\overline{u^{\prime}_{i}u^{\prime}_{j}}\partial\bar{u_{i}}/\partial
x_{j}$ is the production of TKE by mean shear,
$\mathcal{B}=\overline{u^{\prime}_{3}\theta^{\prime}}$ is the conversion of
available potential energy (APE) to TKE by the turbulent buoyancy flux (Gayen
et al., 2013), $\mathcal{D}=\sqrt{Pr/Ra}\ \overline{\partial
u^{\prime}_{i}/\partial x_{j}\partial u^{\prime}_{i}/\partial x_{j}}$ is the
viscous dissipation which converts TKE to internal energy (IE),
$\partial\mathcal{T}_{j}/\partial x_{j}$ is the redistribution of TKE
(Petschel et al., 2015), representing the divergence of the TKE flux
$\mathcal{T}_{j}=\mathcal{T}_{p}+\mathcal{T}_{t}+\mathcal{T}_{v}+\mathcal{T}_{M}$.
The components of the TKE flux are the pressure flux,
$\mathcal{T}_{p}=\overline{u^{\prime}_{j}p^{\prime}}$, the turbulent flux
$\mathcal{T}_{t}=\frac{1}{2}\overline{u^{\prime}_{i}u^{\prime}_{i}u^{\prime}_{j}}$,
the viscous flux $\mathcal{T}_{v}=-\sqrt{Pr/Ra}\partial\mathcal{K}/\partial
x_{j}$, and the magnetic flux $\mathcal{T}_{M}=-\overline{B_{j}}\
\overline{u^{\prime}_{i}{}B^{\prime}_{i}}-\overline{u^{\prime}_{i}B^{\prime}_{i}B^{\prime}_{j}}$.

In equation 8, the term $\mathcal{P}$ represents the production of
$\mathcal{K}$ due to work done by the Lorentz force on the flow field. It can
be further divided into three components
$\mathcal{P}=-\mathcal{P}_{1}+\mathcal{P}_{2}-\mathcal{P}_{3}$, where
$\mathcal{P}_{1}=\overline{B}_{j}\overline{B^{\prime}_{i}\partial
u^{\prime}_{i}/\partial x_{j}}$ is the production of TME from TKE due to work
done by the fluctuating strain rate on the mean magnetic field,
$\mathcal{P}_{2}=\overline{u^{\prime}_{i}B^{\prime}_{j}}\partial\overline{B}_{i}/\partial
x_{j}$ signifies the production of TKE due to mean magnetic field gradient and
$\mathcal{P}_{3}=\overline{B^{\prime}_{i}B^{\prime}_{j}\partial
u^{\prime}_{i}/\partial x_{j}}$ represents the amplification (or attenuation)
of the magnetic energy, due to work done by stretching (or squeezing) of
small-scale magnetic field lines by the fluctuating velocity gradients.

The mean kinetic energy (MKE) budget is expressed as,

$d\mathfrak{K}/dt=-\mathcal{S}+\mathfrak{B}-\mathfrak{D}-\partial\mathfrak{T}_{j}/\partial
x_{j}+\mathfrak{P}$ (9)

where $\mathfrak{K}=1/2\overline{u}_{i}\overline{u}_{i}$ is the MKE,
$\mathfrak{B}=\overline{u}_{3}\overline{\theta}$ is the mean buoyancy flux,
$\mathfrak{D}=\sqrt{Pr/Ra}\partial\overline{u}_{i}/\partial
x_{j}\partial\overline{u}_{i}\partial x_{j}$ is the mean viscous dissipation,
and $\partial\mathfrak{T}_{j}/\partial x_{j}$ is the divergence of the MKE
flux
$\mathfrak{T}_{j}=\overline{u_{j}}\overline{p}+1/2\overline{u^{\prime}_{i}u^{\prime}_{j}}{\overline{u}_{i}}-\sqrt{Pr/Ra}\partial\mathfrak{K}/\partial
x_{j}-\overline{B^{\prime}_{i}B^{\prime}_{j}}{\overline{u}_{i}}-\overline{B}_{i}\
\overline{B}_{j}\overline{u}_{i}$. Here
$\mathfrak{P}=-\mathcal{P}_{4}-\mathcal{P}_{5}$, where,
$\mathcal{P}_{4}=\overline{B^{\prime}_{i}B^{\prime}_{j}}\partial\overline{u}_{i}/\partial
x_{j}$ represents the work done by the mean shear on the fluctuating component
of the magnetic fields whereas, the work done on the mean component of the
magnetic fields is represented by
$\mathcal{P}_{5}=\overline{B}_{i}\overline{B}_{j}\partial\overline{u}_{i}/\partial
x_{j}$.

The equations for the evolution of the turbulent and the mean magnetic
energies are presented below. As the Maxwell equation (Naskar & Pal, 2022a, b)
is similar to the vorticity transport equation, the magnetic energy equations
resemble enstrophy transport equations(Lumley & Tennekes, 1997). The turbulent
magnetic energy (TME) budget equation is given below.

$d\mathcal{M}/dt=-\mathcal{D}^{M}-\partial\mathcal{T}^{M}_{j}/\partial
x_{j}+\mathcal{P}^{M}$ (10)

where, $\mathcal{M}=1/2\overline{B^{\prime}_{i}{}B^{\prime}_{i}{}}$ is the
TME, $\mathcal{D}^{M}=1/Pm\sqrt{Pr/Ra}\ \overline{\partial
B^{\prime}_{i}/\partial x_{j}\partial B^{\prime}_{i}/\partial x_{j}}$ is the
Joule dissipation, $\partial\mathcal{T}^{M}_{j}/\partial x_{j}$ is the
redistribution of TME flux given by
$\mathcal{T}^{M}_{j}=1/2\overline{B^{\prime}_{i}B^{\prime}_{i}u^{\prime}_{j}}-1/Pm\sqrt{Pr/Ra}\partial\mathcal{M}/\partial
x_{j}$. Here, the energy exchange terms can be expressed as
$\mathcal{P}^{M}=\mathcal{P}_{1}+\mathcal{P}_{3}+\mathcal{P}_{4}-\mathcal{P}_{6}$,
where
$\mathcal{P}_{6}=\overline{u^{\prime}_{j}B^{\prime}_{i}}\partial\overline{B}_{i}/\partial
x_{j}$ exchange energy between the turbulent and mean magnetic fields. We can
anticipate its appearance in the mean magnetic energy (MME) equations with an
opposite sign.

$d\mathfrak{M}/dt=-\mathfrak{D}^{M}-\partial\mathfrak{T}^{M}_{j}/\partial
x_{j}+\mathfrak{P}^{M}$ (11)

where, $\mathfrak{M}=1/2\overline{B}_{i}\overline{B}_{i}$ is the MME,
$\mathfrak{D}^{M}=1/Pm\sqrt{Pr/Ra}\partial\overline{B}_{i}/\partial
x_{j}\partial\overline{B}_{i}/\partial x_{j}$ is the mean Joule dissipation,
and $\partial\mathfrak{T}^{M}_{j}\partial x_{j}$ is the divergence of MME flux
$\mathfrak{T}^{M}_{j}=\overline{B_{i}}\
\overline{u^{\prime}_{j}{}B^{\prime}_{i}}-\overline{B_{i}}\
\overline{u^{\prime}_{i}{}B^{\prime}_{j}}-1/Pm\sqrt{Pr/Ra}\frac{\partial\mathfrak{M}}{\partial
x_{j}}$. Energy exchange terms in the MME equation can be expressed as
$\mathfrak{P}^{M}=-\mathcal{P}_{2}+\mathcal{P}_{5}+\mathcal{P}_{6}$.

Figure 1: Energy pathways between the kinetic and magnetic energies in a
convection-driven dynamo. The energetic terms shown here are volume-averaged
as indicated by angular brackets $\langle.\rangle$. The energy pathways marked
in grey are negligible in the absence of a mean flow.

We plot an energy pathway diagram by averaging the terms in the energy budget
equations in the vertical direction to obtain the volume averaged quantities,
as shown in figure 1. The transport terms become negligible due to this
volume-averaging, and we can demonstrate the conversion paths between KE and
ME sustaining the dynamo action. The APE is converted to MKE and TKE via the
mean and turbulent part of the buoyancy flux (Gayen et al., 2013)
respectively. The shear production term, $\mathcal{S}$, may also produce TKE
in the presence of a mean shear. The small-scale turbulent flow can exchange
energy with the small-scale magnetic field through $\mathcal{P}_{1}$ or
$\mathcal{P}_{3}$ or both, or with the large-scale mean magnetic field through
$\mathcal{P}_{2}$. The large-scale flow can exchange energy with the small-
scale and large-scale magnetic field through $\mathcal{P}_{4}$ and
$\mathcal{P}_{5}$, respectively. The term $\mathcal{P}_{6}$ may produce TME by
extracting energy from the mean magnetic field. It may also produce MME by
transferring energy from the turbulent magnetic field to the mean magnetic
field. Finally, the viscous dissipation components ($\mathfrak{D}$ and
$\mathcal{D}$) convert KE to IE, while the ME is transformed to IE via Joule
dissipation components ($\mathfrak{D}^{M}$ and $\mathcal{D}^{M}$). In the
absence of a horizontally-averaged mean velocity field, as in the present
problem, the energy pathways associated with MKE, marked in grey, are
insignificant.

case | $\mathcal{R}$ | $Pm$ | $Ro_{C}$ | $\widetilde{Ra}$ | $\widetilde{Rm}$ | $\langle\mathfrak{M}\rangle/M$ | $M/E$ | $Nu$ | type
---|---|---|---|---|---|---|---|---|---
R10Pm1N | 10 | 1 | 0.069 | 76.0 | 33.4 | 0.0006 | 0.1749 | 55.0 | small
R10Pm1F | 10 | 1 | 0.074 | 87.0 | 25.9 | 0.0097 | 0.9350 | 64.7 | small
R10Pm0.1F | 10 | 0.1 | 0.074 | 87.0 | 3.2 | 0.2811 | 0.7605 | 60.7 | large
R2Pm0.2N | 2 | 0.2 | 0.031 | 15.2 | 1.5 | 0.6129 | 0.4866 | 8.5 | large
Table 1: Volume-averaged diagnostic quantities for the dynamo simulations at
$E=5\times 10^{-7}$ and $Pr=1$. The last column indicates the dynamo types.

(a) (b) (c) (d)

Figure 2: The structure of the magnetic field generated by the dynamos for the
cases (a) R10Pm1F, (b) R10Pm1N, (c) R10Pm0.1F,(d) R2Pm0.2N as visualized by
the isosurface $B_{1}=\pm 0.03$ (olive-positive, blue-negative)

###  2.3 Numerical Details

To characterize the different types of dynamos based on the energy pathway
diagram in figure 1, we perform DNS of RBC-driven dynamos in a doubly-periodic
domain of unit aspect ratio. We use $1024$ uniform grids in each horizontal
direction ($x_{1}$ and $x_{2}$) and $256$ grids in the vertical direction
($x_{3}$) that are clustered near the boundaries. A finite-difference solver
has been used for the simulations that have been validated extensively for
studies on rotating stratified flow (Pal & Chalamalla, 2020), and various
transitional and turbulent shear flows (Pal et al., 2013; Pal & Sarkar, 2015;
Pal, 2020). Details of the solver, the validation studies, the rationale
behind the choice of the grids, and the domain size are given in (Naskar &
Pal, 2022a, b). We choose two dynamos at $\mathcal{R}=Ra/Ra_{c}=10$ and $Pm=1$
where $Ra_{c}$ is the Rayleigh number at the onset of non-magnetic rotating
convection at $E=5\times 10^{-7}$ (Naskar & Pal, 2022b). At this Ekman number,
the critical Rayleigh number has the value $Ra_{c}=3.830\times 10^{9}$ for no-
slip and $Ra_{c}=2.192\times 10^{9}$ for free-slip boundary conditions. In
table 1, the case R10Pm1N is simulated using no-slip and electrically
conducting boundaries, whereas free-slip and pseudo-vacuum boundaries (Naskar
& Pal, 2022b) are incorporated for the R10Pm1F case. The instantaneous
snapshots of the magnetic field in the $x_{1}$-direction, as depicted in
figures 2a and b, illustrate the small-scale nature of the magnetic field
produced by these turbulent dynamos. Another turbulent dynamo case has been
simulated, by lowering $Pm$ to $0.1$, following (Yan & Calkins, 2022), as
denoted by R10Pm0.1F in table 1. In this case, lowering $Pm$ leads to large-
scale magnetic field generation, as evident from figure 2c. Additionally, we
simulate a case with $\mathcal{R}=2$ and $Pm=0.2$ and designate it as
R2Pm0.2N. This case is also a large-scale dynamo with weakly-nonlinear
convection (Stellmach & Hansen, 2004) as demonstrated by the large-scale
magnetic field in figure 2d.

The volume-averaged diagnostic parameters reported in table 1 outline the
global behaviour of these dynamos. The convective Rossby number,
$Ro_{C}=E(Ra/Pr)^{1/2}$, representing the ratio of inertia to Coriolis force,
is of the order $10^{-2}$, indicating the dominant role of the Coriolis force.
Therefore, all the dynamos are produced by rapidly rotating convection with
comparatively small inertia (Naskar & Pal, 2022b). The reduced Rayleigh number
$\widetilde{Ra}=RaE^{4/3}$ indicates that the dynamos operate in a turbulent
state of the flow for $\mathcal{R}=10$, whereas weakly-nonlinear columnar
convection can be observed for $\mathcal{R}=2$ (Naskar & Pal, 2022a; Stellmach
& Hansen, 2004). The reduced magnetic Reynolds number
$\widetilde{Rm}=RmE^{1/3}$ represents the strength of electromagnetic
induction relative to ohmic diffusion at the convective scales. Large-scale
dynamos are expected to be found for $\widetilde{Rm}\lesssim O(1)$ (Yan &
Calkins, 2022). The distinction between large- and small-scale dynamos
(Tilgner, 2012) is apparent from the mean energy fraction
$\langle\mathfrak{M}\rangle/M$, where
$M=\langle\mathfrak{M}\rangle+\langle\mathcal{M}\rangle$. For the small-scale
dynamos, the volume-averaged MME is three orders of magnitude smaller than the
TME, while they are of the same order for the large-scale dynamos.
Additionally, the fraction of magnetic energy $M/E$, where $E=M+K$ is the
total energy, can be regarded as the efficiency of the dynamo action.

##  3 Results

Figure 3 demonstrates the vertical variation of the horizontally averaged TKE
terms in equation 8 for the four cases. All the terms in this equation have
been averaged over time, and normalized by $(RaPr)^{1/2}/(Nu-1)$ in the
figure, so that the volume average of the source term ($\mathcal{B}$) is
unity(Kerr, 2001). The Nusselt number $Nu=qd/k\Delta T$, where q is the total
vertical heat flux, is a non-dimensional measure of convective heat transport
through the fluid layer, as reported in table 1. At a statistically stationary
state, there is a primary balance among the turbulent buoyancy flux
($\mathcal{B}$), TKE transport ($\partial\mathcal{T}_{j}/\partial x_{j}$),
viscous dissipation ($\mathcal{D}$) and the conversion to magnetic energy
($\mathcal{P}$) for all the dynamos. The individual components of
$\mathcal{P}$ and $\partial\mathcal{T}_{j}/\partial x_{j}$ are shown in the
top and the bottom insets, respectively. For instance, in figure 3a, the TKE
is generated by $\mathcal{B}$ in the bulk, which is partly transported by
$\partial\mathcal{T}_{j}/\partial x_{j}$ towards the boundaries where
$\mathcal{D}$ dominates. Among the transport components, the pressure
transport ($\partial\mathcal{T}_{P}/\partial x_{j}$) is the primary mechanism
that transfers TKE towards boundaries while the viscous transport is higher
near the boundaries as observed in the bottom inset, similar to non-rotating
non-magnetic RBC (Petschel et al., 2015). The vertical variation of all the
terms, except $\mathcal{P}$, are qualitatively similar with non-magnetic
rotating convection (Kunnen et al., 2009; Guzmán et al., 2020).

$\color[rgb]{0,0,0}{\boldsymbol{\mathcal{B}}},$
$\color[rgb]{0.72,0,0}{\boldsymbol{\epsilon_{v}}},$
$\color[rgb]{0,0.88,0}{\boldsymbol{\frac{\partial\mathcal{T}_{j}}{\partial
x_{j}}}},$
$\color[rgb]{1,0,1}{\boldsymbol{\frac{\partial\mathcal{T}_{p}}{\partial
x_{j}}}},$
$\color[rgb]{0.5,0.5,0}{\boldsymbol{\frac{\partial\mathcal{T}_{t}}{\partial
x_{j}}}},$
$\color[rgb]{0.68,0.36,1}{\boldsymbol{\frac{\partial\mathcal{T}_{v}}{\partial
x_{j}}}},$
$\color[rgb]{0,1,1}{\boldsymbol{\frac{\partial\mathcal{T}_{M}}{\partial
x_{j}}}},$ $\color[rgb]{0,0,1}{\boldsymbol{\mathcal{P}}},$
$\color[rgb]{0,0,1}{\boldsymbol{-\mathcal{P}_{1}(\circ)},}$
$\color[rgb]{0,0,1}{\boldsymbol{-\mathcal{P}_{2}(\square)},}$
$\color[rgb]{0,0,1}{\boldsymbol{-\mathcal{P}_{3}(\diamond)}}$ (a) (b)
(c) (d)

Figure 3: Vertical variation of the terms in TKE budget for (a) R10Pm1F, (b) R10Pm1N, (c) R10Pm0.1F ,(d) R2Pm0.2N , cases. The insets in the top left corners show the components of the energy exchange $\mathcal{P}$. The insets in the bottom left corners shows a breakup of the transport terms $\partial\mathcal{T}_{j}/\partial x_{j}$ case | $\langle\mathcal{P}_{1}\rangle$ | $-\langle\mathcal{P}_{2}\rangle$ | $\langle\mathcal{P}_{3}\rangle$ | $-\langle\mathcal{P}_{6}\rangle$ | $\langle\mathcal{D}\rangle$ | $\langle\mathcal{D}^{M}\rangle$ | $\langle\mathfrak{D}^{M}\rangle$ | type
---|---|---|---|---|---|---|---|---
R10Pm1N | 0.0002 | 0.0000 | 0.3491 | 0.0000 | 0.6232 | 0.3507 | 0.0000 | small
R10Pm1F | 0.0004 | 0.0014 | 0.7549 | 0.0013 | 0.1627 | 0.7612 | 0.0001 | small
R10Pm0.1F | 0.1397 | 0.0093 | 0.5696 | 0.0069 | 0.2457 | 0.7189 | 0.0026 | large
R2Pm0.2N | 0.3399 | 0.0068 | 0.0538 | -0.0055 | 0.6252 | 0.3878 | 0.0120 | large
Table 2: Volume-averaged diagnostic quantities for the dynamo simulations at
$E=5\times 10^{-7}$ and $Pr=1$. The last column indicates the dynamo types.

The small-scale dynamos R10Pm1N and R10Pm1F in figure 3a and 3b differ only in
the boundary conditions. The choice of a no-slip boundary condition in R10Pm1N
results in an Ekman layer near the boundaries. Moreover, the perfectly
conducting magnetic boundary condition constrains the magnetic field to be
horizontal near the boundaries as compared to a vertical magnetic field at the
boundaries for a pseudo-vacuum boundary condition in R10Pm1F (Naskar & Pal,
2022b). For both of these dynamos, the small-scale production of magnetic
energy ($\mathcal{P}_{3}$) is the dominant component of $\mathcal{P}$. For
R10Pm1N, the amount of TKE converted to IE via $\mathcal{D}$ is higher than
the production of magnetic energy ($\mathcal{P}$), whereas the latter is
higher for R10Pm1F (see also table 2). Another difference between these
dynamos arises from the transport of TKE due to magnetic energy, which is
dominated by the small-scale flux
$\overline{u^{\prime}_{i}B^{\prime}_{i}B^{\prime}_{j}}$, while the large-scale
flux $\overline{B_{j}}\overline{u^{\prime}_{i}{}B^{\prime}_{i}}$ remains
small. The small-scale magnetic field transports TKE towards the interior from
the boundaries in R10Pm1F, while the same term is small for R10Pm1N. The
viscous transport dominates the other transport terms near the boundaries for
R10Pm1N. However, this term is small near the boundaries in R10Pm1F owing to
the absence of an Ekman layer.

A comparison between the small-scale dynamo R10Pm1F with the large-scale
dynamo R10Pm0.1F reveals significant differences in the energy pathways. The
mean magnetic field plays an important role in converting TKE to TME through
the term $\mathcal{P}_{1}$ in R10Pm0.1F, while this term is insignificant for
R10Pm1F. However, for both cases, the redistribution of TKE by the magnetic
field plays a part in the budget with a major contribution from the small-
scale flux $\overline{u^{\prime}_{i}B^{\prime}_{i}B^{\prime}_{j}}$ in R10Pm1F,
while the large-scale flux
$\overline{B_{j}}\overline{u^{\prime}_{i}{}B^{\prime}_{i}}$ is significant for
R10Pm0.1F. Additionally, we can contrast the large-scale turbulent dynamo
R10Pm0.1F against a large-scale dynamo with weakly non-linear convection
R2Pm0.2N in figure 3. For R2Pm0.2N, the conversion of TKE to TME occurs
through $\mathcal{P}_{1}$ while the small-scale production of TME,
$\mathcal{P}_{3}$ remains small. Among these four dynamos, the part of TKE
that converts to IE due to $\mathcal{D}$ is higher, as compared with the
conversion to magnetic energy via $\mathcal{P}$, when no-slip boundary
condition is used (see table 2). The conversion to magnetic energy
$\mathcal{P}$ is higher with free-slip conditions, which makes R10Pm1F and
R10Pm0.1F more efficient dynamos with higher magnetic energy fraction $M/E$,
as compared to the no-slip boundary condition. Also, the transport of TKE by
the magnetic field is significant only for pseudo-vacuum magnetic boundary
conditions.

solid lines: $\color[rgb]{0,0.88,0}{\boldsymbol{\mathcal{P}_{1}}},$
$\color[rgb]{0,0,0}{\boldsymbol{\mathcal{P}_{3}}},$
$\color[rgb]{1,0.39,0.13}{\boldsymbol{\mathcal{D}^{M}}},$
$\color[rgb]{0.68,0.36,1}{\boldsymbol{-\frac{\partial\mathcal{T}^{M}}{\partial
x_{j}}}}$ dashed lines: $\color[rgb]{0,0,1}{\boldsymbol{-\mathcal{P}_{2}}},$
$\color[rgb]{1,0,1}{\boldsymbol{\mathcal{P}_{6}}},$
$\color[rgb]{0.72,0,0}{\boldsymbol{\mathfrak{D}^{M}}},$
$\color[rgb]{0.5,0.5,0}{\boldsymbol{-\frac{\partial\mathfrak{T}^{M}}{\partial
x_{j}}}}$ (a) (b)
(c) (d)

Figure 4: Vertical variation of the terms in TME and MME budgets for (a)
R10Pm1F, (b) R10Pm1N, (c) R10Pm0.1F,(d) R2Pm0.2N cases. TME(MME) budget terms
are plotted using solid(dotted) lines and are to be read from the bottom(top)
abscissa.

The magnetic energy balance of the dynamos is presented in figure 4. The solid
lines represent the terms in the TME budget and are plotted at the bottom
abscissa. The terms of the MME budget are represented by the dashed lines and
are plotted at the top abscissa. The energy flow direction of each dynamo in
figure 1 becomes apparent with the help of table 2 and figure 4. The terms
$\mathcal{P}_{1}$ and $\mathcal{P}_{3}$ are positive, representing a
conversion of TKE to TME by the action of a large-scale and a small-scale
magnetic field, respectively. The term $\mathcal{P}_{2}$ is negative,
indicating a generation of MME at the expense of TKE. The term
$\mathcal{P}_{6}$ has negative values indicating a conversion of MME to
generate TME, except for the case R2Pm0.2N where the turbulent magnetic field
provides energy to the mean magnetic field, exhibiting an upscale transfer of
energy.

In figure 4a the small-scale dynamo R10Pm1N has a primary balance between the
small-scale production $\mathcal{P}_{3}$ and the Joule dissipation
$\mathcal{D}^{M}$. In comparison to these TME terms, the MME terms are three
orders of magnitude smaller. The part of TKE that converts to MME
($\mathcal{P}_{2}$) again transforms to TME, with a small mean dissipation
$\mathfrak{D}^{M}$. A similar energy conversion is observed in R10Pm1F in
figure 4b, though the MME budget terms are one order of magnitude higher than
R10Pm1N. Unlike R10Pm1N, the transport of TME
$\partial\mathcal{T}^{M}_{j}/\partial x_{j}$ plays a significant role in the
balance in R10Pm1F by redistributing the TME from the boundaries towards the
interior of the domain. For the large-scale dynamos R10Pm0.1 in figure 4c, the
MME terms increase by another order of magnitude compared to the small-scale
dynamo R10Pm1F. Further, the large-scale production of TME term
$\mathcal{P}_{1}$ now makes a significant contribution to the budget in the
interior of the domain. In the MME budget, $\mathcal{P}_{2}$ is partially
balanced by $\mathcal{P}_{6}$ in the interior. However, the rest of the MME is
transported towards the boundary and converted to IE by Joule dissipation
$\mathfrak{D}^{M}$. The large-scale dynamo in figure 4d demonstrates a
transfer of TME to MME by $\mathcal{P}_{6}$ near the boundaries. This is the
only dynamo where both the small-scale velocity and magnetic fields provide
energy to the mean magnetic field through $\mathcal{P}_{2}$ and
$\mathcal{P}_{6}$. Additionally, the large-scale production $\mathcal{P}_{1}$
is the primary source of TME generation whereas $\mathcal{P}_{3}$ remains
small in this case, in contrast to R10Pm0.1F in figure 4c.

##  4 Conclusions

In summary, we have performed direct numerical simulations of four dynamos to
compare their magnetic and kinetic energy budgets. The small-scale dynamos
R10Pm1N and R10Pm1F differ by the relative magnitude of small-scale magnetic
energy production $\mathcal{P}_{3}$ and viscous dissipation $\mathcal{D}$,
with the latter being higher for R10Pm1N. This indicates comparatively
efficient dynamo action with free-slip, pseudo-vacuum boundaries that also
promote the redistribution of TKE by the magnetic field, unlike a dynamo with
no-slip, perfectly conducting boundaries. Nevertheless, the mean Joule
dissipation is small for small-scale dynamos. The mechanism for transforming
TKE to TME differs between a large- and a small-scale turbulent dynamo, with
the large-scale production of TME $\mathcal{P}_{1}$ playing a significant role
in the former. This large-scale production $\mathcal{P}_{1}$ becomes the
dominant mechanism of TME production in the weakly non-linear dynamo R2Pm0.2N.
MME is produced from TKE via the term $\mathcal{P}_{2}$ in the presence of a
mean magnetic field gradient. For R2Pm0.2N, an upscale transfer of energy
occurs through $\mathcal{P}_{6}$, which produces MME at the expense of TME.
The scaling of these energy budget terms in the limit of small viscous and
inertial forces, following (Calkins et al., 2015), should provide valuable
insights into the mechanism of energy conversion in astrophysical dynamos.
Additionally, a shell-to-shell energy transfer analysis (Guzmán et al., 2020)
may elucidate further details on the mechanism of large-scale magnetic field
generation.

[Declaration of interests] The authors report no conflict of interest.

[Author contributions] The authors contributed equally to analyzing data and
reaching conclusions, and in writing the paper.

## References

  * Brandenburg et al. (1996) Brandenburg, A., Jennings, R. L., Nordlund, Å, Rieutord, M., Stein, R.F. & Tuominen, I. 1996 Magnetic structures in a dynamo simulation. J. Fluid Mech. 306, 325–352.
  * Brandenburg & Subramanian (2005) Brandenburg, A. & Subramanian, K. 2005 Astrophysical magnetic fields and nonlinear dynamo theory. Phys. Rep. 417 (1-4), 1–209.
  * Calkins et al. (2015) Calkins, M.A., Julien, K., Tobias, S.M. & Aurnou, J.M. 2015 A multiscale dynamo model driven by quasi-geostrophic convection. J. Fluid Mech. 780, 143–166.
  * Childress & Soward (1972) Childress, S. & Soward, A. M. 1972 Convection-driven hydromagnetic dynamo. Phys. Rev. Lett. 29 (13), 837.
  * Gayen et al. (2013) Gayen, B., Hughes, G. O. & Griffiths, R. W. 2013 Completing the mechanical energy pathways in turbulent rayleigh-bénard convection. Phys. Rev. Lett. 111 (12), 124301\.
  * Guzmán et al. (2020) Guzmán, A. J. A., Madonia, M., Cheng, J. S., Ostilla-Mónico, R., Clercx, H. J. H. & Kunnen, R. P. J. 2020 Competition between ekman plumes and vortex condensates in rapidly rotating thermal convection. Phys. Rev. Lett. 125 (21), 214501.
  * Kerr (2001) Kerr, R.M. 2001 Energy budget in Rayleigh–Bénard convection. Phys. Rev. Lett. 87, 244502–1–244502–4.
  * Kunnen et al. (2009) Kunnen, R.P.J., Geurts, B.J. & Clercx, H.J.H. 2009 Turbulence statistics and energy budget in rotating Rayleigh–Bénard convection. Eur. J. Mech. B Fluids 28 (4), 578–589.
  * Lumley & Tennekes (1997) Lumley, L.J. & Tennekes, H. 1997 A Fisrt course in turbulence. The MIT Press.
  * Naskar & Pal (2022a) Naskar, S. & Pal, A. 2022a Direct numerical simulations of optimal thermal convection in rotating plane layer dynamos. J. Fluid Mech. 942, A37.
  * Naskar & Pal (2022b) Naskar, S. & Pal, A. 2022b Effects of kinematic and magnetic boundary conditions on the dynamics of convection-driven plane layer dynamos. J. Fluid Mech. 951, A7.
  * Pal (2020) Pal, Anikesh 2020 Deep learning emulation of subgrid-scale processes in turbulent shear flows. Geophys. Res. Lett. 47 (12), e2020GL087005.
  * Pal & Chalamalla (2020) Pal, A. & Chalamalla, V.K. 2020 Evolution of plumes and turbulent dynamics in deep-ocean convection. J. Fluid Mech. 889, A35.
  * Pal & Sarkar (2015) Pal, A. & Sarkar, S. 2015 Effect of external turbulence on the evolution of a wake in stratified and unstratified environments. J. Fluid Mech. 772, 361–385.
  * Pal et al. (2013) Pal, A., de Stadler M.B. & S., Sarkar 2013 The spatial evolution of fluctuations in a self-propelled wake compared to a patch of turbulence. Phys. Fluids 25, 095106.
  * Petschel et al. (2015) Petschel, K., Stellmach, S., Wilczek, M., Lülff, J. & Hansen, U. 2015 Kinetic energy transport in Rayleigh–Bénard convection. J. Fluid Mech. 773, 395–417.
  * Stellmach & Hansen (2004) Stellmach, S. & Hansen, U. 2004 Cartesian convection driven dynamos at low Ekman number. Phys. Rev. E. 70 (5), 056312\.
  * Tilgner (2012) Tilgner, A. 2012 Transitions in rapidly rotating convection driven dynamos. Phys. Rev. Lett. 109 (24), 248501\.
  * Yan & Calkins (2022) Yan, M. & Calkins, M. A. 2022 Strong large scale magnetic fields in rotating convection-driven dynamos: The important role of magnetic diffusion. Phys. Rev. Res. 4 (1), L012026.


# Phase behaviour of active particles in block copolymer melts

Javier Diaz  CECAM, Centre Européen de Calcul Atomique et Moléculaire, EPFL,
École Polytechnique Fédérale de Lausanne,
Batochime - Avenue Forel 2, 1015 Lausanne, Switzerland  Ignacio Pagonabarraga
<EMAIL_ADDRESS>CECAM, Centre Européen de Calcul Atomique et
Moléculaire, EPFL, École Polytechnique Fédérale de Lausanne,
Batochime - Avenue Forel 2, 1015 Lausanne, Switzerland  Departament de Física
de la Matèria Condensada, Universitat de Barcelona, Martí i Franqués 1, 08028
Barcelona, Spain  Universitat de Barcelona Institute of Complex Systems
(UBICS), Universitat de Barcelona, 08028 Barcelona, Spain

(August 27, 2024)

###### Abstract

Block copolymer melts offer a unique template to control the position and
alignment of nanoparticles due to their ability to self-assemble into periodic
ordered structures. Active Brownian particles are shown to co-assemble with
block copolymers leading to emergent organised structures. The block copolymer
acts as a soft confinement that can control the self-propulsion of the active
Brownian particles, both for interface-segregated and selective nanoparticles.
At moderate activity regimes, active Brownian particles can form organised
structures such as polarised trains or rotating vortexes. At high activity,
the contrast in the polymeric and colloidal time scales can lead to particle
swarms with distorted block copolymer morphology, due to the competition
between the polymeric self-assembly and the active Brownian self-propulsion.

††preprint: APS/123-QED

Block copolymer (BCP) melts are ideal matrices to template the location of
passive nanoparticles (NPs), due to their ability to self-assembly into
ordered mesophases. [1, 2, 3, 4] such as lamellae of hexagonally ordered
circles in two dimensions (2D). This is due to the heterogeneity of the BCP
chain, where monomers of chemically different types are grouped together into
blocks. Passive particles have been shown to form ordered structures when
miscible with one of the BCP phases[5, 6, 7], or decorating the interface[4].
Furthermore, NPs can modify the BCP morphology by changing the effective
composition of the BCP melt, leading to a rich phase behaviour of the hybrid
co-assembled system[8, 9]. This work addresses both the structures that active
particles can form mediated by the BCP template, as well changes in the BCP
mesophase due to activity.

Active particles are intrinsically out of equilibrium, as they consume energy
from the surrounding medium to produce work. This work can be used, for
instance, to allow particles to self-propel with a given direction.
Suspensions of self-propelled particles have been shown to form a rich phase
behaviour[10, 11, 12], which has been largely modelled within the active
Brownian particle (ABP) model, where spherical particles self-propel with a
rotational diffusion time and excluded volume interactions[13, 14, 15, 16, 17,
18, 19].

Furthermore, the dispersion of active particles within complex medium has
resulted in emergent organised structures[20, 21], such as the accumulation of
ABPs at hard confining walls or the formation of vortexes of ABPs under
circular confinement[22, 23, 24, 25, 26]. While most of the literature covers
the hard confinement of ABPs near solid and immobile interfaces, the use of
soft walls permits the activity to deform the confining medium, such as the
case of active filaments within vesicles [27, 28]. The geometry of obstacles
or confinement has been shown to play a role in the accumulation of active
particles in curved walls [29, 30, 31]. Furthermore, inhomogeneous media can
lead to localisation of active particles in the absence of hard
confinement[32]. These works motivate the use of BCP melts as templates to
control the dispersion of active particles that can result in emergent co-
assembled BCP/ABPs morphologies.

Active Janus particles at air/water interfaces have been shown to enhance
their active persistence length by reducing their rotational diffusivity[33],
driven by the wetting of the particle[34]. While equilibrated flat air/water
interfaces have been used to study the constrained collective behaviour of
ABPs, in this work we exploit the intrinsic structure of the BCP interface.
NPs with chemically inhomogeneous surfaces have been successfully dispersed
within BCP melts, leading to their achoring at interfaces[35, 36].

In this work we study how the co-assembly of ABPs in BCP melts differs from
passive NPs and the effect of activity in the emergence of structures for
hybrid NP/BCP composites. We use a minimal model of ABPs coupled with a Cahn-
Hilliard description of a BCP melt. The ability of BCP melts to self-assemble
can be used as a case study of ABPs dispersion within mixtures with intrinsic
ordering in the nanoscale.

In order to capture the overall phase behaviour of active NPs in BCP melts, we
use a mesoscopic hybrid model that allows to reach relatively large length
scales. The total free energy of the system is decomposed as
$F_{tot}=F_{pol}+F_{cpl}+F_{cc}$, where $F_{pol}$ is the standard Ohta-
Kawasaki[37] continuous description of a BCP melt via the differences in
concentration of A and B monomers $\psi=\phi_{A}-\phi_{B}+(1-2f_{0})$. The
colloid-colloid contribution is a pairwise additive potential that prevents
overlapping between ABPs. The colloid and the polymeric description are
coupled with the interaction term $F_{cpl}=\sum_{i}\sigma\int
d\textbf{r}\psi_{c}(r)\left[\psi-\psi_{0}\right]^{2}$ which introduces a
coupling energetic scale $\sigma$ and a chemical affinity parameter $\psi_{0}$
that specifies the wetting of the ABP with the BCP. The shape and size of the
particle is controlled by the tagged function[38] $\psi_{c}$.

The dynamics of a system of $N_{p}$ ABPs with a radius $R$ is described via
the overdamped Langevin equations

$\frac{d\textbf{r}_{i}}{dt}=v_{a}\mathbf{\hat{n}}_{i}+\gamma_{t}^{-1}\left(\textbf{f}_{i}^{cc}+\textbf{f}_{i}^{cpl}\right)+\sqrt{2D_{t}}\mathbf{\xi}_{t}$
(1a)
$\frac{d\phi_{i}}{dt}=\gamma_{r}^{-1}\left(M_{i}^{cc}+M_{i}^{cpl}\right)+\sqrt{2D_{r}}\xi_{r}$
(1b)

for the translational $\mathbf{r}_{i}$ and orientational degrees of freedom
$\phi_{i}$ of particle $i$, respectively. Forces $\textbf{f}_{i}$ and torques
$M_{i}$ are derived from the colloid-colloid and coupling contributions to the
free energy. The Einstein relation applies for both the translational
$D_{t}=k_{B}T/\gamma_{t}$ and rotational diffusion constants
$D_{r}=k_{B}T/\gamma_{r}$. The friction constants are
$\gamma_{t}=6\pi\eta_{0}R$ and $\gamma_{r}=8\pi\eta_{0}R^{3}$ where $\eta_{0}$
is the viscosity of the medium. Similarly, both random noise parameters
$\xi_{t}$ and $\xi_{r}$ satisfy the fluctuation-dissipation theorem. Each ABP
self-propels with velocity $v_{a}$ with direction given by
$\mathbf{\hat{n}}_{i}=(\cos\phi_{i},\sin\phi_{i})$, which decorrelates in a
rotational diffusion time scale $t_{rot}=D_{r}^{-1}$. It can be compared with
the time scale in which an ABP moves its diameter $t_{swim}=2R/v_{a}$ to
define the Péclet number $Pe=t_{rot}/t_{swim}$. Similarly, the persistence
length $l_{pers}=v_{a}t_{rot}$ is the distance that the particle moves before
decorrelating its orientation.

Alignment interactions in active matter have been largely observed in bacteria
systems capable of forming swarms. Polar aligning interactions have been
widely modelled using the Vicsek model[39] (VM) which predicts the phase
transition of randomly-oriented active particles into collective swarms. We
introduce a Vicsek-like torque acting on particles $M_{i}^{cc}=K_{cc}/(\pi
R_{cc}^{2})\sum_{j}\sin(\phi_{i}-\phi_{j})$ where particle i interacts with
neighbors $j$ within a cut-off distance which is set to[40] $R_{cc}=4R$.
Similarly, a torque can be introduced to couple the orientation of ABPs with
the BCP interface as $M_{i}^{cpl}=-K_{cpl}/(\pi
R^{2})\partial/\partial\phi_{i}\int
d\textbf{r}\left(\nabla\psi\cdot\mathbf{\hat{n}}_{i}\right)^{2}$, to capture
the alignment of ABPs interacting with walls[41]. We define the dimensionless
torque parameters $g_{cc}=K_{cc}/(\pi R_{cc}^{2}k_{B}T)$ and
$g_{cpl}=K_{cpl}/(\pi R^{2}k_{B}T)$ for the particle-particle and coupling
contributions, respectively. The calibration of the effect of the aligning
torques is shown in figure S1 B and C in the ESI.

On the other hand, the Cahn-Hilliard[42, 43] dynamics control the time
evolution of the BCP order parameter $\partial\psi/\partial
t=M\nabla^{2}\left(\delta F_{tot}/\delta\psi\right)$ which drives the phase
separation into ordered structures. The phenomenological mobility $M$ allows
to define a BCP diffusive time scale $t_{BCP}\propto M^{-1}$ which in turn can
be compared with the swimming time scale of the ABP to define a coupling
Péclet-like dimensionless parameter $Pe^{cpl}=t_{BCP}/t_{swim}$. Additionally,
the active energetic scale $\epsilon_{swim}=v_{a}\gamma_{t}(2R)$ can be
compared with the colloid-colloid repulsive scale
$\tilde{\epsilon}_{act}=\epsilon_{swim}/U_{0}$ and BCP-ABP coupling strength
$\tilde{\epsilon}_{cpl}=\epsilon_{cpl}/\epsilon_{swim}$ where
$\epsilon_{cpl}=\pi R^{2}\sigma$. We select $R$ and $t_{rot}$ as the units of
length and time, respectively. In this work, the colloid-colloid energetic
scale $U_{0}$ is tuned to prevent overlapping between particles by maintaining
$\tilde{\epsilon}_{act}\ll 1$. A complete description of the model used
throughout this work can be found in the ESI.

Several observables are used throughout this work to quantify the phase
behaviour of the system. The modified nematic order parameter $S_{cpl}=\langle
2(\hat{\textbf{n}}_{i}\cdot\nabla\psi)^{2}-1\rangle$ couples the ABP
orientation with the vector pointing perpendicular to the interface
$\nabla\psi$. The spatial inhomogeneity introduced by the BCP mesophase can
prevent global polar ordering in the system. Therefore, the polarisation of
the system is calculated as the average $P_{cc}=\langle P_{cc}^{i}\rangle$ of
the local polar order for particle $i$ defined as
$P_{cc}^{i}=1/N_{i}\sum_{j=1}^{N_{i}}\hat{\textbf{n}}_{i}\cdot\hat{\textbf{n}}_{j}$
where the polarisation of particle $i$ is calculated with respect to the
$N_{i}$ first neighbours within distance $r_{ij}^{*}=1.3(2R)$. Similarly, the
nematic-like particle-particle alignment is characterised as $S_{rel}=\langle
S_{rel}^{i}\rangle$ with
$S_{rel}^{i}=1/{N_{i}}\sum_{j=1}^{N_{i}}2\left(\hat{\textbf{r}}_{ij}\cdot\hat{\textbf{n}}_{j}\right)^{2}-1$.
The combination of $P_{cc}$ and $S_{rel}$ allows to distinguish between
nematic and polar phases in 1D clusters. Additionally, cluster analysis allows
to calculate the fraction $\Phi_{1D}$ of ABP clusters with 1D-like structure
and the fraction $\Phi_{circle}^{BCP}$ of BCP domains with approximately
circular shape. A full list of the observables used in this work -as well as
details on the calculation method- can be found in the ESI.

Figure 1:  Assembly of neutral active ABPs in a diluted regime
($\phi_{p}=0.01$) in the absence of alignment torques
$M_{i}^{cc}=M_{i}^{cpl}=0$. In A the nematic coupling order parameter
$S_{cpl}$ is shown vs the Péclet number $Pe$ with particles sized
$H_{0}/R=9.6$. The sign of $S_{cpl}$ distinguishes $3$ regimes noted I, II and
III with snapshots shown in the bottom-left of the panel. B shows the fraction
$\Phi$ of colloids at the interface. In C the critical Péclet collapses into a
curve given by equation S22, for three particle radius.

NPs with a neutral interaction with both segments of the BCP (surfactant-like,
with $\psi_{0}=0$) anchor at the interface of the BCP domains[4]. This results
in the soft confinement of neutral ABPs at the BCP interface. In figure 1 the
ordering of ABPs is explored in terms of $Pe$, in a symmetric lamellar-forming
BCP with $H_{0}/R=11.2$. No alignment torques are introduced regarding the
particle-polymer coupling $g_{cpl}=0$ or the particle-particle polar alignmet
$g_{cc}=0$. A low concentration $\phi_{p}=0.01$ can acquire both orientational
and translational order, with three distinguishable regimes: I passive-like
ABPs with low activity $Pe<1$ display no defined nematic order with respect to
the interface $S_{cpl}\sim 0$ while anchoring at the interface $\Phi\sim 1$.
II moderately active ABPs with $1<Pe<10$ can self-propel along the interface
before encountering a lamellar defect. ABPs that accumulate at lamellar
defects contribute to positive nematic order, which couples the ABPs direction
and the interface normal $S_{cpl}>0$, while the ABPs remain at the interface
$\Phi\sim 1$. III highly active partices with $Pe>10$ can detach from the
interface when their swimming energy $\epsilon_{swim}$ is large enough to
overcome the BCP coupling $\epsilon_{cpl}$. In this regime, as the activity
increases and more ABPs can escape the interface $\Phi<1$, the orientation of
the ABPs is increasingly tangential to the interface $S_{cpl}<0$, as particles
that approach an interface with a tangential orientation are likely to reside
at the interface for a longer time than particles colliding head-on with the
interface.

The accumulation of ABPs at lamellar defects and their nematic ordering in
regime II is a consequence of the fingertip-like morphology of symmetric BCP,
that permits ABPs to accumulate at high-curvature regions of the interface.
111This is a metaestable state, in which the lamella typically organise in
experiments in the absence of external fields In the absence of defects in the
lamellar morphology, i.e. an equilibrated phase, the nematic order is not
present as shown in figure S2, while BCP mesophases with intrinsic curvature
display an increased peak. These results indicate the role of heterogeneity in
complex medium leading to co-assembly of ABPs within BCP melts.

In figure 1 B a critical Péclet $Pe^{*}$ can be identified as the approximate
value at which $\Phi\approx 0.9$. Larger values of $Pe>Pe^{*}$ lead to an
abrupt decrease of $\Phi$ as neutral particles are de-coupled from the
interface. Exploring a wide range of $\sigma$ and $R$ values we identify the
critical $Pe^{*}$ under different $\epsilon_{cpl}$ conditions. These data is
seen to collapse into a single curve, following the dimensionless parameter
$\tilde{\epsilon}_{cpl}^{*}$ dependence. This suggests that a critical
dimensionless coupling value of the activity can be identified as
$\tilde{\epsilon}_{cpl}^{*}\approx 21.5$, which is the critical value at which
the self-propulsion is able to detach the particle from the interface
trapping.

Figure 2:  Phase behaviour of neutral active particles with concentration
$\phi_{p}$ with coupling alignment between colloids and BCP $g_{cpl}=34.9$ and
polar alignment between particles $g_{cc}=0.06$ ,in A. Phases are marked as
passive-like in black circles, downwards blue triangles for unstructured
clusters, upwards red triangles for polarised trains at the interface, and
diamonds for swarms escaped from the interface. In B and C the observable
curves are shown for the ABPs and the BCP, respectively, for a fixed
concentration $\phi_{p}=0.0875$. Representative snapshots of each are shown in
the bottom-left.

At moderate ABP concentrations collective behaviour in active systems can be
expected to emerge[28]. Figure 2 shows the phase behaviour of finite
concentrations $\phi_{p}$ of ABPs for an activity rate $Pe$, in the presence
of strong particle-interface $g_{cpl}=34.9$ and moderate particle-particle
coupling $g_{cc}=0.06$ alignment. Neutral ABPs segregate towards the interface
of circle-forming asymmetric BCP, with the system being initialised with ABPs
at the interface of an already-equilibrated BCP. Several regimes can be
distinguished in figure 2 A, based on the structures formed by ABPs and the
morphology of the BCP. The curves of the BCP and ABPs observables are shown
respectively in B and C, for a fixed concentration $\phi_{p}=0.0875$.

In figure 2 A for low $Pe$ ABPs can flow within the interface before
encountering an additional ABPs -marked as black circles- , leading to
colloidal aggregation at the interface similar to the accumulation mechanism
described in figure 1. This low activity region is characterised by a 1D
colloidal cluster shape $\Phi_{1D}\sim 1$ and high nematic $S_{rel}\sim 0.8$
but low polar $P_{cc}\sim 0.2$ interparticle alignment, as can be seen in
figure 2 B, due to the low $g_{cc}$ coupling (see ESI video neutral-phase-
I_Pe1.mp4). Low NP activity introduces no impact in the BCP morphology which
remains circular, as shown in C.

Higher activities $Pe$ allow ABPs to overcome the energetic barrier at the
interface and escape into bulk regions -marked as blue and red triangular
symbols . The required activity rate decreases as the ABP concentration
increases, indicating that ABPs escape collectively due to the particle-
particle polar alignment. Furthermore, two regimes can be distinguished:
unstructured clusters -marked as downwards blue triangles- are found for
larger concentrations and moderate activity, where ABPs possess enough active
energy to escape the interface but, upon entering the bulk regions, tend to
form unstructured clusters characterised by a decrease in $P_{cc}$,
$\Phi_{1D}$ and $S_{rel}$, as shown in figure 2 B. The relative high local
concentration and the moderate activity promote the aggregation, which
introduces considerable changes in the BCP mesophase, as shown in the decrease
in circular BCP domains in figure 2 C (see ESI video neutral-phase-II_Pe8.mp4
).

For moderate activities but lower concentrations ABPs can form polarised 1D
clusters that maintain a continuous flow along the interface -marked as red
upwards triangles. In this regime ABPs have enough active energy to not only
escape the interface, but also to self-propel through the bulk phase before
entering an additional BCP domain (see ESI video neutral-phase-III.mp4 ). This
process can be repeated until the formation of polarised train-like clusters,
characterised with high polarisation in B, in contrast to the low activity
regime. The short-time dynamics of both train-like and unstructured cluster
regime are considerably similar, as depicted in figure 3, where a transient
regime of lowering in the number 1D clusters is observed. Nonetheless, for
higher activity rate ($Pe=11$, red upward triangles) ABPs are able to enter
new BCP interfaces and form 1D polarised trains, while lower activity promotes
the aggregation of ABPs into polarised clusters, leading to unstructured
clusters ($Pe=6$, blue downwards triangles).

Figure 3: Dynamic behaviour of the fraction of 1D ABP clusters for four
representative activities corresponding to the four regimes identified in
figure 2. The system is initialised at $t=0$ with ABPs placed at the interface
of equilibrated circle-forming BCP domains. It can be compared with figure S3
for an initially-random condition.

Finally, for considerably large activity , ABPs can be found completely
detached from the interface, at the top of the phase diagram in figure 2. ABPs
can re-orient upon collisions to form polarised clusters -blue diamonds- which
leads to high polarisation but low interface-coupled nematic order, as shown
in B. In this regime the activity is large enough to largely decouple the
dynamics of the BCP and the ABPs, signalled by the relative time scales
$Pe^{cpl}=0.13Pe$ and the energetic coupling $\tilde{\epsilon}_{cpl}=229/Pe$.
As a result, the BCP relaxation time cannot accommodate the fast dynamics of
the ABPs. In figure 2 C the BCP mesophase is largely distorted as NPs are able
to penetrate the interface, leading to a pronounced decrease in the fraction
of circular domains for $Pe\sim 50$, which is followed by an increase in
$\Phi_{circle}^{BCP}$ indicating the recovery of the equilibrium circular
phase as the dynamics of the BCP and NPs are more decoupled.

Contrary to neutral NPs, selective NPs ($\psi_{0}=-1$) are miscible within one
of the BCP phases[45]. When NPs are compatible with the minority phase in a
circle-forming BCP, the BCP interface acts as a soft wall inducing confinement
on the NPs. The strength of the confinement can be roughly estimated comparing
the coupling energy and the swimming one, $\tilde{\epsilon}_{cpl}$. Particle-
polymer and particle-colloid alignment are introduced, respectively with
$g_{cpl}=34.9$ and $g_{cc}=0.06$ .

In figure 4 several co-assembled regimes can be identified, with passive-like
ABPs marked as black circles in the low activity regime , displaying no
ordering. As the activity grows, the self propulsion leads to NPs accumulating
at the domain walls, where they experience alignment torques promoting the
tangential orientation of NPs with respect to the domain walls. This
corresponds to active persistence length $l_{pers}$ comparable to the diameter
of the BCP domains (see figure S5 in the ESI).

For small concentrations, this leads to the additional re-alignment of NPs
after colloid-colloid collisions, while the wall-particle coupling torque
dominates. As a result, polarised 1D train-like colloids rotate tangentially
to the domain walls, while softly confined within the circular phase, marked
as red upside triangles. In this regime the domain wall coupling remain high
enough to prevent NPs penetrating the interface, quantified by
$\tilde{\epsilon}_{cpl}=229.4/Pe$, which assures that the BCP domains remain
approximately circular. The formation of vortexes of active particles under
confinement has been shown for active filaments in deformable vesicles[28] or
under geometric confinement[46, 26, 23, 47].

For higher concentrations $\phi_{p}>0.09$, the crowded environment of NPs
within some BCP domains limits the ability of NPs to rotate in train-like
polarised clusters. Instead, 2D clusters can be formed, which may coexist with
train-like clusters in other domains, depending on the number of NPs within
the local BCP domain. This regime is marked with yellow asterisks which can be
seen to overlap with the train-like vortexes. In the onset of NP escaping,
active colloids can considerably deform the BCP domain shape while ABPs can
accumulate forming caps, in a similar way as described for active
filaments[28].

For higher activity and larger concentrations, NPs can escape the soft
confinement, marked as blue squares, and form swarm structures with internal
polar ordering, similar to the ones shown in figure 2. Similarly, higher
concentrations reduce the required activity $Pe$ to allow ABPs to escape the
soft confinement. In this regime, the BCP dynamic is largely de-coupled with
the ABPs one, as the swimming time scale is considerably smaller than the
relaxation time of the BCP, following $Pe^{cpl}=0.13Pe$.

Figure 4: Phase behaviour of selective active particles with a finite
concentration $\phi_{p}$. Active particles align with respect to the BCP
interface with $g_{cpl}=34.9$ and $g_{cc}=0.06$. Markers indicate the
colloidal regime: disordered colloids within minority domains marked as black
circles, chain/ring structures marked with red triangles, close-packed
structures within minority domains marked as yellow asterisks and free
particles marked as blue squares.

Polarised train-like morphologies have been found for both neutral and
selective ABPs, where the soft confinement of the curved BCP domains induce
the emergence of rotational motion. Figure 5 shows the appearance of chiral
motion controlled by the BCP mesophase quantified by the angular frequency of
ABPs.

Figure 5:  Mean absolute value of the angular frequency $\omega_{c}$ of ABPs
with neutral ($\psi_{0}=0$, $\phi_{p}=0.08$) and selective interaction
($\psi_{0}=-1$, $\phi_{p}=0.05$) with the BCP mesophase.

ABPs have been shown to co-assemble within BCP melts to produce emergent
organised structures displaying a rich phase behaviour. This is a consequence
of both the BCP intrinsic ordering and the self-propulsion of the ABPs subject
to soft confinement by the BCP. Furthermore, high NP activity can lead to new
co-assembled BCP morphologies resulting from the competition of the BCP and NP
time and energetic scales. BCP melts are particularly well-suited to control
the assembly of active particles as an example of nanostructured medium
capable of inducing soft confinement.

###### Acknowledgements.

The authors thank the European Union’s Horizon 2020 Research and Innovation
Programme project VIMMP under grant agreement No 760907. I.P. acknowledges
support from Ministerio de Ciencia, Innovación y Universidades MCIU/AEI/FEDER
for financial support under grant agreement PGC2018-098373-B-100 AEI/FEDER-EU,
from Generalitat de Catalunya under project 2017SGR-884, and Swiss National
Science Foundation Project No. 200021-175719 .

## References

  * Bates and Fredrickson [1990] F. S. Bates and G. H. Fredrickson, Annual Review of Physical Chemistry 41, 525 (1990).
  * Matsen and Bates [1996] M. W. Matsen and F. S. Bates, Macromolecules 29, 1091 (1996).
  * Thompson [2001] R. B. Thompson, Science 292, 2469 (2001).
  * Kim  _et al._ [2007] B. J. Kim, J. Bang, C. J. Hawker, J. J. Chiu, D. J. Pine, S. G. Jang, S.-M. Yang, and E. J. Kramer, Langmuir 23, 12693 (2007).
  * Ploshnik  _et al._ [2010a] E. Ploshnik, A. Salant, U. Banin, and R. Shenhar, Physical Chemistry Chemical Physics 12, 11885 (2010a).
  * Ploshnik  _et al._ [2010b] E. Ploshnik, A. Salant, U. Banin, and R. Shenhar, Advanced Materials 22, 2774 (2010b).
  * Ploshnik  _et al._ [2013] E. Ploshnik, K. M. Langner, A. Halevi, M. Ben-Lulu, A. H. E. Müller, J. G. E. M. Fraaije, G. J. Agur Sevink, and R. Shenhar, Advanced Functional Materials 23, 4215 (2013).
  * Huh  _et al._ [2000] J. Huh, V. V. Ginzburg, and A. C. Balazs, Macromolecules 33, 8085 (2000).
  * Halevi  _et al._ [2014] A. Halevi, S. Halivni, M. Oded, A. H. E. Müller, U. Banin, and R. Shenhar, Macromolecules 47, 3022 (2014).
  * Tailleur and Cates [2008] J. Tailleur and M. E. Cates, Physical Review Letters 100 (2008), 10.1103/PhysRevLett.100.218103.
  * Cates and Tailleur [2015] M. E. Cates and J. Tailleur, Annual Review of Condensed Matter Physics 6, 219 (2015).
  * Telezki and Klumpp [2020] V. Telezki and S. Klumpp, Soft Matter 16, 10537 (2020).
  * Hagen  _et al._ [2011] B. t. Hagen, S. van Teeffelen, and H. Löwen, Journal of Physics: Condensed Matter 23, 194119 (2011).
  * Romanczuk  _et al._ [2012] P. Romanczuk, M. Bär, W. Ebeling, B. Lindner, and L. Schimansky-Geier, The European Physical Journal Special Topics 202, 1 (2012).
  * Cates and Tailleur [2013] M. E. Cates and J. Tailleur, EPL (Europhysics Letters) 101, 20010 (2013).
  * Fily and Marchetti [2012] Y. Fily and M. C. Marchetti, Physical Review Letters 108, 235702 (2012).
  * Redner  _et al._ [2013] G. S. Redner, M. F. Hagan, and A. Baskaran, Physical Review Letters 110, 055701 (2013).
  * Bialké  _et al._ [2013] J. Bialké, H. Löwen, and T. Speck, EPL (Europhysics Letters) 103, 30008 (2013).
  * Digregorio  _et al._ [2018] P. Digregorio, D. Levis, A. Suma, L. F. Cugliandolo, G. Gonnella, and I. Pagonabarraga, Physical Review Letters 121, 098003 (2018).
  * Bechinger  _et al._ [2016] C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Reviews of Modern Physics 88, 045006 (2016).
  * Frangipane  _et al._ [2019] G. Frangipane, G. Vizsnyiczai, C. Maggi, R. Savo, A. Sciortino, S. Gigan, and R. Di Leonardo, Nature Communications 10, 2442 (2019).
  * Bricard  _et al._ [2015] A. Bricard, J.-B. Caussin, D. Das, C. Savoie, V. Chikkadi, K. Shitara, O. Chepizhko, F. Peruani, D. Saintillan, and D. Bartolo, Nature Communications 6, 7470 (2015).
  * Lushi  _et al._ [2014] E. Lushi, H. Wioland, and R. E. Goldstein, Proceedings of the National Academy of Sciences 111, 9733 (2014).
  * Hernandez-Ortiz  _et al._ [2005] J. P. Hernandez-Ortiz, C. G. Stoltz, and M. D. Graham, Physical Review Letters 95, 204501 (2005).
  * Yang  _et al._ [2014] X. Yang, M. L. Manning, and M. C. Marchetti, Soft Matter 10, 6477 (2014).
  * Wioland  _et al._ [2013] H. Wioland, F. G. Woodhouse, J. Dunkel, J. O. Kessler, and R. E. Goldstein, Physical Review Letters 110, 268102 (2013).
  * Kruse  _et al._ [2004] K. Kruse, J. F. Joanny, F. Jülicher, J. Prost, and K. Sekimoto, Physical Review Letters 92, 078101 (2004).
  * Peterson  _et al._ [2021] M. S. Peterson, A. Baskaran, and M. F. Hagan, Nature communications 12, 1 (2021).
  * Angelani and Leonardo [2010] L. Angelani and R. D. Leonardo, New Journal of Physics 12, 113017 (2010).
  * Mallory  _et al._ [2014] S. A. Mallory, C. Valeriani, and A. Cacciuto, Physical Review E 90, 032309 (2014).
  * Wensink  _et al._ [2014] H. H. Wensink, V. Kantsler, R. E. Goldstein, and J. Dunkel, Physical Review E 89, 010302 (2014).
  * Fernandez-Rodriguez  _et al._ [2020] M. A. Fernandez-Rodriguez, F. Grillo, L. Alvarez, M. Rathlef, I. Buttinoni, G. Volpe, and L. Isa, Nature Communications 11, 4223 (2020).
  * Wang  _et al._ [2015] X. Wang, M. In, C. Blanc, M. Nobili, and A. Stocco, Soft Matter 11, 7376 (2015).
  * Wang  _et al._ [2016] X. Wang, M. In, C. Blanc, P. Malgaretti, M. Nobili, and A. Stocco, Faraday Discussions 191, 305 (2016).
  * Yang and Loos [2017a] Q. Yang and K. Loos, Macromolecular Chemistry and Physics 218, 1600451 (2017a).
  * Yang and Loos [2017b] Q. Yang and K. Loos, Polymer Chemistry 8, 641 (2017b).
  * Ohta and Kawasaki [1986] T. Ohta and K. Kawasaki, Macromolecules 19, 2621 (1986).
  * Tanaka and Araki [2000] H. Tanaka and T. Araki, Physical Review Letters 85, 1338 (2000).
  * Vicsek  _et al._ [1995] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Physical Review Letters 75, 1226 (1995).
  * Sesé-Sansa  _et al._ [2018] E. Sesé-Sansa, I. Pagonabarraga, and D. Levis, EPL (Europhysics Letters) 124, 30004 (2018).
  * Palacios  _et al._ [2019] L. S. Palacios, J. Katuri, I. Pagonabarraga, and S. Sánchez, Soft Matter 15, 6581 (2019).
  * Cahn [1959] J. W. Cahn, The Journal of Chemical Physics 30, 1121 (1959).
  * Cahn and Hilliard [1959] J. W. Cahn and J. E. Hilliard, The Journal of Chemical Physics 31, 688 (1959).
  * Note [1] This is a metaestable state, in which the lamella typically organise in experiments in the absence of external fields.
  * Kim  _et al._ [2006] B. J. Kim, J. Bang, C. J. Hawker, and E. J. Kramer, Macromolecules 39, 4108 (2006).
  * Woodhouse and Goldstein [2012] F. G. Woodhouse and R. E. Goldstein, Physical Review Letters 109, 168105 (2012).
  * Wu  _et al._ [2017] K.-T. Wu, J. B. Hishamunda, D. T. N. Chen, S. J. DeCamp, Y.-W. Chang, A. Fernández-Nieves, S. Fraden, and Z. Dogic, Science 355, eaal1979 (2017).