# Tuning Higher Order Structure in Colloidal Fluids

Xiaoyue Wu<EMAIL_ADDRESS>School of Chemistry, University of Leeds,
Woodhouse Lane, Leeds, LS2 9JT, United Kingdom  Katherine Skipper  H. H. Wills
Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom
Yushi Yang  H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8
1TL, United Kingdom  Fergus J. Moore  H. H. Wills Physics Laboratory,
University of Bristol, Bristol BS8 1TL, United Kingdom  Fiona C. Meldrum
School of Chemistry, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT,
United Kingdom  C.Patrick Royall  Gulliver UMR CNRS 7083, ESPCI Paris,
Université PSL, 75005 Paris, France

###### Abstract

Colloidal particles self assemble into a wide range of structures under AC
electric fields due to induced dipolar interactions [Yethiraj and Van
Blaaderen _Nature_ 421 513 (2003)]. Tuning the external field drives
structural transitions in the system. Here we consider colloidal fluids
comprised of such dipolar particles in experiment and simulation and
investigate their 2-body and higher-order structure using a variety of probes.
The higher–order structure we probe using three-body spatial correlation
functions and a many–body approach based on minimum energy clusters of a
related system. These clusters exhibit a cascade of geometric transitions upon
increasing the strength of the dipolar interaction, which are echoed in the
higher–order structure of the concentrated fluids we study here.

††preprint: AIP/123-QED

##  I Introduction

Particles with a dipolar interaction are of great fundamental importance in
the study of fluids and disordered materials. They are among the simplest
models which describe long range directional interactions, which are exhibited
by molecules [1, 2]. Colloidal dispersions provide suitable models of
atomistic and molecular systems as they exhibit phase behavior following the
same rules of statistical mechanics, yet are amenable to real space
observation using optical microscopy [3, 4, 5]. Rather than spontaneous or
intrinsic dipolar interactions as is often the case for atoms and molecules,
in colloidal systems, dipoles are typically induced by an external electric or
magnetic field [4, 5, 6]. This has the consequence that the dipolar
interactions are aligned in the direction of the electric field. In addition
to their fundamental interest, such dipolar colloidal systems may find
application as electrorheological fluids [7, 8], hydraulic valves and photonic
materials [9].

Among the attractions of dipolar colloidal systems is the ease with which the
dipolar interactions can be tuned with the external field. Indeed real space
studies of such systems led to the discovery of many crystal structures
including fcc, hcp, bcc, body-centered tetragonal (bct) and body-centered
orthorhombic (bco) structures [10, 11, 5], along with a transient labyrinthine
structure [8]. Tuning the electric field _in–situ_ enables the control of
phenomena such as a martensitic transition [12].

Adding softness [13] or attractions [14] to the interaction potential further
increases the range of structures into which the system may self–assemble. In
addition to the rich crystalline phase behavior, dipolar colloids feature a
fluid phase at lower colloid volume fraction and electric field strength than
those at which the crystals are found. In the colloidal fluid, as the field
strength is increased, the symmetry–breaking dipolar interaction leads to a
“string fluid” of particles organized in vertical lines [8, 10, 15], which can
form the basis for producing “colloidal polymers” [16]. The emergence of this
string fluid suggests a rather rich structure in the fluid. If characterized,
such fluid structure could provide information on mechanisms by which
nucleation in the polymorphic system might occur, due to the relation between
fluid structure and crystallization [17, 18, 19].

Characterizing structure in the fluid state beyond pair correlations is
challenging, although not impossible [20], and theoretical treatments have
been developed for the higher–order structural of hard spheres [21, 22]. In
experiment, particle–resolved studies of colloidal systems are ideal for
studying higher–order structural correlations since particle–resolved data is
amenable to the same kind of structural analyses as that of computer
simulation, for which a number of methods have been developed [4].

Such methods include three–body correlation functions such as $g_{3}$ [21, 23,
24], and higher–order correlations such as common neighbor analysis (CNA) [25]
and Voronoi face analysis [26] have shown to be successful in studying
structures of fluids and glasses [27, 28]. Another strategy, the bond
orientation order (BOO) parameters developed by Steinhardt et al.
steinhardt1983 [29], focuses on the local symmetry around a central particle.
This method has been shown to be most successful in the study of
crystallization, especially in identification of small crystalline clusters in
a supercooled liquid[30, 31] and also in the characterization of fivefold
symmetric order in amorphous systems [32].

With the popularity of research in machine learning rising, it has been
applied effectively to local structure in amorphous materials, for example by
combining many structural metics such as the local pair correlation function
[33]. Other examples include combining it with local descriptors such as CNA
and BOO to better characterize the local environment around a single particle
in disordered materials [34, 35]. Both supervised [36] and unsupervised
learning [37] have been used to further our understanding of supercooled
liquid and glass forming systems.

The methods listed above are geometric in nature. An alternative approach,
which takes into account the interactions between the constituent particles of
the system, has its roots in the work of Sir Charles Frank [38], who
postulated that since the minimum potential energy configuration of 13
Lennard-Jones atoms corresponds to an icosahedron, that this would be a common
structure in (supercooled) liquids. With the advent of energy landscape
calculations [39], it has become possible to determine the structure of
minimum potential energy clusters for a wide range and size of systems
including the Lennard-Jones [40] and Stockmayer [41] models. The latter is a
Lennard–Jones interaction with an additional dipolar interaction, which is not
constrained to lie in any particular direction and thus corresponds to a
molecular, rather than colloidal dipolar system in the context of the
discussion above. While it is possible to determine minimum energy clusters,
identifying local arrangements of particles in bulk systems whose bond network
is identical to such clusters can be carried out using the _topological
cluster classification_ (TCC) [42, 43]. The TCC has been used to identify
locally favored structures or minimum energy clusters in systems undergoing
dynamic arrest [44], colloid-polymer mixtures interacting via Morse potential
[45], colloidal suspensions undergoing attractive interactions [46], colloidal
gels [24, 47] and the liquid–gas interface [48].

The colloidal dipolar system is interesting from the perspective of using
minimum energy clusters to characterize the local structure because, unlike
the systems mentioned above, the interaction potential is anisotropic and
changes as a function of electric field strength. Some of us have recently
determined the minimum energy clusters for the colloidal dipolar system [43],
which opens the possibility to use this method to probe the higher–order
structure of this system.

Herein, we report a combined experimental and computer simulation study of
dipolar (nearly) hard sphere colloids in 3d with a range of methods of
quantifying structure. We consider pair correlations in the form of radial
distribution functions $g_{2}(r)$ and three-body correlations in the form of
order parameters to determine “string-like” configurations and also a triplet
correlation function $g_{3}(r,r^{\prime},\theta)$. Finally the we use the
topological cluster classification [42, 43] to explore higher–order spatial
correlations. We find a good match between our experiment and simulation
results. We also compared the Lennard-Jones–dipolar cluster populations
present in experimental and simulation data with theoretical predictions from
energy minimization calculations, since the topology of minimum energy
clusters changes as a function of applied field strength [43].

Figure 1: Minimum energy clusters of the dipolar–Lennard-Jones system for
various sizes. Different geometries corresponds to a change in strength of the
dipolar contribution. [43].

##  II Dipolar Interactions in Colloidal Systems

The colloids in our experiments are immersed in an index-matching solvent with
added salt, as described in Sec: III.1. The interaction between such colloids
can be effectively modeled by a combination of a hard core Yukawa interaction
[49, 50] and dipolar interaction [10]. The Yukawa term takes the from:

$\displaystyle\beta u_{\mathrm{yuk}}(r)$
$\displaystyle=\begin{cases}\infty&\mathrm{for}\ r\leq\sigma\\\\[2.00749pt]
\beta\varepsilon_{\mathrm{yuk}}\frac{\exp\left(-\kappa(r_{ij}-\sigma\right)}{r/\sigma}&\mathrm{for}\
r>\sigma\end{cases}$ (1)

where $\beta$ is the inverse of the thermal energy $k_{B}T$ with $k_{B}$ the
Boltzmann constant, $T$ temperature, $u_{\mathrm{yuk}}$ the Yukawa interaction
potential, $r$ the separation between two particles, $\sigma$ the hard-core
particle diameter and $\kappa$ the inverse of the Debye screening length
($\kappa^{-1}=1/\sqrt{8\pi\lambda_{B}c}$ for monovalent
salt).$\beta\varepsilon_{\mathrm{yuk}}$ is the potential at contact and can be
expressed as:

$\beta\varepsilon_{\mathrm{yuk}}=\frac{Z^{2}}{(1+\kappa\sigma/2)}\frac{\lambda_{B}}{\sigma}$
(2)

where Z is the particle charge and
$\lambda_{B}=e^{2}/4\pi\varepsilon_{0}\varepsilon_{m}k_{B}T$, the Bjerrum
length of the suspending medium with dielectric constant $\varepsilon_{m}$.
When colloids are subjected to external electric field, a dipole-dipole
interaction is induced and the dipolar interaction takes the form

$\beta
u_{\mathrm{dip}}=\frac{\gamma_{\mathrm{exp}}}{2}\left(\frac{\sigma}{r}\right)^{3}\left(1-3{\left(\cos{\theta}\right)}^{2}\right)$
(3)

where $u_{\mathrm{dip}}$ is the dipolar interaction, the $\theta$ is the angle
made by $\mathbf{r}$ and the $z$-axis and $\gamma_{\mathrm{exp}}$ is a
dimensionless prefactor that depends on the strength of the local external
field.

$\gamma_{\mathrm{exp}}=\frac{\mathbf{p}^{2}}{2\pi\varepsilon_{m}\varepsilon_{0}\sigma^{3}k_{B}T}$
(4)

where

$\mathbf{p}=\frac{\pi}{2}\beta\varepsilon_{m}\varepsilon_{0}\sigma^{3}\mathbf{E}_{\mathrm{loc}}$
(5)

is the dipole moment. $\varepsilon_{m}$ is the dielectric constant of the
suspending medium, $\varepsilon_{0}$ is the permittivity of vacuum and
$\mathbf{E}_{\mathrm{loc}}$ is the local electric field.

Combining with the Yukawa interaction Eq. 1 the total interaction between two
colloids under the electric field becomes:

$\beta u_{\mathrm{total}}=\beta u_{\mathrm{dip}}+\beta u_{\mathrm{yuk}}.$ (6)

##  III Methods

###  III.1 Experimental

The colloidal suspension used in this experiment was prepared by adding
sterically-stabilized polymethyl methacrylate (PMMA) spheres (synthesized
following reference [51, 52]) ($\rho$ = 1.196 gcm-3) [53] of diameter 1.73
$\mu$m (polydispersity=6.8%) in a mixture of density and refractive index
matched organic solvents. The organic solvents consist of cis-decalin ($\rho$
= 0.897 gcm-3) and cyclohexyl-Bromide (CHB) ($\rho$ = 1.32 gcm-3).
Tetrabutylammonium bromide (TBAB) was dissolved in the solvent to make up for
a solution with TBAB salt concentration of 260 $\mu$M. This corresponds to a
Debye length $\kappa^{-1}$ of around 100 nm [54]. While more sophisticated
treatments may be carried out to match the interaction potential [54, 55],
here we use a slightly soft potential in the computer simulations and presume
this to be sufficient to match the experimental system, noting that the
effects we seek to study are dominated by the dipolar interactions, rather
than the hard core or precise colloid volume fraction (which we determine by
weighing out the samples) [56].

In order to construct the electric cell to hold the colloidal suspension, two
indium tin oxide glass slides were separated with spacer silica particles of
approximately 60 $\mu$m in diameter to create a transparent, electrically
conductive cell. The electrodes were connected to a signal generator that
supplies alternating currents across the electrical cell.

A Leica SP8 confocal microscope was used to monitor colloidal assembly under
the applied electric field. During each measurement, an entire stack of 3D
confocal image of at least 200 “slices” of $xy$ images along the $z$ direction
were taken with 256$\times$256 pixels with at least ten pixels per particle
diameter in all directions. Only particles at least ten diameters from the
wall were analyzed, to ensure that there were no significant wall effects. In
each measurement, the applied voltage and the thickness of the electrical cell
was measured in order to allow electric field strength comparison across
different experiments. Before each measurement, the system is allowed to
equilibrate for at least 20min. To obtain sufficient statistics at least 50
independent 3d images were taken.

###  III.2 Particle Tracking

The particle tracking was done via a custom written script to improve tracking
accuracy of small particles in 3D (especially along the $z$-axis). It is
effective in improving resolution of images that suffered from blurring in the
axial direction [57]. The algorithm works by first finding some particles from
a 3D image and obtaining their coordinates. From their coordinates, the
average shape of the particles can be measured. With information on both
positions and shape, the algorithm simulates some artificial images of the
particles and compare them with the real images to identify previously unfound
particles. Further details may be found in Ref. [58]. can be tracked.

Figure 2: The colloidal dipolar system. (a) Representative images using
confocal microscopy in the horizontal $xy$ and vertical $yz$ planes with no
field, $\mathbf{E}=0$. Here volume fraction $\phi$=0.1. (b) Confocal
microscopy images of a system at the same volume fraction at the maximum
electric field strength 200 VMm-1. (c) Bond order parameters (bop)
(${\cos{\theta}}^{2}$) as a function of external electric field strength $E$.
Data are shown for experiments and simulations as indicated in the legend.

###  III.3 Computer Simulation

We employed molecular dynamics simulations using the LAMMPS package at
constant volume, in the NVT ensemble [59]. Similarly to reference [11], Ewald
summation was used to compute long range dipole interactions in periodic
boundary conditions. The system was evolved according to Brownian dynamics
[60, 61]. Here we quote simulation results in reduced Lennard-Jones units.
Each simulation run includes at least 3000 particles and was equilibrated for
1000 Lennard–Jones time units and sampled a further 100 time units. Effective
volume fractions were determined with the Barker–Henderson effective hard
sphere diameter.

We use a Weeks-Chandler-Anderson(WCA) interaction and added a dipole-dipole
interaction shown in Eq. 3 to reproduce interactions in the experimental
system. The WCA interaction takes the form:

$\displaystyle\beta u_{\mathrm{wca}}(r_{ij})$
$\displaystyle=\begin{cases}\beta
4\varepsilon_{\mathrm{wca}}[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^{6}]+\varepsilon_{\mathrm{wca}}&\
r\leq 2^{\frac{1}{6}}\sigma\\\\[2.00749pt] 0&\
r>2^{\frac{1}{6}}\sigma\end{cases}$ (7)

where $\varepsilon_{\mathrm{wca}}$ is the interaction energy. The interaction
potential used in the computer simulations is then

$\beta u_{\mathrm{sim}}=\beta u_{\mathrm{dip}}+\gamma_{\mathrm{sim}}\beta
u_{\mathrm{wca}}.$ (8)

where $\gamma_{\mathrm{sim}}$ controls the relative strength of the dipolar
contribution and models the electric field strength in the experiments.

###  III.4 Bond order parameters for dipolar colloids

One method to quantify the angular correlation between particles as a function
of external electric field strength is the bond order parameter (BOP). This
has already been shown to be sensitive to variation in field strength by Li et
al. [15]. These are calculated by finding the angle made by a reference
particle with two of its nearest neighbor. For a system consisting of perfect
chains, ${\cos^{2}\theta}$ = 1.

###  III.5 Two– and three–body correlation functions

The method described in the previous section offers a way of quantifying a
change in structure for more dilute fluids. At higher concentration, we appeal
at first to the two–body spatial correlation function, the radial distribution
function $g_{2}(r)$. We also consider the 3–body spatial correlation function
$g_{3}$. Now this is dependent on the correlations in positions of three
particles, i.e. three vectors, eg
$g_{3}(\mathbf{r}_{12},\mathbf{r}_{23},\mathbf{r}_{31})$ with the numbers
reflecting the three particles. Here however we elect to simplify our
representation to the case where we fix two of the distances (to the particle
diameter $\sigma$) and vary the third separation between the particles, so
that the 3–body correlation function is plotted as a function of angle between
them $g_{3}(r_{12}=\sigma,{r}_{23}=\sigma,\eta)$ where $\eta$ is the angle
between $\mathbf{r}_{12}$ and $\mathbf{r}_{23}$.

Figure 3: Radial distribution function ($g_{2}(r)$). Data are shown for both

experiment and computer simulations at volume fraction (a) $\phi=0.3$ and (b)
$\phi=0.1$. Lines show different field strengths expressed through the
parameter $\gamma$ as indicted. Data are offset for clarity.

###  III.6 Topological Cluster Classification

As discussed in the introduction, the topological cluster classification
identifies local geometric motifs whose bond topology (defined here through a
modified Voronoi decomposition) is identical to that of minimum energy
clusters of a specific size [42]. These clusters are calculated from the
energy optimization algorithm GMIN which uses basin-hoping to find local
energetic minimum that corresponds to a specific configuration for an isolated
number of particles[40]. Now such minimum energy clusters require an
attractive interaction, and therefore to investigate the effect of the dipolar
interaction, clusters were determined for a dipole added to a Lennard–Jones
interaction [43]. That is to say, the interaction potential for which the
minimum energy clusters were determined was

$\beta u_{\mathrm{tcc}}=\beta u_{\mathrm{li}}+\gamma_{\mathrm{tcc}}\beta
u_{\mathrm{dip}}.$ (9)

where $u_{\mathrm{lj}}$ is the Lennard–Jones interaction.

Those resulting clusters which turned out to be rigid [43] are shown in Fig. 1
The clusters ending with PAA consist of 6Z clusters plus additional particles
(9PAA and 10PAA). Whereas the S-clusters are built based on the 7A clusters
(9S,10S,11S and 12S). The dipolar rigid clusters 8O, 9S, 9PAA, 10S, 10PAA,
11S, 11SB, 11O, 12S, 12SB, 12O and finally 13S are the minimum energy rigid
clusters of particles interacting according to Eq. 9. In our analysis of the
TCC clusters, we set the Voronoi parameter $f_{c}=0.82$ [42].

###  III.7 Orientation of anisotropic clusters

Since the dipolar interaction is anisotropic, the clusters found by the TCC
may have a preferred orientation with respect to the direction of the electric
field. We first isolated individual clusters found in both experiments and
simulations. We calculated the principle axis of each cluster and determine
the angle made by its principle axis with respect to the direction of the
electric field. To quantify this angle distribution, we use an order parameter
commonly used for liquid crystals,

$\langle
P_{2}(\cos\alpha)\rangle=\left\langle\frac{3}{2}\cos^{2}\alpha-\frac{1}{2}\right\rangle.$
(10)

where $\alpha$ is the angle made by the principle axis of each cluster with
the direction of the electric field [62]. In the isotropic case when field is
switched off, there should be no preferred orientation. In the perfectly
aligned state, all the clusters should lie parallel to the electric field so
this order parameter is $1$.

##  IV Results

We now present both experimental and simulation results for colloidal dipolar
fluids at volume fractions $\phi=0.1$ and $\phi=0.3$. We investigate the bond
order parameters, two and three body correlation functions, and populations of
minimum energy clusters and the orientations of these clusters. These
quantities are considered as a function of dipole strength $\gamma$. We
convert external field strength to the dipole strength $\gamma$ using Eq. 4,
which we use across experiments (Eq. 5), simulations (Eq. 8) and minimum
energy clusters (Eq. 9).

###  IV.1 Bond order parameter analysis of the string fluid

In our study, with the bond order parameter (see Sec. III.6) we explored
slightly higher volume fraction (0.1 and 0.3) than some previous work [15].
Our results are shown in Fig. 2 and we find rather good agreement between
experiment and simulation. These show that at the range of volume fraction we
are exploring the bond–order parameter analysis is not massively sensitive
towards the change in volume fraction, with slightly higher values for the
lower volume fraction system. However the BOP is highly sensitive to changes
in external field strength. The general trend follows a sigmoid curve that
exhibits a plateau at value close to unity.

###  IV.2 Two and three body correlation functions

Figure. 3 shows the $g(r)$ of both the computer simulations and experiments.
We see that there is a good match between the two. As the dipole strength
increases, the $g(r)$ reveals the emergence of relatively long range order.

In Fig. 4, we again show experimental and simulation results for the
three–body correlation function $g_{3}$. Here the agreement between simulation
and experiment is not as good as for the 2-body data described above, but we
note that $g_{3}$ is of course higher order and more sensitive to small
differences between the two.

At low field strength, a large and broad peak appears at approximately 60∘
which is consistent with an isotropic system where interaction is angle
independent[63]. As the field strength increases, peaks can be seen at
approximately 60∘, 120∘ and 180∘, with most triplets having angle close to
180∘. This happens when the particles start to form strings.

Figure 4: The three–body correlation function $g_{3}(\eta)$. (a) Schematic
indications of geometries of interest, with values of the bond angle
$\eta=$60∘.120∘ and 180∘ (b) $g_{3}(\eta)$ is plotted for volume fraction
$\phi=0.1$. (c) $g_{3}(\eta)$ for volume fraction $\phi=0.1$. In (b) and (c),
data points are experimental data and lines are simulation data.

###  IV.3 Populations of minimum energy clusters

We now move to still higher–order spatial correlations and consider the
minimum energy cluster geometric motifs identified by the topological cluster
classification [43]. To facilitate comparison between experiment and
simulation, we fix the number of particles in the cluster in question. Now the
topology of the minimum energy cluster changes upon increasing the dipolar
contribution [43]. In Fig. 5, we consider 8, 9 and 10–membered clusters, while
in Fig. 6, data for larger 10 and 11–membered clusters is shown. Naively, one
might expect that zero and low field strength would correspond to minimum
energy clusters for the Lennard–Jones interaction as shown in Fig. 1 and known
from studies with hard spheres [54, 47], and that upon increasing the field
strength might lead to a cascade of clusters of increasing elongation as
indicated in Fig. 1(b). Remarkably, this turns out to be the case.

For the smallest size of cluster which exhibits a non–trivial cascade of such
clusters, $m=8$, indeed we see this trend with the Lennard–Jones minimum
energy cluster 8B giving way to the dipolar cluster 8O for both volume
fractions [Fig. 5(a,b)]. Notably, experiment and simulation appear rather
well–matched here.

In the case of 9-particle clusters, a similar trend is found. In Fig. 5(c,d),
we see clusters of 9B (the cluster expected for zero dipole strength) from
$\gamma=0$ to $\gamma=29$, 9S from ($\gamma=29$ to $\gamma=50$), 9PAA from
($\gamma=50$ to $\gamma=87$ and finally strings at $\gamma>150$. The
experimental and simulation work follow the same trends, with 9B peaking at
$\gamma=0$, followed by 9S and finally 9PAA. Strings (which of course are
non–rigid) overtake as the dominant cluster when $\gamma>150$. The analysis
for strings is already considered in Sec. IV.1 where the maximum populations
of strings occurred at the highest field strength with $\gamma>150$. Here, we
see rather more clusters in the experimental data.

For the 10–membered clusters, again a cascade of cluster topologies is seen as
a function of reduced field strength. In particular in Fig. 5(e,f), we see a
transformation from the Lennard–Jones cluster 10B through 10S to 10PAA. The
change in population correlates well with the values of dipole strength at
which the topology of the minimum energy cluster varies. Here the agreement
between experiment and simulation is less good, however the poorer statistics
for these larger clusters as a source for this discrepancy cannot be ruled
out.

Figure 5: Populations of smaller minimum energy clusters detected by the
topological cluster classification as a function of dipole strength. The
number of particles detected in a given cluster $N_{c}$ is scaled by the
number of particles in a cluster of that size. Shading denotes the change in
cluster topology which minimizes the energy at different vales of the dipole
strength $\gamma$ as indicated [43]. White regions of graphs denote values of
$\gamma$ where the minimum energy clusters are not rigid. Data are shown for
different cluster sizes $m$ and volume fractions as follows. (a) Cluster size
$m=8$, volume fraction $\phi=0.1$. (b) $m=8$, $\phi=0.3$. (c) $m=9$,
$\phi=0.1$. (d) $m=9$, $\phi=0.3$. (e) $m=10$, $\phi=0.1$. (f) $m=10$,
$\phi=0.3$.

We now consider larger clusters in Fig. 6(a,b) for volume fraction 0.1 and 0,3
respectively. Data for $m=11$ are shown and here again the cascade of the
Lennard–Jones 11C cluster being favored at zero and small field strength is
found. This cluster topology gives way to 11S, 11SB and 11O upon increasing
the dipole strength. For $m=11$ clusters, slightly more statistics are
obtained for the experiments. Finally, for the $m=12$ data, the statistics are
rather better for volume fraction $\phi=0.3$ in Fig. 6(d) than for $\phi=0.1$
in Fig. 6(c) and therefore we focus on $\phi=0.3$. Upon increasing the field
strength, our simulations show a cascade from 12B (Lennard–Jones), through
12S, 12SB to 12O at higher field strength.

Figure 6: Populations of larger minimum energy clusters detected by the
topological cluster classification as a function of dipole strength. As in
Fig. 6, the number of particles detected in a given cluster $N_{c}$ is scaled
by the number of particles in a cluster of that size. Shading denotes the
change in cluster topology which minimizes the energy at different vales of
the dipole strength $\gamma$ as indicated [43]. White regions of graphs denote
values of $\gamma$ where the minimum energy clusters are not rigid. Data are
shown for different cluster sizes $m$ and volume fractions as follows. (a)
Cluster size $m=11$, volume fraction $\phi=0.1$. (b) $m=11$, $\phi=0.3$. (c)
$m=12$, $\phi=0.1$. (d) $m=12$, $\phi=0.3$.

###  IV.4 Cluster Orientation

It is an interesting observation that even at zero dipole strength some
minimum energy of dipolar clusters (such as 10PAA at $\phi=0.3$) are present
in our system (Fig. 7). This enables us to investigate further into the
orientation of these clusters using our method described above in Sec. III.7.
Since the dipolar interactions of the individual particles are aligned along
the direction of the electric field it is reasonable that the anisotropic
clusters found by the TCC algorithm also show preferred orientation after the
field is switched on. This we find for all the anisotropic clusters for which
we have sufficient statistics, 8O, 9PAA, 9S, 10PAA, 11O and 12O as shown in
Fig. 7.

Figure 7: Orientation of anisotropic clusters with the electric field. (a)
shows a 3D plot of the clusters (10PAA) found in the experimental dipolar
colloids at $\phi=0.3$ at zero field, where the principal axis (shown in black
line) of each cluster do not aligned with the electric field. Whereas in (b)
these 10PAA clusters showed higher degree of alignment along E-field (z-axis)
as field is switched on to $\gamma=240$. (c,d) $\langle
P_{2}(\cos\alpha)\rangle$ of the principle axis is plotted as a function of
reduced field strength, for the anisotropic clusters 8O, 9PAA, 9S , 10PAA, 11O
and 12O. for the two volume fractions under consideration. (c) $\phi=0.1$ and
(d) $\phi=0.3$. Data points are experimental data and lies are computer
simulation.

##  V Discussion

We now discuss our findings figure by figure.

(_i_) Figure 2 shows confocal microscopy images of colloidal suspension of
volume fraction, $\phi$=0.1, taken along the $xy$ plane (perpendicular to the
direction of the electric field) and $yz$ plane (along the direction of field)
for both zero dipole strength and at maximum dipole strength at $\gamma$=200.
The dipolar colloids at $\gamma$=200 shows some structures with colloids
forming strings along the direction of the field. Across the $xy$ plane,
dipolar colloids show signs of aggregation, forming an almost labyrinth-like
structure. Such a labyrinth-like structure has been found in some dipolar
systems, especially in higher volume fraction, but is transient [8], while our
systems are in equilibrium to the best of our knowledge. Figure 2(c) shows a
plot of bond order parameters of string from both our experiment and
simulation results, similar to the study published by Li et al. [15]. Again,
our simulations (line) and experiments (data points) show excellent agreement.
As field strength increases, more dipolar colloids form strings and the angle
made by colloids tends towards 180∘. As indicated in Fig. 2(c), the degree of
string formation increased gradually as opposed to a sharp increase like a
step function.

(_ii_) In Fig. 3 we plot $g_{2}(r)$. This is generally in excellent agreement
between computer simulations and experiments at for both $\phi$=0.3 and
$\phi$=0.1 (with the only exception at $\phi$=0.1, $\gamma$=200). We can
therefore be rather confident that the simulation model used in our work is a
good reflection of our experimental system. The graphs also show the emergence
of long range order as field strength is increased. This is expected since the
confocal images show that as the fluid becoming more structured as the
colloids aligned along the field when it is switched on. The height of the
first peak of $g_{2}(r)$ also increases as field strength is increases, this
means that the number of neighbors of each colloidal particle also increases
as field is switched on. However, the above-mentioned changes in the plots are
only applicable to higher field strength. At low field strength, between
$\gamma$=0 to $\gamma$=50, there is barely any change in $g_{2}(r)$. The
$g_{2}(r)$ is not able to distinguish minute changes to fluid. This is an
important for our discussion below concerning the higher–order correlations
detected by on $g_{3}$ and the TCC.

(_iii_) We now consider three–body correlations in Fig. 4. As the dipolar
strength increases, the peak at 180∘ increases as expected since more dipolar
colloids form strings. However, we also observe peaks at 60∘ and 120∘
increasing with respect to field strength. This could correspond to the
particle arrangements across the $xy$ plane where the labyrinth-like structure
emerged. Even though $g_{3}$ is more sensitive than $g_{2}$ towards small
structural variations in fluids, there is still very little variations between
graphs at lower field strength. To identify small structural variation at
lower field strength, we turn to the topological cluster classification.

(_iv_) The plots in Figures 5 and 6 show the population of dipolar clusters of
different geometries and sizes analyzed with the TCC. In particular Fig. 5(a)
and (b) show the population of clusters made of 8 particles; (c) and (d) show
the population clusters made of 9 particles; (e) and (f) for 10 particle
clusters. Figure 6 (a) and (b) shows data for 11 particle clusters while (c)
and (d) concerns 12 particle cluster. In most cases, the cluster population
found in experiment and simulation match.

Another observation which can be made from the graph is that the cluster
topology changes at the dipole strength which corresponds to the minimum
energy calculation [43]. Minimum energy dipolar clusters are by definition at
zero temperature, and are determined by the interaction energy. However, at
finite temperature, entropy plays a role in many colloidal systems and its
phase behavior is determined by a combination of energy and entropy. The fact
that the cluster population trends largely follow the minimum energy clusters
indicates that the behavior of dipolar colloids is influenced by energetics.
This is in stark – and surprising – contrast to earlier work which suggested
that energy plays only a very limited role in cluster populations [45].
Presumably the strength of the dipolar interactions (which are much larger
than eg Lennard–Jones interactions [45]) is important here.

(_v_). Figure 7 shows that the anisotropic dipolar clusters tend to align
along the $z$-axis (parallel with direction of the field) when it switched on.
The $P_{2}(\cos{\theta})$ plot in Fig. 7(c) shows that as the field strength
increases, so does the orientational order parameter $P_{2}$. $P_{2}$ has a
maximum value of 1 which indicates when principle axes of clusters are
parallel to the electric field. Our result also shows that more anisotropic,
elongated clusters (e.g. 11O) have higher orientational order parameter
$P_{2})$ than shorter clusters (e.g. 9S) at the same field strength.

Finally, we wish to discuss the importance of our results on the higher order
fluid structures. A small variation in dipolar strength between $\gamma=0$ and
$\gamma=30$ does not show up in $g_{2}(r)$ or even $g_{3}(r)$. Whereas, even
at low field strength, dipolar clusters such as 8B and 8O populations vary
drastically between different field strength (see Fig. 5). We can therefore
conclude that higher order analysis such as that presented here is much more
sensitive to small variation in structure and interactions. For instance, the
TCC has already shown to be useful in analysis of glass transition and gel
formation [20, 44, 64].

Coupling our study with other previously published work on TCC analysis of
gels and glasses [64, 65], we can conclude that higher order structural
analysis may be better at capturing the onset of phase transition than the
standard $g_{2}(r)$. Furthermore the results showed that the string formation
is not instantaneous. In fact, the degree of string formation increases
gradually with field strength. This finding could be relevant for studies on
electrorheological fluid where its structure is directly correlated with its
physical properties such as viscosity and yield stress [66]. Our result shows
that potentially the external field strength could be used to control this
fluid’s physical property to produce smart materials that respond to
environmental changes reversibly.

##  VI Conclusion

We have performed a detailed analysis of the fluid structure in dipolar
colloids and found good agreement between our experimensl and computer
simulation data across a wide range of interactions tuned with the electric
field. We found both bond–order parameter analysis of strings and the
three–body correlation function $g_{3}$ to be suitable to quantify the degree
of string formation in dipolar colloids but with $g_{3}$ offering more
detailed information and can be used as a form of “colloidal finger-print”
(analogous to infrared spectroscopy). With the topological cluster
classification methods, not only can we identify clusters relevant to the
dipolar colloids system but also isolate them to quantify their orientations
with respect to field strength. Finally, our experimental and simulation work
agrees with expectations from minimum energy clusters of a dipolar-Lennard-
Jones system [43].

###### Acknowledgements.

This work was supported by funding from the European Research Council (ERC)
under the project DYNAMIN, grant agreement number 788968 (XW and FCM). YY
graetefully acknowledges the China Scholarship Council. F.J.M. was supported
by a studentship provided by the Bristol Centre for Functional Nanomaterials
(EPSRC Grant No. EP/L016648/1). The authors would like to thank Mark Miller,
Josh Robinson and Peter Crowther for the help in building the TCC software.
The authors would also like to thank Mike Allen and Didi Derks for the helpful
discussions.

## Data Availability Statement

Data and code available upon reasonable requests.

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#  The puzzling properties of the MACS1149-JD1 galaxy at z=9.11

Massimo Stiavelli Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA  Takahiro Morishita IPAC, California Institute of
Technology, MC 314-6, 1200 E. California Boulevard, Pasadena, CA 91125, USA
Marco Chiaberge Space Telescope Science Institute for the European Space
Agency (ESA), ESA Office, 3700 San Martin Drive, Baltimore, MD 21218, USA  The
William H. Miller III Department of Physics and Astronomy, Johns Hopkins
University, Baltimore, MD 21218, USA  Claudio Grillo Dipartimento di Fisica,
Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy  INAF
- IASF Milano, via A. Corti 12, I-20133 Milano, Italy  Nicha Leethochawalit
National Astronomical Research Institute of Thailand (NARIT), Mae Rim, Chiang
Mai, 50180, Thailand  Piero Rosati INAF - OAS, Osservatorio di Astrofisica e
Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129 Bologna, Italy
Dipartimento di Fisica e Scienze della Terra, Università degli Studi di
Ferrara, Via Saragat 1, I-44122 Ferrara, Italy  Stefan Schuldt Dipartimento di
Fisica, Università degli Studi di Milano, Via Celoria 16, I-20133 Milano,
Italy  INAF - IASF Milano, via A. Corti 12, I-20133 Milano, Italy  Michele
Trenti School of Physics, University of Melbourne, Parkville 3010, VIC,
Australia  ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions
(ASTRO 3D), Australia  Tommaso Treu Department of Physics and Astronomy,
University of California, Los Angeles, 430 Portola Plaza, Los Angeles, CA
90095, USA

(Accepted October 8, 2023)

###### Abstract

We analyze new JWST NIRCam and NIRSpec data on the redshift 9.11 galaxy
MACS1149-JD1. Our NIRCam imaging data reveal that JD1 comprises three
spatially distinct components. Our spectroscopic data indicate that JD1
appears dust-free but is already enriched,
$12+\log{\rm(O/H)}=7.90^{+0.04}_{-0.05}$. We also find that the Carbon and
Neon abundances in JD1 are below the solar abundance ratio. Particularly the
Carbon under-abundance is suggestive of recent star formation where Type II
supernovae have already enriched the ISM in Oxygen but intermediate mass stars
have not yet enriched the ISM in Carbon. A recent burst of star formation is
also revealed by the star formation history derived from NIRCam photometry.
Our data do not reveal the presence of a significant amount of old
populations, resulting in a factor of $\sim 7\times$ smaller stellar mass than
previous estimates. Thus, our data support the view that JD1 is a young
galaxy.

††journal: ApJ Letters

##  1 Introduction

The high sensitivity of the NIRSpec spectrometer onboard JWST (Böker et al.,
2023) has enabled us to study emission line diagnostics of galaxies at
redshift $z>8$ with unprecedented details. Already the ERO observations of
SMACS J0723.3–7327 S04590 at $z=8.496$ (Curti et al., 2023; Heintz et al.,
2023) enabled the first detection at high redshift of the auroral line
[OIII]$\lambda 4363$ and its use to derive an electron temperature $T_{e}\geq
24,000$ K. The derived metallicity for this object is $12+\log{\rm(O/H)}\sim
7$. Comparison with strong emission line diagnostics showed that this object
was outside the validity range of most calibrations. A broader look at a small
sample of objects with auroral lines by Laseter et al. (2023); Sanders et al.
(2023) finds that abundances derived from the $T_{e}$ method are generally not
consistent with those from locally calibrated strong emission lines.

The galaxy MACS1149-JD1 (Zheng et al., 2012) (hereafter JD1) was one of a
handful of confirmed $\geq 9$ galaxies known before the launch of JWST and,
being substantially lensed, it was a credible candidate for searching for a
potentially very low metallicty proto-galaxy. The high-redshift nature of
MACS1149-JD1 was confirmed thanks to ALMA detection of [OIII]$\lambda
88\micron$ (Hashimoto et al., 2018; Tokuoka et al., 2022). The previous
measurements by Spitzer indicate the presence of old stellar populations,
making it a relatively massive system among other galaxies at similar
redshifts. These exceptional properties of JD1 led us to make it as one of the
prime candidates for deep spectroscopy with JWST.

It is worthwhile noticing that another bright, pre-JWST high-$z$ galaxy,
GN-z11 (Oesch et al., 2016), while extremely interesting (see e.g. Cameron et
al., 2023a; Tacchella et al., 2023; Bunker et al., 2023; Charbonnel et al.,
2023; Maiolino et al., 2023), is considered to possibly host an AGN and is at
a redshift where [OIII]$\lambda$5007 is beyond the NIRSpec sensitivity range
and therefore makes it less likely to be able to carry out a direct
metallicity measurement using the auroral lines. As such, JD1 is currently one
of a few luminous galaxies at $z>9$ that allow a reliable auroral line
analysis.111Boyett et al. (2023) recently reported Gz9p3, another luminous
galaxy at $z=9.3$, that appears interacting and starforming. Unfortunately,
the observations did not cover the ${\rm H\beta}$ and [O iii] lines,
preventing us from conducting a comparable analysis.

In this paper, we present a comprehensive analysis of the properties of JD1
based on new observations by JWST. We describe our data on JD1 in Section 2.
Section 3 derives the H$\beta$ based star formation rate. Section 4 is devoted
to deriving an estimate for dust extinction and presence of dust as well as
constraints on the electron density. Section 5 derives the metallicity of JD1
through the direct method based on auroral lines and discusses implications
from the other lines. Section 6 discusses the non-solar abundance ratios for
JD1. The star formation history and stellar mass are described in Section 7.
Section 8 discusses our conclusions.

Throughout the paper, we adopt the AB magnitude system (Oke & Gunn, 1983;
Fukugita et al., 1996), cosmological parameters of $\Omega_{m}=0.3$,
$\Omega_{\Lambda}=0.7$, $H_{0}=70\,{\rm km~{}s^{-1}}\,{\rm Mpc}^{-1}$, and the
Chabrier (2003) initial mass function. We express all measurements as a
function of $\mu/10$ where relevant.

##  2 Data and Analyses

###  2.1 NIRCam photometry

NIRCam imaging observations (GTO #1199) were executed on June 6 – 8, with six
filters configured (F090W, F115W, F150W, F200W, F277W, F356W, F444W), with
$\sim 1.1$ hrs exposure each. We conduct photometry on the newly taken NIRCam
images, along with the archival HST images that were originally taken in
several HST programs (CLASH, HFF, GLASS, BUFFALO; Postman et al., 2012; Treu
et al., 2015; Lotz et al., 2017; Kelly et al., 2018; Steinhardt et al., 2020).
We follow the same procedure presented in Morishita & Stiavelli (2023) for the
image reduction and photometry. We hereby provide a high level description of
our workflow: raw NIRCam images are reduced by using the official jwst
pipeline (ver1.10.0, with context pmap #1069 Bushouse et al., 2023), with
several customized steps included to effectively remove artifacts and improve
cosmic ray rejection. The final drizzled images are aligned to the WCS of
GAIA. Source fluxes are calculated by using SExtractor (Bertin & Arnouts,
1996) in PSF-matched images to the psf of F444W, in an aperture of
$r=0.\\!^{\prime\prime}16$, reaching to $5\,\sigma$ limiting magnitude of
$\sim 27.9$–28.8 mag.

The NIRCam images of JD1 (see Figure 1) reveal the presence of three main
components, two of which are included in our NIRSpec MSA slits (Sec. 2.2).
This means that geometry might play a significant role in the interpretation
of the observations. However, we do not observe significant color differences
between these components and therefore in our following analysis we will not
consider them separately.

###  2.2 NIRSpec MSA spectroscopy

Our NIRSpec observations were executed over four different visits. In the
first two visits, #20 and #22 respectively for G235M and G395M, were taken at
PA=257.766. In the second group of visits, the same grating pair was used, but
with a slightly different PA, namely PA=259.660 for visit #21/G235M and
PA=256.766 for visit #23/G395M. Slightly offset pointings ensure that roughly
the same area of JD1 is covered by all exposures.

We reduce the MSA spectra using msaexp222https://github.com/gbrammer/msaexp
(ver0.6.13), following the procedure presented in Morishita et al. (2023a,
also, Morishita et al., in prep). The two-dimensional sky background is
estimated by nodding the stacked spectrum for $6$ pixels and then it is
subtracted. The one-dimensional spectrum is optimally extracted (Horne, 1986)
by using the one-dimensional source profile derived from the 2-dimensional
stacked spectrum as weight. For the extracted 1-dimensional spectrum, we fit
each line of interest with a Gaussian, after subtracting the underlying
continuum spectrum, estimated by scaling the best-fit SED template (see Sec.
7). We note that our flux measurements include a small correction for
absorption in the Balmer series, which are inferred by the stellar template of
the best-fit SED model. The correction is 1% for ${\rm H\beta}$, 3% for ${\rm
H\gamma}$, and 4% for ${\rm H\delta}$. For each line, the total flux is
estimated by integrating the flux over the wavelength range of 2$\times$FWHM,
the latter derived from the Gaussian fit. For [O iii]-doublet, we fix the
ratio to 1:3 and adopt a single parameter for the widths of both lines.
Lastly, we determined an aperture correction by scaling the difference between
the continuum measured around individual spectral lines and the one inferred
by the best-fit SED template. We find that this correction is well described
by the slit-loss correction one derives for perfectly centered source (Jdox,
2016, NIRSpec MOS Operations Slit Losses) rescaled by the factor $\sim 2.36$
we find for ${\rm H\beta}$ and [O iii] . This correction is applied to
measured line fluxes. However, most of our results are relatively unaffected
by this wavelength dependent aperture correction. The aperture correction
allows us to apply measurements such as the Star Formation Rate (SFR) to the
whole galaxy. In what follows, when necessary we adopt the redshift $z=9.114$
obtained for the ${\rm H\beta}$+[O iii] lines.

We have checked that the line ratios obtained in all visits are consistent.
Therefore, in the following we coadd the spectra and focus our analysis on the
combined spectra. The measured line fluxes are reported in Table 1.

Figure 1:  Stamp images of MACS1149-JD1 in HST F160W and NIRCam filters
($2.\\!^{\prime\prime}4\times 2.\\!^{\prime\prime}4$). The MSA slits of the
four visits (#20–23) are overlaid in the F277W stamp image. The pseudo 2-color
image using the F356W (blue) and F444W (red) filters, which corresponds to the
Balmer break, is shown (right bottom).  Figure 2:  ($Left$): Spectral energy
distribution of JD1 inferred by gsf (orange solid line; Sec. 7). Observed
NIRCam + HST photometric fluxes (red circles; inverted triangles for
$2\,\sigma$-upper limits) and NIRSpec MSA G235M+G395M spectra (black lines)
are included in the fit. Note that the observed fluxes are presented with
$2\,\sigma$ uncertainties and not corrected for magnification. ($Right$):
Zoom-in plots of the continuum-subtracted spectra around strong emission
lines. Fitted Gaussian profiles are shown (orange solid lines).

##  3 Star formation rate

The measured flux in ${\rm H\beta}$ gives us an estimate of the star formation
rate in JD1. Following the same approach as Heintz et al. (2023), we assume
the Case B ratio for $f_{\rm H\alpha/H\beta}=2.80$ for $T_{e}=1.6\times
10^{4}$ K as derived from the direct analysis described in Section 5. We
derive the star formation rate as:

${\rm SFR_{H\beta}}=5.5\times 10^{-42}L_{\rm H\beta}\times f_{\rm
H\alpha/H\beta}\,{\rm(erg/s)},$ (1)

From the measured ${\rm H\beta}$ flux of $3.7_{-0.1}^{+0.1}\times 10^{-18}$
erg s-1 cm-2 and assuming a gravitational magnification factor $\mu=10$
(Hashimoto et al., 2018)333Grillo et al. (2016) model predicts a median
magnification of $12^{+11}_{-5}$ at the position of JD1, where the $\pm
1\sigma$ values are obtained by extracting 200 random sets of parameter values
from the MCMC model chains. , we obtain SFR${}_{\rm
H\beta}=5.9_{-0.2}^{+0.2}\,(10/\mu)$ M⊙ yr-1. Note that this is the aperture-
corrected value of the SFR (Sec. 2.2). The estimated ${\rm H\beta}$-based SFR
is consistent with the one derived with [O iii] 88 $\mu$m
($4.2_{-1.1}^{+0.8}\,(10/\mu)\,{\rm M_{\odot}\,yr^{-1}}$; Hashimoto et al.,
2018). Also, the estimate is roughly consistent with the one from our SED
analysis, $\sim 3.2\,(10/\mu)\,{\rm M_{\odot}\,yr^{-1}}$, derived from the
rest-frame UV luminosity by following Morishita et al. (2023b).

Given the large uncertainties in the magnification values at the location of
JD1 (e.g. Grillo et al., 2016; Finney et al., 2018) and the relative
insensitivity of our results on the specific magnification we are expressing
our results as a function of $\mu/10$ rather than adopting a specific value.

##  4 Dust content and electron density

We have attempted to determine the presence of dust by looking at the Balmer
decrement. We measure ${\rm H\gamma/H\beta}=0.46\pm 0.03$ as compared to the
Case B recombination value of 0.47 for $T_{e}=1.6\times 10^{4}$ K. This value
would suggest a small $A_{V}=0.12$ but is also compatible with no dust. We
also find ${\rm H\delta/H\beta}=0.33\pm 0.03$ as opposed to the Case B value
of 0.26. The measured value is incompatible at 2.3+$\sigma$ with dust. These
conclusions are supported also by analysing both pointings separately.

In Figure 3, we show the measured values of ${\rm H\gamma/H\beta}$ and ${\rm
H\delta/H\beta}$ (with their 1-$\sigma$ error band as a function of the
electron density $n_{e}$ for a number of Cloudy (Ferland et al., 2017) models
that we have run as well as for Gutkin et al. (2016) models. The value we
measure is incompatible with the presence of dust for gas temperatures below
$2\times 10^{4}$ K. The absence of significant amounts of dust is also in
agreement with the non-detection of FIR continuum for this object (Hashimoto
et al., 2018; Tokuoka et al., 2022).

Figure 3:  Ratio of ${\rm H\gamma/H\beta}$ (lower panel) and ${\rm
H\delta/H\beta}$ (upper panel) as a function of the electron density. The red
dashed line shows the measured value of the ratio for JD1 with the dotted
lines showing the 1$\sigma$ error bars. The green dashed line shows the Case B
value of the ratio for $10^{4}$ K with the dotted lines showing the $5\times
10^{3}$ K and $2\times 10^{4}$ K Case B values. Black squares represent a
multitude of Cloudy models with varying properties, while the blue squares
show the values for the ratio in Gutkin et al. (2016) models.

An intriguing possibility arising from Figure 3 is that JD1 might be
characterized by a higher value of the electron density than the typically
assumed values in the range 100-1000. Unfortunately, our spectra cannot
resolve the [OII] doublet and therefore do not allow us to directly probe the
electron density. As an alternative, we can derive an estimate for the
electron density from the [OIII]$\lambda 88\micron/{\rm[OIII]}\lambda 5007$
ratio. This approach has been adopted by, e.g., Fujimoto et al. (2022) for the
galaxy ERO S04590 at $z=8.5$. In order to compare the ALMA 88$\micron$
measurement with our [OIII]$\lambda\lambda 4959,5007$ measurements, we need to
apply a significant aperture correction which will entail a relatively large
uncertainty. We have determined an aperture correction using two different
methods. The first approach is to assume that the ALMA flux is distributed
across an area similar to the NIRCam image of JD1, with which we have
estimated the aperture correction factor in Sec. 2.2 (i.e. 2.36).
Alternatively, we can derive an aperture correction by comparing the SFR
derived from ALMA [OIII]$\lambda 88\micron$ to that derived from ${\rm
H\beta}$ (Sec. 3), to find an aperture correction of 1.69. 444A magnification
gradient across JD1 would introduce a difference between the ALMA and the
MIRSpec aperture. In this paper we ignore this effect. In the following we
will adopt the average of these two values and adopt as error their semi-
difference, i.e., $2.02\pm 0.33$.

Armed with an estimate of the ALMA 88$\micron$ emission, we can derive the
${\rm[OIII]\lambda 88\micron/[OIII]\lambda\lambda}$4959,5007 ratio as
$0.055\pm 0.012\pm 0.009$, where the first error is the measurement one and
the second the aperture correction one. We adopt the same approach as Fujimoto
et al. (2022) except by using Cloudy models instead of the PyNeb ones (which
gives very similar results). Our Cloudy models (see Section 3.2 of Oesch et
al., 2007) are based on a wide variety of electron densities, and use both a
range of blackbody temperatures and a range of constant star formation rate of
various metallicities modelled by GALAXEV (Bruzual & Charlot, 2003). Our
results are shown in Figure 4. For JD1 we find $\log
n_{e}=2.60^{+0.25}_{-0.27}$, or $n_{e}\simeq 400$. As a consistency check, for
ERO S04590 we recover a value intermediate between 100 and 300 as found by
Fujimoto et al. (2022).

On this basis, in the following we assume that JD1 has a density of
$n_{e}\simeq 400$ and no dust.

Figure 4:  Ratio of $\log{\rm([OIII]\lambda 88\micron/[OIII]\lambda 5007)}$ as
a function of the electron temperature for a set of our Cloudy models.
Triangles are for $n_{e}=100$, squares for $n_{e}=300$, stars for
$n_{e}=1000$. The red square with an error bar is for JD1 and the blue square
for ERO S04590. The inset shows the change of $\log{(\rm[OIII]\lambda
88\micron/[OIII]\lambda 5007)}$ as a function of electron density for an
electron temperature of 16,000K. The red lines give the measured value for JD1
and its errors and the blue lines highlight the inferred values of the
electron density.

##  5 Emission line analysis

In the following, we will adopt the standard notation for line ratios, namely:

${\rm R23=\log(([OIII]\lambda\lambda 4959,5007+[OII]\lambda\lambda
3726,3729)/H\beta})$

${\rm O32=\log([OIII]\lambda 5007/[OII]\lambda\lambda 3726,3729})$

With robust measurements of the auroral line [OIII]$\lambda$4363 and a value
of the electron density, we can now apply the direct method to derive the
electron temperature and the oxygen metallicity. The values derived at the two
pointings are within the 1-$\sigma$ error bar and thus we will use the value
from the combined spectrum. Using Aller (1984) iterative method (see also
Izotov et al., 2006) and $n_{e}=400$, we derive $T_{e}({\rm O^{++}})=1.6\times
10^{4}$ K and infer $12+\log{\rm O^{++}/H}=7.88^{+0.04}_{-0.05}$. The error
bars quoted here are those related to the measurement error. Varying the
density between 100 and 1000 would contribute a 0.002 error. We have estimated
$T_{e}({\rm O^{+}})$ using a variety of methods (Izotov et al., 2006; Campbell
et al., 1986; Pilyugin & Grebel, 2016; Laseter et al., 2023; Sanders et al.,
2023) and find values similar to or lower than $T_{e}({\rm O^{++}})$. Using
these values we derive a contribution from ${\rm O^{+}}$ to metallicity
$12+\log{\rm O^{+}/H}=6.39-6.55$ which is in any case small compared to the
${\rm O^{++}}$ contribution. For the total Oxygen metallicity, we adopt
$12+\log{\rm(O/H)}=7.90^{+0.04}_{-0.05}$. We have also re-derived the direct
metallicity for ERO S04590 and found values in agreement with the published
ones.

Using the methods outlined in Izotov et al. (2006) we also derive an estimate
for the Neon abundance $\log({\rm Ne/H})=-5.03\pm 0.04$. We correct the
measurement with the Ionization Correction Factor (ICF) prescription by Amayo
et al. (2021), to find $\log({\rm Ne/H})=-5.02\pm 0.04$ or $\log({\rm
Ne/O})=-0.69$. Our measurements suggest that JD1 is under-abundant in Neon
compared to solar by $\Delta\log({\rm Ne/O})\simeq-0.23$.

We can also derive a Carbon over Oxygen ratio using the direct method (Aller,
1984; Izotov & Thuan, 1999). Here we find $\log({\rm C^{++}}/{\rm
O^{++}})=-1.09$. We can compute an ionization correction factor following Berg
et al. (2019). From the measured ${\rm O32}=1.535\pm 0.025$ and using the
$Z=0.1Z_{\odot}$ fit, we find $\log U=-1.647\pm 0.026$ and a Carbon ICF of
$1.216\pm 0.013$. This gives us $\log({\rm C/O})=-1.08$, implying
$\Delta\log({\rm C/O})=-0.78$ or a C/O ratio of 0.16 of the solar value.

Figure 5 compares the metallicity and the R23 values we derive for JD1 with
those derived for other auroral line measurements from the literature (Laseter
et al., 2023; Sanders et al., 2023).

Figure 5:  Direct oxygen metallicity measurement vs the $R_{23}$ ratio for
JD1, ERO S04590 (blue) and other objects (cyan points) with auroral based
measurements (Nakajima et al., 2023; Laseter et al., 2023) For reference the
vertical color lines are for Gutkin et al. (2016) models with increasing
ionization parameter $U$ from bottom to top.

In order to explore strong line diagnostics, we considered Gutkin et al.
(2016) models and found that the best fitting models correspond to 0.1 solar
metallicity with $n_{e}=100$ and upper mass function cutoff $m_{\rm
up}=100M_{\odot}$ and the highest values of the ionizing parameter explored in
the models, $U=-1.5/-1$. This is in broad agreement with the direct
measurement of the Oxygen metallicity. The high level of ionization is also
hinted at by the very high ${\rm[OIII]\lambda 5007/H\beta}=10.03\pm 0.31$
which would correspond to the AGN-star formation boundary in the BPT (Baldwin
et al., 1981; Lamareille et al., 2009; Lamareille, 2010) diagram. Indeed, this
diagram appears insufficient to separate star formation from AGN
photoionization at $z\geq 6$ (Cameron et al., 2023b; Übler et al., 2023).
However, we do not see any evidence of a broad component in the Balmer lines.

The measurement of (C/O) for JD1 is consistent with Gutkin’s (C/O) $\sim 0.38$
solar models. Thus, the broad conclusions of the direct measurements, namely
that JD1 has sub-solar metallicity, high-ionization, and Carbon-
underabundance, could be derived from a strong emission line analysis, even
though the details might not be precisely the same.

##  6 On the Non-solar abundance ratio

We have seen that the direct method indicated that JD1 has C/O ratio of 0.16
the solar value. We should note the large difference in wavelength between
CIII]$\lambda 1909$ and ${\rm H\beta}$, and the fact that they fall on
different NIRSpec gratings. The measurement of (C/O) depends also on an
estimation of an ionization correction based on the O32 ratio which is itself
vulnerable to the presence of dust. This opens up the possibility of
uncertainties due to dust (i.e. an additional correction that we have not
applied) and uncertainties in the instrumental calibration. The latter,
however, should be less than 5 %. Acknowledging this potential source of
uncertainty, we now proceed to examine the implication of the derived under-
abundance of Carbon.

A similar under-abundance has been found in a galaxy at $z=6.23$ by Jones et
al. (2023). However, that galaxy has an overall lower metallicity than JD1,
which may make a very young age and enrichment dominated by core-collapse
supernovae less surprising. Comparing JD1 with the results by Arellano-Córdova
et al. (2022), looking at the top panels of their Figure 4, one notices that
JD1’s (C/O) is lower than what they find for their $z>7$ galaxies but is not
anomalous compared to $z\sim 2$ galaxies. The same is true for the (Ne/O)
value of JD1 is not observed locally for objects of its metallicity.

The under-abundance of Carbon exceeds the under-abundance of Neon (see Section
5). It should be noted that while a low C/O is a common feature of core-
collapse supernova yields (e.g., Nomoto et al., 2013), the same is not true
for (Ne/O), even though there are models where this is the case (e.g.,
Rauscher et al., 2002, model S20). Massive, Population III, PISN supernovae
can also lead to significantly under-abundant (Ne/O) (Heger & Woosley, 2010).

##  7 Star formation history and absence of old stellar populations

The sensitive photometry over 1–5 $\mu$m by NIRCam enables us to constrain the
stellar component in JD1. The F356W and F444W data points are critical, as
they cover, for the redshift of JD1, the rest-frame $3800$–$4200$ Å where the
Balmer break, a characteristic break for relatively older (i.e. B, A, and
F-type) stars, is located. Previous studies using Spitzer Ch1 and Ch2 reported
red color ($m_{\rm ch1}-m_{\rm ch2}\sim 0.9$ mag; Huang et al., 2016; Zheng et
al., 2017; Hoag et al., 2018), speculating the presence of such old
populations formed at $z\sim 15$ (Hashimoto et al., 2018). Here, using our
JWST data we report $m_{356}=25.76\pm 0.02$ and $m_{444}=25.63\pm 0.02$ i.e.
the absence of such characteristic features in JD1. As the F356W magnitude is
consistent with the previous IRAC Ch1 measurements, we suspect that image
confusion might have affected the Spitzer IRAC Ch2 measurements.

The NIRCam photometry reveals a blue color, with $m_{\rm F356W}-m_{\rm
F444W}=0.13\pm 0.01$ mag. This is considerably smaller than the previous
measurement. We note that the filter curves of the corresponding bands of the
two telescopes are only slightly different. For the redshift of JD1, [O
iii]$\lambda$5007 line is within the red edge of F444W, but a hair outside the
Spitzer Ch2. As such, the absence of strong color in F356W/F444W (despite the
strong [O iii] included in the redder filter) indicates the dominance of young
($<100$ Myr) stellar populations.

Our spectrum is deep enough to reveal the continuum, and thus allows us to
directly measure the Balmer break. Following the definition of Balogh et al.
(1999), we measure $D(4000)\,\sim 0.5$, meaning that the continuum at the blue
side is brighter than the red side. We also follow the procedure presented in
Curtis-Lake et al. (2023) and measure the strength of the Balmer break $0.5\pm
0.1$, supporting the dominance of young populations.

Lastly, we perform a joint SED analysis combining photometric and
spectroscopic data using gsf (Morishita et al., 2019). Briefly, we use fsps
(Conroy et al., 2009), with the default MIST isochrone (Choi et al., 2016) and
MILES stellar library (Falcón-Barroso et al., 2011). Both stellar- and gas-
phase metallicity are fixed to 1/5 the solar, and dust is set to zero
($A_{V}=0$), based on the results from our ISM analyses above. We adopt a
binned star formation history, which offers flexible inference on star
formation history, with a set of ages [0.001,0.003,0.01,0.03,0.1,0.3,0.5] /
Gyr. We find that JD1 experienced two distinct, major star formation phases in
the past — one $\sim 300$ Myr old ($z\sim 14$) and a more intense one 10 Myr
old. The presence of a more recent burst is consistent with the Carbon deficit
that we have found in our spectral line analysis. We note that the absence of
a significant amount of old stars results in a much smaller stellar mass in
JD1, by a factor of $\sim\times 6.8$ from previous measurements, namely a
stellar mass of $1.6_{-0.3}^{+0.3}\times 10^{8}(10/\mu){\rm M}_{\odot}$.
Combined with the SFR from Section 3 we find a specific star formation rate of
$37_{-7}^{+10}{\rm Gyr}^{-1}$.

To check for consistency among various assumptions in SED fitting, we perform
another SED analysis with BAGPIPES (Carnall et al., 2018; Hsiao et al., 2023)
using BPASS v2.2.1 stellar population models (Stanway & Eldridge, 2018) and
CLOUDY c17.03 photoionisation code (Ferland et al., 2017). We also fix the
metallicities to 0.15 solar and dust to null. We assume a smooth, Gaussian
process-based nonparametric SFH (Iyer et al., 2019). We set the SFH to be
controlled by four parameters: stellar mass, SFR, and two shape parameters,
which essentially divide the SFH into three lookback time intervals in which
the galaxy formed equal mass. The derived SFH agrees reasonably well with the
binned SFH from above. The resulting stellar mass is
$1.42_{-0.04}^{+0.05}\times 10^{8}(10/\mu){\rm M}_{\odot}$. This stellar mass
gives us a specific star formation rate of $40_{-2}^{+3}{\rm Gyr}^{-1}$.

Table 1:  Observed properties of JD1.

|
---|---
Redshift and Line Flux Measurements
$z^{\dagger}$ | $9.114\pm 0.001$
${\rm H\beta_{\lambda 4861}}$ | $375\pm 10$
${\rm H\gamma_{\lambda 4340}}$ | $174\pm 6$
${\rm H\delta_{\lambda 4102}}$ | $125\pm 6$
${\rm[OII]_{\lambda\lambda 3727,3729}}$ | $110\pm 5$
${\rm[OIII]_{\lambda 4363}}$ | $59\pm 5$
${\rm[OIII]_{\lambda\lambda 4959,5007}}$ | $3764\pm 14$
${\rm[NeIII]_{\lambda 3869}}$ | $154\pm 5$
${\rm CIII]_{\lambda\lambda 1907,1909}}$ | $139\pm 8$
Physical properties
$M_{\rm UV}/{\rm mag}$ | $-19.33_{-0.01}^{+0.01}-2.5\log(10/\mu)$
SFR${}_{\rm H\beta}/M_{\odot}\,{\rm yr^{-1}}$ | $5.9\pm 0.2\,(10/\mu)$
SFR${}_{\rm SED}/M_{\odot}\,{\rm yr^{-1}}$ | $3.37_{-0.07}^{+0.07}\,(10/\mu)$
$M_{*}/10^{8}\,M_{\odot}$ | $1.61_{-0.25}^{+0.25}(10/\mu)$
$\beta_{\rm UV}$ | $-2.20_{-0.01}^{+0.01}$
$A_{V}/{\rm mag}$ | [0.0]
$Z/{Z_{\odot}}$ | [0.2]

Note. — $\dagger$: Redshift measured for the ${\rm H\beta}$+[O iii]λλ4959,5007
lines. Flux measurements are corrected for aperture loss and expressed in
units of $10^{-21}\,(10/\mu)$ erg s-1 cm-2.

##  8 Discussion and Summary

We find that JD1 is an actively star forming galaxy characterized by
metallicity about 1/5 the solar value and with emission line ratios best
explained by a very strong ionizing continuum suggestive of a very young
stellar population. The youth of the object is also supported by the star
formation history we derived, showing a major burst $\sim 10$ Myrs ago, and by
our finding of a low (C/O) ratio suggestive of Oxygen enrichment by
supernovae, while intermediate mass stars have not yet had the time to
increase the Carbon fraction. Such indications of a very active, recent and
on-going star formation are somewhat surprising when coupled with the lack of
evidence for dust as indicated by the measured Balmer decrement as well as the
non-detection in the ALMA continuum reported in the literature. While geometry
could conspire to give rise to grey dust (see e.g. Witt & Gordon, 1996, 2000),
it is interesting that this object seems also qualitatively compatible with
the dust ejection scenario described by Ferrara et al. (2023, see also ).
Indeed, the value of specific star formation rate we measure exceeds the value
of $\sim 32{\rm Gyr}^{-1}$ that we derive from Eq. 3 of Ferrara et al. (2023).
It should be noted that the complexities of how supernova driven winds would
affect the observed C/O ratio would need to be evaluated in more detail (see,
e.g., Berg et al., 2019).

It is worth noticing that the (C/O) ratio of JD1, as well as its metallicity
and high ionization (as inferred from the O32 ratio) are comparable to what is
observed in some Green Peas galaxies at $z<0.3$ (Ravindranath et al., 2020;
Rhoads et al., 2023).

We have seen in Section 2.1 that JD1 is characterized by multiple components.
This could be an indication of recent interaction or merger. However, we see
no evidence of color gradients, and the line ratios at the two slightly offset
slit positions are compatible. Given the absence of a color gradient, it is
hard to assess how general our conclusions can be once one opens up geometry
and substructure. For completeness we have also considered two component
Cloudy models. The significant number of additional degrees of freedom
prevents us from obtaining a unique solution but our best two-component fits
share similar properties of hard ionizing flux, sometimes with AGN or an
additional stellar population with $8\times 10^{4}$ K of effective
temperature, and oxygen enriched models. Given the high level of degeneracy,
we do not consider it useful to discuss these solutions in greater detail
except to say that they are broadly compatible with the conclusions of single
component modeling.

While completing this paper we were sent a pre-publication copy of a paper on
JD1 by Bradač et al., we find their results broadly compatible with ours given
the differences in JWST data for the two papers. Furthermore, JD1 has been
observed using the NIRSpec IFU by another program (PID: 1262, PI: N.
Luetzgendorf) but we have currently no access to these data. It will be
interesting to see whether the spatial resolved spectroscopy modifies the
picture presented here and how a combined analysis of both data and of our
Cycle 3 observations (#4552) aiming at observing with high resolution the rest
frame UV of JD1 will affect the picture.

## Acknowledgements

This work is based in part on observations made with the NASA/ESA/CSA James
Webb Space Telescope. The data were obtained from the Mikulski Archive for
Space Telescopes at the Space Telescope Science Institute, which is operated
by the Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-03127 for JWST. The specific observations analyzed can be
accessed via https://doi.org/10.17909/2q9q-mw78 (catalog DOI:
10.17909/2q9q-mw78). These observations are associated with program JWST-
GTO1199. Support for program JWST-GTO1199 was provided by NASA through grant
80NSSC21K1294. We thank S. Charlot for sharing a broader set of emission lines
for the Gutkin et al. (2016) models. The authors wish to thank M. Bradač, R.
Ellis, A. Ferrara, T. Hashimoto, C. Leitherer, R. Maiolino, G. Roberts-
Borsani, S. Suyu for discussions. We thank the anonymous referee for comments
that helped improve the paper.

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