Structure of Blue Phase III of Cholesteric Liquid Crystals

Abstract

We report large scale simulations of the blue phases of cholesteric liquid crystals. Our results suggest a structure for blue phase III, the blue fog, which has been the subject of a long debate in liquid crystal physics. We propose that blue phase III is an amorphous network of disclination lines, which is thermodynamically and kinetically stabilized over crystalline blue phases at intermediate chiralities. This amorphous network becomes ordered under an applied electric field, as seen in experiments.

Full text

On the structure of blue phase III
O. Henrich1, K. Stratford2, M. E. Cates1, D. Marenduzzo1
1SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, UK
2EPCC, The University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, UK
We report large scale simulations of the blue phases of cholesteric liquid crystals. Our results suggest a
structure for blue phase III, the blue fog, which has been the subject of a long debate in liquid crystal physics.
We propose that blue phase III is an amorphous network of disclination lines, which is thermodynamically and
kinetically stabilised over crystalline blue phases at intermediate chiralities. This amorphous network becomes
ordered under an applied electric field, as seen in experiments.
PACS numbers: 61.30.Mp, 61.30.Jf, 64.70.M-
Liquid crystals (LCs) offer prime examples of systems with
spontaneously broken symmetry that support topological de-
fects of various types [1]. Alongside their technological util-
ity in various optical devices, LCs offer testing grounds for
fundamental theories whose counterparts in other fields (such
as cosmology, particle physics, and more recently exotic su-
perconductivity [2]) are less directly testable. While many
aspects of LC physics are by now understood, some, such as
the character of the so-called ‘blue fog’ (blue phase III) have
remained unresolved despite efforts spanning several decades.
The simplest liquid crystalline phase is the nematic, in
which molecules have a preferred orientational axis (the di-
rector) but no translational order. Introduction of molecular
chirality causes the director to precess in space. In the sim-
plest resulting phase (the cholesteric) it does so about a single
helical axis, thereby creating a 1-D periodic structure whose
wavelength is the helical pitch. Locally, however, the ordering
remains nematic and the cholesteric therefore supports topo-
logical line defects, known as disclinations, in which the di-
rector executes a rotation of πunder a full 2πrotation around
the defect line. Within the cholesteric phase itself, these exci-
tations are absent from the ground state.
In many chiral liquid crystals the transition between
the cholesteric and isotropic phases occurs through a cas-
cade of weakly first-order transitions to intermediate struc-
tures, known as ‘blue phases’ (BPs) and consisting of self-
assembled disclination networks [1, 3]. At high enough chi-
rality, a simple helical structure is less stable locally than a
so-called double-twist cylinder (DTC), in which the director
field rotates simultaneously in two directions perpendicular to
a cylinder axis. However DTCs cannot be smoothly patched
together to fill the whole of space; disclination lines are re-
quired at the interstices between the cylinders. If an external
field is used to align the DTCs, the disclinations can form a
simple linear array [4], but more generally they meet at junc-
tions forming a multiply connected network. The resulting
structure is highly colored (‘blue’) since the spacing between
defects lies in the optical range. BPs are stable when the free
energy gained by creating DTCs instead of a simple helix is
enough to compensate the free energy loss due to defect for-
mation. They therefore occur at high chirality (where DTCs
are favored) and for Tclose to but below TIC (the transition
temperature to the isotropic phase) where defect energies are
low. Long viewed by many as little more than a curiosity [1],
BPs have recently emerged as promising materials for pho-
tonics and display devices following their stabilisation over a
much larger temperature range than previously thought pos-
sible (about 60 K compared to <1 K) [5]. This has revived
interest in these fascinating hierarchical materials, in which
molecular-scale interactions stabilize topological defects that
self-organize into a micron-scale superstructure.
Three BPs have been observed experimentally (at zero elec-
tric field). Two, BPI and BPII, are highly ordered: their discli-
nation networks form a regular cubic lattice, and their physics
is well understood. (Another ordered phase, O5, is predicted
by theory at high chirality, but not found experimentally.) The
last one is known as BPIII, and its structure is one of the un-
solved puzzles of liquid crystal physics. Theorists have pro-
posed that BPIII may either be a quasicrystal [6], a spaghetti-
like tangle of double-twist cylinders [7], an amorphous state
formed by BPII domains [8], or a metastable phase [9]. Ex-
periments on the thermodynamics, scattering and electric field
response [10–12] remain inconclusive, although electron mi-
crographs appear to favour an amorphous structure and to rule
out the quasicrystal model [3].
In this work we show that computer simulations can help
settle this important physical question. By means of large
scale (supra-unit-cell) simulations, which enable very accu-
rate comparison of free energies for both ordered and amor-
phous structures, we provide strong evidence that BPIII is in-
deed an amorphous network of disclinations. Remarkably we
show that, within a certain window of chirality and with a
standard choice of free energy functional detailed below, in-
dividual aperiodic structures exist that are more stable than
either BPI, BPII, or O5. This narrow window lies within the
observed range of BPIII, and can only become wider, in line
with experiments, if one allows for configurational entropy.
(Such entropy arises when there exist multiple aperiodic struc-
tures of the same free energy, as seems likely here.) Further-
more, we show that an applied electric field orders our aperi-
odic BPIII candidate into a different, much more ordered blue
phase. This concurs with longstanding experiments [11, 12]
which showed evidence of a field-induced transition to a new
phase, BPE, whose structure was not previously identified.
The local order in the BPs can be described by a trace-
less, symmetric, second rank tensor, Qwhose equilibrium
thermodynamics is governed by the Landau–de Gennes free
energy functional F, whose form, within the one-elastic con-
arXiv:1110.6743v1 [cond-mat.soft] 31 Oct 2011

2
stant approximation (discussed later) is standard, and speci-
fied at [13]. Within that approximation, the phase behavior
of BPs depends on just three dimensioness parameters, a re-
duced temperature τ, a reduced chirality κand a reduced field
strength E[14]. Expressions for these in terms of K(the
elastic constant), q0(the cholesteric wave-vector) and other
parameters in Fare given at [13]. Good agreement between
theoretical [15] and experimental [16] phase boundaries is ob-
tained by taking τ∝(T−TIC )and κlinear in the mole-
fraction of a chiral component (with respective proportionality
constants '2K−1and '2for one specific mixture [15]).
We employ a 3D hybrid lattice Boltzmann (LB) algo-
rithm [15, 17–19] to solve the Beris-Edwards equations. The
evolution of the Qtensor [20] is
DtQ= Γ−δF
δQ+1
3TrδF
δQI.(1)
Here, Γis a collective rotational diffusion constant and Dtis
a material derivative for rod-like molecules [20]. The term
in brackets is the molecular field, H, which ensures that Q
evolves towards a minimum of the free energy. The fluid ve-
locity field obeys the continuity equation and a Navier-Stokes
equation with a stress tensor generalised to describe liquid
crystal hydrodynamics, and discussed elsewhere [17].
Though these equations represent the true dynamics, we
use them here simply to find free energy minima. Thus, as
in previous work we additionally allow a so-called ‘redshift’
in which the parameters in Fare dynamically updated at fixed
τ, κ, E[13]. This exploits a scaling among those parameters to
ensure that the system is not frustrated by periodic boundaries:
in particular, for any cubic BP, a lattice parameter emerges
that truly minimizes F. The accuracies of their computed free
energies F≡min(F)are (at least for E= 0) thus limited
only by discretization. This is chosen to fully resolve the de-
fects [13], whose core energy is finite, and set by Fitself.
In computing the free energy of periodic structures (BPI,
BPII and O5) we apply a perturbation of the appropriate sym-
metry to a uniform state and then evolve dynamically [13].
This delivers an accurate Fvalue in each such phase, of which
the lowest can be chosen; but (in common with all other meth-
ods for computing free energies of ordered phases) we cannot
rule out others of still lower F. To address BPIII, we also
need to generate aperiodic candidates. Here our strategy is
similar: we start from various different aperiodic initial con-
ditions, evolve each dynamically, and choose that of lowest
F[13]. As shown below, this beats all three periodic struc-
tures within a certain parameter window. Because we cannot
exhaust all possible initial conditions, our free energy is an
upper bound on aperiodic states; further exploration can thus
only widen that stability window. In practice, all the aperi-
odic candidates we generated look similar: fuller exploration
is thus unlikely to change our conclusions about the character
of BPIII.
To minimize finite size effects, we simulate very large sys-
tems (in contrast to [15]). Typically we used 1283lattices,
which accomodates 8 half pitches in each direction [21]. Se-
lected simulations with 2563lattices confirm the results which
we report below. As shown in [19], it is easy to generate ape-
riodic structures by placing a localized nucleus of BPI or II
in a cholesteric or isotropic matrix. However, the lowest F
values we have so far found are instead achieved by initial-
ising the system in the cholesteric phase in the presence of a
low density (typically about 1-2 % in volume) of randomly
placed doubly twisted droplets. Once initialized, the system
is relaxed dynamically until it reaches a quiescent end-state.
For low chirality (κ < 1.5), the initial defects are washed
out to leave a cholesteric phase. However, for a large regime
of intermediate chiralities (1.75 ≤κ≤3), our simulations
show an intriguing dynamics, through which the dilute dou-
bly twisted regions grow and rearrange dramatically to form
a whole network of disclinations, which very slowly creeps to
an amorphous end-state (Fig. 1, top). Its amorphous charac-
ter is confirmed by the structure factor C(k)(Fig. 1, bottom).
The ring in C(k)is set by the average distance between the
branch-points in the defect network, whereas the small resid-
ual peaks (which break spherical symmetry) are likely due to
residual finite size or periodic boundary effects. The amor-
phous network approaches equilibrium through very slow lo-
cal rearrangements of the disclination junctions; the end-state
in Fig. 1 is very close to kinetic arrest, in at least a local mini-
mum of F. Notably, four disclination lines meet at most junc-
tions, so that, as suggested by experiment [10], the structure
is locally closer to that of BPII than BPI.
Remarkably, for larger values of the chirality (κ > 3) our
simulations attain a much more regular state, closely resem-
bling O5which (see below) minimizes our chosen Fat very
high κ. (In O5itself, whose free energy we have computed
precisely, eight disclination lines merge at each junction [13].)
Therefore, our methodology is capable in principle of finding
a periodic disclination lattice, if kinetically accessible. Given
that our equations fully incorporate liquid crystal hydrody-
namics, we believe that the kinetic propensity to form a disor-
dered disclination network for intermediate chiralities indeed
reflects a real physical property, although our limited simu-
lation times ('1ms [13]) may well exaggerate the stability
window of amorphous structures in the (κ, τ)plane.
A crucial question is whether our BPIII candidate structure
is only kinetically, or also thermodynamically stable. To an-
swer this, we have carefully compared F(κ, τ)for BPIII with
those of BPII and O5, for a range of chiralities κat selected
values of τ. (BPI is not competitive in the κrange of interest
here.) As an example, curves for F(κ, −0.25) are shown in
Fig. 2. We see that there is a small but finite chirality window
in which the BPIII-network is the thermodynamic equilibrium
phase. We find it remarkable that any single aperiodic struc-
ture can outcompete such periodic ones when minimizing the
relatively simple Landau-de Gennes free energy. In partic-
ular, this minimization takes no account of order-parameter
fluctuations about the local minimum, which might help sta-
bilize BPIII [22], nor the configurational entropy associated
with having many such minima. (These neglected contribu-
tions should be small – of order kBTper unit cell while the
free energy differences in Fig. 2 are typically of order 100-
1000 times larger [13].) However, the thermodynamic stabil-
ity window in BPIII is smaller than the one we find kinetically
– in agreement with the expectation that amorphous struc-
tures should form more easily dynamically than highly or-

3
FIG. 1: Top: End-state disclination network at τ=−0.25, κ = 2.5:
The picture shows the isosurface q(r)=0.12, with qthe largest
eigenvalue of Q. Inside each tube is a disclination line (on which q
takes a minimum value). Bottom: Structure factor C(~
k)≡ |q(k)|2,
on cuts along ky= 0 (left) and kx= 0 (right) with wavevectors
kx/q0, kz/q0∈[−4,4] and ky/q0, kz/q0∈[−4,4] respectively.
dered counterparts, which require long and complicated pro-
cess to annihilate any dislocations and overcome relatively
high energy barriers. The window of thermodynamic stability
might also be broadened by relaxing the one elastic constant
approximation, particularly if this raises the free energy of O5
relative to the other states. This certainly merits further study,
given the experimental absence of that phase.
All the above arguments support our assignment of the
amorphous network seen in Fig. 1 as the theoretically elusive
BPIII, stable at larger κthan BPII, but locally similar to it [3].
Experiments on BPIII in electric fields also point to a field-
induced transition between BPIII and another structure which
has been named BPE [11, 12]. BPE was found to give rise to a
sharper peak in the scattering data consistent with an enhanced
ordering (unfortunately Refs. [11, 12] do not provide further
experimental data on the structure of BPE). It is therefore in-
teresting to ask what happens when our amorphous structure
is subjected to an electric field. In our simulations we can
follow the evolution of the Qtensor and of the disclination
network in an external field E, and also compute C(k); if a
crystalline phase emerges, it will exhibit Bragg peaks.
By stepwise increasing Efrom 0 to 0.65, we found that the
disclination network in Fig. 1 melts away leaving a nematic
state. (This holds for positive dielectric anisotropy [13].) We
then performed a long simulation at E= 0.55, close to but be-
low the threshold beyond which the system becomes nematic.
-0.08
-0.06
-0.04
-0.02
1.8 2 2.2 2.4 2.6 2.8 3
F/ V K q0
2
κ
τ=-0.25
BPIII
O5
BPII
-0.035
-0.03
2.4 2.5 2.6
FIG. 2: Free energy densities (Vis the volume) vs. chirality κat τ=
−0.25 for BPII, our candidate for BPIII, and O5. The inset shows a
blow-up of the region where our BPIII is the equilibrium phase. Our
free energy densities are expressed in units of Kq2
0(see [13] for the
mapping between simulation and real units).
The time evolution of our BPIII, stable at zero field, is shown
in Fig. 3. This shows an interesting rearrangement of the de-
fect texture, through which the junction points rotate and fi-
nally reconstruct to yield a topologically distinct phase, with
helical disclinations lying along layers stacked perpendicular
to the electric field (which is vertical). The disclination lines
in two consecutive layers are turned by 90 degrees, so that
they show a square arrangement when viewed along the field
direction. Within a layer two adjacent disclination lines are
staggered. Again in agreement with experiments, then, we
find that our amorphous BPIII candidate undergoes a field-
induced transition to an ordered disclination network which
we provisionally identify as BPE. It would be interesting to
perform additional experiments to test such an identification.
Our candidate BPE appears related to, but distinct from, other
field- or confinement-induced BPs previously reported [4, 23],
with strong crystalline order (albeit with some defects remain-
ing) as confirmed by its structure factor.
In conclusion, we have presented large scale simulations of
the cholesteric blue phases. We have given strong evidence
that in a region of the temperature–chirality plane, instead of
a periodic BP, an aperiodic one is selected kinetically. More-
over, and strikingly, within the (one-elastic constant) Landau -
de Gennes free energy [13], we have found a finite window of
chirality in which this amorphous network is thermodynami-
cally more stable than the competing crystalline blue phases,
BPI, BPII and O5. These facts suggest that our kinetically
and thermodynamically stable network is none other than the
blue fog, BPIII. This view is strengthened by the study of the
field response of this structure, which reconstructs into an or-
dered phase at intermediate values of the electric field, slightly
smaller than those at which the disclination network melts to
give a field-oriented nematic phase.
We believe our simulations shed important light on the
structure of the blue fog, which we propose is an amorphous
disclination network which is locally close to BPII: this view
is perhaps intermediate between the “BPII domain” [8] and
the “double twist tangle” [7] models which were proposed in

4
FIG. 3: Isosurfaces of q= 0.12 at τ= 0.2, κ = 2.0,E= 0.55: The
sequence shows the transition from the amorphous BPIII network to
a field-induced BPE. The field direction, z, is vertical. Times shown
are t'0.15,0.3,0.6and 1.4ms (taking one timestep '1ns).
Bottow row: Structure factor C(~
k)at the end of the run. The pictures
show a cut through the 3D data at ky= 0 (in the (kx, kz)plane, left)
and kz= 0 (in the (kx, ky)plane, right), with scales as in Fig. 1.
the early 1980’s, and is at odds with some of the subsequent
structures proposed for this phase. Several open questions re-
main or are stimulated by our results. Firstly, one should char-
acterise statistically the phase transition from the blue phase
to the isotropic phase: this may be possible by adding ther-
mal noise to simulations like the ones reported here. Sec-
ondly, use of more accurate scattering and visualisation tech-
niques should allow tests of our candidate structure for BPE
(see Fig. 3). Finally, if the large energy scales that separate
the various BP topologies (of order 100 −1000kBTper unit
cell [13]) also control reconstruction of the defect network
within BPIII itself, one may expect that in some materials,
even when thermal noise is allowed for, such reconstruction
cannot be achieved on any reasonable timescale. If so, BPIII
will represent another elusive entity: an amorphous state of
globally minimum free energy that is kinetically arrested —
or an ‘equilibrium glass’.
We thank A. N. Morozov for useful discussions. This
work was supported by EPSRC grants EP/E045316/1 and
EP/E030173/1. MEC is funded by the Royal Society.
Here we provide further information on the free energy
functional and the control parameters τ, κ, E(section A); on
resolution and on ‘redshift’, i.e., the parameter-dependent unit
cell size of blue phases (section B); on the initial conditions
chosen for runs presented in the main text (sections C and D);
on the conversion of simulation parameters to physical units
(section E); on the free energy comparison between different
blue phases (section F); and on the effect on field-response of
changing the sign of the dielectric anisotropy (section G).
1. Free energy functional, chirality and reduced temperature
The thermodynamics of cholesteric blue phases can be
described via a Landau-de Gennes free energy functional
F[Q] = Rd3rf(Q(r)). The free energy density we use, fol-
lowing [1], is:
f(Q) = A0
21−γ
3Q2
αβ −A0γ
3Qαβ Qβγ Qγα +A0γ
4(Q2
αβ )2
+K
2(εαγδ ∂γQδβ + 2q0Qαβ )2+K
2(∂βQαβ )2
−εa
12πEαQαβ Eβ(2)
Here repeated indices are summed over and Q2
αβ stands for
Qαβ Qαβ etc. The first three terms are a bulk free energy den-
sity whose overall scale is set by A0(discussed further be-
low); γis a control parameter, related to reduced temperature.
Varying the latter in the absence of chiral terms (q0= 0) gives
an isotropic-nematic transition at γ= 2.7with a mean-field
spinodal instability at γ= 3. The next two terms in Eq.2
describe distortions of the order parameter field. As is con-
ventional [1, 24] we have assumed that splay, bend and twist
deformations of the director are equally costly; Kis then the
one elastic constant that remains. The parameter q0is related
via q0= 2π/p0to the pitch length, p0, describing one full
turn of the director in the cholesteric phase.
The remaining term accounts for the coupling to the electric
field, Eα, and εais the dielectric anisotropy. In the text we
considered the case εa>0, whereas below we show what
happens for negative dielectric constant.
The phase behaviour of blue phases (in the absence of noise
due to thermal fluctuations, i.e., as computed by minimization
of F) may be shown to only depend on the following dimen-
sionless parameters [1]:
τ= 27(1 −γ/3)/γ (3)
κ=q108 K q2
0/(A0γ)
E2= (27εaEαEα)/(32πA0γ).
as used in the main text. These are referred to as reduced tem-
perature, chirality and reduced field respectively. Note that if
fis made dimensionless, τappears as prefactor of the term
quadratic in Q, whereas κquantifies the ratio between bulk
and gradient terms.

5
2. Resolution, unit cell size and ‘redshift’
In our simulations, each disclination core spans a few (typ-
ically 3-4) lattice sites. To confirm that the defect core is ade-
quately resolved, we have run selected simulations in which
the resolution was doubled or quadrupled, and found very
similar values for the free energy. On this basis we assess
that the typical error due to discretization for our main runs is
much smaller than the free energy difference between BPs.
In cubic BPs it is known that the equilibrium unit cell size is
slightly different from (and typically slightly larger than) the
half pitch of the cholesteric liquid crystal, π/q0. To account
for this, a ‘redshift’ factor ris introduced [15] whose variation
effectively allows free adjustment of the BP lattice parameter,
Λ→Λ/r, despite the use of periodic boundary conditions in
our simulations. That is, to avoid changing the size of the sim-
ulation box, redshifting is performed by an equivalent rescal-
ing of the pitch parameter and elastic constant, q0→q0/r
and K→K r2. In simulations aimed solely at free energy
minimization, it is legitimate to make ra dynamic parameter
and update it on the fly to achieve this. At each time step one
then selects the value of rfor which the physical free energy
density is lowest. More details on this procedure are given in
Ref. [15] where this scheme was first proposed.
We have used this dynamic updating of the redshift (i) in all
single unit cell runs used to map out the free energy curves,
and (ii) in multi-unit cell dynamical runs after the final net-
work was formed. Redshifting is essential in order to get ac-
curate estimates for the free energy of the cubic blue phases,
whose relative free energy differences between phases is less
than a percent of their typical values. With redshifting, the
accuracy attainable is in principle limited only by discretiza-
tion error; without it, the dominant error is from mismatch
between the preferred unit cell size and the periodic bound-
ary conditions. For amorphous networks, those conditions en-
force a periodic structure (on the box scale) with the aperiodic
structure within that; redshifting likewise avoids any incom-
mensurability between the two length scales. Because of the
constrained periodicity, the computed free energy for amor-
phous structures will always be overestimated, despite which
we find a window of thermodynamic stability for these with
free energies lower than the known ordered phases. (As usual
we cannot rule out a still lower free energy for some other, un-
known, ordered phase.) Our use of redshift on all phases thus
means that the true thermodynamic stability window of BPIII
can only be wider than the one found here.
3. Initial conditions: cubic blue phases
To accurately compute the free energy of cubic blue phases
from single unit cell simulations, we must start from an initial
condition of the correct symmetry and disclination line topol-
ogy. To this end, as discussed in Ref. [14]a, we can adopt
the form for Q(x, y, z)corresponding to the limiting form for
each phase in the high chirality limit [1].
For BPI, with q0
0=√2q0, we then have initially:
Qxx ' −2 cos(q0
0y) sin(q0
0z) + sin (q0
0x) cos (q0
0z)
+ cos (q0
0x) sin (q0
0y)
Qxy '√2 cos (q0
0y) cos (q0
0z) + √2 sin (q0
0x) sin (q0
0z)
−sin (q0
0x) cos (q0
0y)
Qxz '√2 cos(q0
0x) cos(q0
0y) + √2 sin(q0
0z) sin(q0
0y)
−cos(q0
0x) sin(q0
0z)
Qyy ' −2 sin(q0
0x) cos(q0
0z) + sin(q0
0y) cos(q0
0x)
+ cos(q0
0y) sin(q0
0z)
Qyz '√2 cos(q0
0z) cos(q0
0x) + √2 sin(q0
0y) sin(q0
0x)
−sin(q0
0y) cos(q0
0z).(4)
Likewise, for BPII we start from:
Qxx 'cos(2q0z)−cos(2q0y)
Qxy 'sin(2q0z)
Qxz 'sin(2q0y)
Qyy 'cos(2q0x)−cos(2q0z)
Qyz 'sin(2q0x).(5)
Finally, for O5we start from:
Qxx '2 cos(√2q0y) cos(√2q0z)
−cos(√2q0x) cos(√2q0z)
−cos(√2q0x) cos(√2q0y)
Qxy '√2 cos(√2q0y) sin(√2q0z)
−√2 cos(√2q0x) sin(√2q0z)
−sin(√2q0x) sin(√2q0y)
Qxz '√2 cos(√2q0x) sin(√2q0y)
−√2 cos(√2q0z) sin(√2q0y)
−sin(√2q0x) sin(√2q0z)
Qyy '2 cos(√2q0x) cos(√2q0z)
−cos(√2q0y) cos(√2q0x)
−cos(√2q0y) cos(√2q0z))
Qyz '√2 cos(√2q0z) sin(√2q0x)
−√2 cos(√2q0y) sin(√2q0x)
−sin(√2q0y) sin(√2q0z).(6)
This procedure was already discussed in Refs. [15] and
[14]a, where additional details can be found. Fig. 4 shows the
unit cell of O5obtained after a full numerical minimisation
(with redshift) starting from the high chirality limit detailed
above. To ensure that equilibrium was reached, before ending
a run we required that the free energy did not change, in two
successive measurements, by more than an accuracy threshold
(typically chosen as less than 0.1%).
In line with previous theoretical and numerical calcula-
tions [1, 14], we find that O5is the equilibrium structure at
high enough chirality (see Fig. 2 in the main text). This is

6
FIG. 4: Image of the unit cell of O5after free energy minimization.
therefore expected within the free energy functional formal-
ism that we use. Given that O5is not observed experimentally,
it would be interesting to study its window of thermodynamic
stability by relaxing the one elastic constant approximation
employed in this work.
4. Initial conditions: blue phase III
In the main text we presented configurations for our candi-
date blue phase III which were obtained by initialising the sys-
tem in the cholesteric phase in the presence of a low density
(typically about 1-2 % in volume) of randomly placed doubly
twisted droplets. One may therefore wonder what happens
when different initial conditions are selected.
To address this question, we have performed additional runs
starting from a different initial configurations, in which the
system was initialised in the isotropic phase (rather than in
the cholesteric phase as in the main text), again in the presence
of a low density of randomly placed doubly twisted droplets.
Fig. 5 shows the initial and final disclination network in this
case – it can be seen that the final state is very similar to the
one shown in the main text. A minor difference is that, in the
case of an isotropic background, we found that the initial vol-
ume density of doubly twisted impurities needs to be slightly
larger for the network to grow. In no case did we find a free
energy by this route that was lower than the one resulting from
the cholesteric initial condition. As shown in [19], finally, it
is also possible to generate locally stable aperiodic structures
by placing a localized nucleus of BPI or II in a cholesteric or
isotropic matrix. These were found to have higher free ener-
gies than those reported here (but comparable with them).
To summarise, the kinetic stabilisation of amorphous net-
works occurs from a wide range of initial conditions, includ-
ing doubly twisted impurities inside a cholesteric (main text)
or isotropic (background), and cubic blue phase nuclei inside
cholesteric or isotropic phases. Such a predicted stabilisation
is therefore a very robust result of our simulations.
5. Parameter mapping to physical units
Here we describe how simulation parameters are related
to physical quantities in real BP materials. In order to get
from simulation to physical units, we need to calibrate scales
of length, energy and time. We follow the methodology
of [17, 19]. First we define a set of lattice Boltzmann units
(LBU) in which the lattice parameter `, the time step ∆t, and
a reference fluid mass density ρ0are all set to unity. This
is the set of units in which our algorithm is actually written.
The length scale calibration is straightforward, and fixed by
the cholesteric pitch p0, which is typically in the 100-500 nm
range [1]. More precisely, in our simulations we set the unit
cell of BPI/II to be 16 LBU; this gives good resolution with-
out wasting resource. Therefore the LBU of length (one lattice
site) corresponds to, say, 10nm in physical space.
To get an energy scale, we use the measurements cited in
Appendix D of [1] which, as shown in [19] suggest:
27
2A0γ∼2−5×10−6J−1m3.(7)
From this relation (given that γ'3close to the
isotropic/cholesteric transition) we obtain that A0'106Pa.
On the other hand, our simulations typically use a value of
A0'0.01 LBU. This requires that the LBU of free energy
density is about 108Pa in SI units. Therefore our choice of
elastic constant, K= 0.01 LBU, equates to K= 10−10 N,
corresponding to a Frank elastic constant K/2 = 50 pN which
is sensible.
For dynamical studies one can use the above results to cali-
brate the lattice Boltzmann time unit, and then crosscheck that
the resulting fluid density gives acceptable Reynolds numbers.
This is indeed confirmed in [19] for our simulations, which
choose an order parameter mobility Γ=0.3LBU in the no-
tation of that paper, and a fluid viscosity η= 1 −2LBU.
However, such considerations are not crucial in the current
context where the dynamics need not be realistic, so long as
they accurately minimize F. To summarize the results above
(and the additional discussion in [19]): our simulations faith-
fully represent experimentally realisable BP-forming materi-
als, subject to the interpretation of the LBU for length, energy
density and time are close to 10 nm, 100 MPa and 1 ns respec-
tively.

7
FIG. 5: Initial (left) and final (right, after 300,000 lattice Boltzmann steps) configuration for a run which is started with randomly placed and
oriented doubly twisted droplets, inside an isotropic background, as opposed to a cholesteric one as presented in the main text. Also in this
case we observe a final amorphous network, very similar to the one shown in Fig. 1 of the main text.
6. Free energy comparisons
Table I shows the free energy obtained for our candidate
blue phase III, blue phase II and O5, in a temperature–chirality
range somewhat larger than the one presented in the main
text. (BPI is not competitive throughout this region.) At two
points in parameter space, the free energies (in bold) are lower
for BPIII than any other structure. For all other parameter
sets in the Table, with the exception of one marked “CHOL”,
the BPIII candidate remains kinetically stable (when initiated
from double-twist droplets in a cholesteric phase) even though
it has higher free energy than one or more ordered phases.
A difference in the tabulated values of ∆˜
f= 0.1be-
tween phases (that is, ∆f= 10−6LBU) corresponds, with
the parameter mapping detailed above, to a free energy dif-
ference per unit cell of around 10kBTper unit cell. This
assumes a cell size of 160 nm, which is at the low end for
blue phases; for a cell of say 500 nm the difference would be
several hundreds kBTper cell. Adding thermal noise in the
simulations (thereby weighting states by the Boltzmann factor
exp[−F/kBT]rather than simply minimizing F) would be
expected to create entropic shifts in the free energies of order
kBTper unit cell, reflecting configurational entropy and/or
local order parameter fluctuations. These shifts would likely
favor the amorphous BPIII structure, but, by these estimates,
will do so only marginally.
The above scaling of free energy densities via kBTper unit
cell is useful when discussing fluctuation effects but extrane-
ous at the level of free energy minimization using the Landau
- de Gennes functional, in which kBTdoes not appear. In that
context, it is more natural to express the differences in free
energy density between phases in units of either A0or Kq2
0.
Since it relates to more easily measured quantities the latter
scaling is preferable, and is followed in Fig. 2 of the main
text.
7. Dielectric anisotropy effect
In the main text, we have considered the response of our
candidate blue phase III structure to an electric field, assum-
ing that the medium had a positive dielectric constant. We
have also considered the case of negative dielectric constant,
which favors an ordering of the director field normal to the
electric field direction. Also in this case we observe ordering,
although the final state is different, and consists of double he-
lical disclinations parallel to the direction of the field. This
state is shown in Fig. 6.
[1] D. C. Wright and N. D. Mermin, Rev. Mod. Phys. 61, 385
(1989).
[2] M. B. Hindmarsh and T. W. B. Kibble, Repts. Prog. Phys. 58,
477 (1995); R. A. Borzi et al., Science 315, 5809 (2007), V.
Oganesyan et al., Phys. Rev. B 64, 195109 (2001).
[3] P. P. Crooker, Liq. Cryst. 5, 751 (1989).
[4] O. Henrich et al, Comp. Math. with Apps.,59, 2360 (2010);
Phys. Rev. E. 81 031706 (2010).
[5] H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang and T. Kajiyama,
Nat. Mater. 1, 64 (2002); H. J. Coles and M. N. Pivnenko, Na-
ture 436, 997 (2005); Y. Hisakado, H. Kikuchi, T. Nagamura
and T. Kajiyama, Adv. Mater. 17, 96 (2005).
[6] R. M. Hornreich and S. Shtrikman, Phys. Rev. Lett. 56, 1723
(1986); D. Rokhsar and J. Sethna, Phys. Rev. Lett. 56, 1727
(1986).
[7] R. M. Hornreich, M. Kugler and S. Shtrikman, Phys. Rev. Lett.
48, 1404 (1982).
[8] V. A. Belyakov, E. I. Demikhov, V. E. Dmitrienko and V. K.
Dolganov, Sov. Phys. JETP 62, 1173 (1986); P. J. Collings,
Phys. Rev. A 30, 1990 (1984).
[9] P. L. Finn and P. E. Cladis, Mol. Cryst. Liq. Cryst. 84, 159
(1982).
[10] E. P. Koistinen and P. H. Keyes, Phys. Rev. Lett. 74, 4460
(1995); Z. Kutnjak, C. W. Garland, J. L. Passmore and P. J.
Collings, Phys. Rev. Lett. 74, 4859 (1995).
[11] H.-S. Kitzerow, P. P. Crooker and G. Heppke, Phys. Rev. Lett.
67, 2151 (1991); H.-S. Kitzerow, Mol. Cryst. Liq. Cryst. 202,
51 (1991).
[12] D. K. Yang and P. P. Crooker, Phys. Rev. A 37, 4001 (1988).
[13] See Appendix for additional discussion on free energy parame-
ter, redshift, behaviour with negative dielectric anisotropy, and
three additional figures.

8
FIG. 6: Field-induced structure obtained when blue phase III is subjected to an electric field along the zaxis, in the regime analogous to the
one explored in Fig. 3 of the main text, but this time with negative dielectric constant.
[14] A. Dupuis, D. Marenduzzo, and J. M. Yeomans, Phys. Rev. E
71, 011703 (2005); H. Grebel, R. M. Hornreich, and S. Shtrik-
man, Phys. Rev. A 28, 1114 (1983); R. M. Hornreich, M. Ku-
gler, and S. Shtrikman, Phys. Rev. Lett. 54, 2099 (1985).
[15] G. P. Alexander and J. M. Yeomans, Phys. Rev. E 74, 061706
(2006).
[16] D. K. Yang and P. P. Crooker, Phys. Rev. A 35, 4419 (1987).
[17] C. Denniston et al.,Phil. Trans. Roy. Soc. A 362, 1745 (2004);
G. P. Alexander and D. Marenduzzo, EPL 81, 66004 (2008).
[18] M. E. Cates et al, Soft Matter 5, 3791 (2009).
[19] O. Henrich et al, Proc. Natl. Acad. Sci. USA,107, 13212 (2010).
[20] A. N. Beris and B. J. Edwards, Thermodynamics of Flowing
Systems (Oxford University Press, Oxford, 1994).
[21] Our runs were parallel, e.g. 105LB time steps for a 1283lattice,
on 512 nodes, required 3 hours (on the Hector supercomputer).
[22] J. Englert et al, Phys. Rev. Lett. 81, 1457 (1998).
[23] J Fukuda and S. Zumer, Phys. Rev. Lett. 104, 017801 (2010).
[24] P.-G. de Gennes, J. Prost, The Physics of Liquid Crystals (Ox-
ford Univ. Press, New York, 1993).

9
τ κ γ A0·103r- BPIII ˜
f(r)- BPIII r- O5 ˜
f(r)- O5 r- BPII ˜
f(r)- BPII
0 2 3 6.9396 1.097 -3.104123 1.009 -3.047634 0.908 -3.161573
0 2.5 3 4.4413 1.090 -1.804606 1.016 -1.828106 0.914 -1.791576
0 2.75 3 3.6705 1.049 -1.415262 1.018 -1.465741 0.917 -1.376351
0 3 3 3.0843 1.097 -1.144493 1.020 -1.192417 0.919 -1.062554
-0.25 1.75 3.0857 8.8122 1.100 -6.003057 1.000 -5.784407 0.903 -6.154037
-0.25 2 3.0857 6.7468 1.103 -4.42299 1.007 -4.292183 0.907 -4.508668
-0.25 2.25 3.0857 5.3308 1.093 -3.3654 1.012 -3.302209 0.911 -3.400327
-0.25 2.5 3.0857 4.3180 1.085 -2.632778 1.015 -2.597435 0.914 -2.602943
-0.25 2.6 3.0857 3.9922 1.091 -2.39750 1.016 -2.376423 0.916 -2.351800
-0.25 2.75 3.0857 3.5686 1.043 -2.08964 1.017 -2.092873 0.917 -2.046106
-0.25 3 3.0857 2.9986 1.096 -1.672261 1.019 -1.714777 0.919 -1.614218
-0.5 1.75 3.1765 8.5604 1.026 -5.101 (CHOL) 0.996 -7.662615 0.901 -8.166828
-0.5 2 3.1765 6.5540 1.130 -5.872775 1.004 -5.690254 0.907 -6.016180
-0.5 2.5 3.1765 4.1946 1.069 -3.501575 1.013 -3.472278 0.913 -3.546119
-0.5 2.75 3.1765 3.4666 1.050 -2.770071 1.016 -2.812765 0.915 -2.798638
-0.5 3 3.1765 2.9129 1.045 -2.250650 1.018 -2.319368 0.918 -2.2350
TABLE I: Final free energy densities ˜
f(r)≡105fin LBU. These are fully equilibrated (with redshift ras shown) for the ordered phases,
and are redshifted also for the runs initialized from double twisted droplets in a cholesteric background for BPIII. Bold entries mark the free
energies where BPIII is more stable than any of the ordered phases. (BPI is not competitive in this region of parameter space.) Note that for all
parameters shown here BPIII is kinetically stable when initialized as described, with the exception of one low chirality run (marked CHOL)
which reverted to a pure cholesteric state.