Invitation to Random Tensors

Invitation to Random Tensors\star\star$\star$This paper is a contribution to
the Special Issue on Tensor Models, Formalism and Applications. The full
collection is available at
http://www.emis.de/journals/SIGMA/Tensor_Models.html

Razvan GURAU

R. Gurau

CPHT, Ecole Polytechnique, 91128 Palaiseau cedex, France
<EMAIL_ADDRESS>https://www.cpht.polytechnique.fr/?q=fr/node/85

Received September 21, 2016; Published online September 23, 2016

This article is preface to the SIGMA special issue “Tensor Models, Formalism
and Applications”, http://www.emis.de/journals/SIGMA/Tensor_Models.html. The
issue is a collection of eight excellent, up to date reviews on random tensor
models. The reviews combine pedagogical introductions meant for a general
audience with presentations of the most recent developments in the field. This
preface aims to give a condensed panoramic overview of random tensors as the
natural generalization of random matrices to higher dimensions.

random tensors

83C45

## Why random tensors?

General relativity, or classical gravity, is a theory of the ambient space-
time geometry. The electroweak and strong interactions, that is the other
three fundamental forces in nature, are described by perturbatively
renormalizable quantum field theory [68, 72, 73, 74, 124, 132, 142, 145]
living on this geometric background. The main lesson of general relativity is
that the ambient geometry is dynamical, and the main lesson of quantum field
theory is that all the dynamical fields must be quantized. Thus, classical
gravity predicts its own demise: a more fundamental theory, “quantum gravity”,
must come into play at some high energy scale.

While classical general relativity is a field theory, it cannot be quantized
the same way the standard model is: general relativity is _not_ perturbatively
renormalizable [70, 143]. The lack of perturbative renormalizability of
general relativity is a clear indicator that “quantum gravity” is quite
different from the classical theory of gravity. This is why minimalistic
approaches should be taken with a grain of salt: gravity is weakly coupled in
the infrared, hence strongly coupled in the ultraviolet. This suggest that the
fundamental degrees of freedom of quantum gravity are quite different from the
geometric degrees of freedom perceived by low energy observers. It is far more
likely that the geometric infrared degrees of freedom are just bound states of
the genuine quantum gravity ultraviolet degrees of freedom.

Over the years several candidate quantum gravity theories have been developed,
most notably string theory. While much remains to be learned about the elusive
fundamental theory of quantum gravity, one thing is certain: whatever this
theory may be, it must make sense of an expression like:

$\displaystyle\sum_{\text{topologies}}\int[{\cal D}g][dX]\,e^{-S_{\rm
EH}-S_{\rm SM}-S_{\text{other}}}.$ (1)

This is how quantum gravity should look like [51]: a path integral over matter
fields $X$ and over geometric degrees of freedom (for instance metrics $g$ and
topologies) of the exponential of the Einstein–Hilbert action plus the
standard model action plus possibly other terms, describing physics yet to be
discovered. The sum over topologies in equation (1) can be reduced to a single
topology by fine tuning $S_{\text{other}}$, but _a priori_ there is no reason
to do this: one lesson of quantum field theory is that whatever can happen
will happen, and this includes topology change.

As already hinted, the precise meaning of equation (1) is unclear: the sum
over topologies and metrics is ill defined and the naive attempts to use this
equation as the starting point for a theory of quantum gravity fail. A
strategy to make sense of this equation is to replace the sum over (continuum)
geometric degrees of freedom with a sum over triangulations:

$\displaystyle\sum_{\text{topologies}}\int[{\cal
D}g]\quad\rightarrow\quad\sum_{\text{triangulations}},$

and to replace the continuum actions by their discretized versions. However,
when passing to a discrete setting, several questions arise, most notably:

  * •

what is the weight (probability distribution) one should use to sum over
triangulations?

  * •

how does one go back from discrete to continuum geometries?

The answers to these questions are intertwined. In order to go back from
discrete to continuum geometries one needs some kind of phase transition. The
precise features of this phase transition strongly depend on the probability
distribution chosen. In contrast to two or three dimensions, in dimension four
information about the smooth structure is lost when going to a discretization.
Some of this information could perhaps index the available phases of
continuous geometry.

Geometry and topology become increasingly complicated when increasing the
topological dimension and it seems reasonable to first address these questions
in dimension two, and only afterwards pass to the more realistic dimensions
three or four.

In two dimensions the answers to these questions are provided by the theory of
random matrices [4, 51, 115, 146, 147]. Random matrices are probability
distributions for $N\times N$ random variables $\mathbb{M}_{ab}$, which are
invariant under the conjugation of $\mathbb{M}$ by the unitary group. The
moments and partition function of a random matrix model can be evaluated as
sums over ribbon Feynman graphs, with weights fixed by the Feynman rules.
These ribbon graphs are dual to topological surfaces. As the probability
distribution of the surfaces (weights of the graphs) is fixed by the Feynman
rules, random matrices yield a _canonical_ theory of random two-dimensional
topological surfaces and provide an answer to the first question. One still
has some freedom in assigning metrics to the triangulated topological
surfaces. The simplest choice is to consider the triangulations as
equilateral.

As always in quantum field theory, the perturbative Feynman series diverges.
However, the situation is much more subtle in matrix models than in usual
quantum field theory. A matrix model is endowed with a natural small
parameter, $1/N$ (where $N$ is the size of the matrix), which does not exist
in usual quantum field theory, and one can reorganize the perturbative
expansion of a matrix model as a series in $1/N$. In his seminal work [141] ’t
Hooft showed that the $1/N$ series is indexed by the genus. This is the
fundamental feature of matrix models: the $1/N$ expansion reorganizes the
perturbative series into nontrivial but manageable packages of graphs of fixed
genus.

At leading order in $1/N$ the planar graphs [36, 48] dominate. While planar
graphs can be arbitrarily large (i.e., they can have an arbitrary number of
edges), they can be explicitly enumerated [44, 45, 139] and form an
exponentially bounded family. This holds order by order in $1/N$: the family
of graphs of any fixed genus is exponentially bounded. Tuning the coupling
constants of a matrix model to some critical values, infinite graphs with
fixed genus will dominate at any order in $1/N$. At this critical point the
model undergoes a phase transition to a continuum theory of random, infinitely
refined, surfaces [49, 101]. This answers the second question above. Assuming
an equilateral metric assignment, the infinitely refined random geometry
emerging at criticality, the Brownian map [109, 110, 111], has average
Hausdorff dimension 4 and is widely believed to have average spectral
dimension 2.

The behavior of conformal matter coupled to Liouville gravity in two
dimensions is very well captured by matrix models [3, 34, 35, 49, 51, 52, 54,
56, 60, 102, 103, 105, 113]. The double scaling limit of matrix models [37,
55, 71] corresponds to a continuum gravity theory with finite renormalized
Newton’s constant. Matrix models extend naturally to matrix field theories,
which are intimately related to noncommutative quantum field theories [76,
93], in particular to the Grosse Wulkenhaar model, which has been shown
asymptotically safe at all orders [53] and solved in the planar sector [77,
78, 79, 80].

Matrix models can be studied using the eigenvalue decomposition or the
Schwinger–Dyson equations [8, 57, 58, 59, 81, 114], avoiding the divergent
perturbative expansion. However, the link with random surfaces must be
revisited in this case. In order to obtain a theory of random surfaces one
first performs the perturbative expansion and subsequently reorganizes it in
powers of $1/N$. As the perturbative expansion is not summable, it is a non
trivial task [100] to establish the connection between random surfaces and
matrix models rigorously. Even more tantalizing, the critical point where the
continuum limit is reached is at _negative_ values of the coupling constants,
in a range of parameters where the models are unstable. This is a feature, not
an accident: the critical point must correspond to a regime where all the
surfaces add up, and not to a point where the sum over surfaces is alternated.
The tuning to criticality is meaningful after restricting to a fixed order in
$1/N$, but what is the meaning of this tuning to criticality beyond
perturbation theory?

Although many non perturbative questions about random matrices are still open,
by and large, matrix models are a success story. One lesson can be drawn from
their example: in two dimensions, in most cases, _minimal choices suffice_.
The relation between random matrices and quantum gravity is already apparent
if one studies the simplest matrix model and considers the simplest metric
assignment (equilateral) for the triangulations.

The development of matrix models over the past decades is one of the most
impressive achievements of modern theoretical and mathematical physics. This
success inspired their generalization in the 1990s to random tensor models [1,
75, 136, 137, 138] intended to describe random geometries in higher
dimensions. However, for twenty years random tensors essentially failed to
match the success of random matrices because, for a long time, a $1/N$
expansion for tensors could not be found.

Random tensors generate Feynman graphs that can be interpreted as topological
spaces. However, the spaces generated in this way are quite nontrivial: one
obtains not only all the manifolds, but also all the pseudo manifolds of a
fixed dimension111A further complication arose from the fact that the first
models proposed did not generate genuine $D$-dimensional complexes, but only
2-complexes.. Several models [6, 7, 33, 116, 117, 118, 121], mainly under the
guise of “Group Field Theories”, which are tensor models decorated by extra
data, have been proposed in the attempt to tackle this problem. Although
different in some important respects, these models did not bring any insight
into the problem of the $1/N$ expansion.

Starting in 2009 [82, 83, 85], new results and techniques led to the discovery
of the $1/N$ expansion for tensors [31, 84, 88, 94]. These results form the
backbone of the modern theory of random tensors. In this modern point of view,
random tensors are probability distributions for $N^{D}$ random variables
$\mathbb{T}_{a_{1}\dots a_{D}}$, which are invariant under the independent
action of the unitary group on each tensor index:

$\displaystyle\mathbb{T}_{a_{1}\dots a_{D}}\to\mathbb{T}^{\prime}_{b_{1}\dots
b_{D}}=\sum_{a_{1},\dots a_{D}}U^{(1)}_{b_{1}a_{1}}\cdots
U^{(D)}_{b_{D}a_{D}}\mathbb{T}_{a_{1}\dots a_{D}}.$

The main building blocks of a random tensor model are the invariants one can
build out of the tensor $\mathbb{T}$ and its complex conjugate. While for
matrices there is essentially one such invariant at any degree,
$\operatorname{Tr}[({\mathbb{M}}{\mathbb{M}}^{\dagger})^{p}]$, there are
numerous possible choices for tensors. Each of these invariants (generalized
traces) can be represented by a bipartite $D$-regular edge colored graph, and
there are as many independent invariants at degree $2p$ as there are non
isomorphic such graphs with $2p$ vertices. This is the source of all the
richness of random tensors.

The graphs of the new tensor models always encode genuine $D$-dimensional
cellular complexes hence $D$-dimensional topological spaces. The $1/N$ series
[25, 47, 84, 88, 90, 94] is indexed by a positive integer called the _degree_
, which plays in higher dimensions the same role the genus played for matrix
models. Unlike the genus, however, the degree is _not_ a topological
invariant, but mixes topological and triangulation dependent information. This
is not a drawback, but a quality. Topology is quite complicated in dimensions
three and four (or higher): although topologies are not fully classified in
arbitrary dimension, the graphs at fixed degree can be [98].

In the large $N$ limit the graphs of degree zero, called _melonic_ [29, 91,
97], dominate. The melonic graphs triangulate the $D$-dimensional sphere in
any dimension [84, 88, 94] and are an exponentially bounded family. Like
matrix models, tensor models undergo a phase transition to a theory of
continuous random $D$-dimensional topological spaces [29, 31] when tuning to
criticality and posses a double scaling limit [32, 46, 98]. In stark contrast
to matrix models, for $3\leq D\leq 5$, in the double scaling regime only an
exponentially bounded family of triangulations of the sphere contribute. For
$D\geq 6$ (as for $D=2$) the family selected in the double scaling limit is
neither exponentially bounded nor restricted to the spherical topology. The
$1/N$ expansion of tensor models exhibits very strong universality properties
[87, 89, 91] and has been put on a firm mathematical footing [90] in the sense
of constructive field theory [69] by means of the loop vertex expansion [95,
112, 125, 126]. The ensuing theory of random geometries in higher dimensions
has been extensively studied [11, 12, 15, 17, 22, 24, 26, 28, 30, 40, 41, 61,
86, 87, 89, 96, 99, 106, 131, 144].

Random tensor models generalize random matrix models and provide a framework
for the study of random geometries in any dimension. However, as gravity is
quite peculiar in two dimensions (and it is peculiar in a different way in
three dimensions, and it is peculiar in yet another different way in four),
one should keep an open mind for alternatives.

In their simplest form, tensor models can be seen, like matrix models, as
generators of Euclidean dynamical triangulations [2, 50, 92]. There exists a
tensor model [29] whose free energy $W_{\rm TM}(\lambda,N)$ equals an
Einstein–Hilbert gravity partition function discretized on an equilateral
triangulation with edge length $a$:

$\displaystyle\sum_{\text{topologies}}\int[{\cal D}g]\,e^{\frac{1}{16\pi
G}\int
d^{D}x\sqrt{g}(R-2\Lambda)}\rightarrow\sum_{\genfrac{}{}{0.0pt}{}{\text{Triangulations}}{\text{edge
length }a}}e^{-S_{\rm EH}^{\rm discr.}(G,\Lambda;a)}=W_{\rm TM}(\lambda,N),$

where the coupling constant $\lambda$ and the parameter $N$ of the tensor
model are related to the edge length $a$, the dimensionfull cosmological
constant $\Lambda$ and Newton’s constant $G$ respectively the dimensionless
cosmological constant $\tilde{\Lambda}$ and Newton’s constant $\tilde{G}$ by
the relations:

$\displaystyle\tilde{G}=\frac{G}{a^{D-2}}=c_{1}\frac{1}{\ln
N},\qquad\tilde{\Lambda}=\Lambda
a^{2}=c_{2}\frac{G}{a^{D-2}}\ln\left(\frac{1}{\lambda}\right)+c_{3}=\tan\alpha\tilde{G}+c_{3},$

with $c_{1},c_{2},c_{3}>0$ some constants of order $1$ (and $c_{3}=0$ for
$D=2$) and $\tan\alpha=c_{2}\ln\big{(}\frac{1}{\lambda}\big{)}$.

Figure 1: The $(\tilde{G},\tilde{\Lambda})$ plane.

The $(\tilde{G},\tilde{\Lambda})$ plane is represented in Fig. 1. The angle
$\alpha$ is the _slope_ of the (solid) line
$\tilde{\Lambda}(\tilde{G})=\tan\alpha\tilde{G}+c_{3}$ with respect to the
vertical axis. The coupling constant $\lambda$ controls $\alpha$, while $N$
controls the height of the dashed line in Fig. 1. The physical constants
$\tilde{G}$ and $\tilde{\Lambda}$ are the coordinates of the intersection
point of the two lines. For $\lambda\in(0,\infty)$,
$\alpha\in\big{(}\frac{\pi}{2},-\frac{\pi}{2}\big{)}$ and, as $N>1$, the two
parameters $\lambda$ and $N$ allow one to explore the entire upper half plane
of the physical constants.

As long as $a$ is kept fixed, the dimensionfull constants are just a finite
rescaling of the dimensionless ones. The correspondence between the various
regimes of tensor models and the physical coupling constants can be read out
from this figure, observing that $\lambda\to 0$ at fixed $N$ moves the
intersection point to the right $\big{(}\alpha\to\frac{\pi}{2}\big{)}$ at
constant height, while $N\to\infty$ at fixed $\lambda$ descends the
intersection point along the line $\tilde{\Lambda}(\tilde{G})$ at fixed slope
$\alpha$.

The perturbative expansion.

We have $a$ and $N$ fixed, $\lambda\to 0$. The perturbative expansion is an
expansion at _large cosmological constant_ and _fixed Newton’s constant_ :

$\displaystyle\tilde{G}\ \text{constant
(small)},\qquad\alpha\to\frac{\pi}{2},\qquad\tilde{\Lambda}\to\infty.$

This is an expansion around the zero volume state corresponding to
$\tilde{\Lambda}=\infty$.

The $1/N$ expansion.

We have $a$ and $\lambda$ fixed, $N\to\infty$. For $D\geq 3$, the $1/N$
expansion is an expansion at _finite cosmological constant_ , _small Newton’s
constant_ and constant slope:

$\displaystyle\tilde{G}\to 0,\qquad\alpha\
\text{constant},\qquad\tilde{\Lambda}\to c_{3}.$

This is an expansion around geometries of maximal positive curvature at fixed
volume. The situation is different in $D=2$. In that case the $1/N$ expansion
is an expansion at _small cosmological constant_ and _small Newton’s constant_
with constant slope.

The large $N$ limit.

We have $a$, $\lambda$ fixed, $N=\infty$. In this limit one has a _finite
cosmological constant_ and _zero Newton’s constant_ , but this regime is
approached along a line with fixed slope:

$\displaystyle\tilde{G}=0,\qquad\alpha\ \text{
constant},\qquad\tilde{\Lambda}=c_{3}.$

The large $N$ limit projects onto geometries with maximal positive curvature
at fixed volume. Among these geometries, one has a second expansion governed
by the slope $\alpha$ around the zero volume state. That is, once projected
onto geometries with maximal positive curvature, the slope $\alpha$ plays the
role of an effective cosmological constant which is large and positive.

The critical regime.

We have $a$ fixed, $\lambda\to\lambda_{c}$, $N=\infty$. One still has a
_finite cosmological constant_ and _zero Newton’s constant_ , but the slope
approaches a critical value:

$\displaystyle\tilde{G}=0,\qquad\alpha\to\alpha_{c},\qquad\tilde{\Lambda}=c_{3}.$

We represented the line with critical slope $\alpha_{c}$ in red in Fig. 1. The
critical value $\alpha_{c}$ corresponds to a divergent number of $D$-simplices
hence, as $a$ is kept fixed, these geometries have infinite volume.

The continuum limit.

We have $a\to 0$, $\lambda\to\lambda_{c}$, $a^{D}(\lambda_{c}-\lambda)^{-1}$
fixed, $N=\infty$. The number of $D$-simplices diverges but the physical
volume is kept fixed by sending $a$ to zero while keeping
$a^{D}(\lambda_{c}-\lambda)^{-1}$ fixed. The dimensionless and dimensionfull
constants have a different behavior:

$\displaystyle\tilde{G}=0,\qquad\alpha\to\alpha_{c},\qquad\tilde{\Lambda}=c_{3}\qquad\Big{|}\qquad
G=0,\qquad\Lambda=\infty.$

The double scaling regime.

We have $a$ fixed, $\lambda\to\lambda_{c}$, $N\to\infty$,
$N^{D-2}(\lambda_{c}-\lambda)$ fixed. This regime corresponds to a well
defined trajectory in the plane of physical coupling constants:

$\displaystyle\tilde{G}\to
0,\qquad\alpha\to\alpha_{c},\qquad\tilde{\Lambda}\to c_{3}.$

The continuum double scaling limit.

We have $a\to 0$, $\lambda\to\lambda_{c}$, $N\to\infty$, with
$N^{D-2}(\lambda_{c}-\lambda)$ fixed and $a^{D}(\lambda_{c}-\lambda)^{-1}$
fixed. In this regime we have:

$\displaystyle\tilde{G}\to
0,\qquad\alpha\to\alpha_{c},\qquad\tilde{\Lambda}\to c_{3}\qquad\Big{|}\qquad
G\to 0,\qquad\Lambda\to\infty.$

The analytically accessible regimes of this tensor model explore regions with
_small positive $($or zero$)$_ Newton constant and _large $($or very large$)$
positive_ cosmological constant. Care should be taken when interpreting these
values. This theory is supposed to describe the ultraviolet behavior of
gravity, therefore the cosmological and Newton’s constant we are discussing
here are the ultraviolet ones. In order to obtain the infrared coupling
constants one needs to follow a renormalization group flow, and both $\Lambda$
and $G$ will vary substantially as they are dimensionfull in $D\geq 3$.

The geometries of maximal positive curvature at fixed volume are rather simple
and fall into the universality class of the continuous random tree (branched
polymers). On the one hand this is good news, as one can treat analytically
statistical systems in random geometry [22, 24, 27, 28, 30], hence the
coupling of gravity with matter fields. On the other, one can only do so much
with branched polymers and it is important to find regimes in which the
emergent geometries are richer.

The double scaling limit is one attempt to go beyond branched polymers. In
this limit larger families of graphs are included and, more importantly, below
$D=6$ it seems likely that the double scaling regime can be followed by a
triple scaling regime, revealing an even larger family of graphs. The strategy
one employs in this context is to keep the metric interpretation of a graph as
dual to an equilateral triangulation, and to extend (in a controlled manner)
the family of graphs included in the statistical ensemble.

A second option is to encode metric degrees of freedom in additional data
associated to Feynman graphs. This strategy is sometimes called group field
theory (GFT) [5, 6, 7, 33, 104, 116, 117, 118, 121] and has proven quite
successful. In GFT one considers tensors over some Lie group that are
furthermore invariant under the diagonal action of the group. With minimal
adaptations the $1/N$ expansion holds for GFTs [84, 88]. Due to the diagonal
gauge invariance, the amplitude of a GFT graph is the discretized BF action on
the dual triangulation. At least in three dimensions, this is exactly the
gravity partition function on the dual triangulation (in higher dimensions the
BF action must be supplemented by constraints). The tantalizing fact is that,
choosing for instance the group ${\rm SU}(2)$ in three dimensions, the melonic
triangulations (which still dominate) are endowed with a _flat connection_.
This, most definitely, is _not_ a branched polymer geometry. However, the
reader should be aware of the following fact. While a metric on a (pseudo)
manifold is uniquely encoded in the holonomies around all the closed loops in
that (pseudo) manifold, when discretizing one loses a _local conformal
factor_. While the fact that melonic geometries have a flat connection is very
encouraging, before fixing this conformal factor issue one cannot control the
emerging random geometry in GFTs. Be that as it may, GFT has successfully been
applied to quantum cosmology [62, 63, 64, 65, 66, 67, 119, 120, 122, 123,
140], in what is the most relevant phenomenological application of tensor
models to date.

A third option is to extend the framework of tensor models to tensor field
theories. As matrix models extend naturally to matrix field theories, tensor
models extend naturally to tensor field theories (TFT) [127, 128, 130] by
breaking the unitary invariance of the Gaussian part of the measure. The
covariance of such models possesses a nontrivial spectrum, which in turn is
naturally divided into scales. The integration of the high scales leads to a
genuine renormalization group flow [9, 18, 19] and the TFTs explore the
_tensor theory space_ [129] spanned by all the unitary invariant polynomial
interactions. One can combine both tensors with a diagonal gauge invariance
and a nontrivial covariance in what could be called tensor group field theory.
Such models are also perturbatively renormalizable (which is highly nontrivial
in this case, and relies crucially on the ultraviolet dominance of the melonic
graphs), and exhibit a very rich phase portrait. While much remains to be
done, one could hope that this class of models can fix the local conformal
factor problem of the usual GFTs.

The TFTs are examples of _renormalizable, nonlocal_ field theories. A regime
parallel to the large $N$ limit of tensor models is reached in TFTs via a
genuine renormalization group flow [9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21,
23, 38, 39, 42, 43, 107, 108, 133, 134, 135], and not by sending the parameter
$N$ to infinity. The large $N$ behavior of tensor models corresponds to the
ultraviolet limit of TFTs. Typically the couplings grow in the infrared which
suggests that TFTs develop bound states at low energy. This low energy
behavior remains largely to be explored either by analytic [38, 39] or by
numeric [21] methods. We emphasize that the natural metric assignment for the
triangulations dual to graphs in TFTs is not yet understood. However, the fact
that TFTs develop bound states in the infrared is an encouraging sign and
establish them as potential ultraviolet completions of gravity.

Random tensors are the straightforward generalization of random matrices in
higher dimensions. It should however be stressed that the $D=2$ case of
matrices is very special. Indeed, random tensors behave by and large quite
differently from random matrices. This is due to the fact that the melonic
family, which dominates in tensor models, is very different from (and in fact
much more restricted than) the planar family dominating matrix models.

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# A Continuum Model for Morphology Formation from Interacting Ternary
Mixtures: Simulation Study of the Formation and Growth of Patterns

Rainey Lyons<EMAIL_ADDRESS>Stela Andrea Muntean  Emilio N. M. Cirillo
Adrian Muntean

###### Abstract

Our interest lies in exploring the ability of a coupled nonlocal system of two
quasilinear parabolic partial differential equations to produce phase
separation patterns. The obtained patterns are referred here as morphologies.
Our target system is derived in the literature as the rigorous hydrodynamic
limit of a suitably scaled interacting particle system of Blume–Capel–type
driven by Kawasaki dynamics. The system describes in a rather implicit way the
interaction within a ternary mixture that is the macroscopic counterpart of a
mix of two populations of interacting solutes in the presence of a background
solvent. Our discussion is based on the qualitative behavior of numerical
simulations of finite volume approximations of smooth solutions to our system
and their quantitative postprocessing in terms of two indicators (correlation
and structure factor calculations). Our results show many similar features
compared to what one knows at the level of the stochastic Blume–Capel dynamics
with three interacting species. The properties of the obtained morphologies
(shape, connectivity, and so on) can play a key role in, e.g., the design of
the active layer for efficient organic solar cells.

Keywords: Continuum model, interacting ternary mixture, Blume–Capel model with
Kawasaki dynamics, phase separation, morphology formation, domain growth,
numerical simulation

PACS Subject Classification: 64.75.Qr, 64.75.St, 02.60.Lj
AMS Subject Classifications: 35K40, 35Q70, 65N08

##  1 Introduction

We are interested in exploring the capability of interacting mixtures of
populations of particles, typically two different solutes mixed within a
solvent, to allow for phase separation when evaporating the solvent. Such
situations are rather typical in modern materials science. In particular, we
have in mind two specific materials science applications: the formation of
internal morphologies for organic solar cells (cf. e.g. [20]); and the
formation of rubber–based zones in acrylate environments related to the design
of thin adhesive bands (cf. e.g. [11]). The evaporation process plays a key
role in both of these contexts. The presence of such a non–equilibrium process
makes this scenario different compared to the more classical phase separation
settings arising in crystal growth, metallurgy, and so on, normally treated by
Allen–Cahn–type or Cahn–Hilliard–type equations. Essentially, by controlling
the evaporation mechanism one has the possibility to freeze a local
equilibrium, i.e. one has the possibility to actively select specific
morphologies as the end configuration. We refer the reader to the recent works
[5, 26, 32], where this setting has been explored using Monte Carlo
simulations for suitable lattice–based models. With such final and stable
morphologies at hand (see e.g. Figure 1 in [5]), one can then ask key
questions about their efficiency with respect to the effective charge
transport in the case of the organic solar cells or their efficiency as a
global (effective) measure of stickiness in the case of adhesive bands.

To this end, it is crucial to have sufficiently rich models posed at the
continuum level that inherit, from the interacting particle systems, the
ability to produce physically meaningful phase separation. This would set the
stage for process and shape optimization operations to reach optimal
macroscopic transport and reaction properties, mediated by optimal morphology
shapes. However, quite interestingly, in spite of huge efforts in the
statistical mechanics and applied mathematics communities, deriving rigorously
the corresponding hydrodynamic limit equations for mixtures of interacting
particle systems having the possibility of evaporation at one of the
boundaries is currently out of reach. The main technical difficulties seems to
arise because of the evaporation component of the process. Therefore, we lay
foundation in this manuscript by looking at a simplified case where the
evaporation is not taken into account. The problem remains interesting and
relevant, with a rich phenomenology, for the presence of three coexisting
phases undergoing the separation process.

In [5, 26], and follow–up papers, a generalized version of the nearest
neighbor Blume–Capel model in $2D$ with Kawasaki dynamics powered with the
Metropolis algorithm was employed as working tool to simulate the phase
behaviour of a ternary mixture upon evaporation of one of its components. Here
we are genuinely interested in devising the corresponding hydrodynamics limit
in absence of evaporation. The components with spin $+1$, $-1$, and $0$ in the
Blume–Capel dynamics correspond to two distinct solute phases and,
respectively, to solvent222When handling the active layer for an organic solar
cell, the mixture is composed of electron–acceptor, electron–donor and solvent
molecules, respectively.. In particular, in [26] it was used as governing
Hamiltonian the usual nearest neighbor Blume–Capel Hamiltonian with zero
chemical potential and magnetic field, see [9].

Although we are using working techniques from [27] to investigate the
situation, we are unaware of the corresponding hydrodynamic limit equations
arising from this precise structure of the stochastic model. However, for a
modified Hamiltonian structure involving Kac potentials much is known.
Specifically, the authors of [24] rigorously identify the continuum limit
model for both the Kawasaki and Glauber dynamics in the infinite volume case
(so, the evaporation processes is not taken into account). We refer the reader
also to [17, 16] for related discussions on the same class of models.

The phase ordering dynamics of the two–field model derived in [24] will be the
object of our study. We remark that studying equilibrium and non–equilibrium
properties in connection with phase transitions for statistical mechanics
models with multiple degenerate ground states is a natural extension of the
theories developed for two–state systems, such as the paradigmatic Ising
model, which, independently from possible real-world applications, has a great
theoretical value. In particular, when dynamical phenomena are concerned, new
interesting behaviors emerge. We refer, for instance, to [9, 7, 8] for the
study of metastability and to [28] for the phase ordering problem.

Phase ordering dynamics of Ising–like two–phase mixtures has been widely
studied and it is well understood; compare e.g. [4, 2, 19]. The interval of
times in which the domain morphology does not change is characterized by the
dynamical scaling of the correlation function $G(\vec{x},t)$, where $\vec{x}$
and $t$ are the space and time variable, respectively. This can be expressed,
in the isotropic case, by saying that $G(\vec{x},t)=g(|\vec{x}|/L(t))$, where
$L(t)$ is the time dependent length scale, which is interpreted as the typical
size of the growing domains. Both for conservative and non–conservative
dynamics, the length scale is characterized by a power law $L(t)\sim
t^{\alpha}$. Simulations suggest the approximate value of exponent $\alpha$ is
$1/2$ for the non–conservative and $1/3$ for the conservative dynamics. While
there is evidence that the scaling hypothesis holds in many situations, it has
only been shown to be true for certain models; see, for instance, [1, 3, 10,
13] as well as [31] for a closely related context.

For systems with multiple phases the understanding of the phase separation
processes is far from being complete. For some specific models, such as the
clock and the Potts model with conservative dynamics, it is well established
that the late times growth exponent is $1/2$, see, e.g., [28] and references
therein. Note though that some controversies have been more recently pointed
out in [21]. What concerns those dynamics conserving the order parameter, not
many studies are available. We mention, again, [28] for the case of the clock
model. Here the authors have shown a crossover between exponents $1/4$ and
$1/3$. A particular version of the nearest neighbor Blume–Capel model was
investigated in [26]. The reported simulations indicate that the $1/3$ growth
behavior can be reached, but results are not yet conclusive.

The main focus of this work is to explore, in the context of the two–field
model derived in [24] and by means of numerical simulations, three main ideas:

  * –

to which extent the two-field continuum model reported in [24] for the
Kawasaki dynamics is able to produce phase separation;

  * –

how the obtained patterns (called here morphologies) relate to what is known
for the classical Ising model in 2D and to what we have observed ourselves in
our quantitative study [26] for the three–component mixture driven by the
standard Blume–Capel model

$H(\sigma)=\frac{1}{2}\sum_{\begin{subarray}{c}x\neq x^{\prime}\in V:\\\
|x-x^{\prime}|=1\end{subarray}}[\sigma(x)-\sigma(x^{\prime})]^{2},$ (1)

where $V\subset\mathbb{Z}^{2}$ is a finite square, with or without the
evaporation of one of the phases;

  * –

to which extent the scaling hypothesis holds in this setting.

This work is organized as follows: in Section 2 we define the model and
describe the finite volume scheme we use to compute decent approximations of
solutions to the target model. In the main section of the manuscript, Section
3, we report the observed morphologies for selected parameter regimes borrowed
from [26] so that direct comparisons between the two settings can be done. The
analysis of the growth of morphologies on intermediate and long time scales is
done in Section 3.1 using common measures applied to interacting particle
systems modified so they are applicable to problems posed at the continuum
level. We illustrate numerically that such methods are able to approximate the
characteristic length of the domains. Although this is rather common knowledge
in the statistical mechanics–type exploration of patterns obtained e.g. with
lattice models, our type of quantitative analysis of patterns seems to be
novel in the context of phase–field models of Cahn–Hilliard type. In the same
section, we present evidence explaining to which extent model (2) satisfies
the scaling hypothesis. Finally, we close the paper with a conclusion and
outlook Section 4.

##  2 The model

Let $\emptyset\neq\Omega\subset{\mathbb{R}}^{2}$ be a bounded open set with
smooth boundary (e.g., $\partial\Omega$ Lipschitz) describing the spatial
domain for the processes to take place. Let $T>0$ represent the final time of
the overall dynamics. The time $T$ takes a large positive value, which in the
current setting333If the evaporation process is included in the model, then
one can relate the value of the final time of the overall process $T$ to the
amount of evaporated solvent. is picked arbitrarily. We set $\beta>0$,
$m_{0}:(0,T)\times\bar{\Omega}\to[-1,+1]$, and
$\phi_{0}:(0,T)\times\bar{\Omega}\to[0,+1]$. We are interested in the
following two-field continuum model: find the pair $(m,\phi)$ solving the
system

$\left\\{\begin{split}\partial_{t}m&=\nabla\cdot\left[\nabla
m-2\beta(\phi-m^{2})(\nabla J*m)\right]\mbox{ in }(0,T)\times\Omega\\\
\partial_{t}\phi&=\nabla\cdot\left[\nabla\phi-2\beta m(1-\phi)(\nabla
J*m)\right]\mbox{ in }(0,T)\times\Omega\end{split}\right.,$ (2)

where $J\in C_{+}^{2}(\mathbb{R}^{2})$, symmetric, compactly supported, and
$\int_{\mathbb{R}^{2}}J(r)dr=1$, together with the initial data

$m(t=0)=m_{0}\mbox{ and }\phi(t=0)=\phi_{0}\mbox{ in }\bar{\Omega}$ (3)

and with periodic boundary conditions. We refer to this system as problem
$P(\Omega)$.

The problem $P({\mathbb{R}}^{2})$ is derived in [24] as the rigorous
hydrodynamic limit of the Kawasaki dynamics with inverse temperature $\beta$
for the Blume–Capel model with Kac potential with range of interaction
$\gamma^{-1}$, magnetic field $h_{1}$, and chemical potential $h_{2}$, whose
Hamiltonian in a finite square $V\subset\mathbb{Z}^{2}$ is

$H_{\gamma}(\sigma)=\frac{1}{2}\sum_{\begin{subarray}{c}x\neq x^{\prime}\in
V\end{subarray}}J_{\gamma}(x-x^{\prime})[\sigma(x)-\sigma(x^{\prime})]^{2}-\sum_{x\in
V}h_{1}\sigma(x)-\sum_{x\in V}h_{2}\sigma^{2}(x),$ (4)

where $J_{\gamma}:\mathbb{R}^{2}\to\mathbb{R}$ is such that

$J_{\gamma}(r)=\gamma^{2}J(\gamma r)$ (5)

for all $r\in\mathbb{R}^{2}$.

We are studying here the framework offered by $P(\Omega)$ and have in view the
situation when one can think of the diameter of the set $\Omega$, say
$\text{diam}(\Omega)$, to be large. However, we are not currently concerned
with the case $\text{diam}(\Omega)\to\infty$.

In [24], $m$ is referred to as magnetization, while $\phi$ is called
concentration. Their precise physical meaning seems to be best understood at
the level of the stochastic dynamics. In other words, if the spin variable at
the site $i\in\mathbb{Z}^{2}$ is denoted by $\sigma(i)$, then for some set
$\Lambda\subset\mathbb{Z}^{2}$ the empirical measures
$\sum_{i\in\Lambda}\sigma(i)$ and $\sum_{i\in\Lambda}\sigma^{2}(i)$ would
recover in a certain way444To obtain the structure of our continuum model, a
scaling similar to [15] is employed in the arguments presented in [24]. (see
Section 3 in [24]) as weak–star limits the densities $m$ and $\phi$,
respectively. In our interpretation of the model for a subset
$A\subset\Omega$, $\int_{A}m(t,x)dx$ represents the net spin in the set $A$
and $\int_{A}(1-\phi(t,x))dx$ represents the solvent fraction in $A$. In our
visualizations for $\phi$ shown in Section 3, we will make use of the color
red to point out the time–space distribution of the solvent fraction. If the
values taken by the image of $m$ are close to $-1$, and respectively $+1$,
then we point out the the time–space distribution of the other two competing
phases. We visualize these phases with yellow and blue colors, respectively.

###  2.1 Finite Volume Scheme

We begin by discretizing the domain with a mesh of sizes $\Delta x,\Delta y$.
We denote the uniformly spaced nodes of the mesh by the pair $(x_{i},y_{j})$
and we denote the mesh cells by $\Lambda_{i,j}:=[x_{i}-\frac{1}{2}\Delta
x,x_{i}+\frac{1}{2}\Delta x)\times[y_{j}-\frac{1}{2}\Delta
y,y_{j}+\frac{1}{2}\Delta y)$ with the natural modifications for the boundary.
The initial pair $(m_{0},\phi_{0})$ is made discrete with the approximations

$m^{0}_{i,j}:=\frac{1}{|\Lambda_{i,j}|}\int_{\Lambda_{i,j}}m_{0}(x,y)dxdy\quad\text{
and
}\quad\phi^{0}_{i,j}:=\frac{1}{|\Lambda_{i,j}|}\int_{\Lambda_{i,j}}\phi_{0}(x,y)dxdy.$

We then approximate the solution to (2) by a fully explicit finite–volume
(difference) scheme given by

$\left\\{\begin{split}\frac{m^{k+1}_{i,j}-m^{k}_{i,j}}{\Delta
t}&=\frac{1}{\Delta x^{2}}D^{2}_{i}[m^{k}_{i,j}]+\frac{1}{\Delta
y^{2}}D^{2}_{j}[m^{k}_{i,j}]\\\ &-\frac{\beta}{\Delta
x}D^{1}_{i}[(\phi^{k}_{i,j}-(m^{k}_{i,j})^{2})\tilde{J}^{k}_{x,i,j}]-\frac{\beta}{\Delta
y}D^{1}_{j}[(\phi^{k}_{i,j}-(m^{k}_{i,j})^{2})\tilde{J}^{k}_{y,i,j}]\\\
\frac{\phi^{k+1}_{i,j}-\phi^{k}_{i,j}}{\Delta t}&=\frac{1}{\Delta
x^{2}}D^{2}_{i}[\phi^{k}_{i,j}]+\frac{1}{\Delta
y^{2}}D^{2}_{j}[\phi^{k}_{i,j}]\\\ &-\frac{\beta}{\Delta
x}D^{1}_{i}[m^{k}_{i,j}(1-\phi^{k}_{i,j})\tilde{J}^{k}_{x,i,j}]-\frac{\beta}{\Delta
y}D^{1}_{j}[m^{k}_{i,j}(1-\phi^{k}_{i,j})\tilde{J}^{k}_{y,i,j}]\end{split}\right.,$

where

$D^{2}_{l}[f_{l}]:=f_{l+1}-2f_{l}+f_{l-1},\quad
D^{1}_{l}[f_{l}]:=f_{l+1}-f_{l-1},$

and $\tilde{J}^{k}_{x,i,j}$ (respectively, $\tilde{J}^{k}_{y,i,j}$) denotes
the approximation of $\partial_{x}J*m^{k}(x_{i},y_{j})$ (respectively,
$\partial_{y}J*m^{k}(x_{i},y_{j})$) using a fast Fourier transform method
similar to [34] with modifications for the periodic domain. We note that the
periodic boundary conditions are implemented implicitly during the
calculations. Since the scheme here is fully explicit, we make use of a
Courant–Friedrichs–Lewy (CFL) condition similar to standard conditions for
advection–diffusion processes, $\frac{\beta\Delta t}{\Delta x\Delta
y}<\frac{1}{4}$. We do not claim this choice of CFL condition is sharp, but it
has held up through numerical experiments. Further rigorous analysis of the
scheme studying its convergence and stability will illuminate more optimal
conditions of relevance particularly when the parameter $\beta$ varies (i.e.
when there are temperature changes in the system). There are many choices of
numerical schemes for such equations. Since we are not interested in the
particular geometry of the domain, we opt for the case where $\Omega$ is a
square with periodic boundary conditions. Therefore, the choice of finite
volume schemes is natural due to the conservative nature of the model. We also
have in mind to eventually follow existing literature where schemes of similar
flavor are used in the study of Cahn–Hillard equations (see e.g. [22, 23,
12]).

##  3 Observed Morphologies

We use the finite difference scheme described above to simulate the phase
separation process according to the dynamics of (2). In all simulations, the
initial conditions are chosen in a random manner such that
$m^{0}_{i,j}\in\\{-1,0,1\\}$ with $\phi^{0}_{i,j}=|m^{0}_{i,j}|$. The
proportion of positive spin ($m=1$) and negative spin ($m=-1$) particles is
kept equal in each simulation, while the proportion of zero spin solvent
particles (sometimes denoted by $c_{0}$) varies over the ranges 0%–80%. In all
simulations, we take the parameter $\beta=10$ and $J$ is taken to be a
standard ‘bump’ function such that $\int_{\mathbb{R}^{2}}J(r)dr=1$.

Figure 1: Morphologies produced by $m$ on a domain $\Omega:=[-10,10]^{2}$ with
512 nodes in both the $x$ and $y$ axes. From top to bottom, each row
represents an initial solvent concentration ($c_{0}$) of 0, 0.2, 0.4, 0.6, and
0.8, respectively. Regions where $m$ is positive are colored blue, negative
regions are colored yellow, and regions $m$ is near zero are colored red.

Figure 2: Morphologies produced by $\phi$ on a domain $\Omega:=[-10,10]^{2}$
with 512 nodes in both the $x$ and $y$ axes. From top to bottom, each row
represents an initial solvent concentration ($c_{0}$) of 0, 0.2, 0.4, 0.6, and
0.8, respectively. Regions where $\phi$ is near one are colored blue whereas
regions near zero are colored red.

Figure 3: Morphologies produced by $m$ on a domain $\Omega:=[-10,10]^{2}$ with
512 nodes in both the $x$ and $y$ axes. From top to bottom, each row
represents an initial solvent concentration ($c_{0}$) of 0, 0.2, 0.4, 0.6, and
0.8, respectively. The color scale is as in Figure 1.

Figure 4: Morphologies produced by $\phi$ on a domain $\Omega:=[-10,10]^{2}$
with 512 nodes in both the $x$ and $y$ axes. From top to bottom, each row
represents an initial solvent concentration ($c_{0}$) of 0, 0.2, 0.4, 0.6, and
0.8, respectively. The color scale is as in Figure 2.

In this section, we present the heat maps of the solutions $m$ and $\phi$ at
different times and different solvent concentration levels in Figures 1 and 2,
respectively. Interestingly, one can see that if the solvent concentration
$c_{0}$ is close to zero, then the produced bi–connected morphologies resemble
to what one would obtain when simulating numerically the Ising model. The
resemblance to the Ising model is best seen in Section 3.3, where we study the
domain growth at large characteristic timescales. The growth exponents that we
obtain in the case $c_{0}=0$ are around $1/3$ as for the Ising model in $2D$,
see, e.g., [4, 2, 19].

If the level of solvent increases, then the connectivity in the system
decreases. Finally, a different type of morphology arises (ball–like
structures). This happens when the solvent concentration is so large
(approximatively larger than $0.5$) that the solvent component percolates in
the lattice: in this regime a unique connected solvent component is observed
in the lattice [18]. In particular, looking at our simulation results shown in
Figures 6, 7, and 5, we see that indeed the growth rate of morphologies is
decreasing with increasing $c_{0}$. A solvent fraction at the level
$c_{0}=0.4$ seems to offer some kind of separating threshold: If $c_{0}$ lies
below $0.4$, then both the morphologies and their growth rates are somewhat
similar to what we would expect from the classical Ising scenario; If $c_{0}$
takes values greater than $0.4$, then the morphologies and their growth rates
resemble more to what we would expect from standard percolation scenarios.

In the next section, we will see that the indicators of the characteristic
length scale oscillate during early times. To observe how these patterns
initially form, we also present the heat maps of the simulation at early times
in Figures 3 and 4. For these early time simulations, one can see that neither
the magnetization function, $m$, or the concentration function, $\phi$, are
well organized. This demonstrates that there are at least two different time
regimes under consideration in the continuum model. One at the early times
before the morphologies are well formed and one at later times after the
phases become separated. Multiple time regimes have been observed both
experimentally and theoretically [4]. One also sees the beginnings of the
morphologies for small concentrations (e.g. $c_{0}=0.2$) while for larger
concentrations (e.g. $c_{0}=0.8$), the mixture is still cloud like in
resemblance. This behavior seems to indicate different concentrations require
different amounts of time for phases to separate into robust morphologies. As
one can see from Figure 4, the solvent concentration increasingly slows down
the formation of morphologies.

###  3.1 Analysis on Domain Growth

In this section, we analyse the domain growth of the simulations presented in
Section 3. Common measures of the characteristic length of the domains include
the two–point correlation function and the structure factor [4]. Since we are
interested in a continuous setting, we present the continuous formulation of
these metrics here. Their discrete counterparts can be found in [26]; see also
[33, 14]. The two–point correlation function is given by the equation

$G(t,s)=\frac{1}{|\Omega|}\int_{\Omega}m(t,x,y)\,m(t,x+s_{x},y+s_{y})dxdy,$

where $s=(s_{x},s_{y})$ and $|\Omega|$ denotes the volume of the set $\Omega$.
In particular, the horizontal and vertical two–point correlation functions are
defined as $G_{x}(t,s_{x})=G(t,(s_{x},0))$ and
$G_{y}(t,s_{y})=G(t,(0,s_{y}))$, respectively. These functions generally have
a maximum at $s=0$, decrease below some cut–off value where the function
begins to oscillate before returning back to the maximum (due to the periodic
boundary conditions on the model). We can then estimate the horizontal and
vertical diameters by calculating the horizontal intercept of
$G_{\alpha}(t,\cdot)$ ($\alpha\in\\{x,y\\})$ using standard methods.

The structure factor of $m(t,\cdot)$ is defined as

$C(t,(k_{x},k_{y}))=\frac{1}{|\Omega|}\left|\int_{\Omega}m(t,x,y)e^{i(k_{x}x+k_{y}y)}dxdy\right|^{2},$

where $(k_{x},k_{y})\in[-\pi,\pi]^{2}.$ Notice the quantity inside the
absolute value is the two–dimensional Fourier transform of $m(t,\cdot)$. We
then estimate the horizontal and vertical diameters of the domains by
calculating

$R_{\alpha}(t)=\frac{\int_{[-\pi,\pi]^{2}}C(t,(k_{x},k_{y}))dk_{x}dk_{y}}{\int_{[-\pi,\pi]^{2}}|k_{\alpha}|C(t,(k_{x},k_{y}))dk_{x}dk_{y}},$

where $\alpha\in\\{x,y\\}$.

In Figure 5, we use the above indicators $G(\cdot,\cdot)$ and $C(\cdot,\cdot)$
to plot the domain sizes in time for all concentrations presented in Figures
1–4. In these plots, we observe an early time oscillatory behaviour and a
later time linear (in a log scaling) growth which corresponds to the power
laws of the scaling hypothesis. We observe in Figure 6, that these two time
regimes correspond to the time intervals where the domains are ill formed and
well formed, respectfully. We analyse these two situations in more detail in
the next section.

Figure 5: Structure factor (left) and correlation function (right)
calculations for the simulations presented in Figure 1 during the time
interval $(0,9]$. Each marker represents $4,000$ time steps in the finite
volume scheme. Lines of the form $y=Ct^{1/p}$ are plotted to demonstrate the
approximate growth law of each concentration. The value of $p$ of each line is
places in the legend underneath its corresponding solvent ratio level.

Figure 6: Demonstration of the different time regimes. Notice at early times
(top/blue box), the $m$ and $\phi$ fields have not yet manifested well formed
morphologies and so the growth of the structure factor does not fit a power
law structure. On the other hand, for later times (bottom/red box), the growth
is more consistent with a power law.

###  3.2 Domain Growth at Intermediate Time Scales

In this section, we examine the behavior of domains during the early stages of
the simulation, i.e. before morphologies are well developed. Initially, close
to the starting moment of the overall process, the randomness of the initial
distribution of phases is robust. As time elapses, morphologies tend to form,
while their precise shape still evolve until stable shapes are reached. In
both the structure factor and correlation function calculations, an
oscillating behavior is observed during these early stages in simulations
where the morphologies are ill-formed; we emphasize particularly Figure 6 as
well as Figure 7. This behavior has been observed experimentally as well and
is discussed in [25].

Figure 7: A zoomed in view of Figure 5 at early times. Here, each marker
represents 1,000 time steps in the finite volume scheme. Note how each
concentration level has different peaks indicating the dynamics are slowed by
the solvent.

###  3.3 Domain Growth at Large Time Scales

We examine now the domain growth during a time frame where the morphologies
are well formed. In this time frame, we observe from the indicator functions
the domains are growing in accordance with power laws. In other words, we
observe evidence that the simulations obey the scaling hypothesis [4]. The
accuracy of these power laws is demonstrated in Figure 8 by rescaling the
correlation functions as discussed in the introduction. The observed behavior
for accurate scaling regimes would be a ‘collapse’ of these curves into a
single curve in the variable $z:=|s|/t^{\alpha}$, as shown in Figure 8. In
other words, this visualization suggests there exists a value of $\alpha$ so
that the function $G(t,z)$ is independent of time. We particularly point out
from Figure 8 the power 1/3 from the Ising model appears to be close to the
true scaling power of the $c_{0}=0$ case. This type of demonstration has been
used for other phase separation models, see e.g. [4, 6] and the references
within.

As illustrated in the right-hand plot of Figure 5, the late time behavior of
the system at solvent fraction $c_{0}=0,0.2$ is very much compatible with the
typical $1/3$ growth exponent characterizing conservative two–state systems
dynamics. This is confirmed also by the morphologies shown in the first two
rows of Figure 1, where the bicontinuous phase typical of Ising–like systems
is observed.

The situation is very much different at large values of solvent concentration,
that is to say for the case $c_{0}=0.6,0.8$ where the growth exponent is much
smaller. Correspondingly, the growth process is much slower, and hence, the
morphology is characterized by approximately spherical domains inside a
percolating sea of solvent.

Finally, at the intermediate value $c_{0}=0.4$ of solvent fraction, a sort of
crossover between the small and large solvent fractions is observed. Figure 5
suggests an exponent droping from $1/4$ to $1/12$ around time $t=3$. The
corresponding morphologies are coherent. Indeed, the initial bicontinuous
phase progressively transforms into a picture characterized by well separated
domains inside the sea of solvent.

Figure 8: Correlation functions for the time interval $[6,9]$ plotted against
the numerical spacial steps scaled according to the power law $t^{1/p}$, where
$p=3$ for $c_{0}=0$ (left) and $p=18$ for $c_{0}=0.8$ (right). The colors of
the plots represent the time of the simulation with green representing $t=6$
and red representing $t=9$. All other concentrations are similar with their
respective powers.

###  3.4 Behavior for Small Concentrations

It is natural to contemplate the behavior of the growth power laws discussed
in the previous sections as the limit of the solvent concentration tends to 0.
We plot the results of the structure factor and correlation function
indicators in Figure 7. We observe that the limiting behavior of the
simulation non–monotonically tends toward the behavior of the two state model
with $0$ solvent concentration level. Assuming the behavior changes
continuously with the initial solvent concentration, this would imply the
existence of an optimal initial solvent proportion maximizing the power,
$1/p$. We will also point out that since the mesh size $\Delta x,\Delta y$ is
fixed for all simulations, there may be issues tracking the solvent at small
(relative to the mesh size) concentrations.

Figure 9: Structure factor (left) and correlation function (right)
calculations for solvent concentrations $c_{0}=0.1,0.05,0.01$ during the time
interval $(0,9]$. Each marker represents $4,000$ time steps in the finite
volume scheme.

##  4 Conclusion and Outlook

Taking into consideration the results presented, it is clear that the behavior
of model (2) is rich with complex dynamics and possesses the aptitude for
phase separation. In particular, as the model was derived as the hydrodynamic
limit of the Blume–Capel model with Kawasaki dynamics and Kac interactions, it
may prove to be a useful continuum model for the Monte–Carlo simulations
addressed in [26, 32] without evaporation. Of course, the model currently has
no machinery to capture the evaporation effect on the solvent particles and
there is less flexibility (with respect to the Monte–Carlo simulation) in the
tuning of interaction parameters. To mend these differences, potential
modifications to the model, including for example the addition of an
evaporation boundary or interaction tensor, must be investigated.

From a mathematical perspective, the well–posedness and regularity of
solutions to this model remains to be studied. While [24] provides existence
of solutions via a hydrodynamic limiting process and uniqueness in
$L^{\infty}$, the parabolic structure of the equations in (2) suggests that
more regularity is expected. Once a suitable space for solutions is
established, convergence of the finite volume scheme can be rigorously shown.
Numerical simulations point out that we can also expect certain properties on
the relationship between $m$ and $\phi$ (e.g. $|m|\leq\phi$ for all $t>0$).
Showing these relationships hold for model (2) may have strong implications
for the different applications.

The analysis on the domain growth done throughout Section 3.1 suggests this
model satisfies the scaling hypothesis presented in [3]. Moreover, it appears
that solvent can behave as a type of catalyse in the growth of the
morphologies. This behavior has also been observed in the stochastic model
[26]. It remains to be shown if the powers presented in Section 3.1 can, under
some assumptions, be analytically calculated from model (2) as a function of
the initial solvent concentration. One may follow some of the arguments in [4]
to formulate potential ideas for calculating approximate expressions of the
growth laws, i.e. to get insight into the so–called coarsening rates at least
for simple ball–like morphologies. It is worth noting that our numerical
simulations indicate that for $c_{0}=0$, the model recovers domain growths as
expected in the two–species Ising model in two–space dimensions. As $c_{0}$
increases up to some value around $0.5$ a percolation regime is noticed, while
if $c_{0}$ takes relatively high values, then the growth of morphologies is
slowed down accordingly, as in fact expected from the physical point of view.

Last but not least, the continuum model is superior to its stochastic
microscopic description in terms of computation time. This feature keeps open
the possibility of fast estimation of model parameters especially if one
thinks of involving more complex physics, like accounting for solvent
evaporation, transport and recombination of charges along the obtained
morphologies, polycrystallization, degradation – typical scenarios
encountered, for instance, in the case of organic solar cells; see [29, 30]
for recent related works.

## Acknowledgments

RL and AM are grateful to Carl Tryggers Stiftelse for the financial support
via the grant CTS 21–1656. ENMC acknowledges the Mathematics Department of the
Karlstad University for its warm hospitality as well as INDAM–GNFM for
support. AM thanks SNIC for projects nr. 2020/9–178, 10–94 (HPC2N), and
2022/22-1171 Multiscale simulations of hybrid continuum–discrete–stochastic
systems for providing computational resources and storage capacity. SAM
acknowledges the funding from the Swedish National Space Agency (Grant 174/19)
and the Knut and Alice Wallenbergs Stiftelse (Grant 2016.0059)

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