# The Abrikosov vortex structure revealed through near-field radiative heat
exchange

S. G. Castillo-López, R. Esquivel-Sirvent, G. Pirruccio, and C. Villarreal
Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México
C.P. 04510 Mexico.

###### Abstract

One of the signatures of superconductivity is the formation of the Abrikosov
vortex lattice in type-II superconductors in the presence of an external
magnetic field. Here, we study the near-field radiative heat transfer between
a spherical nanoparticle and a nearby planar substrate, both made of
optimally-doped YBa2Cu3O7-δ. We show that the heat flux displays a periodic
spatial pattern congruent with the material optical response modulated by the
Abrikosov lattice. Our results enrich the toolbox of methods available to
study non-conventional superconductivity.

Understanding the fundamental mechanisms governing thermal emission is of
primary importance both from a fundamental point of view and practical
applications such as thermal management Ben-Abdallah and Biehs (2014), thermal
rectification Yuan et al. (2022); Yang et al. (2013); Wen et al. (2019); Otey
et al. (2010) and energy conversion devices Lucchesi et al. (2021). At
submicron distances, for body separations well below the Wien thermal
wavelength, $\lambda_{T}=\hbar c/k_{B}T$, the near-field radiative heat
transfer (NFRHT) is dominated by the electromagnetic evanescent modes
supported by each material Hargreaves (1969); Polder and Hove (1971);
therefore, thermal radiation exceeding the black-body limit between
contactless materials may constitute the main heat flux channel. In past
years, many systems have been considered based on geometric or physical
properties Messina et al. (2017); Moncada-Villa et al. (2015), giving rise to
fundamental questions about the transport properties Francoeur (2015), sample
size effects St-Gelais et al. (2014, 2016), and coherence effects occurring at
the micro and nanoscale Jones et al. (2013); Yu and Fan (2023).

Superconducting (SC) systems yield the possibility of assessing NFRHT
modifications associated to the shift from normal to quantum coherent behavior
for temperatures above and below the critical value $T_{c}$. Recently, NFRHT
suppression between parallel plates of BCS superconductors has been
experimentally shown. Similarly, a pronounced suppression or enhancement of
NFRHT has been predicted for non-conventional superconductors, particularly,
setups involving YBa2Cu3O7-δ (YBCO), depending on whether bulk or film
configurations are considered, as well as on changes of the optical response
due to the superconducting transition Castillo-López et al. (2020, 2022).
HTSCs offer the opportunity of achieving superfluid charge carrier motion at
temperatures considerably higher than conventional BCS superconductors,
opening the possibility of their use in heat flux modulation for applications
involving thermal switches or thermal emitters.

Under the action of the applied field, conventional BCS superconductors
exhibit a perfect diamagnetic response, –the Meissner effect-, such that for
field intensities, $B<B_{c}$, surface supercurrents expel the magnetic field
lines from their interior within a penetration length $\lambda(T)=(4\pi
e^{*2}n_{s}(T)/m^{*})^{-1/2}$, where $e^{*}=2e$, $m^{*}$, and $n_{s}(T)$
represent the charge, mass and number density of Cooper pairs (CPs) at a
temperature $T$. In contrast, HTSCs involve a mixed diamagnetic phase,
delimited by two critical field intensities, $B_{c1}<B_{c2}$, in which normal
charge carriers with density $n_{n}(T)$ coexist with SC pairs, such that the
charge carrier density can be split as $n=n_{n}(T)+n_{s}(T)$, In this phase,
local screening supercurrents, $\mathbf{J}_{s}$, force the magnetic flux to
penetrate the sample in the form of quantum vortexes with an intensity
$\Phi_{0}=h/e^{*}$ Annett (2004). Even though a field increase beyond $B_{c2}$
causes the disappearance of superconductivity, Abrikosov found that for
intensities of the field just below $B_{c2}$, the quantum vortexes align
displaying a lattice with periods
$\mathcal{L}_{x}=\mathcal{L}_{y}=\mathcal{L}$ (see Fig. 1) Abrikosov (2004).

In this Letter, we tackle the fundamental question on whether the Abrikosov
lattice influences the local thermal properties of HTSCs. Specifically, we
study the NFRHT between a spherical YBCO nanoparticle of radius $R$ in the
normal state ($T>T_{c}$) and a planar YBCO substrate in the SC state
($T<T_{c}$), in the presence of an external magnetic field. This setup is
sketched in Fig. 1. By assuming that $R\lesssim\mathcal{L}$, this setup allows
a local scanning of the heat flux. We show that, within a dipolar
approximation for the sphere-substrate coupling, the heat flux acquires a
spatial structure inherited from the Abrikosov lattice. Surprisingly, we find
that, while for temperatures $T\lesssim T_{c}$ the heat flux maxima coincide
with the vortex core centers, for temperatures $T\ll T_{c}$ they radially
migrate, forming a ringed structure over the vortex core borders.

To characterize the dielectric permittivity of cuprate ceramics, like YBCO, we
put forth a variant of the Ginzburg-Landau (GL) superconductivity theory that
considers specific attributes of these materials. In the GL theory, the
transit to the SC state is described as a second-order phase transition
determined by a complex order parameter, $\Psi(\mathbf{r})$, null in the
normal phase, but finite in the SC phase, implying a long-range order
specified by the number density of SC pairs:
$n_{s}(\mathbf{r})=|\Psi(\mathbf{r})|^{2}$ Tinkham (2004); Annett (2004). In
presence of an applied magnetic field, $\mathbf{B}=\nabla\times\mathbf{A}$,
the order parameter satisfies the non-linear GL equation

$\left[\frac{1}{2m^{*}}\left(-i\hbar\nabla-e^{*}\mathbf{A}/c\right)^{2}-(a_{T}+b|\Psi_{0}|^{2})\right]\Psi_{0}=0,$
(1)

where $a_{T}=a_{0}(T-T_{c})$, and $b>0$ in the standard GL approach. In
absence of external fields or boundaries, this equation predicts a homogeneous
SC ordering, $|\Psi_{\infty}|^{2}=-a_{T}/b$. Also, for an interface between a
normal metal ($z<0$) and a superconductor ($z>0$), Eq.(1) yields
$\Psi(z)=\Psi_{\infty}\tanh[z/\sqrt{2}\,\xi(T)]$, where the GL coherence
length $\xi(T)=(\hbar^{2}/2m^{*}|a_{T}|)^{1/2}$ is the distance in which the
order parameter acquires its bulk value, $\Psi_{\infty}$ Tinkham (2004). Given
that $\xi(T)$ is microscopically determined by the CP correlation length, the
parameter $\kappa=\lambda(T)/\xi(T)$ represents a ratio of CP separation with
respect to their intrinsic size.

We incorporate into the GL theory two singular features of optimally doped
HTSCs: (a) An extremely short pair coherence length, such that $\kappa\ll 1$,
congruent with a local description of the electromagnetic properties of the
system Abrikosov (1957); moreover, this is indicative of a strongly attractive
inter-fermion interaction, leading to the formation of single-pair bound
states able to undergo Bose-Einstein condensation (BEC) Nozieres and Schmitt-
Rink (1985). (b) A layered crystallographic structure, in which superfluid
charge transport occurs mainly along CuO2 planes -the $ab$-planes-. Together,
these conditions imply that YBCO pairs define a 2D dilute boson gas, with an
energy spectrum $E_{k}=-V_{0}+\mathcal{E}_{k}$, where $V_{0}$ is the pair
binding energy, and $\mathcal{E}_{k}$ the excitation spectrum; in the dilute
regime, this may be approximated by a phonon dispersion relation,
$\mathcal{E}_{k}\approx\hbar c_{s}k$, with $c_{s}$ the sound’s speed. BEC
formation occurs at a temperature $T_{c}$ when the pair chemical potential
$2\mu=-V_{0}$, and the pair occupancy number can be expressed, for $T<T_{c}$,
as follows:

$n^{2D}(\mathbf{r},T)=n_{0}^{2D}(\mathbf{r},T)+\sum_{\mathbf{k}>0}\frac{1}{\exp\left(\hbar
c_{s}k/k_{B}T\right)-1}$ (2)

Since the linear structure of $\mathcal{E}_{k}$ obeys Landau’s criterion for
superfluidity Landau and Lifshitz (2013), it turns out that the density of SC
carriers coincides in this limit with the condensate:
$n_{s}^{2D}(\mathbf{r},T)\approx n_{0}^{2D}(\mathbf{r},T)$. Straightforward
evaluation of the summation in Eq.(2) then leads to:

$n_{s}^{2D}(\mathbf{r},T)=\left(1-T^{2}/T_{c}^{2}\right)N|\Psi_{0}(\mathbf{r})|^{2}.$
(3)

Here,
$T_{c}=\left(2\pi\hbar^{2}c_{s}^{2}n^{2D}/k_{B}^{2}\zeta(2)\right)^{1/2}$
Fujita and Godoy (1996), with $\zeta(2)$ a Riemann’s zeta function, $\int
d^{2}r|\Psi_{0}(\mathbf{r})|^{2}=1$, and $N$ is the number of charge carriers
per unit area. The general relation (3) describes, in absence of magnetic
fields, experimental measurements performed in a number HTSC materials under a
diversity of doping levels Zuev et al. (2005); Chen et al. (2005); Lomnitz et
al. (2013). On the other hand, consistency with the original GL theory is
achieved by assuming that the parameter
$a_{T}=-\widetilde{a_{0}}(1-T^{2}/T_{c}^{2})$ in Eq.(1). This implies in turn
that the $ab$-correlation length
$\xi_{ab}(T)=\xi_{ab}^{2}(0)/(1-T^{2}/T_{c}^{2})$.

In the following, we combine relation (3) with spectroscopic measurements of
the YBCO optical response obtained at the normal ($T=100~{}\mathrm{K}$), and
SC phases ($T=2~{}\mathrm{K}$) Bonn et al. (1988). As a result, we obtain an
expression for the $ab$-plane dielectric permittivity in the SC phase, valid
in the whole temperature range $0<T<T_{c}$:

$\displaystyle\varepsilon^{(s)}_{ab}$ $\displaystyle=$
$\displaystyle\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2}}\left(1-\frac{T^{2}}{T_{c}^{2}}\right)|\Psi_{0}(\mathbf{r})|^{2}\left[1+\frac{i\pi\omega\delta(\omega)}{2}\right]$
$\displaystyle-$
$\displaystyle\left(\frac{T^{2}}{T_{c}^{2}}\right)\frac{\omega_{p}^{2}}{\omega^{2}+i\gamma(T)\,\omega}-\sum_{r=1}^{N_{ph}+1}\frac{\omega_{pr}^{2}}{\omega^{2}-\omega_{r}^{2}+i\gamma_{r}\omega}.$

Here, the temperature-dependent terms represent Drude-like contributions, with
plasma frequency, $\omega_{p}$, and scattering rate, $\gamma(T)$, while the
summation includes contributions associated to a mid-infrared and $N_{ph}$
phonon resonances. The normal state permittivity, $\varepsilon_{ab}^{(n)}$,
may be derived from the limit $T\to T_{c}$ of (The Abrikosov vortex structure
revealed through near-field radiative heat exchange); the respective sphere
and substrate permittivities are then given by
$\varepsilon_{ns}=\varepsilon^{(n)}_{ab}$ and
$\varepsilon_{sub}=\varepsilon^{(s)}_{ab}$ (see the Supplementary Information
(SI) for more details on the YBCO permittivity in the normal and SC phases).
To discuss our results, it is worth mentioning that the main contributions to
the NFRHT at subwavelength separations, $L<\lambda_{T}$, arise form evanescent
surface modes (SMs) that may be excited for frequencies such that
$\varepsilon_{ab}^{(s)}(\omega)=-1$ Ibach and Luth (2009).

Figure 1: Setup consisting of a spherical YBCO nanoparticle of radius $R$ in
the normal state ($T_{sph}>T_{c}$) and a planar YBCO substrate in the SC state
($T_{sub}<T_{c}$), a distance $L$ apart, in presence of an external magnetic
field. The substrate subtends an order parameter
$n_{s}^{2D}(\mathbf{r})=|\Psi_{0}(\tilde{x},\tilde{y})|^{2}$ defining a
periodic lattice with periods $\mathcal{L}_{x}=\mathcal{L}_{y}$.
Superconducting current lines coincide with equiprobability lines of the order
parameter. Vortex cores correspond to the darkest zones. Figure 2: Spatial
distribution of heat flux between a nanosphere and a YBCO substrate with an
Abrikosov vortex structure. Here $R=10$ nm, $L=20$ nm and, (a) $T_{ns}=300$ K,
$T_{sub}=90$ K, (b) $T_{ns}=300$ K, $T_{sub}=2$ K. Accordingly with our
theoretical predictions, $\mathcal{L}_{x}(2\,\mathrm{K})\approx 6$ nm, whereas
$\mathcal{L}_{x}(90\,\mathrm{K})\approx 100$ nm. Notice that the vortex size
can be tuned using the lattice parameter, $\mathcal{L}_{x,y}$, and the
temperature $T_{sub}$. See Supplementary Fig. S1.

The explicit expression of $\Psi_{0}(\mathbf{r})$ is now evaluated by solving
the GL equation. By assuming that the magnetic field points in the $z$
direction, $\mathbf{B}=B\mathbf{e}_{z}$, then the vector potential in the
Landau gauge, $\mathbf{A}=Bx\,\mathbf{e}_{y}$, and the GL equation becomes:

$\frac{-\hbar^{2}}{2m^{*}}\left(\nabla_{\perp}-\frac{ie^{*}}{\hbar}Bx\mathbf{e}_{y}\right)^{2}\Psi_{0}-\left(a_{T}+b|\Psi|^{2}\right)\Psi_{0}=0,$
(5)

where $m^{*}=2m_{ab}$ is the effective CP mass along the $ab$-plane. By taking
into account that the underlying crystal structure induces a square
translational symmetry, Abrikosov showed Abrikosov (2004) that Eq.(5) admits
an analytic solution determined by a Jacobi theta function:

$\Psi_{0}(\tilde{x},\tilde{y})=C\exp[-\frac{1}{2}\kappa^{2}\tilde{x}^{2}]\theta_{3}\left[1;\sqrt{2\pi}\
\kappa i(\tilde{x}+i\tilde{y})\right],$ (6)

where $\tilde{x}=x/\lambda_{ab}$, $\tilde{y}=y/\lambda_{ab}$, and the
$ab$-penetration length,
$\lambda_{ab}(T)\sim(n_{s}^{2D}(\mathbf{0},T)/m_{ab})^{-1/2}$. By writing
$\Psi_{0}(\tilde{\mathbf{r}})=|\Psi_{0}(\tilde{\mathbf{r}})|e^{i\chi(\tilde{\mathbf{r}})}$
it follows that the GL current density is given by

$\mathbf{J}_{s}=(\hbar e/m_{ab})|\Psi_{0}|^{2}(\nabla\chi-(2e/\hbar
c)\mathbf{A}).$ (7)

Fig.(1) depicts the equiprobability contours lines defined by
$|\Psi_{0}(\tilde{\mathbf{r}})|^{2}$, showing the emergence of a periodic
order parameter defining elementary cells with period
$\tilde{\mathcal{L}_{x}}=\tilde{\mathcal{L}_{y}}=\sqrt{2\pi}/\kappa$. SC
current lines coincide with lines of constant $\Psi_{0}$, with the vortex
cores located at the darkest zones. Notice that the vortex size can be tuned
by the substrate temperature $T_{sub}$ which modulates the lattice parameter;
in normal units to $\mathcal{L}_{x}(2\,\mathrm{K})\approx 6$ nm, whereas
$\mathcal{L}_{x}(90\,\mathrm{K})\approx 100$ nm (see Supplementary Fig. S1).

We now assess the effect of the Abrikosov lattice on the NFRHT using Rytov’s
theory of fluctuating electrodynamics Vinogradov and Dorofeev (2009a). In this
theory, the spectral heat flux is determined by the ensemble average of the
Poynting’s vector:
$\langle\mathbf{S}(\mathbf{r})\rangle_{\omega}=(c/8\pi)\langle\mathbf{E}(\mathbf{r})\times\mathbf{B}^{*}(\mathbf{r})\rangle_{\omega}+CC$.
Denoting by $\lambda$ to the relevant electromagnetic wavelengths for the heat
flux, for our system, $R/L<1$ and $R/\lambda\ll 1$; we accordingly assume that
a dipolar interaction can safely describe the electromagnetic coupling between
the sphere and the substrate Tschikin et al. (2012); Manjavacas and García de
Abajo (2012).

For a given separation, $L$, the heat flux, $Q$, is then obtained from the
integrated spectral distribution Vinogradov and Dorofeev (2009b),

$Q(\mathbf{r},T_{sub},T_{ns},L)=\int_{0}^{\infty}d\omega\,S_{\omega}(\mathbf{r},T_{sub},T_{ns},L),$
(8)

where
$S_{\omega}(\mathbf{r},T_{sub},T_{ns},L)=\Delta\Theta_{\omega}(T_{ns},T_{sub})\,\rho_{\omega}(\mathbf{r},L)$,
$\Delta\Theta_{\omega}(T_{ns},T_{sub})=\Theta_{\omega}(T_{ns})-\Theta_{\omega}(T_{sub})$,
and the Planckian distribution,
$\Theta_{\omega}(T)=\hbar\omega/[\exp(\hbar\omega/k_{B}T)-1]$. On the other
hand,
$\rho_{\omega}(\mathbf{r},L,T)=\rho^{prop}_{\omega}(\mathbf{r},L,T)+\rho^{evan}_{\omega}(\mathbf{r},L,T)$
represents the electromagnetic mode density contributions of propagating and
evanescent waves to the heat flux, respectively. They are specified by
$\varepsilon_{ns}$, $\varepsilon_{sub}$, and the nanosphere polarizability,
$\alpha_{ns}=4\pi R^{3}(\varepsilon_{ns}-1)/(\varepsilon_{ns}+2)$. Their
explicit form is presented in SI.

Figure 3: (a) Real part of the YBCO permittivity at $T=2$ K as a function of
the normalized frequency $\omega/\omega_{0}$, $\omega_{0}=10^{14}$ rad/s. Each
curve corresponds to a different position in the Abrikosov lattice, marked by
asterisks in Fig. 2 (b). Vertical dashed lines indicate frequencies
corresponding to the SM excitation condition,
$\varepsilon_{ab}^{(s)}(\omega)=-1$, met for $\omega_{1}=0.322\,\omega_{0}$
and $\omega_{2}=\,0.268\omega_{0}$, respectively. (b) Spatial dependence of
the heat flux $S_{\omega}(y)$ for two different values of $\omega$ indicated
as dashed vertical lines in Fig. 3 (a). Here $x/L_{x}=0.5$, $T_{ns}=300$ K and
$T_{sub}=2$ K.

Figure 2 shows the local radiative heat exchange between the YBCO nanosphere
($R=10$ nm, $T_{ns}=300$ K) and the SC YBCO substrate ($B\lesssim B_{c2}$) at
two different temperature regimes, (a) $T_{sub}=90~{}\mbox{K}$, (b)
$T_{sub}=2~{}\mbox{K}$, for $L=20~{}\mbox{nm}\ll\lambda_{T}$. The heat flux is
normalized to the total heat flux $Q_{B=0}$ arising in the absence of the
magnetic field when the substrate behaves as a homogeneous superconductor with
dielectric function $\varepsilon_{ab}^{(h)}$ ($i.e.$, $|\Psi_{0}|^{2}=1$ in
(The Abrikosov vortex structure revealed through near-field radiative heat
exchange)). This figure reveals that the otherwise homogeneous heat flux
acquires a periodic spatial dependence when the substrate supports the
Abrikosov vortex lattice. It has been formerly shown that the SC transition
causes a drop in the NFRHT as a consequence of the suppression of low-
frequency SMs, which cannot be efficiently excited by thermal photons
Castillo-López et al. (2020, 2022). On the contrary, within the vortexes, the
local density of SC carriers drops, and the presence of unscreened normal
charge carriers causes an enhanced heat exchange.

When $T_{sub}\lesssim T_{c}$, the density of SC carriers is low and the
presence of the Abrikosov lattice is curtailed. As shown in Fig. 2 (a), $Q$
reaches its maximum at the center of the vortexes and exhibits a slight
spatial modulation on the order of few percent. On the other hand, in the low-
temperature regime, at $T_{sub}=2~{}\mbox{K}$, a 6-fold contrast increase
between the minimum and maximum values of $Q$ across the unit cell can be
observed. Notably, in this case the heat flux maximum suffers a radial
displacement towards the vortex core borders forming a local basin, in
contrast with what would be naively expected by solely considering the spatial
distribution of $n_{s}^{2D}$. This puzzling behavior can be understood by
considering that the relative weight of SMs that mainly contribute to the
spectral heat flux can show a strong variation with the lattice location. In
principle, all modes associated to all frequencies participate to the heat
transfer at each spatial point of the Abrikosov lattice unit cell. However, in
order to qualitatively explain the observed spatial structure of the heat
flux, we shall restrict the discussion to those frequencies for which SMs are
excited, expecting them to dominate the radiative properties of the lattice.
Figure (3)(a) portrays the real part of $\varepsilon_{ab}^{(s)}$ as a function
of the normalized frequency $\omega/\omega_{0}$, with $\omega_{0}=10^{14}$
rad/s, for two relevant points corresponding to the vortex center
($y=0.5\mathcal{L}_{y}$), and the vortex core border ($y\approx
0.4\mathcal{L}_{y}$) where, according to Fig. 2, $Q$ is maximum. At these
positions, the condition for SM excitation,
$\varepsilon_{ab}^{(s)}(\omega)=-1$, is met for $\omega_{1}=0.322\,\omega_{0}$
and $\omega_{2}=0.268\,\omega_{0}$, respectively. The corresponding spectral
heat flux along the $y$-direction, $S_{\omega}(y)$, is plotted in Figure
(3)(b) for fixed $x=0.5\mathcal{L}_{x}$, confirming that $S_{\omega 1}(y)$
attains its maximum value at the vortex core, while $S_{\omega 2}(y)$ reaches
it at the core border. We verify in the SI that the contribution to the
spectral heat flux from other SM frequencies is less relevant due to the
increased mode damping. Finally, outside the vortexes $S_{\omega}$ is low
because SMs cannot be excited at low frequencies due to the high reflectivity
of the YBCO substrate.

The former results have been derived from a mean-field approach that neglects
thermal fluctuations. When these are considered, vortex matter theories
predict various states, including liquid, glassy, and nearly ordered
crystalline structures with considerable spatial periodicity. At high
temperatures and near $B_{c2}$ vortex matter may form a liquid state, whereas
by lowering the temperature, it may freeze into a glassy state. Thus, the low-
temperature regime discussed before seems to constitute the most favorable
scenario to observe the spatial modulation of NFRHT.

In conclusion, we calculated the near-field radiative heat transfer properties
of vortex material. The quantum vortexes enhance the heat flux in their
vicinity by breaking the superconducting state. The main result of this letter
is the prediction of the spatial structure of the heat flux arising from the
vortex lattice on the superconducting surfaces. Imaging spatial dependence of
the near-field heat flux is possible using a room-temperature nanoprobe raster
scanning a cold superconducting substrate subject to an external magnetic
field. Our results also suggest that each vortex acts as a heat transfer
channel through which the energy exchange between the nanoprobe and the
substrate in the near-field is selectively enhanced. We expect that by
sweeping the magnetic field, it should be possible to track the evolution of
the vortexes and the formation of the Abrikosov lattice. Abrikosov vortexes
have found several applications leading to the field of vortex matter Blatter
(1997), where Majorana states near vortexes have been predicted Akzyanov and
Rakhmanov (2017); likewise, they have been proposed as quantized information
bits for use in digital cryoelectronics Golod et al. (2015). We envision that
the Abrikosov lattice can serve as a means to create ordered arrays of thermal
sources at the nanoscale, opening a new direction of research in vortex
materials.

##  I Acknowledgments

G.P. acknowledges financial support from UNAM DGAPA PAPIIT Grant No. IN104522,
UNAM DGAPA PAPIME PE101223 and CONACyT Projects No. 1564464 and No. 1098652.
R. E-S acknowledges partial support from CONACyT Project No. A1-S-10537. The
authors acknowledge the valuable comments of R. Gutiérrez-Jáuregui. They also
acknowledge graphical design by R. Pérez González.

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# Design of linear block copolymers and $ABC$ star terpolymers that produce
two length scales at phase separation

Merin Joseph<EMAIL_ADDRESS>Daniel J. Read<EMAIL_ADDRESS>Alastair
M. Rucklidge<EMAIL_ADDRESS>[

###### Abstract

Quasicrystals (materials with long range order but without the usual spatial
periodicity of crystals) were discovered in several soft matter systems in the
last twenty years. The stability of quasicrystals has been attributed to the
presence of two prominent length scales in a specific ratio, which is 1.93 for
the twelve-fold quasicrystals most commonly found in soft matter. We propose
design criteria for block copolymers such that quasicrystal-friendly length
scales emerge at the point of phase separation from a melt, basing our
calculations on the Random Phase Approximation. We consider two block
copolymer families: linear chains containing two different monomer types in
blocks of different lengths, and $ABC$ star terpolymers. In all examples, we
are able to identify parameter windows with the two length scales having a
ratio of 1.93. The models that we consider that are simplest for polymer
synthesis are, first, a monodisperse $A_{L}BA_{S}B$ melt and, second, a model
based on random reactions from a mixture of $A_{L}$, $A_{S}$ and $B$ chains:
both feature the length scale ratio of 1.93 and should be relatively easy to
synthesise.
Macromolecules 2023, 56, 7847–7859, 10.1021/acs.macromol.3c00800 (Open
Access).

###### keywords:

Quasicrystals, Phase separation, block copolymer, Self assembly, Random Phase
Approximation

University of Leeds] School of Mathematics, University of Leeds, Leeds LS2
9JT, UK

##  1 Introduction

Quasicrystals are crystals that have long range order, and so have sharp X-ray
diffraction spectra, and yet do not have the spatial periodicity usually
associated with crystals1. They often have rotation symmetries that are
incompatible with spatial periodicity. The first examples were in metal alloys
and had icosahedral symmetry2. Subsequently Zeng et al.3 reported
quasicrystals in micelles made from wedge-shaped dendrimers: these examples
were quasicrystalline, with twelve-fold rotation symmetry in two dimensions
and periodic in the third. Twelve-fold quasicrystals have recently been
reported in systems as simple as oil–water–surfactant mixtures4. An earlier
discovery by Hayashida et al.5 was an example with the same dodecagonal
rotation symmetry in a three-component $ABC$ star terpolymer blend of
polyisoprene, polystyrene and poly(2-vinylpyridine). In this case, the bulk
properties were as if the three components were immiscible, forming tiles
composed the $A$, $B$ and $C$ block copolymers in two dimensions, with the
junction points aligned in the third. The small-angle X-ray scattering pattern
of the quasicrystal sample had two circles of wavevectors with twelve peaks on
each circle, demonstrating dodecagonal rotation symmetry and the presence of
two length scales, roughly in a ratio of $1:1.93$.

Block copolymers offer versatility in the structures they can form owing to
the wide variety of different monomers, the different ways that the monomers
can interact, and the control of the lengths of the monomer chains6. Several
recent papers use this versatility to choose polymers in such a way as to form
quasicrystal approximants, for example by choosing different monomer types in
diblock micelles7, 8, in giant surfactants9, in polymer liquid crystal
systems10, and in $ABC$ triblock copolymers11. The potential diversity of
resultant mesoscale structures in block copolymer systems is explained in
detail in the review by Huang et al.12. These structures offer the potential
of developing materials with unusual photonic bandgap behavior13, 14

There is experimental evidence that quasicrystals are associated with the
presence of two length scales in the system5, 12 in a wide range of examples
going beyond soft matter and materials science (for example, fluid dynamics15,
16, 17 and nonlinear optics18). The presence of different length scales is
qualitatively clear in the structure of the components for several of the
soft-matter systems that form QCs, including micelles with a soft corona3, 19
and star copolymers with arms of different lengths5, 20. The connection
between having two length scales and the stability of QCs is supported by a
large body of theoretical work, including from the fields of fluid dynamics
and pattern formation15, 21, 22, 23, 1, 24, 25, 26, phase field crystals27,
28, 29, 30, 31, 32, 33, 34, classical density functional theory of interacting
particles35, 36, 37, 38, 39, 40, 41, molecular dynamics42, 43, and self-
assembly of hard particles44, 45 and hard particles with shoulder
potentials46, 47. At the most basic level, the theoretical work attributes the
stability of QCs to the nonlinear three-wave interaction of waves of density
fluctuations on the two length scales. For example, when the ratio of those
length scales is $2\cos 15^{\circ}\approx 1.93$, the nonlinear interactions
between two waves of one length scale and one of the other favor density waves
that are spaced $30^{\circ}$ apart in Fourier space15, 16, 48, 17, 49, 25,
giving twelve-fold symmetry. These arguments suggest that the length scales
should usually be within a factor of two of each other to encourage QCs, with
a ratio of $1.93$ for twelve-fold QCs and $1.618$ for ten-fold or icosahedral
QCs50, 22, 1, 49, 24, 29, with other ratios stabilizing other quasicrystals51.
Nevertheless, stable QCs can also be found with larger ratios, for example, an
eight-fold quasipattern was found in a reaction–diffusion problem with a
length scale ratio of $4$26.

The theoretical arguments attribute the stability of QCs to the presence to
two length scales, but this presence is not sufficient for the formation of
QCs: even with two length scales, hexagons, lamellae or other structures of
different sizes can be stable. Nor is the presence of two length scales
necessary for the formation of QCs: in fluid dynamics, there are examples of
quasipatterns in Faraday wave experiments52 whose stability can be explained
in the context of a single length scale21, 49. Nonetheless, the presence of
two length scales in an appropriate ratio is strongly associated with the
stability of QCs31, though some tuning of parameters is usually needed to
ensure that QCs are favored over competing crystalline phases such as hexagons
and lamellae38.

In this paper, we focus on what features in the polymer design can lead to two
length scales in the instability towards phase separation. We will work in the
weak segregation limit, and use the Random Phase Approximation (RPA) to
characterize the length scales that emerge, concentrating on the point of
phase separation of block copolymer melts. This complements other theoretical
approaches to this problem, for example using self-consistent field theory53,
54, 55. The advantage of using the simpler RPA theory is that it allows a
rapid search through parameter space for likely candidates of two length scale
phase separation. The disadvantage is that the theory does not predict which
structure will ultimately be stable.

Prior work in this area56, 57, 58 considered a limited range of architectures
giving rise to two length scales. Here we extend their work by (i) working
within the same classes of architecture but increasing the explored parameter
space, (ii) extending the investigation to include copolymers formed by random
reaction, and (iii) considering three component star polymers of the form
investigated by Hayashida et al.5. One focus, especially within themes (i) and
(ii) above, has been to find structures that are as simple as possible to
synthesise whilst retaining the two length scale feature.

(a)

(b)

Figure 1: Schematic models for the block copolymers. (a) $A_{L}(BA_{S})_{n}$
with $A$ in black and $B$ in gray, and $L$ and $S$ indicating the long and
short $A$ blocks. The length fraction (of the total polymer length) of the
long $A$ block in part I is $f_{A}$, and within part II, $\phi_{A}$ is the
length fraction of the short $A$ blocks within each of the $n$ $BA_{S}$
diblocks. (b) $ABC$ star block copolymer with $A$ in black, $B$ in light gray
and $C$ in mid-gray, having length fractions $f_{A}$, $f_{B}$ and $f_{C}$
respectively.

We consider two classes of polymer architectures, both chosen to allow two
length scales to emerge, see Figure 1, and ask whether there is one or two
length scales, and in the latter case, how can the ratio of these length
scales be controlled? The first class has two types of monomer ($A$ and $B$),
while the second has three ($A$, $B$ and $C$). Each example is specified by
the proportions of the different components and the strengths of the
interactions between them. The selection of length scales during phase
separation involves a balance between the entropy of stretching the polymer
chain and the energy penalty of having incompatible monomers interacting.
Qualitatively, the phase separation lengthscales are set by the size of
subsections of the chain that repel one another. To achieve phase separation
simultaneously at two lengthscales requires fine tuning of the relative
degrees of repulsion via composition and interaction parameters. Typically,
one requires greater average repulsion (per monomer) to drive phase separation
at the short lengthscale, because the stretching energy is larger; conversely,
less repulsion is needed at the longer lengthscale. These qualitative features
are present in all examples explored below.

Linear block copolymers are much easier to manufacture compared to branched
block copolymers, so our first class of polymer (Figure 1a) explores the
design of linear block copolymers. Block copolymers that are manufactured will
normally exhibit polydispersity, but we start with a discussion of idealized
monodisperse models before introducing some aspects of polydispersity via
random assembly of the blocks. A similar architecture was considered by Nap56,
but they restricted their investigation to cases where the $A_{S}$ and $B$
blocks are of equal length; here we show that relaxing that constraint leads
to greater flexibility in the design space and the possibility of easier
synthesis. The monodisperse chains have a long section $A_{L}$ of $A$-type
monomers followed by $n$ alternating $BA_{S}$ diblocks, with shorter stretches
of $B$-type and $A$-type monomers linked back to back. Microphase separation
of the long $A_{L}$ block and the $(BA_{S})_{n}$ tail gives one length scale,
while the incompatibility between $A$ and $B$ sub-blocks within the tail can
lead to microphase separation on a second length scale. The presence of $A$
monomers in the $BA_{S}$ tail reduces the incompatibility between it and the
$A_{L}$ section. Polydispersity is achieved by starting with a mixture of
$A_{L}$, $A_{S}$ and $B$ blocks. These are allowed to react in such a way that
each $B$ block links to an $A$ block at either end, the $A_{S}$ blocks link to
$B$ blocks at either end, while the $A_{L}$ blocks only react with $B$ blocks
at one end. The result is a mixture of polymers of different lengths, starting
and ending with $A_{L}$ blocks but with different lengths of $BA_{S}\dots
A_{S}B$ blocks in between, consistent with the proportions in the initial
mixture.

The second class of polymers (Figure 1b) is an $ABC$-star structure, with
different lengths of the $A$, $B$ and $C$ arms, inspired by the polymers used
by Hayashida et al.5 Here, two length scales can emerge if one arm ($A$) is
longer than the other two ($B$ and $C$), with microphase separation between
$A$ and $B$ with $C$ together leading to the long length scale, and microphase
separation between $B$ and $C$ leading to the short length scale. Again, this
class has been explored in the literature59; the new perspective here is the
focus on phase separation with two length scales.

In each case, we explore the parameter ranges in which two length scales
emerge, and we indicate the parameters for which the ratio between the two
length scales would favor twelve-fold quasicrystals. This work is part of a
long-term effort to develop design criteria for polymers that will robustly
and spontaneously form quasicrystals.

Our main tool is the Random Phase Approximation (RPA) for polymer blends,
which is the truncation of the free energy functional at the quadratic term in
density fluctuations60, 61. The theory describes the point at which there is a
transition from a polymer melt to a phase separated structure, near the point
of the initial segregation. As a result, the theory does not identify the
final stable phase. The method uses coarse-graining of monomers into monomer
units with effective bond length (or Kuhn length) $b$, which allows the
polymer to be considered as consisting of flexible and freely rotating units
that can be described as a random walk. Microphase separation occurs when
there are density fluctuations that decrease the free energy of the
homogeneous melt, and the wavenumber of these fluctuations gives the preferred
length scale of the resulting phase separated structure. The melt phase is
(meta-)stable when a (small) density fluctuation of any wavenumber increases
the free energy.

In the case of an incompressible melt with two monomer types $A$ and $B$, and
in the absence of any specific interaction between the monomer types other
than incompressibility interactions, the RPA is concerned with the non-
interacting structure factor ${S_{0}(q)}$ of the homogeneous and isotropic
melt. This non-interacting incompressible structure factor depends on
wavenumber $q$ and is expressed in terms of correlations between small random
composition fluctuations, as described in more detail below.

Interactions between the monomer units are parameterised by the Flory
interaction parameter $\chi_{AB}$, which is related to the interaction
energies between the different monomer types. Including these interactions
leads to additional composition correlations that are described in terms of
the structure factor ${S(q)}$. This structure factor also gives the form of
the expected results of scattering in experiments that detect composition
fluctuations in the melt.

In an incompressible two-component ($A$ and $B$) copolymer, consisting of
lengths of $A$ units joined to lengths of $B$ units, possibly with repetition
or branching, and with a Flory interaction parameter $\chi_{AB}$, the
structure factor $S$ is related to the non-interacting structure factor
$S_{0}$ by62

$S(q)=\left(\frac{1}{S_{0}(q)}-\frac{2\chi_{AB}}{\Omega\rho}\right)^{-1},$ (1)

where $\Omega$ is the system volume and $\rho$ is the monomer unit density, so
$\Omega\rho$ is the total number of monomer units. When there are no
interactions ($\chi_{AB}=0$), the two structure factors are the same. The non-
interacting structure factor $S_{0}(q)$ is positive and may have a maximum at
a particular wavenumber, so $S_{0}(q)^{-1}$ may have a positive minimum. As
interactions are introduced (as $\chi_{AB}$ increases), $S(q)$ goes to
infinity at the value of $\chi_{AB}$ for which the term in brackets in eq (1)
first goes to zero, and so phase separation occurs on the length scale
corresponding to the wavenumber at which $S_{0}(q)$ is maximum. If $S_{0}(q)$
has two maxima of equal height at different wavenumbers, eq (1) indicates that
phase separation will happen simultaneously at both wavenumbers.

There are more general expression for compressible systems and for copolymers
with more than two types of monomer. In these cases, the RPA is formulated by
writing the free energy as a quadratic functional of the composition
fluctuations, with multiple Flory interaction parameters. For the melt to be
stable, this quadratic form needs to be positive definite, so phase separation
occurs when eigenvalues of the quadratic form change sign.

Read63 formulated a method to find the noninteracting structure factor for an
arbitrary block copolymer melt, using ‘self-terms’, ‘coterms’ and ‘propagator
terms’ to describe the architecture of the block copolymer, tracing the
arrangement and connections between blocks in the chain. Here, the self-terms
describe the density correlations between monomer units within a single block,
and the coterms describe the density correlations between monomers units from
two different blocks on the same polymer. The chain between two different
blocks may have other blocks in between, depending on the architecture of the
polymer, and the propagator terms specify how the correlations (given by
coterms) are modified by the presence of the intermediate blocks. We use this
method to compute the structure factor for our two classes of copolymer
architecture.

##  2 Two component linear chain with fixed architecture

In our two component block copolymer system shown in Figure 1(a), we index
each polymer chain within the system by $\alpha$, so $1\leq\alpha\leq n_{c}$,
where $n_{c}$ is the number of chains. Within each chain $\alpha$, we index
the location of each monomer unit by $l$, with $1\leq l\leq N$, where $N$ is
the number of monomer units in each chain, assumed (in this section) to be the
same for each chain. The location of the monomer unit is thus
$\bm{r}_{l}^{\alpha}$, and each monomer unit is of type $A$ or type $B$. The
density of $A$ monomer units in physical space is then a sum of delta
functions, summed over all chains and all locations on each chain of the $A$
monomer units, and similarly for $B$. The Fourier transforms of these two
density distributions are then

$\displaystyle\rho_{\bm{q}}^{A}$
$\displaystyle=\sum_{A}\exp(i\bm{q}\cdot\bm{r}_{l}^{\alpha}),$ (2)
$\displaystyle\rho_{\bm{q}}^{B}$
$\displaystyle=\sum_{B}\exp(i\bm{q}\cdot\bm{r}_{l}^{\alpha}),$

where $\bm{q}$ is the wavevector, and the sums are taken over the locations of
the $A$ and $B$ monomer units. The structure factors for the $A$ and $B$
monomer units in the absence of any interactions are given by the correlations
in the densities of the two types, which in Fourier space can be written as

$\displaystyle S_{0}^{AA}(q)$
$\displaystyle=\langle\rho_{-\bm{q}}^{A}\rho_{\bm{q}}^{A}\rangle_{0},$ (3)
$\displaystyle S_{0}^{AB}(q)$
$\displaystyle=\langle\rho_{-\bm{q}}^{A}\rho_{\bm{q}}^{B}\rangle_{0}=S_{0}^{BA}(q),$
$\displaystyle S_{0}^{BB}(q)$
$\displaystyle=\langle\rho_{-\bm{q}}^{B}\rho_{\bm{q}}^{B}\rangle_{0},$

where the angle brackets with subscript $0$ represent the average over all
possible configurations of the polymers in the absence of any interactions
between monomers, within the constraints of the polymer architecture.

The incompressibility constraint requires
$\rho_{\bm{q}}^{A}=-\rho_{\bm{q}}^{B}=\rho_{\bm{q}}$, so we need only use
$\rho_{\bm{q}}$ in place of $\rho_{\bm{q}}^{A}$ and $-\rho_{\bm{q}}$ in place
of $\rho_{\bm{q}}^{B}$. Using the RPA61, we can express the incompressible
structure factor in the absence of interactions as:

$S_{0}(q)=\frac{S_{0}^{AA}S_{0}^{BB}-(S_{0}^{AB})^{2}}{S_{0}^{AA}+S_{0}^{BB}+2S_{0}^{AB}}$
(4)

and the free energy functional up to second order in density fluctuations (in
units of $k_{B}T$) as:

$F\\{\rho_{\bm{q}}\\}=\frac{1}{2}\sum_{\bm{q}}\rho_{\bm{q}}\rho_{-\bm{q}}\left(\frac{1}{S_{0}(q)}-\frac{2\chi_{AB}}{\Omega\rho}\right).$
(5)

If the term in brackets in eq (5) is positive for all wavenumbers, the melt is
(meta-)stable, while if this term changes sign, the melt is unstable. So, as
in the discussion of eq (1), the length scale of phase separation is
associated with a maximum of $S_{0}(q)$.

In eq (5), $\Omega\rho$ is equal to the total number of monomer units, and
with $n_{c}$ chains each having $N$ monomer units, we have
$\Omega\rho=n_{c}N$. We see below that $S_{0}(q)$ is proportional to
$n_{c}N^{2}$, so by treating the Flory interaction parameter $\chi_{AB}$ in
combination with $N$, all dependence on $N$ can be isolated into $N\chi_{AB}$.

Figure 2: We index each polymer chain within the system by $\alpha$, and
$\gamma$ and $\gamma^{\prime}$ are two blocks within the chain (these could be
$A_{L}$, $A_{S}$ or $B$).

We use the method of Read63 to calculate the term ${S_{0}^{AA}(q)}$,
${S_{0}^{AB}(q)}$ and ${S_{0}^{BB}(q)}$ in the absence of any interaction
between the monomers, for the $A_{L}(BA_{S})_{n}$ structure, previously
considered by Nap56, and shown in Figure 1(a). The method treats each polymer
chain $\alpha$ by splitting it into different blocks, with each block being of
a single monomer type. We illustrate this in Figure 2, with $\gamma$ and
$\gamma^{\prime}$ indicating two different blocks. Each block is associated
with a ‘self-term’ $J_{\gamma}$, a ‘coterm’ $H_{\gamma}$ and a ‘propagator
term’ $G_{\gamma}$63. The self-term for an individual block $\gamma$ gives the
contribution to the structure factor from that block, and comes from summing
over monomer pairs within that block. The contribution from the interaction
between two blocks $\gamma$ and $\gamma^{\prime}$ is the coterm for block
$\gamma$ multiplied by the propagator terms of all the blocks on the unique
connecting path between $\gamma$ and $\gamma^{\prime}$ (see Figure 2), and
finally multiplied by the coterm for block $\gamma^{\prime}$ at the end. This
contains all pairwise monomer interactions between monomers in the two blocks
$\gamma$ and $\gamma^{\prime}$. The structure factor for a given polymer chain
$\alpha$ is then a combination of the sum of the self-terms for each block and
a double sum of the product of coterms with appropriate propagators over all
non-identical pairs of blocks in that chain.

Each block $\gamma$ has its own normalised wavenumber $Q_{\gamma}$, dependent
on the number of monomer units $N_{\gamma}$ in the block:

$Q_{\gamma}^{2}=\frac{N_{\gamma}b^{2}}{6}q^{2},$ (6)

where $q=|\bm{q}|$. This wavenumber is scaled by the effective bond length
$b$; for the sake of simplicity we take same length $b$ for all blocks. Each
of the self-, co- and propagator terms are functions of these normalized
wavenumbers $Q_{\gamma}$ for each block $\gamma$. These terms are all based on
Debye functions60 and are given by63

$\displaystyle\begin{aligned} \text{self-
term:}\quad&J_{\gamma}=N_{\gamma}^{2}j(Q_{\gamma}^{2})\hskip 28.45274pt\text{
where }j(Q^{2})=\frac{2}{Q^{4}}(\exp{(-Q^{2})}-1+Q^{2}),\\\
\text{coterm:}\quad&H_{\gamma}=N_{\gamma}h(Q_{\gamma}^{2})\hskip
28.45274pt\text{ where }h(Q^{2})=\frac{1}{Q^{2}}(1-\exp{(-Q^{2})}),\\\
\text{propagator term:}\quad&G_{\gamma}=\exp(-Q_{\gamma}^{2}).\end{aligned}$
(7)

Turning now to the specific $A_{L}(BA_{S})_{n}$ architecture, the polymer
chain is considered in two parts. Part I is the $A_{L}$ block with length
fraction $f_{A}$, and part II is the tail $(BA_{S})_{n}$, comprising $n$
$BA_{S}$ diblocks, with length fraction $1-f_{A}$. Within each $BA_{S}$
diblock, the $A_{S}$ block has a length fraction of $\phi_{A}$ and hence
length fraction of the $B$ block is $1-\phi_{A}$. If the total number of
monomer units in a chain is $N$ then the number of $A$ and $B$ units in each
block of the polymer chain can be written down:

$\displaystyle\text{Number of $A$ monomer units in the }A_{L}\text{ block in
part~{}{I}}:$ $\displaystyle N_{A_{L}}=f_{A}N;$ (8) $\displaystyle\text{Number
of $A$ monomer units in each }A_{S}\text{ block in part~{}{II}}:$
$\displaystyle N_{A_{S}}=\frac{1}{n}(1-f_{A})\phi_{A}N;$
$\displaystyle\text{Number of $B$ monomer units in each }B\text{ block in
part~{}{II}}:$ $\displaystyle N_{B}=\frac{1}{n}(1-f_{A})(1-\phi_{A})N.$

We note that the two types of $A$ chain have the same lengths when
$N_{A_{L}}=N_{A_{S}}$, or

$\phi_{A}=n\frac{f_{A}}{1-f_{A}}.$ (9)

For the specific $A_{L}(BA_{S})_{n}$ polymer, the three normalised wavenumbers
$Q_{A_{L}}$, $Q_{A_{S}}$ and $Q_{B}$ are given by

$\displaystyle Q_{A_{L}}^{2}$ $\displaystyle=f_{A}Q^{2},$ (10) $\displaystyle
Q_{A_{S}}^{2}$ $\displaystyle=\frac{1}{n}(1-f_{A})\phi_{A}Q^{2},$
$\displaystyle Q_{B}^{2}$
$\displaystyle=\frac{1}{n}(1-f_{A})(1-\phi_{A})Q^{2},$

where $Q^{2}=\frac{Nb^{2}}{6}q^{2}$.

To calculate the structure factor, we start by treating parts I and II
separately, and we illustrate the calculation for the case $n=2$ before
discussing the case of general $n$.

In part I, there is only the $A_{L}A_{L}$ self-term:

${J_{AA}^{I}}={J_{A_{L}}}=N_{A_{L}}^{2}\,j(Q_{A_{L}}^{2})=N^{2}f_{A}^{2}\,j(f_{A}Q^{2}).$
(11)

There are no $J_{BB}^{I}$ and $J_{AB}^{I}$ self-terms. We have given the
explicit dependence on $N_{A_{L}}$ and $Q_{A_{L}}$, and on $N$ and $Q$, in
this case, but we will suppress this below. We note that, here and below, all
terms and the final expressions can written as functions of the scaled
wavenumber $Q$, and that the self-terms, and the coterm–propagator term–coterm
combinations, will be proportional to $N^{2}$.

In part II, we work out composite self-terms for the
$(BA_{S})_{2}=BA_{S}BA_{S}$ chain: $J_{AA}^{II}$, $J_{BB}^{II}$ and
$J_{AB}^{II}$. For $J_{AA}^{II}$, each $A_{S}$ block can interact with itself,
yielding a self-term $J_{A_{S}}$ multiplied by $2$ since there are two of
these. Each $A_{S}$ block can also interact with the other $A_{S}$ block: for
this $A_{S}A_{S}$ interaction, we use a coterm ($H_{A_{S}}$ at each end) and a
propagator term ($G_{B}$) to jump across the $B$ block. There is a factor of
$2$ since the interaction since either $A_{S}$ block could be the starting
point. Putting these together results in

$J_{AA}^{II}=2J_{A_{S}}+2H_{A_{S}}G_{B}H_{A_{S}}.$ (12)

Similarly, $J_{BB}^{II}$ is calculated by considering the two self-terms
$J_{B}$ and the coterm–propagator term–coterm chain in each direction:

$J_{BB}^{II}=2J_{B}+2H_{B}G_{A_{S}}H_{B}.$ (13)

The last term within part II is $J_{AB}^{II}$, starting with an $A_{S}$ block
and ending with a $B$ block. In this case, we have coterms $H_{A_{S}}H_{B}$
for every instance of adjacent $A_{S}$ and $B$ blocks, and we have
coterm–propagator term–propagator term–coterm chains for the $A_{S}$ and $B$
blocks separated by the other $A_{S}$ and $B$ blocks:

$J_{AB}^{{II}}=3H_{A_{S}}H_{B}+H_{B}G_{A_{S}}G_{B}H_{A_{S}}.$ (14)

There is an equal expression for $J_{BA}^{{II}}$, starting with a $B$ block
and ending with an $A_{S}$ block.

Finally, we consider interactions between parts I and II. The $A_{L}$ block
has the self-term $J_{A_{L}}$ and coterm $H_{A_{L}}$, and will interact with
the $A_{S}$ blocks and $B$ blocks in part II. The $AA$ interactions lead to
contributions $2H_{A_{L}}G_{B}H_{A_{S}}$ and
$2H_{A_{L}}G_{B}G_{A_{S}}G_{B}H_{A_{S}}$, with the factor $2$ in each case
since the $AA$ interactions can start with $A_{L}$ or $A_{S}$. The $AB$
interactions, starting with $A_{L}$ and ending with $B$, are $H_{A_{L}}H_{B}$
and $H_{A_{L}}G_{B}G_{A_{S}}H_{B}$.

These contributions are combined to give the three terms:

$\displaystyle S_{0}^{AA}$
$\displaystyle=n_{c}\left(J_{AA}^{I}+J_{AA}^{{II}}+2H_{A_{L}}G_{B}H_{A_{S}}+2H_{A_{L}}G_{B}G_{A_{S}}G_{B}H_{A_{S}}\right),$
(15) $\displaystyle S_{0}^{AB}$
$\displaystyle=n_{c}\left(J_{AB}^{{II}}+H_{A_{L}}H_{B}+H_{A_{L}}G_{B}G_{A_{S}}H_{B}\right),$
$\displaystyle S_{0}^{BB}$ $\displaystyle=n_{c}J_{BB}^{{II}},$

where we have multiplied by the number of chains $n_{c}$.

In the case of general $n$, the part I term remains the same. For the other
terms, we end up with longer expressions involving additional powers of
$G_{A_{S}}G_{B}$ propagator terms. After some combinatorics, and summing the
resulting finite geometric series, the general $n$ composite self-terms for
part II are

$\displaystyle J_{AA}^{{II}}$
$\displaystyle=nJ_{A_{S}}+2H_{A_{S}}^{2}G_{B}\left(\frac{n\left(1-G_{A_{S}}G_{B}\right)-\left(1-(G_{A_{S}}G_{B})^{n}\right)}{\left(1-G_{A_{S}}G_{B}\right)^{2}}\right),$
(16) $\displaystyle J_{BB}^{{II}}$
$\displaystyle=nJ_{B}+2H_{B}^{2}G_{A_{S}}\left(\frac{n\left(1-G_{A_{S}}G_{B}\right)-\left(1-(G_{A_{S}}G_{B})^{n}\right)}{\left(1-G_{A_{S}}G_{B}\right)^{2}}\right),$
$\displaystyle J_{AB}^{{II}}$
$\displaystyle=H_{A_{S}}H_{B}\left(\frac{(2n+1)\left(1-G_{A_{S}}G_{B}\right)-2+(G_{A_{S}}G_{B})^{n}+(G_{A_{S}}G_{B})^{n+1}}{\left(1-G_{A_{S}}G_{B}\right)^{2}}\right).$

There is an equal expression for $J_{BA}^{{II}}=J_{AB}^{{II}}$. Then, the
three terms in the total structure factor, for $n_{c}$ chains, are:

$\displaystyle S_{0}^{AA}$
$\displaystyle=n_{c}\left(J_{AA}^{{I}}+J_{AA}^{{II}}+2H_{A_{L}}H_{A_{S}}G_{B}\left(\frac{1-(G_{A_{S}}G_{B})^{n}}{1-G_{A_{S}}G_{B}}\right)\right),$
(17) $\displaystyle S_{0}^{BB}$ $\displaystyle=n_{c}J_{BB}^{{II}},$
$\displaystyle S_{0}^{AB}$
$\displaystyle=n_{c}\left(J_{AB}^{{II}}+H_{A}^{{I}}H_{B}\left(\frac{1-(G_{A_{S}}G_{B})^{n}}{1-G_{A_{S}}G_{B}}\right)\right).$

These three terms are proportional to $n_{c}N^{2}$ and are functions of the
scaled wavenumber $Q$. The three terms are then combined to give the overall
noninteracting incompressible struture factor $S_{0}(q)$ using eq (4), also
proportional to $n_{c}N^{2}$ and a function of $Q$.

Figure 3: The incompressible structure factor $S_{0}$ (scaled by $n_{c}N^{2}$)
from eq (4) as a function of scaled wavenumber $Q$ for $f_{A}=0.39225$,
$\phi_{A}=0.85$ and $n=5$. The two maxima in the plot indicate the two length
scales of phase separation. In this example, the wavenumbers at the maxima are
$Q_{1}=2.53$ and $Q_{2}=6.85$, which have a ratio of $2.70$, and the maxima
are at the same height.

Typically, for phase separation on one length scale, the structure factor plot
will have a single dominant peak. We have chosen the architecture
$A_{L}(BA_{S})_{n}$ in order to allow phase separation with two length scales,
which will manifest as two maxima, at scaled wavenumbers $Q_{1}$ and $Q_{2}$
in the structure factor plot. An example of the resulting structure factor
$S_{0}$ is plotted in Figure 3 with $f_{A}=0.39225$, $\phi_{A}=0.85$ and
$n=5$. In this example, the structure factor has two maxima at the same
height, and the wavenumbers at these two maxima are in the ratio $2.70$. The
two maxima will in general have different heights, with the higher one
indicating the wavenumber that will appear first in phase separation. We
define the ratio of wavenumbers $Q_{r}=Q_{2}/Q_{1}$, with $Q_{1}<Q_{2}$.

Figure 4: Regions of a single maximum (white) and two maxima (shaded), as a
function of $f_{A}$ and $\phi_{A}$ with $n=1.5$, $2$, $5$ and $10$. In the
shaded regions, the darker (resp. lighter) areas are where the maximum with
the smaller (resp. larger) wavenumber is higher. The solid contour line across
the cusp indicates a wavenumber ratio $Q_{r}=3.5$, with dashed, dash-dot and
dotted lines indicating wavenumber ratios of $2.5$, $2.0$ and $1.5$
respectively. The teal line indicates parameter where the wavenumber ratio is
$1.93$. The maroon line is where the two types of $A$ blocks have the same
length, using eq (9).

Whether there is one maximum or two maxima, and their heights and values of
the wavenumbers at the maxima, depends on the model parameters $f_{A}$,
$\phi_{A}$ and $n$. We have computed the structure factor for all possible
combinations of $f_{A}$ and $\phi_{A}$, with $1\leq n\leq 10$. A summary of
the results is presented in Figure 4. As happen in other systems with
transitions between one and two length scales, the boundary between the
regions in parameter space separating one from two length scales are cusp-
shaped56, 58, 64. Within the cusps (shaded), there are two maxima in the non-
interacting structure factor and hence two length scales in the phase
separation.

The architecture with $n=1$, $A_{L}BA_{S}$, does not give any two length scale
phase separation. The smallest model in the $A_{L}(BA_{S})_{n}$ family that
gives a cusp is that with $n=2$ (Figure 4b), which is a linear chain with only
$5$ blocks. From Figure 4(b,c,d), we see that the area of the cusps increases
with $n$, and the maximum ratio between the two length scales increases as
well. This is because the short length scale is set by the size of the
$BA_{S}$ blocks, while the long length scale is set by the overall size of the
polymer, which increases with $n$. The smallest linear chain with two
components that gives two length scale phase separation is $A_{L}BA_{S}B$,
indicated by $n=1.5$ in Figure 4(a), with the structure factor computed along
the same lines as described above.

We also show in Figure 4 the lines given by eq (9), where the two $A$ blocks
have the same lengths. We note that these lines do not intersect the cusps,
which shows that having $A$ blocks of different lengths is needed to have two
length scales at the point of phase separation.

##  3 Two component linear chain with random assembly

The polydisperse model involves the random assembly of $A_{L}$, $A_{S}$ and
$B$ blocks using the Markov chain method proposed by Read63. In this model,
the $A_{L}$ blocks have one reactive end, the $A_{S}$ blocks have two reactive
ends, and the $B$ blocks also have two reactive ends. The $A$ reactive ends
combine with $B$ reactive ends to form linear chains. At the end of
polycondensation, assuming a complete reaction and stoichiometry, the mixture
will contain only chains that have $A_{L}$ blocks at both ends and different
lengths of $BA_{S}\dots A_{S}B$ blocks in between, for example, $A_{L}BA_{L}$,
$A_{L}BA_{S}BA_{L}$, etc., as illustrated in Figure 5.

Figure 5: Schematic representation of the polydisperse model. Starting with a
mixture of $A_{L}$, $A_{S}$ and $B$ (top), the final mixture (bottom) will
contain polymer chains of architecture $A_{L}BA_{L}$, $A_{L}BA_{S}BA_{L}$,
etc.

The polycondensation process starts with initial block fractions
$\beta_{A_{L}}$, $\beta_{A_{S}}$ and $\beta_{B}$ for $A_{L}$, $A_{S}$ and $B$
blocks respectively, with $\beta_{A_{L}}+\beta_{A_{S}}+\beta_{B}=1$. In the
reaction, all the $B$ type ends will react with $A$ type ends, so the initial
mixture must contain the same number of each type. Thus

$2n_{blocks}\beta_{B}=n_{blocks}\beta_{A_{L}}+2n_{blocks}\beta_{A_{S}},$ (18)

where $n_{blocks}$ is the total number of blocks. Eliminating $\beta_{B}$, the
condition for complete reaction is

$3\beta_{A_{L}}+4\beta_{A_{S}}=2.$ (19)

In order to use the RPA for this system, we need the number of monomer units
in each block. We use the number of monomer units in the $A_{L}$ block
($N_{A_{L}}$) to scale the numbers in the $A_{S}$ and $B$ blocks ($N_{A_{S}}$
and $N_{B}$), using scaling factors $\nu_{A_{S}}$ and $\nu_{B}$:

$\begin{split}N_{A_{S}}&=\nu_{A_{S}}N_{A_{L}},\\\
N_{B}&=\nu_{B}N_{A_{L}}.\end{split}$ (20)

Thus the three parameters that describe the model are $\beta_{A_{L}}$,
$\nu_{A_{S}}$ and $\nu_{A_{S}}$.

Polymerization of such a mixture is by random assembly. At each stage in the
polymerization, an $A$ reactive end is always followed by a $B$ block, while a
$B$ reactive end combines with $A$ reactive ends from $A_{L}$ or $A_{S}$
blocks with probability given by the proportion of the two types. In terms of
the model parameters, the probability that a $B$ block will be followed by an
$A_{L}$ block is $P_{A_{L}B}=\frac{\beta_{A_{L}}}{2\beta_{B}}$, and that a $B$
block will be followed by an $A_{S}$ block is
$P_{A_{S}B}=\frac{\beta_{A_{S}}}{\beta_{B}}$, so from eq (18), we have
$P_{A_{L}B}+P_{A_{S}B}=1$. The probabilities that a $B$ block follows $A_{L}$
or $A_{S}$ blocks are both $1$, and the probabilities of other combinations
(for example, $A_{S}$ followed by $A_{L}$) are zero.

This is a Markov process, where the block that gets attached depends only on
the last block, and the structure factor can be computed by the methodology of
Read63. The ideas are an extension of those discussed above, but with infinite
rather than finite geometric series. For example, for the contribution to the
$S_{0}^{AA}$ part of the non-interacting structure factor from the $A_{L}$ to
$A_{L}$ interactions, there are two coterm $H_{A_{L}}$ factors for each end,
as well as sums of propagator terms corresponding to all the possible chains
that can go between the two ends, weighted by the probability of that chain.
So, a single $B$ block contributes $P_{A_{L}B}G_{B}$, a $BA_{S}B$ chain
contributes $P_{A_{S}B}G_{B}G_{A_{S}}P_{A_{L}B}G_{B}$, a $BA_{S}BA_{S}B$ chain
contributes $(P_{A_{S}B}G_{B}G_{A_{S}})^{2}P_{A_{L}B}G_{B}$, and so on. The
infinite geometric series, with common factor $P_{A_{S}B}G_{B}G_{A_{S}}$, can
readily be summed, for this and for the other interactions65. The outcome of
these calculations is

$\begin{split}S_{A_{L}A_{L}}&=N_{A_{L}}\Omega\rho\frac{\beta_{A_{L}}}{\beta_{A_{L}}+\nu_{A_{S}}\beta_{A_{S}}+\nu_{B}\beta_{B}}\left(j_{A_{L}}+h_{A_{L}}^{2}\frac{P_{A_{L}B}G_{B}}{1-P_{A_{S}}BG_{A_{S}}G_{B}}\right),\\\
S_{A_{L}A_{S}}&=N_{A_{L}}\Omega\rho\frac{\nu_{A_{S}}\beta_{A_{L}}}{\beta_{A_{L}}+\nu_{A_{S}}\beta_{A_{S}}+\nu_{B}\beta_{B}}h_{A_{L}}h_{A_{S}}\frac{P_{A_{S}B}G_{B}}{1-P_{A_{S}B}G_{A_{S}}G_{B}},\\\
S_{A_{L}B}&=N_{A_{L}}\Omega\rho\frac{\nu_{B}\beta_{A_{L}}}{\beta_{A_{L}}+\nu_{A_{S}}\beta_{A_{S}}+\nu_{B}\beta_{B}}h_{A_{L}}h_{B}\frac{1}{1-P_{A_{S}B}G_{A_{S}}G_{B}},\\\
S_{A_{S}A_{S}}&=N_{A_{L}}\Omega\rho\frac{\nu_{A_{S}}^{2}\beta_{A_{S}}}{\beta_{A_{L}}+\nu_{A_{S}}\beta_{A_{S}}+\nu_{B}\beta_{B}}\left(j_{A_{S}}+2h_{A_{S}}^{2}\frac{P_{A_{S}B}G_{B}}{1-P_{A_{S}B}G_{A_{S}}G_{B}}\right),\\\
S_{A_{S}B}&=N_{A_{L}}\Omega\rho\frac{\nu_{A_{S}}\nu_{B}\beta_{A_{S}}}{\beta_{A_{L}}+\nu_{A_{S}}\beta_{A_{S}}+\nu_{B}\beta_{B}}2h_{A_{S}}h_{B}\frac{1}{1-P_{A_{S}B}G_{A_{S}}G_{B}},\\\
S_{BB}&=N_{A_{L}}\Omega\rho\frac{\nu_{B}^{2}\beta_{B}}{\beta_{A_{L}}+\nu_{A_{S}}\beta_{A_{S}}+\nu_{B}\beta_{B}}\left(j_{B}+2h_{B}^{2}\frac{P_{A_{S}B}G_{A_{S}}}{1-P_{A_{S}B}G_{A_{S}}G_{B}}\right).\end{split}$
(21)

The overall structure factors $S_{0}^{AA}$, $S_{0}^{BB}$ and $S_{0}^{AB}$ are

$\begin{split}S_{0}^{AA}&=S_{A_{L}A_{L}}+2S_{A_{L}A_{S}}+S_{A_{S}A_{S}}\\\
S_{0}^{BB}&=S_{BB}\\\ S_{0}^{AB}&=S_{A_{L}B}+S_{A_{S}B}\end{split}$ (22)

These are combined into the non-interacting structure factor expression in eq
(4).

Thus, the non-interacting structure factor $S_{0}(q)$ is determined as a
function of the normalized wavenumber and the three parameters: the block
fraction $\beta_{A_{L}}$ and the monomer fractions $\nu_{A_{S}}$ and
$\nu_{B}$. For fixed $\beta_{A_{L}}$, the regions of $(\nu_{A_{S}},\nu_{B})$
where there are two maxima in the structure factor are once again cusp-shaped
(see Figure 6 for $\beta_{A_{L}}=0.2$ and $\beta_{A_{L}}=0.45$). The lines
across the shaded part of the cusps indicate the ratio between the wavenumbers
where the two peaks occur. When $\beta_{A_{L}}$ is $0.2$ (with
$\beta_{A_{S}}=0.35$ and $\beta_{B}=0.45$), there are more $A_{S}$ and $B$
blocks that $A_{L}$ block, so the polymer will form longer chains, and the
region for two length scale phase separation is larger than with
$\beta_{A_{L}}=0.45$ ($\beta_{A_{S}}=0.1625$ and $\beta_{B}=0.3875$). In both
these cases, the length scale ratio corresponding to $12$-fold symmetry is
indicated by the teal line.

Figure 6: Top row: regions of single maxima (white) and two maxima (shaded) as
a function of $\nu_{A_{L}}$ and $\nu_{B}$ with $\beta_{A_{L}}=0.2$ and $0.45$,
for the two component linear chain with random assembly. In the shaded
regions, the darker (resp. lighter) areas are where the maximum with the
smaller (resp. larger) wavenumber is higher. The solid contour line across the
cusp indicates a wavenumber ratio $Q_{r}=3.5$, with dashed, dash-dot and
dotted lines indicating wavenumber ratios of $2.5$, $2.0$ and $1.5$,
respectively. The teal line indicates where the wavenumber ratio is $1.93$.
Bottom row: we plot the results for the monodisperse versions with (c) $n=5$
and (d) $n=2$ in terms of equivalent values of $\nu_{A_{L}}$ and $\nu_{B}$.

Direct mapping between the monodisperse fixed $n$ and polydisperse random
assembly versions of the two component linear chain models is not possible,
because of the range of chain lengths possible in the second case, and because
of the difference in architecture. But, roughly speaking, $\beta_{A_{L}}$ is
inversely proportional to the number of $A_{S}$ and $B$ blocks available in
the polydisperse model, which is related to the number $n$ of $BA_{S}$
diblocks in the monodisperse case. However, for fixed $n$ in the monodisperse
case, it is possible to convert the parameters from $(f_{A},\phi_{A})$ in that
case to $(\nu_{A_{S}},\nu_{B})$, but with keeping the $n$ the same. The two
length scale regions in the polydisperse case (Figure 6, top row, with
$\beta_{A_{L}}=0.2$ and $\beta_{A_{L}}=0.45$) are comparable to the two length
scale regions in the monodisperse case (Figure 6, bottom row, with $n=5$ and
$n=2$). This supports the hypothesis that longer chains and the presence of
$A_{S}$ blocks encourage two length scale phase separation.

##  4 Three component $ABC$ star

For a polymer system with three components and with Fourier transformed
monomer densities $\rho_{{\bm{q}}}^{A}$, $\rho_{{\bm{q}}}^{B}$ and
$\rho_{{\bm{q}}}^{C}$ for $A$, $B$ and $C$ monomer units respectively, the
procedure for calculating the partition function is similar to the two
component system, though imposing the incompressibilty condition is more
involved. The non-interacting (compressible) structure factor matrix $M_{q}$
is a three-by-three matrix whose components are $S_{0}^{AA}$, …, $S_{0}^{CC}$:

$M_{q}=\begin{bmatrix}S_{0}^{AA}&S_{0}^{AB}&S_{0}^{AC}\\\
S_{0}^{AB}&S_{0}^{BB}&S_{0}^{BC}\\\
S_{0}^{AC}&S_{0}^{BC}&S_{0}^{CC}\end{bmatrix}=n_{c}N^{2}\begin{bmatrix}s_{0}^{AA}&s_{0}^{AB}&s_{0}^{AC}\\\
s_{0}^{AB}&s_{0}^{BB}&s_{0}^{BC}\\\
s_{0}^{AC}&s_{0}^{BC}&s_{0}^{CC}\end{bmatrix},$ (23)

where, as discussed in the case of the two component system, the $S_{0}^{IJ}$
terms are proportional to $n_{c}N^{2}=\Omega\rho N$. With this scaling, the
structure factor terms $s_{0}^{IJ}$ depend only on the scaled wavenumber and
on parameters that describe the architecture of the polymer.

We need the inverse of this matrix, which we write as

$M_{q}^{-1}=\frac{1}{\Omega\rho
N}\begin{bmatrix}\Gamma^{AA}&\Gamma^{AB}&\Gamma^{AC}\\\
\Gamma^{AB}&\Gamma^{BB}&\Gamma^{BC}\\\
\Gamma^{AC}&\Gamma^{BC}&\Gamma^{CC}\end{bmatrix}.$ (24)

Then, the free energy functional is:

$F\left(\\{\rho_{{\bm{q}}}^{A},\rho_{{\bm{q}}}^{B},\rho_{{\bm{q}}}^{C}\\}\right)=\frac{1}{2}\sum_{\bm{q}}\begin{bmatrix}\rho_{{\bm{q}}}^{A}&\rho_{{\bm{q}}}^{B}&\rho_{{\bm{q}}}^{C}\end{bmatrix}\left(M^{-1}_{q}+\frac{1}{\Omega}\begin{bmatrix}V_{AA}&V_{AB}&V_{AC}\\\
V_{AB}&V_{BB}&V_{BC}\\\
V_{AC}&V_{BC}&V_{CC}\end{bmatrix}\right)\begin{bmatrix}\rho_{{-\bm{q}}}^{A}\\\
\rho_{{-\bm{q}}}^{B}\\\ \rho_{{-\bm{q}}}^{C}\end{bmatrix},$ (25)

where the three-by-three matrix of $V_{AA}$ etc. expresses the interaction
potential between the different monomer types, and will be written in terms of
$N$ times the Flory interaction parameters below.

The incompressibility in the system provides the constraint

$\rho_{{\bm{q}}}^{C}=-(\rho_{{\bm{q}}}^{A}+\rho_{{\bm{q}}}^{B}),$ (26)

which can be used to reduce the three-by-three matrices in the free energy to
two-by-two:

$F\left(\\{\rho_{{\bm{q}}}^{A},\rho_{{\bm{q}}}^{B}\\}\right)=\frac{1}{2}\sum_{\bm{q}}\begin{bmatrix}\rho_{{\bm{q}}}^{A}&\rho_{{\bm{q}}}^{B}\end{bmatrix}W_{q}\begin{bmatrix}\rho_{{-\bm{q}}}^{A}\\\
\rho_{{-\bm{q}}}^{B}\end{bmatrix}.$ (27)

Here, $W_{q}$ is a $2\times 2$ matrix that contains all the non-interacting
structure factor terms and the interaction potentials between different types
of monomers, and is defined by

$\displaystyle\Omega\rho NW_{q}=$
$\displaystyle\begin{bmatrix}\Gamma^{AA}+\Gamma^{CC}-2\Gamma^{AC}&\Gamma^{AB}-\Gamma^{BC}-\Gamma^{AC}+\Gamma^{CC}\\\
\Gamma^{AB}-\Gamma^{BC}-\Gamma^{AC}+\Gamma^{CC}&\Gamma^{BB}+\Gamma^{CC}-2\Gamma^{BC}\end{bmatrix}+{}$
(28) $\displaystyle\qquad
N\begin{bmatrix}-2\chi_{AC}&\chi_{AB}-\chi_{BC}-\chi_{AC}\\\
\chi_{AB}-\chi_{BC}-\chi_{AC}&-2\chi_{BC}\end{bmatrix}.$

Here, the Flory interaction parameters $\chi$ for the monomer pairs of $A$,
$B$ and $C$ blocks are defined by

$\displaystyle\chi_{AB}$
$\displaystyle=-\frac{\rho}{2}\left(V_{AA}+V_{BB}-2V_{AB}\right),$ (29)
$\displaystyle\chi_{BC}$
$\displaystyle=-\frac{\rho}{2}\left(V_{BB}+V_{CC}-2V_{BC}\right),$
$\displaystyle\chi_{AC}$
$\displaystyle=-\frac{\rho}{2}\left(V_{AA}+V_{CC}-2V_{AC}\right),$

where the $\chi_{AB}$, $\chi_{BC}$ and $\chi_{AC}$ describe the interaction
between pairs of monomer types.

For the $ABC$ star architecture, the $S_{0}^{AA}$, $S_{0}^{AB}$, … terms in
the $M_{q}$ matrix in eq (23) are computed using Read’s method63, though only
self-terms and coterms are needed as the polymer architecture is simpler than
in the two component system. For the $ABC$ star block copolymer system with
$N$ monomer units, we write the length fractions of the $A$, $B$ and $C$ arms
as $f_{A}$, $f_{B}$ and $f_{C}$ respectively, with

$f_{A}+f_{B}+f_{C}=1.$ (30)

The number of monomer units in each arm is then

$\displaystyle\text{Number of monomer units in the }A\text{ block}:$
$\displaystyle N_{A}=f_{A}N;$ (31) $\displaystyle\text{Number of monomer units
in the }B\text{ block}:$ $\displaystyle N_{B}=f_{B}N;$
$\displaystyle\text{Number of monomer units in the }C\text{ block}:$
$\displaystyle N_{C}=f_{C}N=(1-f_{A}-f_{B})N.$

The non-interacting structure factor terms corresponding to the $ABC$ star
architecture are

$\displaystyle\begin{aligned} S_{0}^{AA}&=J_{A}=N_{A}^{2}j_{A},\\\
S_{0}^{BB}&=J_{B}=N_{B}^{2}j_{B},\\\ S_{0}^{CC}&=J_{C}=N_{C}^{2}j_{C},\\\
S_{0}^{AB}&=H_{A}H_{B}=N_{A}N_{B}h_{A}h_{B},\\\
S_{0}^{BC}&=H_{B}H_{C}=N_{B}N_{C}h_{B}h_{C},\\\
S_{0}^{AC}&=H_{A}H_{C}=N_{A}N_{C}h_{A}h_{C}.\end{aligned}$ (32)

Here, $j_{A}$, $j_{B}$, $j_{C}$, $h_{A}$, $h_{B}$ and $h_{C}$ are the self-
terms and coterms for the $A$, $B$ and $C$ branches as in eq (7). These are
functions of the normalized wavenumbers $Q_{A}$, $Q_{B}$ and $Q_{C}$:

$\displaystyle Q_{A}^{2}$ $\displaystyle=f_{A}Q^{2},$ (33) $\displaystyle
Q_{B}^{2}$ $\displaystyle=f_{B}Q^{2},$ $\displaystyle Q_{C}^{2}$
$\displaystyle=f_{C}Q^{2}=(1-f_{A}-f_{B})Q^{2},$

where $Q^{2}=\frac{b^{2}N}{6}q^{2}$ as before. With this, now writing
wavenumber dependence in terms of $Q$, and taking incompressibility into
account, the matrix $W_{Q}$ (and its eigenvalues) is proportional to
$\frac{1}{\Omega\rho N}=\frac{1}{n_{c}N^{2}}$, and can otherwise be expressed
a function of the scaled wavenumber $Q$, the polymer block fractions $f_{A}$
and $f_{B}$, and the interaction parameters $N\chi_{AB}$, $N\chi_{BC}$ and
$N\chi_{AC}$.

Figure 7: The two eigenvalues $\lambda$ of $W_{Q}$ as functions of the
wavenumber $Q$. The two minima of the smaller eigenvalue (solid line) give the
two length scales at phase separation. In this example, the wavenumbers are
$Q_{1}=2.1096$ and $Q_{2}=3.3754$, with the ratio between the two wavenumbers
being $Q_{r}=1.60$. The other parameters are $f_{A}=0.66775$,
$f_{B}=f_{C}=0.166125$, $N\chi_{AB}=N\chi_{AC}=39.335$ and
$N\chi_{BC}=95.070$.

With $W_{Q}$ defined, the quadratic form in eq (27) is positive definite when
the eigenvalues of $W_{Q}$ are both positive, and phase separation occurs when
the smallest eigenvalue of $W_{Q}$ changes from positive to negative. The
wavenumber(s) at which this occurs gives the length scale(s) at phase
separation. We compute the eigenvalues of $W_{Q}$ as functions of the scaled
wavenumber $Q$: see Figure 7 for an example where the smaller eigenvalue of
$W_{Q}$ has two minima at zero, and so is on the verge of phase separation
with two length scales in a ratio of $1.6$.

For the three component model, the $Q$-dependent eigenvalues of $W_{Q}$ depend
on $5$ parameters: $f_{A}$, $f_{B}$, $N\chi_{AB}$, $N\chi_{BC}$ and
$N\chi_{AC}$, with $f_{C}=1-(f_{A}+f_{B})$. Hence, achieving instability to
phase separation simultaneously at two lengthscales depends both on the ratios
of the interaction parameters and on the molecular composition (this is in
contrast to the two component case, where polymer composition alone determines
the lengthscales of instability). We explore this parameter space at several
fixed values of the ratio of $\chi_{AC}$ to $\chi_{AB}$, defining
$\frac{\chi_{AC}}{\chi_{AB}}=\xi$. The first step is to select a desired ratio
of wavenumbers $Q_{r}$ (for example, $Q_{r}=1.6$) and a value of $\xi$ (for
example, $\xi=1$), taking $f_{B}=f_{C}$ as a starting point. If $\lambda(Q)$
is the smaller of the two eigenvalues, and the two minima of $\lambda(Q)$ are
at wavenumbers $Q_{1}$ and $Q_{2}=Q_{r}Q_{1}$, the four equations for a zero
double minimum of $\lambda(Q)$, as illustrated in Figure 7, are:

$\displaystyle\begin{aligned} \lambda(Q_{1})=0;\hskip
28.45274pt&\lambda(Q_{2})=0;\\\
\frac{d\lambda}{dQ}\Bigr{\rvert}_{Q_{1}}=0;\hskip
28.45274pt&\frac{d\lambda}{dQ}\Bigr{\rvert}_{Q_{2}}=0.\end{aligned}$ (34)

We solve these four equations for $Q_{1}$ and the values of the parameters
$f_{A}$, $\chi_{AB}$ and $\chi_{BC}$, with $\chi_{AC}=\xi\chi_{AB}$ and
$f_{B}=f_{C}=\frac{1}{2}(1-f_{A})$. This allows us to identify, for the choice
of $Q_{r}$, for the choice that $f_{B}=f_{C}$ and for the chosen ratio between
$\chi_{AB}$ and $\chi_{AC}$, the other parameters at which phase separation
occurs. We then keep the choice of $Q_{r}$ and the ratio between $\chi_{AB}$
and $\chi_{AC}$ fixed, but allow $f_{B}\neq f_{C}$ to compute nearby solutions
to the same four equations. In this way, we build up curves in the
$(f_{A},f_{B},f_{C})$ space on which phase separation occurs at two length
scales simultaneously. We choose other values of $Q_{r}$ to get several curves
in that space, as illustrated in Figure 8, for three choices of $\xi$.

Figure 8: Curves of constant wavenumber ratio $Q_{r}$, shown as contour plots
in $(f_{A},f_{B},f_{C})$ diagrams, for three choices of $\xi$: $\xi=1$, $0.7$
and $1.3$. The outer contour in the petal shaped regions is $Q_{r}=1.2$, with
$Q_{r}$ increasing to $2.4$ in steps of $0.2$ in the $\xi=1$ case and to $2.5$
in steps of $0.1$ in the other two cases. The purple contour is $Q_{r}=1.93$.

In Figure 8, we focus on the case where the $A$ block has the longer arms, so
we take $f_{A}>\frac{1}{3}$. There will be other curves of two length scale
phase separation in the other corners of the triangle. The outer contour in
the petal shaped regions of two length scale phase separation is $Q_{r}=1.2$,
with $Q_{r}$ increasing on the inner contours, with the larger $Q_{r}$
corresponding to $f_{A}$ closer to $1$. In the case $\xi=1$, when $B$ and $C$
have the same interaction potential with $A$, the petal region is symmetric
under reflections in the $f_{B}=f_{C}$ diagonal (dotted line). With smaller
$\xi$ ($\chi_{AC}<\chi_{AB}$) the petal bends upwards towards the $C$ region
with larger $f_{C}$. Similarly, with larger $\xi$, the petal region bends
downwards towards the $B$ region. The region of having two length scales is
quite narrow, and is located in the parameter region where $A$ is long and the
other two are of similar sizes. Although we have only computed the contours up
to $Q_{r}=2.5$, we expect that larger length scale ratios would be possible
for $f_{A}$ closer to $1$.

##  5 Conclusion

We have investigated the length scales that emerge at the point of phase
separation in two classes of block copolymer models. The first has alternating
lengths of polymers of two different types, and the second has three different
monomer types in a terpolymer star configuration. In both cases, we find that
as well as having a single length scale at phase separation, it is possible to
design the polymers so that two length scales emerge. The transition from one
to two length scales occurs at a point (a cusp) in the parameter space when
the length scale ratio is one. Beyond this cusp, the length scale ratio can be
made much larger than one. In principle, the ideas and techniques developed
here apply to arbitrary configurations of different types of monomers: we have
explored only the simplest examples.

We have identified experimentally accessible architectures that have phase
separation at the length scale ratio 1.93 that favors twelve-fold
quasicrystals. In the $A_{L}(BA_{S})_{n}$ case, the smallest molecule has
$n=1.5$, and the parameters are $(f_{A},\phi_{A})\approx(0.7,0.9)$ (Figure
4a), which corresponds to an $A_{L}BA_{S}B$ structure, with $A_{L}$ being
$70\%$ of the length of the chain, the two $B$’s are about $2.7\%$ each and
the $A_{S}$ is $24.6\%$. For molecules with more repeating $BA_{S}$ units, the
parameters that have phase separation with two length scales have larger
proportions of $B$ and smaller proportions of $A_{L}$, but these will be
harder to manufacture. In the case of random assembly, an example choice of
parameters is a mixture that is $20\%$ $A_{L}$, $35\%$ $A_{S}$ and $45\%$ $B$,
where the lengths of the three chains are in a ratio
$A_{L}:A_{S}:B=1:0.39:0.14$ (see Figure 6a). Of course, attention will also
need to be paid to the values of $N\chi$ that are needed at the point of
instability, and the relative heights of the peaks in the structure factor.
Prior work in this area56 took $\phi_{A}=\phi_{B}$; our work covers parameter
values that would allow considerably easier synthesis of the polymers, in
regimes that ought to favor the formation of quasicrystals.

Up until now, experimental observations in linear block copolymer melts have
mainly found only relatively simple structures, such as hexagons and lamellae,
or hierarchical two length scale structures, such as lamellae-within-lamellae
with several layer thicknesses66, 67. There are indications that more complex
structures can be stable: self-consistent field theory studies of linear
$ABAB$ tetrablock copolymers68 report structures including a lamella–sphere
phase and a gyroid phase. Cylindrical 12-fold quasicrystal approximants in
linear $ABCB$ terpolymer melts have also been found in self-consistent field
theory calculations69, and the Fourier transform images reported in that paper
have two length scales in the characteristic 1.93 ratio.

To our knowledge, ours is the first presentation of phase separation with two
length scales in the $ABC$ star terpolymer system. This is a system where
quasicrystals and close approximants have been found experimentally5. As in
the $A_{L}(BA_{S})_{n}$ case above, values of the three $N\chi$ parameters are
quite large, suggesting that self-consistent field theory or strong
segregation theory would be an appropriate next step.

Of course, our RPA calculations are only the first step: these reveal the
length scales but not the final stable structures. Subsequent steps, involving
weak segregation theory70, self-consistent field theory53, 54, 71, 69, 68,
strong segregation theory72, etc., have been carried out by several authors,
but finding quasicrystals has been challenging, partly because the
calculations in all these cases are quite demanding. The final structure will
be influenced by the length scale ratio and the heights of the peaks in the
structure factor or values of the eigenvalues in the dispersion relation. The
examples in Figures 3 and 7 are where these are equal at the two length
scales, but the shaded regions in Figures 4 and 6 indicate where one or other
peak height is larger. In some (phase field crystal) models of soft matter
quasicrystals, fully developed quasicrystals are found where the peaks have
the same or similar heights29, 31, while in other (density functional theory)
models, quasicrystals are found where the peaks have quite different
heights36, 37. In the models, the stability of fully developed quasicrystals
depends not only on the eigenvalues at the two length scales but also on the
(negative) eigenvalues at other length scales that might feature in regular
crystalline orderings that compete with quasicrystals38. Our work provides
this information and so should provide useful starting parameter values with
two length scale phase separation, which should be a good place to start a
search for quasicrystals or their approximants, experimentally or using more
sophisticated theoretical methods.

The authors thank Profs Andrew Archer, Ron Lifshitz, An-Chang Shi and Bart
Vorselaars for stimulating conversations, and Joanna Tumelty for advice on
numerical methods. MJ is grateful for a PhD studentship from the Soft Matter
and Functional Interfaces (SOFI) Centre for Doctoral Training (EP/L015536/1)
and the School of Mathematics, University of Leeds. AMR is grateful for
support from the Engineering and Physical Sciences Research Council
(EP/P015611/1) and the Leverhulme Trust (RF-2018-449/9).

The data associated with this paper are openly available from the University
of Leeds Data Repository (http://doi.org/10.5518/1333)73.

For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission.

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