Now at ] Département de Physique, Université de Fribourg, Chemin du Musée 3,
CH-1700 Fribourg, Switzerland

# Tunable high-index photonic glasses

Lukas Schertel  Fachbereich Physik, Universität Konstanz, Universitätsstraße
10, D-78457 Konstanz, Germany  Physik-Insitut, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Ilona Wimmer
Fachbereich Physik, Universität Konstanz, Universitätsstraße 10, D-78457
Konstanz, Germany  Fachbereich Chemie, Universität Konstanz,
Universitätsstraße 10, D-78457 Konstanz, Germany  Patricia Besirske
Fachbereich Chemie, Universität Konstanz, Universitätsstraße 10, D-78457
Konstanz, Germany  Christof M. Aegerter  Physik-Insitut, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Georg Maret  Fachbereich
Physik, Universität Konstanz, Universitätsstraße 10, D-78457 Konstanz, Germany
Sebastian Polarz<EMAIL_ADDRESS>Fachbereich Chemie,
Universität Konstanz, Universitätsstraße 10, D-78457 Konstanz, Germany
Geoffroy J. Aubry<EMAIL_ADDRESS>[  Fachbereich Physik,
Universität Konstanz, Universitätsstraße 10, D-78457 Konstanz, Germany

(August 27, 2024)

###### Abstract

Materials with extreme photonic properties such as maximum diffuse
reflectance, high albedo, or tunable band gaps are essential in many current
and future photonic devices and coatings. While photonic crystals, periodic
anisotropic structures, are well established, their disordered counterparts,
photonic glasses (PGs), are less understood despite their most interesting
isotropic photonic properties. Here, we introduce a controlled high index
model PG system. It is made of monodisperse spherical TiO2 colloids to exploit
strongly resonant Mie scattering for optimal turbidity. We report spectrally
resolved combined measurements of turbidity and light energy velocity from
large monolithic crack-free samples. This material class reveals pronounced
resonances enabled by the possibility to tune both the refractive index of the
extremely low polydisperse constituents and their radius. All our results are
rationalized by a model based on the energy coherent potential approximation,
which is free of any fitting parameter. Surprisingly good quantitative
agreement is found even at high index and elevated packing fraction. This
class of PGs may be the key to optimized tunable photonic materials and also
central to understand fundamental questions such as isotropic structural
colors, random lasing or strong light localization in 3D.

##  I Introduction

The interaction of light with matter is of paramount importance for numerous
technologies. While many of them rely on electronic excitation (e.g.,
photovoltaics, photocatalysis), it is also highly relevant to control the
propagation of light using materials structured at the scale of the optical
wavelength. So far, research on photonic structures mainly addressed photonic
crystals (PCs) which are periodic patterns of the refractive index in space
Yablonovitch (1987); *John1987; *Yablonovitch1989; Joannopoulos  _et al._
(1997); López (2003). The lattice structure of PCs implies strongly
anisotropic photonic properties such as wavelength-dependent Bragg-scattering,
angular-dependent structural coloration in biology Vukusic  _et al._ (1999);
*Vukusic2003, and anisotropic band gaps. A well-known strategy for the
generation of PCs involves the self-assembly of monodisperse colloidal
particles which naturally tend to crystallize von Freymann  _et al._ (2013).

Less attention has been paid to photonic glasses (PGs), the disordered
counterpart of PCs Ballato (2000); García  _et al._ (2007); *Garcia2010. PGs
are important because, in many applications such as white paints, coatings,
diffusors, or matrix materials for future photonic devices, a prime
requirement is the isotropy of photonic band gaps, transport, reflectance, and
transmittance. The strongest isotropic scattering materials nowadays,
commercial white paints, are a class of empirically optimized turbid
materials. Moreover, PGs can be used for the exploration of various optical
phenomena such as Anderson localization of light Anderson (1985), random
lasing Gaio  _et al._ (2015), and tunable isotropic structural colors Forster
_et al._ (2010); *Park2014; *Xiao2017; *Shang2018.

Thus, it is of prime interest to understand light transport in high index
disordered photonic materials quantitatively by using an appropriate transport
theory which can be bench-marked by experiments. Such analysis relies on model
materials which have to fulfill a large number of requirements at once. The
size of the colloidal particles should be of the order of the optical
wavelength ($400-800$ nm). To fully exploit the enhancement of scattering
efficiency due to Mie resonances the size-distribution used for the
construction of the materials should be as narrow as possible (polydispersity
$<5\%$) as, otherwise, polydispersity in size and shape smears out resonances
and spoils the targeted optical features García  _et al._ (2008). Although
high particle filling fractions (typically $>50\%$) have to be realized, the
occurrence of crystalline domains needs to be avoided. Because of the strong
tendency of monodisperse hard spheres to form colloidal crystals, it is very
difficult to generate a homogeneous glassy structure. In addition, optical
experiments necessitate macroscopic homogeneity of the material, i.e., crack-
free monoliths, because shortcuts (large voids in which the photons propagate
ballistically) have to be avoided. The material should have a minimum optical
absorption in the visible region of the spectrum, be it intrinsic or due to
impurities. Because the scattering efficiency increases with the index
contrast, compounds with the highest possible refractive index (e.g., titanium
dioxide (TiO2); $n_{\mathrm{rutile}}=2.7$) should be embedded in a low index
matrix (e.g., air; $n_{\mathrm{air}}=1$). Accomplishing all these requirements
at once is of utmost difficulty and a task for materials science.

PGs macroscopic in size have been rarely reported so far. Only lower index
materials such as polystyrene (PS) García  _et al._ (2007, 2008); Sapienza
_et al._ (2007); Chen  _et al._ (2017a); Emoto and Fukuda (2012), poly(methyl
methacrylate) (PMMA) García  _et al._ (2007); Sapienza  _et al._ (2007), or
silica (SiO2) Espinha  _et al._ (2016); Montesdeoca  _et al._ (2016); Rezvani
Naraghi  _et al._ (2015) were studied. Attempts with higher refractive index
materials were done with not perfectly monodisperse zinc sulfide spheres (ZnS,
$n_{\mathrm{ZnS}}=2.4$) dispersed in deionized water or isopropanol Scholz
_et al._ (1998), or with irregularly shaped TiO2 particles Störzer  _et al._
(2006).

Resonant light transport behavior in monodisperse PG has been connected to
resonant multiple Mie scattering García  _et al._ (2008); Sapienza  _et al._
(2007) but no quantitative description of this connection was given. In a
recent publication, we provided a model able to quantitatively describe light
transport in densely packed PGs made of PS spheres Aubry  _et al._ (2017). The
presented model of the transport mean free path $\ell^{\star}$, the important
scattering quantity in multiple light-scattering quantifying inverse
turbidity, uses the energy coherent potential approximation (ECPA) for the
effective refractive index to account for near field coupling and structural
correlations Busch and Soukoulis (1995); *Busch1996. In this study Aubry  _et
al._ (2017), the model was tested against ab initio numerical simulations,
earlier experimental data obtained from transmission experiments García  _et
al._ (2008), and experimental results from spectrally resolved coherent
backscattering experiments on specially synthesized PS ($n_{\mathrm{PS}}=1.6$)
colloidal glasses. In addition, as pointed out recently van Tiggelen  _et al._
(2017), it is difficult to distinguish between situations where the photon
diffusion constant $D$ is small due to Anderson localization effects leading
to small $\ell^{\star}$, or due to a small transport energy velocity
$v_{\mathrm{E}}$ related to Mie resonances. This long-standing issue of the
effect of resonant scattering behavior on dynamic scattering properties—i.e.,
the photon diffusion constant $D$—and static scattering properties—i.e., the
mean-free-path $\ell^{\star}$—awaits experimental clarification.

In this paper we first describe the controlled preparation of crack-free
monolithic high-index colloidal TiO2 PG samples at various refractive indices,
particle sizes, and polydispersities. We then determine their turbidities and
energy velocities over the entire visible spectrum and compare them to
empirically optimized commercial white paints. Very good overall agreement
between measurements and our theoretical model is found.

##  II Fabrication of colloidal photonic glasses

The first step is the synthesis of monodisperse TiO2 colloids. Several reviews
have already addressed the preparation of TiO2 nanoparticles Chen and Caruso
(2013); Xiang and Zhao (2017). Spherical particles made of amorphous titania
have been obtained via a sol-gel route Barringer and Bowen (1982); Jiang  _et
al._ (2003), which is similar to the well-known Stöber method developed for
silica Stöber  _et al._ (1968). Based on previous work from our own group
Eiden-Assmann  _et al._ (2004), we have prepared electrostatically stabilized
TiO2 particles first (see Supporting Information A.1). Despite numerous
attempts it was not possible to prepare samples with sufficiently low
polydispersity (see Fig. 1(a,b)).

Figure 1: Scanning electron microscope (SEM) micrographs of TiO2 colloids. (a)
polydisperse sample (SD = 22%), (c) monodisperse sample (SD = 2.8%), scale
bars 1 $\mu$m. (b) Optimization of polydispersity: Gaussian fits (grey curves)
of the particle size distribution functions of the polydisperse sample
(orange) SD = 22% and the monodisperse sample (red) SD = 2.8%. (d) Adjustment
of the particle size by addition of deionized water (see Supporting
Information A.2).

For instance, the analysis of the sample containing particles with a mean
radius $r_{\mathrm{colloid}}=240$ nm shows a standard deviation (SD) of 22% in
the size distribution function (grey Gaussian fit to bright histogram in Fig.
1(b)).

The quality of the dispersions could be improved significantly by following a
protocol by Tanaka et al. on sterically stabilized TiO2 particles Tanaka  _et
al._ (2009) (see Supporting Information A.2). In agreement with these results,
we achieved $r_{\mathrm{colloid}}=206$ nm particles with a low SD value of
only 5.2%. We were able to improve the size distribution further by
optimization of the reaction parameters while keeping the average size of the
particles constant (see A.2). Finally, particles with $r=209$ nm and SD = 2.8%
were obtained (Fig. 1(c) and dark histogram in Fig. 1(b)). As expected, the
narrower the size distribution function becomes, the higher is the tendency to
form colloidal crystals. For the optical experiments, it is also important to
adjust the mean particle size. We managed to do so (Fig. 1(d) and Fig. S1) by
the variation of the amount of deionized water used in the synthesis protocol.
The mean particle size was also confirmed by dynamic light scattering.

Colloidal crystallization at high packing fraction was suppressed during
ultracentrifugation of the monodisperse TiO2 particles dispersions by
destabilization with Ca2+ (Fig. 2(a)).

Figure 2: (a) Preparation of TiO2 PG monoliths. SEM micrographs of particles
packings with CaCl2 solution added; (b) no addition, (c) 5 mM, (d) 10 mM. (e)
Photographic image and (f) SEM micrograph of the mesoscopic structure of the
final monolith (diameter $\sim 1$ cm) structurally stabilized by
polyacryamide. Scale bars for the micrographs: $2\ \mu$m. (g) PXRD patterns of
materials obtained after sintering at different temperatures. Reference
patterns of TiO2-anatase (blue bars) and rutile (mauve bars) are also shown.

This has been shown in previous papers Chen  _et al._ (2015); *Chen2017a; Chen
_et al._ (2017a) to be a suitable method for the realization of kinetically
disordered packings of PS-latex. It can be seen that without Ca2+,
crystallized zones are present (Fig. 2(b)). The addition of Ca2+ has a marked
influence and heavily disturbs the emergence of order. At a concentration of 5
mM CaCl2, the sample is entirely disordered with no crystalline areas (Fig.
2(c)). Higher salt concentrations can also be used, but the number of voids
increases (Fig. 2(d)), which lowers the filling fraction of the PG. The
resulting monoliths are brittle and difficult to handle. Removal of the
solvent (water) by drying leads to crack formation. Cracks are undesired
because they lead to optical shortcuts. We therefore stabilized the structure
of the monoliths by the in situ polymerization of N,N’-methylene-bis-
acrylamide. The resulting polymer is located in between the titania particles
and acts like a “glue” preserving the structure during drying (see Fig. 2(e,f)
and Fig. S4). The monoliths are crack-free. Because the polymer represents
only a minor fraction of the entire material and its refractive index
($n_{\mathrm{polymer}}=1.4$) is much lower than that of titania, the effect on
optical properties may be negligible. The filling fraction was estimated
experimentally to be $f\approx 0.3\pm 0.1$ by tracking the weight percent of
particles used and the volume of the sample. This value for $f$ is close to
its optimized value for strong scattering Pattelli  _et al._ (2018).

Titania prepared according to sol-gel methodologies is amorphous and should be
described as a titanium-oxo-hydroxo phase. To induce atomic crystallization to
anatase or rutile, the materials were sintered at higher temperatures.
According to thermogravimetric analysis (shown in Fig. S2(a)) the as-prepared
material loses mass in two steps. In the range $T=50-200^{\circ}$C, one sees
the removal of solvents and at $T_{\mathrm{max}}=401^{\circ}$C, surface bound
organics are removed and eventually dehydroxylation takes place Wu  _et al._
(2017). In the FT-IR spectra of a sample treated at $T=400^{\circ}$C, one sees
neither signal for organic compounds nor for remaining $-$OH groups (see Fig.
S2(b)). The treatment at higher temperature has the desired effect on the
crystallinity of the samples (Fig. 2(g)). Signals characteristic for
TiO2-anatase appear in powder x-ray diffraction (PXRD) at a temperature of
400∘C. We see the emergence of first peaks indicating rutile phase TiO2 at
500∘C. The transformation is finished at 600∘C, and the entire sample contains
only the rutile phase. The good purity of the materials is also documented by
UV-Vis spectra (see Fig. S2(c)) and pictures of the anatase and rutile
particles are shown in the Supporting Information Fig. S3. The PG monoliths
can now be prepared with particles made of amorphous TiO2, anatase-TiO2 and
rutile-TiO2 (see Fig. 2(f)). This provides a way to tune systematically the
refractive index of the photonic material ($n=2.0,2.5,2.7$).

##  III Resonant transport in high index photonic glasses

Optical turbidity measurements were performed for the samples described above.
For comparison, we measured reference samples consisting of dense packings of
commercially available TiO2 powders (classical white paints) and also compared
with our previous data from PS spheres Aubry  _et al._ (2017). This was done
to demonstrate the effect on the scattering behavior of both the refractive
index and the monodispersity of the scattering particles. The scattering
strength $\lambda_{0}/\ell^{\star}$ is obtained by analyzing the shape of the
coherent backscattering cone (CBC) Wolf and Maret (1985); *Albada1985 at
different incident wavelengths Aubry  _et al._ (2017) ($\lambda_{0}$ is the
incident optical wavelength in vacuum). The width of the CBC is proportional
to $1/k\ell^{\star}$ ($k=2\pi/\lambda_{0}$). Thus, the width of the CBC
increases dramatically for strongly scattering samples such as TiO2-based PGs.
Therefore, a large angular range needs to be covered in reflection for
determination of $\ell^{\star}$ of such highly scattering samples. We used a
CBC setup recording backscattering angles up to 60∘ as described in refs.
Gross  _et al._ (2007) (see Supporting Information D). A tunable laser system
(Fianium, WL-SC-400-8 and LLTF VIS) was used as a light source. For each
sample, a wavelength scan from 450 nm to 780 nm (limited by the used detectors
and optical components) was performed in 10 nm steps if not mentioned
differently.

In Fig. 3, we plot $\lambda_{0}/\ell^{\star}$ for different samples versus the
size ratio $r/\lambda_{0}$ with $r$ the mean particle radius.

Figure 3: Comparison of the experimental data (points) with the ECPA
scattering model (lines); $f$: filling fraction, $n$: refractive index, poly
refers to $5\%$ polydispersity taking into account in the model. (a) PGs
prepared from amorphous TiO2 spheres with different sizes and low (red) and
high (orange) polydispersity. (b) Data for PGs prepared from particles with
different refractive indices: Polystyrene (grey, data taken from ref. Aubry
_et al._ (2017)), amorphous-TiO2 (red), and anatase-TiO2 (blue). (c) Data for
two rutile-TiO2 based PGs: one prepared from low polydisperse particles
($r=170\pm 5$ nm, mauve), and one composed of strongly polydisperse commercial
TiO2 (R700, green) as a white paint reference. The dashed grey line indicates
the presumed critical value $k\ell^{\star}=1$ between diffusion and
localization.

The resonant behavior in all measured PGs leads to strong scattering at
certain $r/\lambda_{0}$ values. Note that in the whole paper, each material
has its own color (gray: PS spheres, orange/red: amorphous TiO2 spheres
polydisperse/monodisperse, blue: anatase spheres, mauve: rutile spheres,
green: R700). The experimental data are compared with a model developed by us
recently Aubry  _et al._ (2017), which is based on the calculation of the
scattering cross section, taking into account the Mie solution Bohren and
Huffman (1998) and the glass structure factor Percus and Yevick (1958); Fraden
and Maret (1990) (see Supporting Information E). Because of the close
proximity of the individual scatterers, considering the usual far field Mie
solution of a single scatterer with refractive index $n$ in a background with
refractive index $n_{0}$ does not properly describe the multiple scattering
behavior. At high packing fractions, due to the electromagnetic coupling and
to the positional correlations of neighboring particles, the single scattering
Mie resonances are shifted. One way to approximate this a priori complex
scattering is to renormalize $n_{0}$ to an effective refractive index
$n_{\mathrm{eff}}$, which is calculated using the energy-density coherent-
potential approximation (ECPA) introduced by Soukoulis et al. Busch and
Soukoulis (1995); *Busch1996 The key step to describe the effective medium is
to introduce a coated sphere as a basic scattering unit. The particle shell
has a refractive index $n_{0}$ and a thickness related to the average particle
distance in the glass and couples electromagnetically each scatterer with the
surrounding effective medium. The ECPA-based scattering model was already
tested successfully by us on PGs made of PS spheres Aubry  _et al._ (2017).
Unlike other models used so far, it predicts very well the resonant behavior
in low index PGs without any adjustable parameter. We now check the
predictions of this model on TiO2-based high-index PGs. Figure 3(a)
illustrates the influence of the monodispersity of the particles on the
scattering behavior of the PGs. All samples contain amorphous TiO2 spheres but
they differ regarding particle size and polydispersity. The orange data on the
figure shows the spectral measurements of the scattering strength of two PGs
prepared from TiO2 colloids (polydispersity $<5\%$) with $r=209\pm 8$ and
$240\pm 15$ nm, respectively. These data are compared to the ECPA model for
$n_{\mathrm{TiO_{2}}}=2.0$, a filling fraction $f=0.3$ and a polydispersity of
5%. There is good agreement between experimental data and our model. The
positions of the experimentally observed resonances overlap very well with the
predictions from the ECPA scattering model. The amplitudes of the observed
resonances are weaker than expected from the model, but the mean absolute
values match quite well. Note here that in the experiments the amplitude might
be somewhat lowered because of the spectral width of the light source (2.5
nm). Another reason is the polymer network that holds the particles together
which may reduce slightly the index contrast. The data as well as the model
show that for larger $r/\lambda_{0}$ values, the resonances smear out. This is
explained by the residual polydispersity Aubry  _et al._ (2017). The latter
effect is more striking when using a less monodisperse titania dispersions
($r=229\pm 27$, $509\pm 66$ nm) to synthesize polydisperse PGs. Because
ordered structures do not form for such high polydispersity (Fig. 1(b)) these
PGs were prepared by simple compression of the synthesized and dried powders
(see Supporting Information B.2). The red data in Fig. 3(a) show again a
direct comparison between experimental data and the corresponding ECPA model.
As expected, in this case resonances are absent over the entire
$r/\lambda_{0}$ range; however, the data follow the trend of the model.

In the following, we focus on monodisperse samples and analyze the effect of
the refractive index (Fig. 3(b)). The resonances of the material based on
amorphous TiO2 (red) are enhanced at least by a factor two compared to
particles with lower refractive index (PS, grey) (note the logarithmic plot of
the scale). The refractive index is higher for anatase-TiO2 (blue) spheres,
and the resulting PGs show an even more enhanced scattering and more defined
resonances. Again, there is a very good agreement between the experimental
data and the ECPA model.

Rutile-TiO2 particles have the highest refractive index. Because commercial
white paints contain irregular-shaped rutile-TiO2 particles, it is worth using
these as a reference system Störzer  _et al._ (2006); Aegerter  _et al._
(2006); Sperling  _et al._ (2013). PGs were obtained by compacting
commercially available TiO2 particles to tablets with $f\approx 0.5$ (see
B.2). The powders have high polydispersities in the range $25-47\%$ with a
mean size varying from 170 to 283 nm (DuPont R700; see B.2). The data are
plotted in Fig. 3(c) versus the relative size parameter
$rn_{\mathrm{TiO_{2}}}/\lambda_{0}$. This index correction is performed as
these data were obtained by varying the wavelength while the scattering model
is calculated for a fixed wavelength (590 nm) varying the particle radius. As
the refractive index is wavelength dependent, the used wavelength in the
experiments are weighted by the wavelength dependent $n$ taken from ref.
Devore (1951). Note that this correction is small and was neglected for PS,
amorphous, and anatase TiO2 due to a lack of literature data. The scattering
strength of the commercial rutile powders follows the trend of the ECPA
scattering model over a large range of $rn_{\mathrm{TiO_{2}}}/\lambda_{0}$
from 0.5 to 1.0 (green points in Fig. 3(c)). Due to the arguments given above,
resonances are absent. Note that the amplitude of the scattering strength is
higher than expected by the model mean value. Either the model slightly
underestimates the scattering strength or the random shape of the particles
(different from a PG made out of polydisperse spheres) leads to stronger
scattering.

We now discuss the scattering features of the optimum sample (monodisperse
rutile-TiO2\- based PGs). The mauve points in Fig. 3(c) show the results for
$r=170\pm 5$ nm. The scattering behavior of the monodisperse PG follows
remarkably well the predictions from the model without any fit parameter. Mean
free path values of $\ell^{\star}\sim 230-240$ nm are reached corresponding to
$\lambda_{0}/\ell^{\star}\sim 3.13$. This is comparable to the values reported
for the strongest scattering commercial white paints Störzer  _et al._ (2006)
where no indications of Anderson localization were found Sperling  _et al._
(2016). Note that these values are still larger than the critical value
$k\ell^{\star}\approx 1$ were the transition to Anderson localization is
expected from the Ioffe-Regel criterion (Fig. 3(c); dashed line) Ioffe and
Regel (1960). Therefore, an important outcome of our investigation is that the
regime of light localization cannot be reached even with the optimized
spherical scattering systems presented here.

##  IV Energy transport velocity

Yet another unsolved issue in the quest for Anderson localization is the
respective role of the static time averaged scattering strength
$1/\ell^{\star}$ and the dynamic diffusion constant, which depends—in addition
to $\ell^{\star}$—on the energy velocity $v_{\mathrm{E}}$ of the light wave
through $D=v_{\mathrm{E}}\ell^{\star}/3$ in a multiple scattering sample.
$v_{\mathrm{E}}$ can thus be calculated from $D$, which is accessible via
photon time of flight (ToF) measurements Drake and Genack (1989); Sperling
_et al._ (2016). How $v_{\mathrm{E}}$ is affected by resonant scattering,
which of these two quantities ($v_{\mathrm{E}}$ and $\ell^{\star}$) dominates
$D$ and what controls the Anderson transition is not fully understood in the
literature Busch and Soukoulis (1995); van Tiggelen  _et al._ (2017). We
therefore performed measurements of ToF combined with $\ell^{\star}$ to
characterize the resonant transport behavior in a PG formed by amorphous TiO2
spheres with $r=228$ nm. ToF is done by sending a short laser pulse onto the
sample and by measuring the transmitted intensity using a time-resolved
photodetector. For a diffusive sample, the diffusion constant $D$ is extracted
by fitting such ToF curves with diffusion theory, provided the sample
thickness $L$ is known Drake and Genack (1989); Sperling  _et al._ (2016).

Figure 4(a) shows measurements of the diffusion constant $D$ versus the
wavelength $\lambda_{0}$.

Figure 4: Spectral measurements of (a) the diffusion constant $D$, (b) the
transport mean free path $\ell^{\star}$, and (c) the calculated energy
velocity $v_{\mathrm{E}}$, for (red, left scales) an amorphous TiO2 PG with
particle radius $r=228$ nm and a sample thickness $L=0.5$ mm, and (green,
right scales) for a commercial rutile TiO2 powder having a sample thickness
$L=0.4$ mm (DuPont R700). The dashed lines in (c) indicate the energy velocity
calculated by the Maxwell-Garnett effective refractive index (note here the
common left and right scales).

For the amorphous PG (red points), $D$ has a minimum at exactly the same
position where $\ell^{\star}$ has its minimal value (Fig. 4(b)). The
calculated values of $v_{\mathrm{E}}$ (Fig. 4(c)) can be compared to
$v_{\mathrm{E}}=c/n_{\mathrm{MG}}\simeq 2.3\times 10^{8}$ ms-1 (dashed red
line), with $c$ the velocity of light and $n_{\mathrm{MG}}$ the Maxwell-
Garnett effective refractive index Garnett (1904). Accounting for the error in
the measurement of the sample thickness ($L=0.5\pm 0.2$ mm), the observed
energy velocity agrees quite well with that value. Though the data are
somewhat noisy, a minimum in $v_{\mathrm{E}}$ can be identified close to the
minimum in $\ell^{\star}$. Therefore, the behavior observed for
$\ell^{\star}$, $D$ and $v_{\mathrm{E}}$ are strongly correlated.

To reveal the origin of the resonances, we also measured commercial TiO2
powders to study the dynamic transport behavior in samples made out of
randomly shaped, polydisperse particles for comparison. The green points in
Fig. 4 (R700, $L=0.40\pm 0.05$ mm) show one example illustrating that the
diffusion constant $D$ decreases with decreasing incident wavelength. As
expected from the ill-defined shape of the particles, no resonant behavior is
observed. The energy velocity $v_{\mathrm{E}}$ decreases monotonically with
increasing size ratio in contrast to the behavior of $\ell^{\star}$. Moreover,
$v_{\mathrm{E}}$ is lower than the value expected from
$v_{\mathrm{E}}=c/n_{\mathrm{MG}}\sim 1.87\times 10^{8}$ m/s, using $f=0.5$ in
the calculation of $n_{\mathrm{MG}}$ (green dashed line in Fig. 4(c)). All
this shows the complexity of the scattering behavior in the case of multiple
scattering from materials composed of randomly shaped, densely packed
particles, which is clearly beyond the applicability of the model presented
here.

##  V Conclusion

Understanding the optical properties of white paints and appreciating the
potential of PGs relies on materials suitable as model systems for
quantitative comparison of a manageable theoretical model with experiments. In
this paper we describe the preparation of highly monodisperse titania colloids
with adjustable refractive index by controlling the particle’s crystallinity
(amorphous vs anatase vs rutile). A centrifugation method combined with
colloidal destabilization was successful in suppressing any colloidal
crystallization and large monolithic PG samples were obtained. These allowed
for accurate measurements of the scattering strength and of the diffusion
constants over a wide range of particle size parameters covering the whole
visible spectrum. We studied the influence of colloidal particle size,
polydispersity, and refractive index in comparison to our recently proposed
model for resonant light transport in densely packed sphere systems Aubry  _et
al._ (2017). Good agreement of data and model was found, both for the
amplitudes and spectral resonance positions of turbidity and energy velocity.
This paper therefore paves the way toward a class of controlled and
application-optimized photonic materials where isotropy is elemental.

Even for close-to-perfect PGs made from monodisperse spherical rutile-TiO2,
light transport is dictated by diffusion: the conditions for reaching strong
localization of light are not met. To reach this goal, other disordered
photonic structures than just randomly packed spheres have to be invented. One
recently proposed way is to use hollow or coated spheres Escalante and
Skipetrov (2017) to hinder the propagation of longitudinal evanescent fields
responsible for new transport channels Skipetrov and Sokolov (2014); Rezvani
Naraghi  _et al._ (2015). Another possible direction may be suggested by
nature where, despite the low refractive index material in the exoskeleton of
some white beetles (chitin, $n=1.6$), light scattering is optimized in an
unprecedented way Vukusic  _et al._ (2007); *Burresi2014. The Cyphocilus
beetle uses elongated scatterers to achieve highly scattering material,
similar to recently published work on artificial fibrillar networks Toivonen
_et al._ (2018); *Syurik2018. One could imagine reaching Anderson localization
of light with a photonic structure similar to what is found in the white
beetle, but made of high-index materials such as rutile-TiO2.

###### Acknowledgements.

The samples synthesis was developed and carried out by I.W., while the light-
scattering experiments were done by L.S. We thank the Deutsche
Forschungsgemeinschaft for funding within Project No. Ae94/2-4, Ma817/8 and
within the framework of the Collaborative Research Centre SFB-1214, Project
No. A5 and Particle Analysis Center (PAC). The Schweizer Nationalfonds Grant
No. SNF 200020M-162846 is acknowledged as well as the Center for Applied
Photonics (CAP) and the Zukunftskolleg (Independent Research Starting Grant)
of the Universität Konstanz.

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Supporting Information

The Supporting Information contains information on the TiO2 colloid
preparation and characterization, how photonic glasses are prepared from TiO2
colloids and commercial TiO2 powders, details on the coherent backscattering
measurements and on the computation of the scattering strength.

##  Appendix A TiO2 colloid preparation

###  A.1 Electrostatic stabilized particles

For the preparation of the titanium dioxide (TiO2) particles, 20 mL ethanol
and $80-100\,\mu$L of a 0.1 M KCl solution were mixed together and stirred for
several minutes. 0.6 mL of the Ti(O iPr)4 precursor was added to the solution
with 1.5 mL per minute. After 4 hours, the reaction was stopped and the
particles were collected by filtration, washed with ethanol, and dried under
ambient conditions at 60∘C.

###  A.2 Steric stabilized particles

In a typical synthesis, 1 g of dodecylamine was dissolved in a mixture of 106
mL of methanol and 42 mL of acetonitrile. Afterwards, 0.2 to 0.7 mL deionized
water was added, and the solution was stirred for 10 minutes. 1 mL of the Ti(O
iPr)4 precursor was added at once to the solution. After stirring for 8 hours,
the obtained particles were washed 3 times with methanol and for further
analysis dried at 60∘C under ambient conditions. The size of the colloids were
determined by measuring the radius of 200 particles on Scanning electron
microscope (SEM) pictures (Fig. S1).

(a) Synthesis with 0.2 mL H2O: $\left<r\right>=296.8$ nm. Scale bar in the SEM
micrograph: 1 $\mu$m.

(b) Synthesis with 0.5 mL H2O: $\left<r\right>=228.0$ nm. Scale bar in the SEM
micrograph: 2 $\mu$m.

(c) Synthesis with 0.7 mL H2O: $\left<r\right>=204.9$ nm. Scale bar in the SEM
micrograph: 1 $\mu$m.

Figure S1: Variation of mean particle size of TiO2 particles for different
amounts of H2O. Left: SEM micrographs; right: histograms of 200 counted
particles. Bin size in the histograms: 5 nm. The fits are Gaussian fits.

The thermogravimetric analysis (TGA), Fourier-transform infrared (FT-IR)
spectra of a sample treated at different temperatures and UV-Vis spectra are
plotted in Fig. S2. Prior to the heat treatment, different features can be
observed in the FT-IR spectra. The broad signal with its maximum at
$\nu=3200\,\mathrm{cm}^{-1}$ corresponds to the oxygen-hydrogen vibration of
the remaining ethanol and the nitrogen-hydrogen vibration of the dodecylamine.
The two sharp bands at $\nu=2857\,\textrm{cm}^{-1}$ and
$\nu=2925\,\textrm{cm}^{-1}$ are due to the asymmetric and symmetric carbon-
hydrogen vibration. For increasing temperatures these bands are decreasing and
at 200∘C they completely disappear. At $\nu=1620\,\textrm{cm}^{-1}$ the Ti-O-H
stretching vibration is observable, but at 400∘C, it has completely vanished.

(a) TGA

(b) FT-IR

(c) UV-Vis

Figure S2: (a) TGA trace (black) and first derivative (blue). (b) FT-IR
spectra of samples treated at different temperatures. (c) UV-Vis diffuse
reflectance acquired in an integrating sphere containing either amorphous or
rutile TiO2.

Anatase particles were obtained through sintering at 400∘C for 4 hours while
sintering at 700∘C for 1 hour led to rutile particles (see main text Fig.
2(g)). Fig. S3 shows SEM picture of such anatase and rutile particles.

Figure S3: (left) Anatase and (right) rutile particles after sintering. (SEM
micrographs, scale bar 1 $\mu$m)

##  Appendix B Preparation of photonic glasses

###  B.1 Photonic glasses (PGs) of TiO2 colloids

For preparation of a free-standing monolith, 1 mL of the TiO2 dispersion, 0.36
mL of a 30 wt% acrylamide solution, 0.32 mL of a 2 wt%
N,N’-methylenbisacrylamide solution, 0.04 mL H2O and 0.08 mL of a 0.18 M CaCl2
solution were mixed in a glass vial with flat bottom. The solution was
centrifuged for 1 hour, 1000 revolutions per minute (rpm) and 25∘C. Afterwards
the supernatant solution was removed and 0.02 mL
N,N,N’,N-tetramethylethan-1,2diamine and 0.02 mL of a 10 wt% ammonium
persulfate solution were added to initiate the hydrogel’s polymerization. The
monoliths were dried at 80∘C for 1 hour to obtain a free-standing monolith.
Figure S4 shows a wide view of a typical sample.

Figure S4: Polymer-stabilized PG monolith of titania nanospheres (SEM
micrographs, scale bar 10 $\mu$m)

###  B.2 PGs of polydisperse TiO2 powders

Fig. S5 shows the shape of the polydisperse R-700 commercial TiO2 colloids.

Figure S5: Reference materials prepared by compressing commercial DuPont R700
TiO2 powders. (SEM micrographs, scale bars 1 $\mu$m)

To obtain densely packed powders in slab geometry, the polydisperse powders
are pressed to tablets in cylindrical shaped sample holders. A lid can be
screwed on the cylindrical holder before the powder is filled in the holder
and pressed with a stamp fixed by screws. This leads to filling fractions of
approximately $f=0.5$. The powders are pressed between two glass plates to
perform transmission experiments. One glass plate can be removed from the
holder for coherent backscattering cone (CBC) experiments to guarantee for a
flat sample/air surface to avoid additional reflections by the glass.

##  Appendix C Materials structure characterization

  * •

SEM measurements were performed using a Zeiss CrossBeam 1540XB.

  * •

Powder X-Ray Diffraction (PXRD) measurements were performed using a Bruker D8
DISCOVER with Cu Kα radiation.

  * •

TGA and Differential scanning calorimetry measurements were performed using a
Netzsch STA F3 Jupiter.

  * •

FT-IR spectra were measured with a PerkinElmer Spectrum 100 spectrometer with
an attenuated total reflection (ATR) unit.

  * •

Transmission electron microscope (TEM) measurements were performed using a
Jeol JEM 2200 FS.

  * •

UV-Vis measurements were performed using Varian Cary 100 Scan UV-Vis
Spectrometer with Ulbricht-sphere.

  * •

Preparative Ultracentrifugation was done using an Optima XL-I 70 of Beckman
Coulter. The rotor was a SW 55 Ti rotor.

##  Appendix D Coherent backscattering cone measurements

We used a coherent backscattering cone setup recording angles from
$-60^{\circ}$ to $60^{\circ}$. A white light laser (Fianium, WL-SC-400-8 )
with a total output power of $P\approx 8$ W in a wavelength range of
$400-2500$ nm was used as a light source. The white light pulses (repetition
rate of 80 MHz) are coupled in a spectral tunable filter (Fianium, LLTF VIS)
that uses two spectral overlapping Bragg gratings to tune the wavelength
between $400-1000$ nm with a spectral width of 2.5 nm and an average output of
$4\,\mathrm{mW}/\mathrm{nm}$. The beam is focused through an entrance hole in
the half circle mount (1.2 m in diameter) of the 256 photodiodes which are the
detectors Gross  _et al._ (2007). The beam is focused with the focal point at
the diodes position to minimize the non-measurable angles in the
backscattering direction. At small angles ($<9.75^{\circ}$) photodiode arrays
(Hamamatsu, S5668) are used for a higher resolution of $0.15^{\circ}$ at the
cone tip while at larger angles ($<19.55^{\circ}$) single photodiodes
(Hamamatsu, S4011) are used with a resolution of $0.7^{\circ}$. For even
larger angles ($<60^{\circ}$) the same diodes are positioned with a resolution
of $\approx 1^{\circ}$. The illuminating beam passes a circular polarizer
before impinging onto the sample. The filter wavelength range ($380-780$ nm)
limits the spectral measurements. Another circular polarizer foil in front of
the photo diodes filters single scattering events. The sample is mounted on a
rotating motor to average out the static speckle pattern during the
measurement. The illumination spot is 1 cm in diameter. A teflon sample is
used for calibration as it has such a narrow cone that only the incoherent
background is detected by the diodes.

##  Appendix E Computation of the scattering strength

As explained in our previous paper Aubry  _et al._ (2017), the transport mean
free path $\ell^{\star}$ is computed from

$\ell^{\star}=\frac{\ell_{\mathrm{s}}}{1-\langle\cos\theta\rangle}\overset{\text{sphere}}{=}\frac{1}{1-\langle\cos\theta\rangle}\frac{4\pi
r^{3}}{3f\sigma_{\mathrm{s}}}.$ (1)

where $\ell_{\mathrm{s}}$ is the photon scattering mean free path, $r$ is the
radius of the colloids forming the photonic glasses, $f$ is the filling
fraction of the colloids in the glass, and $\sigma_{\mathrm{s}}$ and
$\langle\cos\theta\rangle$ are computed from the standard formula Fraden and
Maret (1990)

$\sigma_{\mathrm{s}}=\frac{\pi}{k^{2}}\int_{0}^{\pi}F(\theta)S(\theta)\sin\theta\,\mathrm{d}\theta,$
(2) $\langle\cos\theta\rangle=\dfrac{\int_{0}^{\pi}\cos\theta
F(\theta)S(\theta)\sin\theta\,\mathrm{d}\theta}{\int_{0}^{\pi}F(\theta)S(\theta)\sin\theta\,\mathrm{d}\theta},$
(3)

where $k=2\pi/\lambda$ with $\lambda$ the wavelength of the light in the
surrounding medium, $F(\theta)$ is the Mie form factor Bohren and Huffman
(1998) and $S(q)$ is the Percus-Yevick structure factor Percus and Yevick
(1958) of the glass (with $q=2k\sin\theta/2$). In these formula, the near
field effects are taken into account by replacing all wave vectors $k$ by
$k_{\mathrm{eff}}=2\pi n_{\mathrm{eff}}/\lambda_{0}$ with $n_{\mathrm{eff}}$
the energy coherent potential approximation (ECPA) refractive index Busch and
Soukoulis (1995); *Busch1996. The ECPA effective refractive index depends on
$r$, $f$ and on the refractive indices of the particles $n_{\mathrm{p}}$ and
of the surrounding matrix (air, $n_{0}$), and is computed iteratively as
described in our previous paper Aubry  _et al._ (2017).

In the end, the so-computed transport mean free path $\ell^{\star}$ depends on
$\lambda_{0}$ (wavelength in vacuum), $r$, $f$, $n_{\mathrm{p}}$ and $n_{0}$.
As already shown in our previous work Aubry  _et al._ (2017), the positions of
the resonances do not depend on $f$.