# Heterogeneous diffuse interfaces: a new mechanism for arrested coarsening in binary mixtures. Heterogeneous diffuse interfaces.

**ABSTRACT** We discuss the dynamics of binary fluid mixtures in which surface tension density is allowed to become locally negative within the interface, while still preserving positivity of the overall surface tension (heterogeneous diffuse interface). Numerical simulations of two-dimensional Ginzburg-Landau phase field equations implementing such mechanism and including hydrodynamic motion, show evidence of dynamically arrested domain coarsening. Under specific conditions on the functional form of the surface tension density, dynamical arrest can be interpreted in terms of the collective dynamics of metastable, non-linear excitations of the density field, named compactons, as they are localized to finite-size regions of configuration space and strictly zero elsewhere. Aside from compactons, the heterogeneous diffuse interface scenario appears to provide a robust mechanism for the interpretation of many aspects of soft-glassy behaviour in binary fluid mixtures.

**0**Bookmarks

**·**

**87**Views

Page 1

DOI 10.1140/epje/i2011-11093-8

Regular Article

Eur. Phys. J. E (2011) 34: 93

THE EUROPEAN

PHYSICAL JOURNAL E

Heterogeneous diffuse interfaces: A new mechanism for arrested

coarsening in binary mixtures

Heterogeneous diffuse interfaces

R. Benzi1, M. Bernaschi2, M. Sbragaglia1, and S. Succi2,a

1Physics Department, University of Roma, Tor Vergata, via della Ricerca Scientifica, 1, 00133, Roma, Italy

2Istituto Applicazioni Calcolo-CNR, via dei Taurini 19, 00185, Roma, Italy

Received 18 April 2011 and Received in final form 15 June 2011

Published online: 23 September 2011 – c ? EDP Sciences / Societ` a Italiana di Fisica / Springer-Verlag 2011

Abstract. We discuss the dynamics of binary fluid mixtures in which surface tension density is allowed to

become locally negative within the interface, while still preserving positivity of the overall surface tension

(heterogeneous diffuse interface). Numerical simulations of two-dimensional Ginzburg-Landau phase field

equations implementing such mechanism and including hydrodynamic motion, show evidence of dynami-

cally arrested domain coarsening. Under specific conditions on the functional form of the surface tension

density, dynamical arrest can be interpreted in terms of the collective dynamics of metastable, non-linear

excitations of the density field, named compactons, as they are localized to finite-size regions of configura-

tion space and strictly zero elsewhere. Aside from compactons, the heterogeneous diffuse interface scenario

appears to provide a robust mechanism for the interpretation of many aspects of soft-glassy behaviour in

binary fluid mixtures.

Introduction

Recent work, both analytical and numerical, has evi-

denced that binary mixtures supporting a locally negative

cost of building an interface, the surface tension being still

positive, exhibit many signatures of soft-glassy behavior,

such as ageing, arrested coarsening and non-linear rheol-

ogy under shear forcing [1,2]. As a special case of the afore-

mentioned Heterogeneous Diffuse Interface (HDI) sce-

nario, it has been shown that soft-glassy behaviour can

be associated with the onset and collective interactions

of coherent excitations of the fluid density, named “com-

pactons”, because of their finite support in configura-

tion space. Such compactons, whose existence was first

suggested by numerical simulations of Lattice Boltzmann

(LB) models with competing attractive/repulsive inter-

actions [1], have been analytically proven to be exact

metastable solutions of a suitable Ginzburg-Landau (GL)

phase field model, derived from the LB kinetic model un-

der the assumption of constant density of the overall mix-

ture [3]. Compactons purport a very appealing picture of

self-glassiness, for they provide a direct link between the

ruggedness (coexistence of many competing local minima

of the free-energy landscape) and the complexity of den-

sity field in configuration space. As a result, there is sig-

nificant scope for analyzing their properties in depth.

In this paper we investigate the HDI scenario, and the

role of compactons, on domain coarsening. The study is

ae-mail: s.succi@iac.cnr.it; succi@iac.rm.cnr.it

performed through direct numerical simulation of a suit-

able two-dimensional Ginzburg-Landau phase field model,

with conserved parameter dynamics and hydrodynamic

interactions. The simulations show that, also under these

hitherto unexplored conditions, the HDI mechanism does

lead to arrested domain coarsening. This further corrobo-

rates the idea of HDI scenario as a potentially new mecha-

nism to promote slow-relaxation phenomena and arrested

coarsening in binary fluid mixtures.

1 Ginzburg-Landau phase field model

We begin by considering a binary mixture of fluids, say

A and B, whose dynamics is described by the space-time

evolution of the following order parameter:

φ(x;t) =ρA− ρB

ρA+ ρB

,

(1)

where ρkdenotes the density of fluid k = A,B. By defini-

tion φ = ±1 in the A(B) bulk phase (with zero minority

species), and φ = 0 at the interface.

The order parameter is assumed to obey the following

Ginzburg-Landau (GL) phase field equation [4]:

∂tφ(x;t) = −δF[φ]

δφ

+√?η(x,t),

(2)

F[φ] =

?

dx

?

V (φ)+1

2D(φ)|∇φ|2+κ

4(Δφ)2

?

. (3)

Page 2

Page 2 of 7 The European Physical Journal E

In the above, V (φ) is the bulk free-energy density, which,

with no loss of generality, we shall take in the standard

double-well form V (φ) = −1

jumps between the two stable bulk phases, φ = ±1, with a

false vacuum φ = 0 in between. Finally, η is a white noise

δ-correlated in space and time, whose variance ? fixes the

temperature of the system.

The key ingredient of our model is the stiffness func-

tion D(φ), describing the lowest-order approximation to

the energy cost of building an interface between the two

fluids. In the standard Ginzburg-Landau formulation, this

is a constant parameter D0, fixing the value of the surface

tension through the relation

?

x being the coordinate across the (flat) interface.

Positive values of γ promote coarsening, as a result of

the surface tension tendency to minimize the surface/vol-

ume ratio of the fluid. Negative values of γ, on the other

hand, trigger unstable growth of the interface, which is

typically responsible for morphological changes and pat-

tern formation phenomena [5–10]. Such an instability is

usually tamed at short scales by higher-order “bending”

terms of the form ∼ κ(Δφ)2, where κ is usually referred

to as bending rigidity. It is readily seen that this stabilized

scenario is realized by the choice D0< 0 and κ > 0.

More complex scenarios are opened up by allowing the

rigidity D to acquire a dependence on the local value of

the order parameter φ. Among others, [6–10], Gompper

and coworkers studied the case with piece-wise constant

D(φ) to describe micro-emulsions [11,12].

Our model belongs to the same class as Gompper’s

one, with a two-parameter quadratic dependence,

2φ2+1

4φ4, so as to support

γ ∼ D0

(∂xφ)2dx,

(4)

D(φ) = D0+ D2φ2.

(5)

The key feature of such functional form is to allow the

rigidity D(φ) to become locally negative, while still pre-

serving the positivity of the overall integral defining the

surface tension, according to eq. (4). A necessary condi-

tion for this to happen is that D0and D2be of opposite

sign, which reflects the underlying competition between

repulsive and attractive interactions. More specifically, the

cross-over occurs at a critical value Φc= ∓

the condition for such cross-over to take place within the

interface, i.e. |Φc| ? 1, reads

????

and D2< 0, or vice versa. In the former case, the local

instability is confined to the peripheral regions of the in-

terface (|Φ| > |Φc|), while in the latter, it takes place in

the internal region (|Φ| < |Φc|). The latter is chosen in the

sequel. To be noted that, at variance with previous work,

we insist on keeping the overall surface tension positive,

namely

D0+ D2?φ2? > 0,

?

−D0

D2, so that

D2

D0

????? 1.

(6)

The above relation can be realized in two ways; D0> 0

(7)

where brackets stand for average over the density gradi-

ent squared. Based on the previous definition of critical

order parameter, this also writes as ?φ2? < Φ2

tions (6), (7) provide a quantitative definition of the HDI

scenario, as purported in this paper.

Since the surface tension remains globally positive, the

HDI scenario hereafter considered does not require any

extra stabilization mechanism via the bending term. As a

result, we shall set κ = 0.

We also wish to point out that the quadratic form (5)

can be obtained bottom-up, from an asymptotic treat-

ment of a lattice kinetic model for a two-component fluid

with competing short-range attraction and mid-range re-

pulsion. More specifically, it derives from an expansion of

the free energy of the kinetic model around φ = 0, with

the assumption that the total density ρ = ρA+ ρB be

constant in space and time, so that the free energy is a

function(al) of the order parameter only.

In the following, we shall show that, under the HDI

conditions introduced above, the fluid mixture ultimately

exhibits a dynamic arrest of domain coarsening.

c. The condi-

2 Compactons as analytical solutions of the

GL model

As a special case of the HDI scenario, we shall show that

the choice D0= 0 and D2> 0 leads to the onset of very

peculiar structures, named hereafter “compactons”, ow-

ing to their property of being confined to finite-support

regions of configuration space. We present our analytical

study by inspecting one-dimensional, stationary solutions

of eqs. (2), (3), at zero temperature (? = 0). The standard

Euler-Lagrange equations,

dx

d

?

∂F

∂(∂xφ)

?=∂F

∂φ, yield

d

dx(D(φ)∂xφ) =1

2D?(φ)(∂xφ)2+ V?(φ),

(8)

where prime stands for d/dφ. In the standard limit D2→0,

i.e. D?(φ) → 0, a simple quadrature delivers the following

prime integral:

1

2D0(∂xφ)2+1

2φ2−1

4φ4= E,

(9)

where E = −V (±1) = 1/4, as per the boundary condition

∂xφ = 0 at infinity, is an arbitrary integration constant.

By denoting with K(x) ≡ K(φ,φx) the inhomoge-

neous (“kinetic energy density”) term in the free-energy

functional, we readily identify the integration constant as

E = K(x) − V (x), where we have set V (x) ≡ V (φ(x)).

As a result, based on the expression (3), the free energy

of the configuration is given by

?

In the false vacuum,φ(x)=0, we have K(x)=V (x)=0,

hence E = F = 0, while in the true ground state,

φ(x) = ±1, K(x) = 0, V (x) = −1/4, so that E = 1/4 and

F = L/4−2L/4 = −L/4, which corresponds to the global

minimum.

F =(K(x) + V (x))dx = EL + 2

?

V (x)dx.

(10)

Page 3

R. Benzi et al.: Heterogeneous diffuse interfaces: A new mechanism for arrested coarsening ...Page 3 of 7

The expression (9) yields the familiar soliton/antisol-

iton solution, φ(x) = ±1tanh−1(x/ld), where ld=

is the typical width of the soliton interface. Clearly, sta-

bility of the soliton requires D0> 0.

In the presence of a generic D(φ), the situation is

slightly more complicated (it is the analogous of a par-

ticle moving in a potential with a space-dependent mass).

However, little algebra shows that, upon multiplying both

sides by ∂xφ, the basic equation still reads formally the

same, i.e. (8) rewrites as

?1

This admits the same prime integral, yet with a generic

D(φ).

In the limit D0→ 0, this prime integral reads as fol-

lows:

1

2D2φ2(∂xφ)2+1

where the arbitrary integration constant E can no longer

be set to 1/4 because in the limit φ → 0, the space deriva-

tive φxcan develop a discontinuity.

A further quadrature delivers the analytical solution

of (12) in the form

?D0/2

d

dx

2D(φ)(∂xφ)2+1

2φ2−1

4φ4

?

= 0.

(11)

2φ2−1

4φ4= E,

(12)

φE(x) = ±

?

1 − cosh(ξ) + esinh(ξ) χ

?x − x0

le

?

, (13)

where ξ = (x − x0)/ld, e =

Note that the diffusive length is now defined as ld=

?D2/2. Here x0 is an arbitrary location and χ is the

of the segment x0− le/2 ≤ x ≤ x0+ le/2, where

?E/E0and E0= 1/4.

characteristic function (χ = 1 inside and χ = 0 outside)

le= ldtanh−1

?

2e

1 + e2

?

(14)

(see fig. 1).

Several comments are in order. First, this solution is

compact, i.e., it is identically zero outside the segment

[x0−le/2 < x < x0+le/2], the support of the compacton

of “energy-density” E. This property is crucially related

to the vanishing of the prefactor in front of the differ-

ential operators, which allows discontinuity in the slope

of φ(x). The location of the segment x0 is arbitrary be-

cause of translation invariance, while its extension le is

dictated by the specific value of the prime integral E. Un-

der the condition that ldbe real, i.e., D2> 0, a positive

E > 0 corresponds to the nucleation of a compacton of

size le> 0. The “compacton” can eventually invade the

system, le? L, L being the size of the domain, provided

E → E0= 1/4, since le→ ∞ as E → 1/4. More interest-

ing, however, and entropically favored as well, is the pos-

sibility that a gas of N “compactons” may invade the sys-

tem at lower values E < 1/4. This can be realized through

a superposition of N compactons, each of size li (sub-

script e relaxed for simplicity), centered upon a different

value of x0, with the global constraint LN=?N

i=1li= L.

Fig. 1. A sketch of a multi compacton-anticompacton config-

uration. The empty segments in between, at φ = 0, correspond

to false (unstable) vacua.

The resulting “energy” is EN =?N

responding to φ(x) = 0 (no compacton), and EN = 0,

associated to the two extremal values φ(x) = ±1. Given

that lidiverges in the limit E → 1/4, the latter condition

can only be attained in the thermodynamic limit L → ∞.

The above configurations represent the unstable vacuum

and the true ground state of the field theory, respectively.

In between, there lies a full spectrum of metastable states

with intermediate “energy” EN, each associated with a

given partition Λ = {li, i = 1,N}. For each given set

of parameters, the time-asymptotic value of EN attained

by the system is dictated by the initial conditions. The

possibility of such a linear superposition of elementary

solutions of a highly non-linear field theory is a direct

consequence of compactness. Since “compactons” do not

overlap, they obey a non-linear superposition principle,

(?

alized GL equation. By appealing to the above non-linear

superposition principle, a stability analysis shows that, as

long as the overall surface tension is positive, γ > 0, the

gas of “compactons” is stable against arbitrary (square-

integrable) perturbations of the order parameter, hence it

represents a local minimum of the free-energy landscape.

Since this analysis is not completely standard, here

we sketch the basic underlying ideas, leaving full details

to a lengthier publication. The first step is to consider

the variation, δF ≡ F[φ + δφ] − F[φ], of the free-energy

functional under a perturbation δφ(x) confined within the

support [x0,x0+ lE] of the compacton ΦE(x − x0).

We have

i=1Eili=¯ENL, with

0 ≤¯EN ≤ 1/4. The limiting cases are EN = L/4, cor-

iφi)n=?

iφn

i, for any power n. As a result, an arbi-

trary superposition of “compactons” still obeys the gener-

δF =F[φE(x−x0) + δφ]−F[φE(x−x0)] =

D2

2

+D2

2

?

dx?φ2

dx?∇(φE)2∇(δφ)2?+1

E(∇δφ)2+ (δφ)2(∇φE)2?

?

2

?

dx[3φ2

E−1](δφ)2,

Page 4

Page 4 of 7The European Physical Journal E

where φE is shorthand for φE(x − x0). The second term

on the r.h.s. of the above expression can be integrated by

parts, to give

?

Putting all terms together we obtain

?

+dx

2(3φ2

D2

2

dx?∇(φE)2∇(δφ)2?= −D2

2

?

dx(δφ)2Δ(φE)2.

δF =D2

2

dx?φ2

E(∇δφ)2+ (δφ)2(∇φE)2?

E−1)−D2(∇φE)2−D2φEΔφE

?

?1

?

(δφ)2.

(15)

Since φEis a stationary solution, it obeys the steady-state

equation of motion D2φEΔφE = −D2(∇φE)2− 1 + φ2

so that the eq. (15) simplifies to

?

+1

2

This expression is clearly positive definite for D2> 0,

thus proving the stability of the compacton. To be noted

that, in performing the integration by parts, terms have

been discarded which only vanish on the assumption that

the perturbation δφ be identically zero outside the com-

pacton support. This requirement stems from the fact that

φE = 0 is an unstable vacuum, and can be removed by

generalizing the stability analysis to the case where φEis

explicitly given by a superposition of multiple compactons.

This result qualifies “compactons” as relevant collec-

tive degrees of freedom responsible for slow relaxation of

the binary mixture. Therefore, we are led to a very elegant

and intuitive picture of self-glassiness, as the nucleation

of a “gas of compactons”, each of which corresponds to

a local minimum of the free-energy associated with the

LG eqs. (2), (3). Since “compactons” can be added to-

gether, each different combination generates a distinct dy-

namical partition of physical space. This provides a direct

map between the complexity of the free-energy landscape

(coexistence and competition of multiple minima, some-

times referred to as “ruggedness”) and the morphological

complexity of the fluid density in configuration space. If

only as a mere analogy, this picture is reminiscent of the

inherent-structures emerging from numerical and analyt-

ical studies of glass-forming fluids [13–15]. To be noted

that the fact that they do not overlap does not config-

ure compactons as non-interacting structures. Quite on

the contrary, since φ = 0 is an unstable solution, the

empty regions between the compactons are constantly

filled with new compactons, which interact via contact-

like forces, as a sort of bootstrap rigid-spheres of varying

size (le).

E,

δF =D2

2

dx?φ2

[1 + φ2

E(∇δφ)2+ (δφ)2(∇φE)2?

E](δφ)2dx.

?

(16)

2.1 Discrete compactons

Since the collective properties of the “gas of compactons”

are demonstrated via numerical simulations, it is impor-

tant to prove that compactons survive discreteness, as we

shall show in the sequel. In particular, we analyze under

what condition on D0and D2one can still find compact

solutions on a lattice. To this aim, we consider stationary

solutions of the discretized version of (2) which become

0 at x = 0 and find a symmetry in the solution, namely

x → −x implies φ → −φ (compacton-anticompacton du-

ality). This symmetry is clearly broken by the solution

defined in (13) and the condition for the existence of a

symmetry-breaking, non-zero, solution of the discrete LG

equation, reads D0−2D2

tice spacing. In the limit of small Δx and large D2, the

latter yields

of competing scales, as l2

to D0/Δx and ldhas been defined previously. This way,

the limit D0→ 0, where the system shows self-glassiness,

now reads as ld? l0. To be noted that, with D0> 0, the

continuum limit Δx → 0 would suppress the compactons,

in line with the fact that compactons in the continuum

require D0→ 0.

Summarizing, the present analysis highlights the fol-

lowing natural hierarchy of scales associated with the dis-

crete gas of lattice compactons

0

Δx2+ 2D2E > 0, Δx being the lat-

D2

0

D2Δx2 < E and can be rephrased in terms

0/l2

d< 1, where l0is proportional

l0< ld< le< L.

(17)

The above hierarchy of scales hints at the possibility of

developing a statistical mechanics of the compacton gas.

In principle, this requires the evaluation of the partition

function Z(β) = ?e−βH[φ]?, H being the GL free-energy

functional and brackets standing for functional integration

upon all compacton configurations, plus thermal fluctua-

tions on top of them at non-zero temperature. The calcu-

lation of such functional integral appears fairly challeng-

ing, due to i) the singularities which develop in ∂xφ in the

limit φ → 0, and ii) the fact that the sum over all possible

dynamical partitions of physical space as induced by the

set of all possible compacton configurations, needs to be

computed. This is a very interesting problem for future

research.

3 Numerical simulations

Phase-field equations with non-conserved parameter, such

as (2), (3), have been recently simulated, and shown neat

evidence of arrested coarsening and ageing phenomena [3].

We next proceed to show that such soft-glassy behaviour

is indeed observed also in numerical simulations of gen-

eralized LG eqs. (18), (19), with conserved parameter

and hydrodynamic effects. In particular, we consider a

convective-diffusion equation of Cahn-Hilliard type (com-

putational details can be found in [16])

∂tφ + (u · ∇)φ = MΔμ(φ),

(18)

where M is an order parameter mobility (here assumed in-

dependent of concentration φ) that controls the strength

Page 5

R. Benzi et al.: Heterogeneous diffuse interfaces: A new mechanism for arrested coarsening ... Page 5 of 7

Fig. 2. The coarsening length for cases (a), (b) and (c). The following plots are shown: (a) D0 = 0.03, D2 = 0.0; (b) D0 = −0.003,

D2 = 0.03, (c) D0 = 0.03, D2 = 0.03. In all cases, we have used a constant mobility M = 0.09 and η = 0.16666 and resolution

Nx× Ny = 256 × 256. The hydrodynamic scaling prediction, R ∼ t3/2is also reported for comparison. Three snapshots of the

order parameter φ(x,y) at the intermediate time step t = 20000 are also reported.

of the diffusion, and u is the fluid velocity. The fluid ve-

locity obeys the Navier-Stokes equation, which for an in-

compressible fluid at constant density ρ reads

ρ(∂tu + (u · ∇)u) = −∇P − φ∇μ + ηΔu,

where all symbols are standard. The chemical potential

is derived from the free energy described in the previous

sections, namely

(19)

μ = −δF

δφ,

F = V (φ) +1

2(D0+ D2φ2)|∇φ|2.

We take a constant mobility M = 0.09 and η = 0.16666

and a 2d system with resolution Nx× Ny = 256 × 256.

Boundary conditions are periodic, and initial conditions

are chosen randomly, φ(x,y;t = 0) = r, where r is a ran-

dom number uniformly distributed in [−0.1,0.1].

In fig. 2 we plot the coarsening length for three dif-

ferent cases: (a) D0= 0.03, D2= 0.0; (b) D0= −0.003,

D2 = 0.03, (c) D0 = 0.03, D2 = 0.03. Case (a) is the

standard one, whereby the cost of building an interface

is positive everywhere across the interface. Case (b) cor-

responds to the HDI scenario discussed in the previous

sections. Finally, case (c) is a variant of (a), in which the

cost of extending the interface is higher in the periphery

than in the central region.

As first observation, we note that, in case (a), the

coarsening length obeys the hydrodynamical scaling pre-

diction R ∼ t2/3[4], until saturation is reached. Such sat-

uration, however, is not related to any dynamical arrest

mechanism, but is simply due to the finite size of the sim-

ulation box. Case (c) shows a similar growth, yet with a

much less well-defined scaling exponent, supporting the

conclusion that the additional rigidity brought about by

D2> 0 spoils the 2/3 hydrodynamic scaling. Finally, case

(b) shows clear evidence of dynamical arrest, after about

t = 10000 time steps. In fig. 3, we show three color plates

of the order parameter φ(x,y;t) at initial (t = 0), inter-

mediate (t = 20000) and late stage (t = 100000) condi-

tions, for the case D0= 0.03 and D2= 0, without noise.

From this figure, it is apparent that coarsening proceeds

all along, and only saturates due to the finite size of the

simulation box. A typical sequence of snapshots for case

(b) is shown in fig. 4. From this sequence, it is appar-

ent that domain coarsening is basically suppressed since

the early stage of the evolution. A quite natural ques-

tion is why the previous results do not show any visual

evidence of compacton configurations. To investigate this

point, we performed further simulations with D2= 0.03

and two values of D0= 0.03 and D0= 0.001, both with-

out hydrodynamics. From fig. 5, it is appreciated that the

latter gives rise to standard separation with the expected

1/3 Lifshitz exponent, whereas the former shows a dis-

tinct arrest (rightmost panel). Visual inspection of the

corresponding configurations now shows clear evidence of

compacton structures (leftmost panel), as opposed to the

phase-separating case (mid panel). One may argue that

the onset of compactons is indeed related to the small

value of D0= 0.001, as opposed to D0= 0.03. However,

even taking D0 = 0.001, the simulations with hydrody-

namic interactions did not show visual evidence of com-

pactons, in spite of the arrested phase separation. We can

conclude that the HDI scenario does indeed lead to ar-

rested phase separation under fairly general conditions,

Page 6

Page 6 of 7The European Physical Journal E

Fig. 3. Three snapshots of the density configuration for case (a): D0 = 0.03, D2 = 0, at t = 0, t = 20000 and t = 100000.

Parameters are the same as in fig. 2.

Fig. 4. Three snapshots of the order parameter configuration for case (b): D0 = −0.003, D2 = 0.03, at t = 0, t = 20000 and

t = 100000.

Fig. 5. Order parameter configuration for the case D0 = 0.001, D2 = 0.03 (left) and D0 = 0.03, D2 = 0.03 (middle) at

t = 200000. The right panel shows the corresponding evolution of the coarsening length.

i.e. with conserved or non-conserved order parameter and

with or without hydrodynamic interactions. However, so

far, we have not been able to detect compactons when

hydrodynamic interactions are present.

4 Discussion and outlook

The present study lends further support to the role of

the HDI scenario as an effective mechanism for promot-

ing dynamical arrest of domain coarsening in binary fluid

mixtures. The basic condition is that the surface tension

density be allowed to become locally negative within in-

terface, while still preserving the positivity of the overall

surface tension. Clearly, such condition can only be real-

ized in the presence of competing attractive and repulsive

microscopic interactions, and can hopefully be tested on

experimental grounds. The effects of the HDI mechanism

appear to hold for both conserved and non-conserved pa-

rameter dynamics and irrespectively of whether or not hy-

drodynamic motion is included. This points to HDI as a

very robust mechanism to interpret soft-glassy behaviour

in binary mixtures. Under special conditions, D0→ 0 and

D2> 0, the HDI materializes through the emergence of

a gas of collective excitations of the density field, named

compactons, which are numerically and analytically shown

to associate with local minima of the free-energy land-

scape. Compactons provides an elegant account for many

observed signatures of soft-glassy behaviour, including

ageing and dynamical arrest. However, so far, we have not

been able to detect compactons in the presence of hydro-

dynamic interactions. The study of the HDI mechanism

and the role of compactons on the rheology of binary mix-

tures makes an interesting object for future investigation.

Page 7

R. Benzi et al.: Heterogeneous diffuse interfaces: A new mechanism for arrested coarsening ... Page 7 of 7

References

1. R. Benzi, S. Chibbaro, S. Succi, Phys. Rev. Lett. 102,

026002 (2009).

2. R. Benzi et al., Europhys. Lett. 91, 14003 (2010).

3. R. Benzi, M. Sbragaglia, M. Bernaschi, S. Succi, Phys.

Rev. Lett. 106, 164501 (2011).

4. A. Bray, Adv. Phys. 43, 357 (1994).

5. D. Seul, D. Andelman, Science 267, 476 (1995).

6. A. Lamura, G. Gonnella, J.M. Yeomans, Europhys. Lett.

45, 314 (1999).

7. S. Wu et al., Phys. Rev. B 70, 024207 (2004).

8. S.A. Brazovskii et al., Sov. Phys. JETP 66, 625 (1987).

9. P.L. Geissler, D.R. Reichman, Phys. Rev. E 71, 031206

(2004).

10. M. Tarzia, A. Coniglio, Phys. Rev. Lett. 96, 075702 (2006).

11. G. Gompper, M. Schick, Phys. Rev. Lett. 65, 1116 (1990).

12. G. Gompper, S. Zshocke, Phys. Rev. A 46, 4836 (1992).

13. F.H. Stillinger, J. Weber, Science 225, 983 (1984).

14. F. Sciortino et al., Phys. Rev. Lett. 83, 3214 (1999).

15. T.S. Grigera et al., Phys. Rev. Lett. 88, 055502 (2002).

16. V. Kendon, M. Cates, I. Pagonabarraga, J.-C. Desplat, P.

Blandon, J. Fluid Mech. 440, 147 (2001).