INAUGURALARTICLE
APPLIED PHYSICAL SCIENCES
Mechanical theory of nonequilibrium coexistence and
motility-induced phase separation
AhmadK.Omar
a,b,1,2
ID
,HyeongjooRow
c,1
ID
,StewartA.Mallory
d,1
,andJohnF.Brady
c,2
ThiscontributionispartofthespecialseriesofInauguralArticlesbymembersoftheNationalAcademyofScienceselectedin2020.
ContributedbyJohnF.Brady;receivedNovember21,2022;acceptedMarch24,2023;reviewedbyHartmutLoewen,M.CristinaMarchetti,
andIgnacioPagonabarraga
Nonequilibrium phase transitions are routinely observed in both natural and synthetic
systems. The ubiquity of these transitions highlights the conspicuous absence of a
general theory of phase coexistence that is broadly applicable to both nonequilibrium
and equilibrium systems. Here, we present a general mechanical theory for phase
separation rooted in ideas explored nearly a half-century ago in the study of
inhomogeneous fluids. The core idea is that the mechanical forces within the interface
separating two coexisting phases uniquely determine coexistence criteria, regardless of
whether a system is in equilibrium or not. We demonstrate the power and utility of
this theory by applying it to active Brownian particles, predicting a quantitative phase
diagram for motility-induced phase separation in both two and three dimensions. This
formulation additionally allows for the prediction of novel interfacial phenomena,
such as an increasing interface width while moving deeper into the two-phase region,
a uniquely nonequilibrium effect confirmed by computer simulations. The self-
consistent determination of bulk phase behavior and interfacial phenomena offered
by this mechanical perspective provide a concrete path forward toward a general theory
for nonequilibrium phase transitions.
nonequilibrium
|
phase diagram
|
active matter
|
driven assembly
|
coexistence
The diversity of phase behavior and pattern formation found in far-from-equilibrium
systems has brought renewed focus to the theory of nonequilibrium phase transitions.
Intracellular phase separation resulting in membraneless organelles (1, 2) and pattern
formation on cell surfaces (3) are just a few instances in which nonequilibrium phase
transitions are implicated in biological function. Colloids (4) and polymers (5–8) subject
to boundary-driven flow can experience shear-induced phase transitions and patterns
that profoundly alter their transport properties. Microscopic self-driven particles, such
as catalytic Janus particles, motile bacteria, or field-directed synthetic colloids, exhibit
phase transitions eerily similar to equilibrium fluids despite the absence of traditional
equilibrium driving forces (9–14).
A general predictive framework for constructing phase diagrams for these driven
systems is notably absent. For equilibrium systems, the formulation of a theory for phase
coexistence was among the earliest accomplishments in thermodynamics. Maxwell (15),
building on the work of van der Waals, derived what are now familiar criteria for phase
equilibria for a one-component system: equality of temperature, chemical potential, and
pressure. These criteria are rooted in the fundamental equilibrium requirements that
free energy be extensive and convex for any unconstrained degrees of freedom within a
system. The lack of such a variational principle for nonequilibrium systems has limited
the theoretical description of out-of-equilibrium phase transitions.
The absence of a general theory for nonequilibrium coexistence has been particularly
evident in the field of active matter. The phenomenon of motility-induced phase
separation (MIPS)—the occurrence of liquid–gas phase separation among repulsive active
Brownian particles (ABPs)—has motivated a variety of perspectives (16–26) in pursuit
of a theory for active coexistence. These perspectives range from kinetic models (27),
continuum and generalized Cahn–Hilliard approaches (16, 18, 28), large deviation
theory (29, 30), and power functional theory (24, 25). Some of these approaches appeal to
equilibrium notions such as free energy and chemical potential (19), concepts which lack
a rigorous basis for active systems. Without a first-principles nonequilibrium coexistence
theory, one cannot compare or assess the various perspectives. Despite the significant
progress, a closed-form theory for the coexistence criteria for MIPS, which makes no
appeals to equilibrium ideas, remains an outstanding challenge in the field.
Significance
Phase separation, the coexistence
between distinct macroscopic
phases (such as oil coexisting
with water), is ubiquitous in
everyday life and motivated the
development of the equilibrium
theory of coexistence by Maxwell,
van der Waals, and Gibbs.
However, phase separation
is increasingly observed in
both synthetic and living
nonequilibrium systems, where
thermodynamic principles are
strictly inapplicable. Here, we
develop a mechanical description
of phase separation, offering a
route for constructing phase
diagrams without presuming
equilibrium Boltzmann statistics.
We highlight the utility of our
approach by developing a
first-principles theory for
motility-induced phase
separation and the uniquely
nonequilibrium interfacial
phenomena that accompany
this transition.
Author contributions: A.K.O., H.R., S.A.M., and J.F.B.
designed research; performed research; contributed new
reagents/analytic tools; analyzed data; and wrote the
paper.
Reviewers: H.L., Heinrich-Heine-Universität Düsseldorf;
M.C.M., University of California, Santa Barbara; and I.P.,
University of Barcelona.
The authors declare no competing interest.
Copyright
©
2023 the Author(s). Published by PNAS.
This article is distributed under Creative Commons
Attribution-NonCommercial-NoDerivatives License 4.0
(CC BY-NC-ND).
1
A.K.O., H.R., and S.A.M. contributed equally to this work.
2
To whom correspondence may be addressed. Email:
aomar@berkeley.edu or jfbrady@caltech.edu.
This article contains supporting information online
at http://www.pnas.org/lookup/suppl/doi:10.1073/pnas.
2219900120/-/DCSupplemental.
Published April 24, 2023.
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Mechanics is a natural choice for describing the behavior of
both equilibrium and nonequilibrium systems as it is agnostic
to the underlying distribution of microstates. In this article,
we construct an entirely mechanical description of liquid–gas
coexistence, relying only on notions such as forces and stresses.
This formulation is an extension of the mechanical perspective
developed decades ago to describe coexistence and interfacial
phenomena for equilibrium systems (31–33). We highlight the
utility of this framework by developing a theory for the coexis-
tence criteria of MIPS and comparing our theory’s predictions
to results from computer simulation. Our formulation further
allows for the prediction of novel nonequilibrium interfacial
behavior, such as a nonmonotonic interfacial width, as the system
is taken deeper into the coexistence region.
The Mechanics of Nonequilibrium Coexistence
We briefly review the thermodynamics of phase separation
for a one-component system undergoing a liquid–gas phase
transition. The order parameter distinguishing the liquid and
gas phases is the number density
ρ
≡
N
/
V
, where
N
and
V
are the number of particles and volume, respectively.
For simple substances at a uniform temperature
T
below a
critical temperature
T
c
, the mean-field Helmholtz free energy
F
(
N, V, T
)
becomes concave for a range of densities, in violation
of thermodynamic stability. The system resolves this instability
by separating into coexisting macroscopic domains of liquid and
gas with densities
ρ
liq
and
ρ
gas
, respectively. The free energy
of the phase-separated system (neglecting interfacial free energy)
is now
V
liq
f
(
ρ
liq
, T
) +
V
gas
f
(
ρ
gas
, T
)
where we have defined
the free energy density
f
(
ρ
, T
)
≡
F
(
N, V, T
)
/
V
. The volumes
occupied by the liquid (
V
liq
)
and gas (
V
gas
) phases sum to the
total system volume
V
. We now obtain the coexistence criteria
by minimizing the total free energy with respect to
ρ
liq
and
ρ
gas
subject to the conservation of particle number constraint
(i.e.,
V
liq
ρ
liq
+
V
gas
ρ
gas
=
V
ρ
). This results in the familiar
coexistence criteria:
μ
(
ρ
liq
, T
) =
μ
(
ρ
gas
, T
) =
μ
coexist
(
T
)
,
p
(
ρ
liq
, T
) =
p
(
ρ
gas
, T
) =
p
coexist
(
T
)
,
[1a]
where
μ
(
ρ
, T
) =
∂
f
(
ρ
, T
)
/∂ρ
is the chemical potential,
p
(
ρ
, T
) =
−
f
(
ρ
, T
)+
ρμ
(
ρ
, T
)
is the pressure, and
μ
coexist
(
T
)
and
p
coexist
(
T
)
are the coexistence values for the chemical
potential and pressure, respectively, at the temperature of interest.
It is straightforward to show that Eq.
1a
can be equivalently
expressed as
μ
(
ρ
liq
) =
μ
(
ρ
gas
) =
μ
coexist
,
∫
ρ
liq
ρ
gas
[
μ
(
ρ
)
−
μ
coexist
]
d
ρ
=
0
,
[1b]
or similarly
p
(
υ
liq
) =
p
(
υ
gas
) =
p
coexist
,
∫
υ
liq
υ
gas
[
p
(
υ
)
−
p
coexist
]
d
υ
=
0
,
[1c]
where we have defined the inverse density
υ
≡
1
/ρ
and have
dropped the dependence on
T
in Eqs.
1b
and
1c
for convenience.
The integral expressions in Eqs.
1b
and
1c
are often referred to
as equal-area or Maxwell constructions (15) in the
μ
−
ρ
and
p
−
υ
planes, respectively. These expressions are equivalent to Eq.
1a
and can be used to compute the coexistence curve or binodal as
a function of
T
. The spinodal boundaries enclose the region of
the phase diagram in which thermodynamic stability is violated,
i.e.,
(
∂
2
f
/∂ρ
2
)
T
<
0 or equivalently when
(
∂
p
/∂ρ
)
T
<
0 or
(
∂μ/∂ρ
)
T
<
0. These boundaries can thus be determined by
finding the densities at which
(
∂
p
/∂ρ
)
T
=
0 or
(
∂μ/∂ρ
)
T
=
0
for a specified temperature.
Interestingly, the coexistence criteria presented in Eq.
1c
con-
tains only the mechanical equation-of-state, a quantity which is
readily defined for nonequilibrium systems (unlike, for example,
chemical potential). In fact, Eq.
1c
has been used in previous
studies (19, 34) to obtain the phase diagram of active systems.
However, its validity for nonequilibrium systems is questionable
as its origins are clearly rooted in a variational principle that holds
only for equilibrium systems.
We are now poised to construct a theory of coexistence based
purely on mechanics. As previously noted, the order parameter for
liquid–gas phase separation is density. The evolution equation for
the order parameter is therefore simply the continuity equation:
∂ρ
∂
t
+
∇
·
j
ρ
=
0
,
[2]
where we are now considering a density field
ρ
(
x
;
t
)
that is
continuous in spatial position
x
(with
∇
=
∂/∂
x
) and
j
ρ
(
x
;
t
)
is
the number density flux. A constitutive equation for the number
density flux follows directly from linear momentum conservation.
This connection can be appreciated by noting that
j
ρ
(
x
;
t
)
≡
ρ
(
x
;
t
)
u
(
x
;
t
)
(where
u
(
x
;
t
)
is the number average velocity of
particles) and is therefore proportional to the momentum density
by a factor of the particle mass
m
. Expressing linear momentum
conservation with
j
ρ
(rather than the more traditional
u
),
∂
(
m
j
ρ
)
∂
t
+
∇
·
(
m
j
ρ
j
ρ
/ρ
) =
∇
·
흈
+
b
,
[3]
where
흈
(
x
;
t
)
is the stress tensor and
b
(
x
;
t
)
are the body
forces acting on the particles. In simple systems, Eqs.
2
and
3
may constitute a closed set of coupled equations describing the
temporal and spatial evolution of the density profile. However,
the precise form of the stresses and body forces may depend on
other fields, which will require additional conservation equations
to furnish a closed set of equations.
As we are interested in scenarios in which phase separation
reaches a stationary state of coexistence, the continuity equation
reduces to
∇
·
j
ρ
=
0, and linear momentum conservation
is now
∇
·
(
m
j
ρ
j
ρ
/ρ
) =
∇
·
흈
+
b
. While
j
ρ
=
0
for
systems in equilibrium, nonequilibrium steady-states may admit
nonzero fluxes
*
. However, a phase-separated system with a planar
interface will satisfy
j
ρ
=
0
due to the quasi-1d geometry
and no-flux boundary condition. We restrict our discussion
to macroscopic phase separation. Therefore, both equilibrium
and nonequilibrium systems will adopt a density flux-free
state, reducing the linear momentum conservation to a static
mechanical force balance:
0
=
∇
·
흈
+
b
.
[4]
*
Phase-separated nonequilibrium systems with interfaces of finite curvature (i.e., if the
domain of one of the coexisting phases is of nonmacroscopic spatial extent) may exhibit
nonzero density fluxes (35).
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Eq.
4
is the mechanical condition for liquid–gas coexistence and
can be used to solve for
ρ
(
x
)
with constitutive equations for
흈
and
b
. The nature of these constitutive equations will also determine
whether other conservation equations will be required.
Let us now demonstrate that the equilibrium coexistence
criteria are recovered from this mechanical perspective. In
principle, for any system, whether it is in or out of equilibrium,
microscopic expressions for Eqs.
2
and
3
can be obtained precisely
through the
N
-body distribution function and its evolution
equation. It will later be necessary to follow such an approach
to obtain stresses and body forces when considering the phase
coexistence of active particles. However, in equilibrium, the
stresses and body forces can also be obtained variationally through
a free energy functional. Consider the following free energy
functional:
F
[
ρ
] =
∫
V
[
f
+
ρ
U
ext
+
κ
2
|
∇
ρ
|
2
]
d
x
,
[5]
where
f
(
ρ
)
is the mean-field free energy density,
κ
(
ρ
)
is a
(positive) coefficient such that the square-gradient term penalizes
density gradients (36) and
U
ext
(
x
)
represents all externally applied
potential fields. Minimizing
F
[
ρ
]
with respect to
ρ
(
x
)
(36–38)
results, after some straightforward manipulations (
SI Appendix
for details), in Eq.
4
, allowing us to identify the reversible stress
and body forces as
흈
=
−
p
I
+
(
1
2
∂
(
κρ
)
∂ρ
|
∇
ρ
|
2
+
κρ
∇
2
ρ
)
I
−
κ
∇
ρ
∇
ρ
,
[6a]
b
=
−
ρ
∇
U
ext
,
[6b]
where the pressure is again
p
(
ρ
) =
−
f
(
ρ
) +
ρ∂
f
/∂ρ
and
I
is
the second-rank identity tensor. Note that the gradient terms
appearing in Eq.
6a
are the so-called Korteweg stresses (39).
The equilibrium coexistence criteria can now be obtained from
Eqs.
4
and
6
.
Without loss of generality, we take the
z
-direction to be normal
to the planar interface and neglect any external potential (i.e.,
b
=
0
). In this case, the static force balance, Eq.
4
reduces to
d
σ
zz
/
dz
=
0, where we have exploited the spatial invariance
tangential to the interface. The stress is therefore constant across
the interface resulting in
−
σ
zz
=
p
−
1
2
(
∂κ
∂ρ
ρ
−
κ
)(
d
ρ
dz
)
2
−
κρ
d
2
ρ
dz
2
=
C,
[7]
where
C
is a to-be-determined constant.
The complete density profile
ρ
(
z
)
can now be determined
by solving Eq.
7
with the appropriate boundary conditions.
For a macroscopically phase-separated system, the density profile
approaches constant values
ρ
liq
and
ρ
gas
as
z
→ ±∞
. In these
regions of constant density, the gradient terms in Eq.
7
vanish and
the pressure in the two phases is equal:
p
(
ρ
liq
) =
p
(
ρ
gas
) =
C
.
We now recognize the constant
C
as the coexistence pressure
p
coexist
and recover the first of the two expected coexistence criteria
in Eq.
1c
. Before proceeding to the second coexistence criteria,
we rearrange Eq.
7
:
p
(
ρ
)
−
p
coexist
=
a
(
ρ
)
d
2
ρ
dz
2
+
b
(
ρ
)
(
d
ρ
dz
)
2
,
[8]
where
a
(
ρ
) =
κρ
and
b
(
ρ
) = [(
∂κ/∂ρ
)
ρ
−
κ
]
/
2. To recover
the second coexistence criteria in a form similar to Eq.
1c
, we seek
to integrate Eq.
8
with a variable such that the right-hand-side
vanishes. Aifantis and Serrin (32) recognized that the gradient
terms can be eliminated by multiplying Eq.
8
by a weighting
function
E
(
ρ
)
d
ρ/
dz
, where
E
(
ρ
) =
1
a
(
ρ
)
exp
(
2
∫
b
(
ρ
)
a
(
ρ
)
d
ρ
)
,
[9]
and spatially integrating the result across the interface. This
operation eliminates the gradient terms, resulting in a coexistence
criterion purely in terms of equations-of-state:
∫
ρ
liq
ρ
gas
[
p
(
ρ
)
−
p
coexist
]
E
(
ρ
)
d
ρ
=
0
.
[10]
Aifantis and Serrin further established that Eq.
10
has a
unique coexistence solution, provided
a
(
ρ
)
>
0 and
p
(
ρ
)
is
nonmonotonic in
ρ
(32).
Eq.
10
is no longer an equal-area construction, but such a form
can be readily obtained through a simple change of variables (21,
22)
E
(
ρ
)
≡
∂
E
/∂ρ
resulting in
∫
E
liq
E
gas
[
p
(
E
)
−
p
coexist
]
d
E
=
0
.
[11]
Eq.
11
now has the form of an equal-area construction in the
p
−
E
plane. For the equilibrium system of interest, one finds that
E
(
ρ
) =
1
/ρ
2
=
υ
2
and
E
(
ρ
) =
υ
(multiplicative and additive
constants in
E
(
ρ
)
and
E
(
ρ
)
do not affect the coexistence criteria),
recovering the expected equilibrium coexistence criteria, Eq.
1c
from our mechanical perspective.
We emphasize that, for equilibrium systems, retaining higher-
order gradient terms in the free energy functional would not affect
the resulting coexistence criteria, i.e.,
E
(
ρ
) =
υ
would remain
the integration variable independent of the order of truncation.
This can be verified by adding higher-order terms [e.g., ref. 40] to
Eq.
5
(they must be even with respect to spatial gradients to satisfy
the spatial inversion symmetry of the free energy) and confirming
that, for the resulting stress, integration with respect to
E
(
ρ
) =
υ
also eliminates the additional higher-order interfacial stress
terms. This should not be surprising as, for equilibrium systems,
the coexistence criteria can be derived without referencing the
interface (as done at the beginning of this section) and thus should
not depend on the precise details of the interface, including the
truncation order.
We further note that in order to define the spinodal without
invoking thermodynamic stability, a linear stability analysis on
Eqs.
2
and
3
using the reversible stress Eq.
6a
can be performed to
determine whether small density perturbations to a homogeneous
base state will grow in time. In doing so (
SI Appendix
for details),
we recover the mechanical spinodal criteria
(
∂
p
/∂ρ
)
<
0.
This completes our discussion of the mechanics of equilibrium
coexistence and stability.
For a nonequilibrium system, an additional complexity arises:
the possibility of spontaneously generated internal body forces.
The absence of applied external fields does not exclude the
possibility of body forces for nonequilibrium systems. A gen-
eral nonequilibrium coexistence criterion for liquid–gas phase
separation must therefore account for these internal body forces.
To understand this physically, let us consider a steady-state force
balance on a collection of particles in a control volume Fig. 1.
Application of an external force field on the particles results in
a net volumetric force acting on the particles: a body force. By
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Fig. 1.
Force balance on the particles within a control volume at steady
state. Application of an external force field
F
ext
(
Top
) to a passive system with
conservative reciprocal interaction forces
F
C
and a system with no external
forces but with active forces
F
A
in addition to
F
C
(
Bottom
).
Newton’s third law, interparticle interactions do not give rise to
a net volumetric force within the volume interior. It is only at the
surface of the control volume that interparticle forces (exerted
by particles outside the volume on the interior particles) are
nonvanishing, resulting in stresses. The polarization of active
forces (
Bottom
of Fig. 1) results in a net active force within the
volume, behaving similarly to an external force field (41).
At steady state, the self-generated body force density due to
nonequilibrium forces must balance a stress difference across
the volume. In this case, the steady-state one-dimensional (1d)
mechanical balance is
d
σ
zz
/
dz
+
b
z
=
0. For a one-dimensional
system, the body force can always be expressed as
b
z
=
d
σ
b
/
dz
,
and the mechanical balance can now be expressed as
d
(
σ
zz
+
σ
b
)
/
dz
=
0. This newly defined effective stress
6
≡
σ
zz
+
σ
b
is,
just as before, constant spatially. Expressing
6
as a second-order
gradient expansion in density,
−
6
=
P
(
ρ
)
−
a
(
ρ
)
d
2
ρ
dz
2
−
b
(
ρ
)
(
d
ρ
dz
)
2
=
C,
[12]
where
P
(
ρ
)
is a dynamic or effective pressure. We again recognize
that, as the gradients must vanish in the bulk phases,
P
(
ρ
liq
) =
P
(
ρ
gas
) =
C
, where we identify the constant as the coexistence
effective pressure
P
coexist
. The second coexistence criteria can be
found analogously as before through the use of an integrating
factor
E
(
ρ
)
d
ρ/
dz
, where
E
(
ρ
)
is defined in Eq.
9
. The two
coexistence criteria are then:
P
(
E
liq
) =
P
(
E
gas
) =
P
coexist
,
[13a]
∫
E
liq
E
gas
[
P
(
E
)
−
P
coexist
]
d
E
=
0
,
[13b]
where
∂
E
∂ρ
=
1
a
(
ρ
)
exp
(
2
∫
b
(
ρ
)
a
(
ρ
)
d
ρ
)
.
[13c]
Eq.
13
is the general nonequilibrium coexistence criteria for
liquid–gas phase separation.
The powerful idea that coexistence criteria can be extracted
from knowledge of interfacial mechanics was, to the best of our
knowledge, first proposed by Aifantis and Serrin (32) in the
context of equilibrium systems. Solon and coworkers proposed
a similar gradient-expansion-based approach beginning with a
generalized Cahn–Hilliard model (21, 22). The criterion derived
herein, Eq.
13
makes clear that for nonequilibrium phase
separation, one criterion is always equality of dynamic pressure,
while the other is obtained from knowledge of the interfacial
stresses and body forces.
Application of this criterion to determine the phase diagram
will require expressing the dynamic pressure
P
(
ρ
)
as a second-
order density gradient expansion in order to identify the equal-
area construction variable
E
(
ρ
)
. Furthermore, provided that a
timescale exists such that this dynamic pressure can also be
defined for time-dependent states, the spinodal criterion is now
(
∂
P
/∂ρ
)
<
0, as shown in
SI Appendix
. We now proceed to
obtain the dynamic pressure of active Brownian particles and
apply this nonequilibrium coexistence criterion.
The Mechanical Theory of MIPS
For a theoretical prediction of the phase diagram of active Brow-
nian particles, our mechanical perspective requires expressions
for the dynamic pressure,
P
(
ρ
)
, and the coefficients of the
leading gradient terms,
a
(
ρ
)
and
b
(
ρ
)
. These quantities are
needed to calculate the appropriate integration variable
E
(
ρ
)
such that Eq.
13
is satisfied. To derive these quantities, we require
expressions for the stress
흈
and body forces
b
without invoking
a variational principle. These constitutive equations can be
obtained systematically, beginning with the equations-of-motion
describing the motion of the microscopic degrees of freedom. We
consider active Brownian particles with overdamped translational
and rotational equations-of-motion describing the position
r
α
and orientation
q
α
(
|
q
α
|
=
1) of particle
α
as
̇
r
α
=
U
0
q
α
+
1
ζ
F
C
α
,
[14a]
̇
q
α
=
훀
R
α
×
q
α
,
[14b]
where
ζ
is the translational drag coefficient and
F
C
α
is the inter-
particle force on particle
α
. The orientation of a particle evolves
under the influence of a stochastic angular velocity
훀
R
α
, which
follows the usual white noise statistics with a mean of
〈
훀
R
α
(
t
)
〉
=
0
and a variance of
〈
훀
R
α
(
t
)
훀
R
β
(
t
′
)
〉
= (
2
/τ
R
)
δ
αβ
δ
(
t
−
t
′
)
I
, where
τ
R
is the reorientation time and
δ
αβ
is the Kronecker delta. We
aim to describe the strongly active (athermal) limit of hard active
disks and spheres where the phase diagram for these systems
are fully described by two geometric parameters: the volume (or
area) fraction
φ
≡
v
p
ρ
(where
v
p
is the area (
d
=
2) or volume
(
d
=
3) of a particle) and the dimensionless intrinsic run length
`
0
/
D
, where
`
0
≡
U
0
τ
R
, with
D
being the particle diameter
and
U
0
being the intrinsic active speed. We therefore choose a
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