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Nonlocal Elasticity Yields Equilibrium Patterns in Phase Separating Systems
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Nonlocal Elasticity Yields Equilibrium Patterns in Phase Separating Systems
Yicheng Qiang , Chengjie Luo , and David Zwicker
Max Planck Institute for Dynamics and Self-Organization, Am Fa?berg 17, 37077 G?ttingen, Germany
(Received 1 August 2023; revised 16 December 2023; accepted 5 March 2024; published 12 April 2024)
Recent experiments demonstrated the emergence of regular mesoscopic patterns when liquid droplets
form in an elastic gel after cooling. These patterns appeared via a continuous transition and were smaller in
stiffer systems. We capture these observations with a phenomenological equilibrium model describing the
density field of the elastic component to account for phase separation. We show that local elasticity theoriesare insufficient, even if they allow large shear deformations. Instead, we can account for key observations
using a nonlocal elasticity theory to capture the gel ’s structure. Analytical approximations unveil that the
pattern period is determined by the geometric mean between the elastocapillary length and a nonlocalityscale. Our theory highlights the importance of nonlocal elasticity in soft matter systems, reveals the
mechanism of this mesoscopic pattern, and will improve the engineering of such systems.
DOI: 10.1103/PhysRevX.14.021009 Subject Areas: Soft Matter
I. INTRODUCTION
Phase separation in elastic media is a ubiquitous phe-
nomenon, which is relevant in synthetic systems to control
micropatterning [1–3]and in biological cells, where drop-
lets are embedded in the elastic cytoskeleton or chromatin[4–6]. While biological systems are typically dynamic and
involve active processes, the simpler synthetic systems can
exhibit stable regular structures. These patterns harborpotential for metamaterials and structural color, particularly
since they are easier to produce and manipulate than
alternatives like self-assembly by block copolymers [7]
or chemical cross-linking [8]. In these applications, it is
crucial to control the length scale, the quality, and the
stability of the pattern.
Recent experiments found stable regular mesoscopic
patterns and demonstrated remarkable control over these
structures [1]. However, the mechanism underlying their
formation is unclear, complicating further optimization.
The experiment proceeds in two steps [Fig. 1(a)][1].F i r s t ,
a polydimethylsiloxane gel is soaked in oil at hightemperatures for tens of hours until the system is equili-
brated. When the temperature is lowered in the second
step, the sample develops bicontinuous structures, remi-niscent of spinodal decomposition. However, in contrast
to spinodal decomposition, the length scale of thestructure does not coarsen but stays arrested at roughly
1–10μm, depending on the gel ’s stiffness. Interestingly,
this transition is reversible and the pattern disappears upon
reheating, suggesting a continuous phase transition.
Moreover, the resu lting pattern is independent of the
cooling rate, in contrast to earlier experiments on similarmaterials [3,9] . Consequently, the experiments might be
explainable by an equilibrium theory that captures elastic
deformations in the polydimethylsiloxane gel due to oildroplets formed by phase separation.
The experimental observations are reminiscent of micro-
phase separation, e.g., observed in block copolymers
[11,12] and interpenetrating polymer networks [13,14] .
However, phase transitions in such models are typicallyfirst order, e.g., in the seminal Ohta-Kawasaki model [15].
Moreover, in these theories, the size of the involved
molecules is similar to the size of the patterns they form,whereas the patterns in the experiment are much larger than
the oil molecules and the typical mesh size of the elastic gel
[1]. Alternatively, spinodal decomposition of a phase
separating system augmented with elasticity might describethe experiments [1]. However, typical local elasticity theory
can only account for slowed coarsening [16,17] , and we
will show that it does not yield stable equilibrium patterns.Consequently, these conventional models cannot explain
the qualitative features of the experiments.
In this paper, we propose an equilibrium theory that
explains the experimental observations [1]. Using a phe-
nomenological approach, we describe the system by acontinuous density field of the elastic component to
describe phase separation and elastic deformations with
a single free energy. We show that local elastic theories,based on the deformation gradient tensor, cannot account
for equilibrium patterns. Consequently, we consider a
david.zwicker@ds.mpg.de
Published by the American Physical Society under the terms of
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Further distribution of this work must maintain attribution to
the author(s) and the published article ’s title, journal citation,
and DOI. Open access publication funded by the Max Planck
Society.PHYSICAL REVIEW X 14,021009 (2024)
2160-3308 =24=14(2)=021009(11) 021009-1 Published by the American Physical Societyhigher order of the phenomenological approximation,
yielding a nonlocal elasticity theory that takes into accountthe structure of the gel [18–22]. We find a continuous phase
transition to a mesoscopic patterned phase, consistent with
experimental observations. We predict that the equilibriumperiod is governed by the geometric mean between the
elastocapillary length and the nonlocality scale, which
captures the stiffness dependence of the experimentallyobserved pattern length scale.
II. RESULTS
We aim to explain the experimental results [1]using a
phenomenological equilibrium theory for an isothermal
system. We thus define a free energy comprising entropic
and enthalpic contributions that can induce phase sepa-ration as well as contributions from elastic deformation.
While the former contributions can be captured by the
volume fraction density ??x?defined in lab coordinates x,
deformations are described by the deformation gradient
tensor F?X??dx=dX, which quantifies how material
points at position xhave been moved from the reference
positions Xwhere the gel is undeformed. Note that
volume conservation implies det ?F???
0=?,w h e r e ?0
denotes the fraction in the relaxed homogeneous initial
state [16].
A. Local elasticity models cannot explain periodic
equilibrium patterns
We start by investigating a broad class of elastic models,
where the elastic energy density is only a function of thedeformation gradient tensor F. The free energy Fof the
entire system can then be expressed as
F
local?F;?/C138?kBT
νZ
?fel?F??f0????κj??j2/C138dx; ?1?
where kBis Boltzmann ’s constant, Tis the constant
absolute temperature of the system, and νis a relevant
molecular volume, e.g., of the solvent molecules. In theintegral, the first term captures the elastic energy, f
0
accounts for molecular interactions and translational
entropy associated with ordinary phase separation, whilethe last term proportional to the positive parameter κ
penalizes volume fraction gradients, thus causing surface
tension [23,24] . Equilibrium states then correspond to
functions ??x?andF?X?that minimize F
localand obey
the compatibility constraint and volume conservation.
Can local elasticity models permit periodic equilibrium
states? To test this, we assume that such a state, described by
periodic functions ?/C3?x?andF/C3?x?, exists. We then show
that scaling this state by a factor λ>1in all spatial directions,
?/C3?λ?1x?, lowers the free energy Flocalgiven by Eq. (1),
implying that it could not have been an equilibrium state.
While we present the mathematical details in the
Supplemental Material [10], the gist of the argument canbe seen by considering the free-energy density ?f?Flocal=V
of a unit cell of volume Vof the periodic pattern. Scaling does
not affect the contribution of the second integrand in Eq. (1)to
?f, precisely because it averages a local function over one
period. Similarly, the first integrand stays invariant since thescaling factor λaffects the lab coordinates xand the reference
coordinates Xequally, so the values of the deformation
gradient tensor F?dx=dXare invariant and the scaled
tensor field reads F
/C3?λ?1x?. In contrast, the last term
contributes less for the scaled pattern since scaling reduces
the gradient term to λ?1??/C3, consistent with a lower penalty
for shallower interfaces. Taken together, we thus showed thatthe free energy of any periodic state can be reduced by scaling,
implying such states cannot be at equilibrium and instead
would eventually evolve toward longer length scales. Inessence, this is because only the interfacial parameters κ
carries dimensions of length whereas length scales associatedwith the structural details of the elastic material do not appearin local elastic theories.
B. Mesh structure suggests nonlocal
elasticity theory
Realistic elastic meshes exhibit length scales like the
mesh size ( ?10nm[22,25,26] ) and correlation lengths
of spatial heterogeneities ( ?100nm[27–29]), which
are comparable to the pattern length scale (several100 nm to several micrometers [1]). We thus hypothesize
that a characteristic length of the mesh is key forexplaining the observed patterns. Such a characteristiclength can be systematically included in our phenom-enological theory by expanding the elastic energy in
terms of the displacement field; see Sec. IIIof the
Supplemental Material [10]. This approach generically
leads to nonlocal elasticity theory , where a nonlocality
scale ξquantifies at what length scale nonlocal effects
become relevant [18–22].
The origin of nonlocal elasticity theory can be illustrated
in the simple case where the elastic mesh is described
as a collection of elastic elements; see Fig. 1(b). These
elements can represent either molecules forming the meshor structures on the larger correlation length scale ofheterogeneities. In any case, the elastic elements connectmaterial points separated by a finite distance, implying thestress at a particular material point results from summingover the interactions with all connected material points.Consequently, stresses are never strictly local, and theassociated elastic energy cannot be expressed as a localfunction of the strain. Instead, in a continuous field theory,the nonlocal stress is expressed as a convolution [18,21] .
The familiar local elasticity theory then emerges as alimiting case when considering phenomena on scales largeto the nonlocality scale ξ.
To develop a simple description of phase separation with
nonlocal elasticity, we focus on one-dimensional systems,where the deformation of the elastic mesh is captured by theQIANG, LUO, and ZWICKER PHYS. REV . X 14,021009 (2024)
021009-2scalar strain ?, which is directly connected to the only
component Fxxof the deformation gradient tensor,
??Fxx?1. Volume conservation then implies
??X???0
??X??1; ?2?
where the fraction ??x?in the lab frame follows from the
coordinate transform d x=dX???X??1. This connection
between strain ?and volume fraction ?permits a theory in
terms of only one scalar field in this one-dimensional case.Using a simple linear elastic model for the local stress,
σ?E?with elastic modulus E, we obtain the nonlocal
stress ,
σ
nonlocal ?X??EZ
??X0?gξ?X0?X?dX0; ?3?
where we choose a Gaussian convolution kernel [21,30] ,
gξ?X?????????
2
πξ2s
exp/C18
?2X2
ξ2/C19
; ?4?with a characteristic length ξ, which quantifies the non-
locality scale of the mesh [18,21,30] . This nonlocal model
can also be derived more rigorously, either generically (seeSupplemental Material [10]) or from a more explicit model
[18,31] . Note that the convolution is performed in the
reference frame since the topology of the network, gov-erning which material points interact with each other, isdetermined in this unperturbed state. The elastic energy
density is then given by the product of strain and nonlocal
stress, so the free energy of the entire system reads
F
nonlocal ??/C138?1
2Z
??X?σnonlocal ?X?dX
?kBT
νZ
?f0????κ????2/C138dx; ?5?
where the first term captures the nonlocal elastic energy
expressed in the reference coordinates X, whereas the second
term describes the free energy associated with phase sepa-
ration, expressed in lab coordinates x. We capture the essence
of phase separation using a Flory-Huggins model for the
local free-energy density [32–34],
f0?????log???1???log?1????χ??1???;?6?
where 1??is the solvent fraction. Here, the first two
terms capture entropic contributions, while the last term
describes the interaction between the elastic and solvent
component, quantified by the Flory parameter χ. Taken
together, Eqs. (2)–(6)define the free energy Fnonlocal as a
functional of the fraction ?of the elastic component.
C. Nonlocal elasticity enables periodic
equilibrium patterns
We start by analyzing equilibrium states of the model
by determining profiles ??x?that minimize Fnonlocal using
a numerical scheme described in the Supplemental
Material [10]. Here, we use the nonlocality scale ξas
the length unit and kBTas the fundamental unit of energy.
Consequently, we consider interfacial parameters κ<ξ2
since the interfacial width, which is typically of molecular
size, should be smaller than ξ. Our choice of the stiffness
Eis directly motivated by experimentally measured
moduli, which are on the order of 100 kPa. Using theseparameters, we find typical macroscopic phase separation,
but also periodic patterns for some parameter sets; see
Fig. 1(c) herein and Fig. S1 in Supplemental Material
[10]. In soft systems (small stiffness E), dilute regions,
corresponding to solvent droplets, alternate with dense
regions, where the elastic mesh is hardly strained ( ??1).
In contrast, harmonic profile can emerge for stiff systems(large E). Taken together, the nonlocal elastic theory
supports periodic patterns t hat qualitatively resemble
the patterns observed in experiments [1].
(a)
(b) (c)
FIG. 1. Nonlocal elasticity yields regular equilibrium patterns.
(a) Schematic picture of the experiment [1]: A relaxed elastic gel
is swollen in a solvent at high temperature; after cooling, a regularpattern emerges. (b) Schematic of a network of elastic elements(curly lines) connecting material points (red dots). Arrowsindicate the displacement of material points from the referencestate (transparent, positions X) to the deformed state (opaque,
positions x). The energy of the highlighted elastic element
depends on the distance between the two connected points,revealing its nonlocal nature. Coarse graining this system yieldsthe nonlocal convolution kernel (blue density), whose size ξis
roughly given by the length of the elastic elements. Note that theelastic elements need not correspond to molecules, but couldcapture the interaction of dense mesh regions since realisticmeshes are heterogeneous. (c) Equilibrium profiles ??x?for
various stiffnesses Eand interaction parameters χfor?
0?1,
???0.5, and κ?0.05ξ2. Profiles were obtained by numerically
minimizing Fnonlocal ; see Supplemental Material [10].NONLOCAL ELASTICITY YIELDS EQUILIBRIUM PATTERNS … PHYS. REV . X 14,021009 (2024)
021009-3To understand when periodic patterns form, we next
investigate the simple case where components can freelyexchange with a surrounding reservoir kept at fixedchemical potential μ; see Supplemental Material [10].
This situation allows solvent molecules to rush in and
out of the system, adjusting the average fraction ??of the
elastic component. Figure 2shows two phase diagrams of
this grand-canonical ensemble at different stiffnesses E.I n
the soft system [Fig. 2(a)], the phase diagram mostly
resembles that of ordinary phase separation: For weakinteractions ( χ<2), we find only a homogeneous phase
andμsimply controls ??. In contrast, above the critical point
atχ≈2(black disk), we observe a first-order phase
transition (brown line) between a dilute phase ( μ?0)
and a dense phase ( μ?0). However, at even stronger
interactions ( χ?3.3), an additional patterned phase
(denoted by P) emerges, where the periodic patterns exhibit
the lowest free energy. The lines of the first-order phasetransitions between the patterned phase and the dilute or
dense homogeneous phase (blue and brown dashed curves)
meet the line of the phase transition between the twohomogeneous states at the triple point (gray star), wherethese three states coexist.
The grand-canonical phase diagram of soft systems
[Fig. 2(a)] qualitatively resembles simple pressure-
temperature phase diagrams, e.g., of water. Assuming thatthe chemical potential μplays the role of pressure and that
the interaction χis negatively correlated with temperature,the dilute and dense homogeneous phases respectively
correspond to the gas and liquid phases. They become
indistinguishable at the critical point at low interaction
strength (corresponding to high temperatures). In contrast,the patterned phase, with its periodic internal structure,resembles the solid phase.
The general form of the grand-canonical phase diagram
persists for stiff systems [Fig. 2(b)], although the parameter
region of the patterned phase is much larger. However, thefirst-order transition between the dilute and dense homo-geneous phases disappears together with the normal criticalpoint of phase separation. Instead, we now find a continu-
ous phase transition (dotted red line) between the homo-
geneous and the patterned phases, which we will discuss inmore detail below. Taken together, these phase diagramssuggest that stable patterned phases emerge for sufficiently
large stiffness Eand interaction χfor intermediated ??.
The grand-canonical ensemble that we have discussed so
far is suitable when the timescale of an experiment is longcompared to the timescale of particle exchange with the
reservoir. In the experiments [1], the initial swelling takes
place over tens of hours with a measurable increase in sizeand mass, indicating that solvent soaks the sample until it isequilibrated with the surrounding bath. In contrast, thetemperature quench, during which the patterned phase isobserved, takes place on a timescale of minutes without thesolvent bath. This suggests that this process is better
described by a closed system.
D. Patterned and homogeneous phases
coexist in closed systems
In the closed system, corresponding to a canonical
ensemble, the average fraction ??
of elastic components,
and thus also the average fraction of solvent, is fixed. In this
situation, we find that multiple different phases can coexistin the same system; see Fig. 3. This is again reminiscent of
phase separation, where the common-tangent constructionreveals the fractions in coexisting homogeneous states.Indeed, we find exactly this behavior in soft systems [left-
hand panel of Fig. 3(a)], where a dilute and dense phase
coexist for fractions between the two vertical dotted lines,while the free energy of the patterned phase (blue line) isalways larger and thus unfavorable. The picture changes forlarger stiffness [right-hand panel of Fig. 3(a)], where the
patterned phase has lower energy and we can construct twoseparate common tangents, which respectively connect the
dilute and dense homogeneous phase with the patterned
phase. Analogously to phase separation, we thus expectsituations in which a patterned phase coexists with a
homogeneous phase (when ??is in the region marked with
H?PorP?H). Figure 3(b)corroborates this picture and
shows various coexisting phases as a function of the
stiffness Eand the interaction strength χ. Taken together,
the main additional feature of the canonical phase diagramsis the coexistence of multiple phases, which was only(b) (a)
FIG. 2. Grand-canonical phase diagrams reveal patterned phase.
(a) Phase diagram as a function of the chemical potential μand the
interaction strength χforE?0.01kBT=ν. Homogeneous phases
(region H) coexist on the brown line between the critical point of
phase separation (black disk) and the triple point (gray star), whilethe patterned phase (region P) coexists with the homogeneous
phase on the blue and brown dashed line. (b) Phase diagram as afunction of μandχforE?0.2k
BT=ν. The binodal line separating
the homogeneous and patterned phase exhibits either a first-ordertransition (blue and brown dashed line) or a continuous transition(red dotted line with associated critical points marked by red disks;see details in the Supplemental Material [10]). (a),(b) Model
parameters are ?
0?1andκ?0.05ξ2.QIANG, LUO, and ZWICKER PHYS. REV . X 14,021009 (2024)
021009-4possible exactly at the phase transition in the grand-
canonical phase diagram.
E. Higher stiffness and interaction
strength stabilize patterned phase
The canonical phase diagrams shown in Fig. 3(b) are
complex, but they generally preserve three crucial aspects
of the grand-canonical phase diagram shown in Fig. 2:
Higher stiffness (i) slightly favors the homogeneous phases,
(ii) greatly expands the parameter region of the patterned
phase, and (iii) induces a continuous phase transition. The
first point is illustrated by the binodal line of the homo-
geneous phase (thick brown lines and red dotted lines),which moves up with increasing stiffness E, implying that
larger interaction strengths χare necessary to stabilize
inhomogeneous systems. Inside the binodal line the systemexhibits various behaviors, which can be categorized by χ.
At a critical value χ
/C3, the patterned phase (blue star)
coexists with the dilute and dense homogeneous phase(brown stars), and the associated tie line corresponds to
the triple point in Fig. 2. For weaker interactions ( χ<χ/C3),
we mostly observe coexistence of a dilute and dense
homogeneous phase (region H?H), which corresponds
to normal phase separation. For stronger interactions(χ>χ
/C3), the system exhibits the patterned phase, either
exclusively (colored region) or in coexistence with a
homogeneous phase (regions H?PandP?H). Larger
stiffness Elowers the critical value χ/C3, thus expanding
the parameter region where the patterned phase exists.
Eventually, for sufficiently large E,χ/C3approaches the
critical point of the binodal (gray point), a tiny region
with patterned phase appears, and part of the binodal line
becomes a continuous phase transition (red dotted line),
reproducing the behavior predicted by the grand-canonical
phase diagram of stiff systems [Fig. 2(b)].
The influence of stiffness Eand interaction strength χ
becomes even more apparent in the three-dimensionalphase diagram shown in Fig. 3(c): With increasing E,
theχassociated with the critical point of phase separation
(a)
(c)(b)
FIG. 3. Closed systems exhibit phase coexistence. (a) Schematic free energy of homogeneous and patterned phases with common-
tangent construction (thin gray lines) for two stiffnesses E. Figure S2 in Supplemental Material shows corresponding numerical results
[10]. (b) Phase diagram as a function of the average fraction ??of the elastic component and interaction strength χfor various E. Only the
homogeneous phase (region H) is stable outside the binodal (brown line; black disk marks critical point) with a continuous phase
transition at the red dotted part. Only the patterned phase (region P) is stable inside the blue lines with color codes indicating length scale
and amplitude in the left- and right-hand column, respectively. Two indicated phases ( H?P,P?H,H?H) coexist in other regions.
The triple point corresponds to the tie line (thin gray line), where fractions ??of coexisting homogeneous and patterned phases are
marked by brown and blue stars, respectively. (c) Phase diagram as a function of ??,χ, and E. The binodal of the homogeneous phase
(brown surface) and the patterned phase (blue surface) overlap in the continuous phase transition (red surface). The critical points in(b) now correspond to critical lines, which all merge in the tricritical point (large black disk). A rotating version of the diagram is
available as a movie in Supplemental Material [10]. (a)–(c) Model parameters are ?
0?1andκ?0.05ξ2.NONLOCAL ELASTICITY YIELDS EQUILIBRIUM PATTERNS … PHYS. REV . X 14,021009 (2024)
021009-5(black line) increases slightly, whereas the states of three-
phase coexistence (blue line and brown lines) shift to lower
χ. All lines meet at the tricritical point (black sphere) for
E≈0.037kBT=ν,??≈0.54, and χ≈2.14. Increasing E
further, a part of the binodal line exhibits a continuous
phase transition, which expands with larger E. The phase
diagram thus summarizes three main aspects of our model.
First, the binodal line of phase separation, which is only
weakly affected by E, determines whether the system can
exhibit nonhomogeneous states. Second, if the system canbe inhomogeneous, the stiffness Edetermines at what value
ofχpatterned phases emerge. Third, for sufficiently large
E, these patterned phases form immediately due to the
continuous phase transition.
F. Continuous phase transition explains
experimental measurements
The continuous phase transition that we identified at
sufficiently large stiffness Eimplies that the system can
change continuously from a homogeneous phase to a
patterned phase when the interaction strength χis increased
(corresponding to cooling). Indeed, the amplitude of thepredicted pattern vanishes near the transition [right-hand
panel of Fig. 3(b)], while the length scale stays finite [left-
hand panel of Fig. 3(b)]. This behavior is not expected for
typical phase separating systems with first-order transi-
tions, where the order parameter changes discontinuously
during the phase transition [see gray line in Fig. 4(a)for an
example].
The continuous phase transition was already hypoth-
esized for the experiments [1], based on a lack of hysteresis
and a continuous change of the contrast measured by
light intensity. To connect to experiments, we mimic the
contrast using the square of the amplitude of the optimalvolume fraction profile. Figure 4(a)and the right-hand panel
of Fig. 3(b)show that the contrast changes continuously from
zero when the interaction strength χis increased for suffi-
ciently stiff systems. Moreover, Fig. 4(b) shows that the
associated pattern length scale changes only slightly, con-
sistent with the experiments. Note that deviations in theform of the curves could stem from thermal fluctuations,
finite resolution in the experiment, and also deviations in
model details.
G. Stiffness and interfacial
cost control pattern length scale
We next use the numerical minimization of the free energy
F
nonlocal to analyze how the length scale Lof the patterned
phase depends on parameters. Figure 5shows that L
decreases with larger stiffness Eand increases with the
interfacial cost parametrized by κ. The data in Fig. 5(a)
suggest the scaling L=ξ∝E?1=2over a significant parameter
range, which matches the experimental observations [1].
Moreover, Fig. 5(b) suggests L=ξ∝ξ?1=2κ1=4, which has
not been measured experimentally. Taken together, the twoscaling laws suggest that the equilibrium length scaleemerges from a competition between elastic and interfacial
energy.
The two scaling laws emerge qualitatively from a simple
estimate of the elastic and interfacial energies: Since shorterpatterns have more interfaces, the interfacial energy per
unit length is proportional to γL
?1, with surface tension
γ∝κ1=2[23]. In contrast, the elastic energy of a single period
originates from stretching a part of material from initiallength ξto final length L, resulting in an elastic energy
density proportional to ELξ
?1. Minimizing the sum of
these two energy densities with respect to Lresults in
L=ξ∝ξ?1=2E?1=2κ1=4, which explains the observed scalings
qualitatively.
(a) (b)
FIG. 4. Continuous phase transition recovers experimental
measurements. Squared amplitude (a) and length scale (b) ofperiodic patterns as a function of interaction strength χfor various
parameters indicated in (b), ?
0?1, and κ?0.05ξ2. The ampli-
tude indicates a continuous (colored data) and first-order (graydata) transition.(a) (b)
FIG. 5. Pattern length scale exhibits scaling laws. Length scaleLas a function of stiffness E(a) and interfacial parameter κ(b) for
various parameters. Putative scaling laws are indicated and the
prediction by Eq. (9)is shown for ?
0?1,???0.5,χ?4, and
γ≈kBTκ1=2=ν(green line).QIANG, LUO, and ZWICKER PHYS. REV . X 14,021009 (2024)
021009-6H. Approximate model predicts length scale
To understand the origin of the length scale Lin more
detail, we consider the limit of strong phase separation,
where the interfacial width is small compared to L; see
Fig.1(c). We thus approximate the volume fraction profile
??x?of the elastic component by a periodic step function
with fixed fractions ??and??; see dotted lines in Fig. 6(a).
Material conservation implies that the relative size of these
regions is dictated by the average fraction ??in the swollen
state, so we can only vary the period ?Lof the profile. The
stable period Lthen corresponds to the ?Lthat minimizes
Fnonlocal given by Eq. (5), implying F0
nonlocal ?L??0. Since
changing ?Ldoes not affect the local free energy f0,
we investigate only the average free energy of the
interface, ?fint??L?≈2γ?L?1, and the average elastic free
energy, ?fel??L??1
2?L?1R?L0
0σnonlocal ?X???X?dX, where ?L0?
???=?0??Lis the period in the reference frame. Figure 6(b)
shows the derivatives of these contributions with respect to
?L, indicating that they sum to zero for ?L?L. We show in
the Supplemental Material [10] that
??fel
??L≈E
ξ8
>>><
>>>:0 ?Lmin
1????
2πp/C16
1???
??/C172Lmin1????
8πp/C16
?0
????0
??/C172ξ2
?L2?L>L max;?7?
indicating three regimes bounded by
Lmin????π
2r?0
??ξand Lmax????
1
2r
?0
???????
?????ξ: ?8?
Figure 6(b)shows that this approximation of ??L?felcaptures
the main features of the full numerical data. Figure 6(b)
suggests that stable patterns are mainly possible in the gray
region ( LminIn this region, we use Eq. (7)to solve ??L?fel???L?fint?0
for ?L, resulting in
L≈?8π?1=4??
?????/C18ξγ
E/C191=2
; ?9?
consistent with numerical results; see transparent green
lines in Fig. 5. This expression shows that the stable period
Lis governed by the geometric mean of the elastocapillary
length γ=Eand the nonlocality scale ξ. Moreover, L
increases with a larger average fraction ??of the elastic
component, i.e., less swelling. In contrast, the fraction ??
has only a weak influence since it is close to 1 in the case of
strong phase separation, implying that the interaction
strength χaffects Lonly weakly.I. Patterned phase is governed by reference state
Finally, we use the approximate model to understand
when the patterned phase emerges. Here, it proves useful
to interpret Eq. (8)in the reference frame, where the
convolution of the nonlocal elastic energy takes place. De-
fining the length L0????=?0?Lin the reference frame and
the associated fraction α0????=?0????????=???????
occupied by the solvent droplet [Fig. 6(a)], we find
L>L min?L0>???π
2r
ξ; ?10a?
L1
2r
ξ; ?10b?
where the numerical prefactors arevery close to one. The first
condition ( L0?ξ) suggests that two solvent droplets need to
be separated by more than ξin the reference frame since L0
roughly estimates their separation; see Fig. 6(a).I fd r o p l e t s
were closer, they would feel each other ’s deformations,
which is apparently unfavorable. In the extreme case(a)
(b)
FIG. 6. Approximate model explains scaling laws. (a) Example
for a volume fraction profile (pink lines) and the correspondingpiecewise approximation (dotted gray lines) in the reference (top)and lab frame (bottom). (b) Derivatives of the average energydensity (in units of k
BTξ=ν) as a function of the pattern period ?L.
Shown are data from full numerics (symbols), numerics for thepiecewise profile (solid lines), and asymptotic functions (dashedlines) for the elastic (gray, disks) and negative interfacial energy(violet, squares). The stable length Lcorresponds to the crossing
point of the elastic (black) and the interfacial terms (violet).
Model parameters are E?0.02k
BT=ν,κ?0.05ξ2,?0?1,
???0.5, and χ?4.NONLOCAL ELASTICITY YIELDS EQUILIBRIUM PATTERNS … PHYS. REV . X 14,021009 (2024)
021009-7(L0?ξ), the average elastic energy is almost constant,
essentially because short-ranged variations are averaged by
the comparatively large nonlocal kernel. In contrast, thesecond condition implies that the droplet size in the reference
frame ( α
0L0) must be smaller than the nonlocality scale ξ.
Assuming ξcorresponds to the mesh correlation length, this
suggests that the droplet can at most deform the correlatedpart of the mesh, which we will discuss below. If droplets
were larger ( α
0L0?ξ), nonlocal features would only be
relevant at interfaces, so the system would behave as if it hadonly local elasticity and coarsen indefinitely.
This analysis highlights that the existence of the periodic
pattern depends on the reference frame, while its length
scaleLalso depends on the different stretch of the dilute and
dense region; see Fig. 6(a). This observation suggests an
intuitive explanation for the influence of the interaction χ:
Assuming that ?
?and??correspond to equilibrium volume
fractions and ???1
2for simplicity, we find α0∝??, which
decreases with larger χ. Consequently, the lower bound Lmin
is unaffected, while Lmaxincreases, consistent with our
observation that the patterned phase forms easier at higher χ
and the scaling law given by Eq. (9)holds for broader
parameter range with higher interaction strength (Fig. 5).
J. Spatial heterogeneity could cause nonlocality
Since the periodic equilibrium patterns crucially depend
on the nonlocality scale ξ, we hypothesize that such a
length scale is relevant in the experiments [1]. However, it
is unlikely that the mesh size (typically below 100 nm)
directly controls ξsince the observed droplets are larger
(several hundred nanometers). Instead, we propose that ξis
governed by spatial heterogeneities. The correlation length
of these heterogeneities, roughly measuring the size of soft
regions, can be much larger than the mesh size [27–29].
Figure 7illustrates the difference between the two
interpretations.Realistic meshes exhibit multiple length scales, which
could all affect the behavior. To elucidate this, we brieflyconsider the impact of a convolution kernel that consists oftwo parts, g?X??? 1?ζ?g
ξ1?X??ζgξ2?X?, where ξ1rep-
resents the mesh size, ξ2denotes the scale of spatial
heterogeneities, and the nondimensional weight ζ<1
determines the relative contributions. If the mesh size ξ1is
much smaller than the pattern length scale, the correspond-ing convolution reduces to local elasticity, so the elastic
energy following from Eq. (3)can be approximated as
F
el≈?1?ζ?E
2Z
??X?2dX
?ζE
2ZZ
??X???X0?gξ2?X?X0?dXdX0:?11?
The part corresponding to ξ1effectively changes only the
local free-energy density. In the simple case of one-dimen-sional systems, the first integrand can then be expressed interms of the volume fraction ?, and absorbed in a rescaled
free-energy density f
0???. In contrast, the nonlocality scale
ξ2provided by the second term will control the periodic
equilibrium patterns. The effective stiffness correspondingto this term is ζE, whereas the experimentally measured bulk
modulus of a uniform deformation will include both terms
and thus remain E. Taken together, this increases the
effective elastocapillary length to γ=?ζE?, implying larger
pattern length scales L; see Eq. (9).
III. DISCUSSION
We propose a phenomenological theory that explains the
experimentally observed patterns [1]based on nonlocal
elasticity, which captures aspects of the mesh ’s structure.
Within our equilibrium theory, regular periodic patterns
appear for sufficiently strong phase separation (largeenough χ) and stiffness E, while surface tension γopposes
the trend. Essentially, solvent droplets inflate a region of theelastic mesh of the size of the nonlocality scale ξ. The
pattern period Lthen results from a balance of elastic and
interfacial energies, so that Lscales as the geometric mean
between ξand the elastocapillary length γ=E. In contrast,
the interaction strength χ, leading to phase separation in the
first place, affects Lonly weakly, but it determines whether
the patterned phase is stable, similar to ordinary phaseseparation. However, the transition between the homo-geneous and heterogeneous phase, which is normally firstorder, can now be continuous. Consequently, the patterned
phase can appear with arbitrarily small amplitude in a
reversible process.
Our model captures the main features of the experiment
[1], including the continuous phase transition leading to
reversible dynamics. This suggests that the experiment is in
quasiequilibrium, which would also explain why thepattern is independent of the cooling rate. Moreover, ourtheory explains why the pattern length scale Lis only
(a) (b)
FIG. 7. Heterogeneity could explain nonlocality scale ξ.
(a) Schematic of a relatively regular mesh, whose nonlocalityscale ξis linked to the mesh size. (b) Schematic of a hetero-
geneous mesh, where ξis given by the correlation length of
spatial heterogeneities.QIANG, LUO, and ZWICKER PHYS. REV . X 14,021009 (2024)
021009-8weakly affected by the final temperature and decreases with
stiffness E. Importantly, our model predicts that a structural
length ξof the mesh is essential for the emergence of the
observed L. Our numerics indicate that Lcan be an order of
magnitude larger than ξ, suggesting that ξcould relate to
observed correlation lengths of the order of a few hundred
nanometers [27]. Since ξis small compared to the distance
between droplets [see Eq. (10)], the nonlocal effects of
elasticity do not affect droplet positioning. Furthermore,
we found that a coexisting homogeneous phase does
not affect the free energy of the patterned phase strongly(see Supplemental Material [10]), suggesting that the two
phases can be interspersed, which would contribute to
irregularity of the droplet placement in real systems. In
contrast, the observed variation in droplet size [1]likely
originates from local heterogeneity in material properties,likeξ,E, and γ.
To capture the mesh ’s structure, we employ nonlocal
elasticity [18–22]based on a convolution of the strain field.
Such nonlocal elasticity emerges naturally for phenomeno-logical theories and from coarse-graining microscopic theo-ries. The description only converges to local elasticity theorywhen the length scales of phenomena are large compared tothe nonlocality scale. While such length scale separation isoften feasible in macroscopic elastic problems, nonlocalelasticity is required to explain microscopic phenomena,e.g., in fracture mechanics [35]. Indeed, the convolution
kernel given by Eq. (4)can be interpreted as a Green ’s
function of a diffusion process in the reference frame,suggesting that the nonlocal elasticity is similar to the damagefield introduced in fracture mechanics [36]. Taken together,
nonlocal elasticity theories are crucial to describe elasticphenomena in microscopic systems, e.g., biological cells.
Our work complements related theories of phase sepa-
ration in elastic media, which either modeled poresexplicitly [26,37 –40]or resorted to particle-based methods
[41,42] . Nonlocality is generally responsible for the emer-
gence of structure in multiple physical systems, such as theOhta-Kawasaki model [43], phase separation with electro-
static interaction [44], and also nonlocal elasticity [45,46] ,
e.g., to study polymeric materials [47,48] .H o w e v e r ,i nt h e
first two models, the convolution acts directly on thedescribed field ?, whereas we convolved the strain field
?, which is inversely related to ?; see Eq. (2). Moreover, the
Coulomb form of the convolution kernel prohibits macro-phase separation in these two models. Another difference is
that we use a convolution in the reference frame, capturing
the quenched microscopic topology of the elastic mesh.These differences are a consequence of the fact that we
build our phenomenological model systematically for the
case of elastic meshes.
We developed our model for the simple case of one
spatial dimension, which surprisingly already accounts forthe key experimental observations. However, to capture
more details, including various morphologies, we will need
to generalize the model to higher dimensions, which willrequire a tensorial convolution kernel
[19]. Additionally,
nonlocal elasticity can be viewed as a phenomenological
model containing the correlation length of spatial hetero-geneities of the polymer network [49]. In realistic meshes,
heterogeneities would imply disorder in the convolution
kernel, but our theory suggests that only the long-range
behavior is crucial for forming patterns. More generally,
disorder might affect the phase transition and the morphol-ogies of patterns, potentially explaining the lack of long-
range order. Incorporating quenched disorder would
require a proper averaging based on statistical mechanics,e.g., by applying a random convolution kernel or combin-
ing simulation techniques [49,50] . Moreover, to describe
details of the experiments, we might require more realistic
models of phase separation (including different molecular
sizes and higher-order interactions terms) and elasticity(involving finite extensibility, viscoelasticity [51],a sw e l l
as plastic deformation, like fracture [52,53] and cavitation,
which can lead to regular droplet patterns [54]). Finally,
experimental systems exhibit heterogeneities in key model
parameters including ξ,E,a n d γ, which will contribute to
uncertainty and might even induce large-scale rearrange-
ments [9,55] . Such extended theories will allow us to
compare the full pair correlation and scattering functionsto experiments, shedding light on how we can manipulate
this pattern forming system to control microstructures.
ACKNOWLEDGMENTS
We thank Carla Fernández-Rico, Robert W. Style, Eric
R. Dufresne, Marcus Müller, and Stefan Karpitschka forhelpful discussions. We gratefully acknowledge funding
from the Max Planck Society and the European Union
(ERC, EmulSim, 101044662).
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