PHYSICAL REVIEW E 104, 054103 (2021)

Phase-ordering kinetics in the Allen-Cahn (Model A) class: Universal aspects elucidated
by electrically induced transition in liquid crystals
Renan A. L. Almeida1

1,2,3,*and Kazumasa A. Takeuchi

3,†

Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan2
Departmento de Física, Universidade Federal de Viçosa, 36570-900 Viçosa, MG, Brazil
3
Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Received 20 July 2021; accepted 27 September 2021; published 5 November 2021)

The two-dimensional (2D) Ising model is the statistical physics textbook example for phase transitions
and their kinetics. Quenched through the Curie point with Glauber rates, the late-time description of theferromagnetic domain coarsening finds its place at the scalar sector of the Allen-Cahn (or Model A) class, which
encompasses phase-ordering kinetics endowed with a nonconserved order parameter. Resisting exact results
sought for theoreticians since Lifshitz’s first account in 1962, the central quantities of 2D Model A—most scaling
exponents and correlation functions—remain known up to approximate theories whose disparate outcomes urge
experimental assessment. Here we perform such assessment based on a comprehensive study of the coarseningof 2D twisted nematic liquid crystals whose kinetics is induced by a superfast electrical switching from a spatiotemporally chaotic (disordered) state to a two-phase concurrent, equilibrium one. Tracking the dynamics via
optical microscopy, we first show the sharp evidence of well-established Model A aspects, such as the dynamic
exponent z = 2 and the dynamic scaling hypothesis, to then move forward. We confirm the Bray-Humayun
theory for Porod’s regime describing intradomain length scales of the two-point spatial correlators and show that
their nontrivial decay beyond the Porod’s scale can be captured in a free-from-parameter fashion by Gaussian
theories, namely the Ohta-Jasnow-Kawasaki (OJK) and Mazenko theories. Regarding time-related statistics, we
corroborate the aging hypothesis in Model A systems, which includes the collapse of two-time correlators into amaster curve whose format is, actually, best accounted for by a solution of the local scaling invariance theory: thesame solution that fits the 2D nonconserved Ising model correlator along with the Fisher-Huse conjecture. We
also suggest the true value for the local persistence exponent in Model A class, in disfavor of the exact outcome
for the diffusion and OJK equations. Finally, we observe a fractal morphology for persistence clusters and extract
their universal dimension. Given its accuracy and possibilities, this experimental setup may work as a prototype
to address further universality issues in the realm of nonequilibrium systems.
DOI: 10.1103/PhysRevE.104.054103

I. INTRODUCTION

Since the seminal studies [1,2] by Lifshitz, Slyozov, andWagner on the Ostwald ripening, and those [3,4] own to
Lifshitz, Allen, and Cahn on curvature-driven domain growth,
investigations of phase ordering continue very alive as a center of statistical physics and interdisciplinary sciences for a
myriad of natural, social, and artificial phenomena [5–11].
Over the past decade, we have witnessed solid developments
not only in traditional subjects but also in the discovery on
the pivotal roles that phase separation and domain coarsening
play in biology and artificial active matter. The traditionalroad includes the continued interest on spinodal decomposition of alloys and glasses [12–14], evolving patterns of foams[15,16], growing crystalline films [17], not to mention theevergreen simulations of Ising (e.g., Refs. [18–22]), Potts[23,24], and voter [25] kinetics. The second path is directed
to a quantitative understanding of emergent behaviors in bio-*†

renan.lisboaufv@gmail.com
kat@kaztake.org

2470-0045/2021/104(5)/054103(18)logical systems, which range from the intracellular assembly
kinetics and cellular functionalization—notably liquid-liquid
phase separations involved in the formation of membranelesscell compartments [11,26,27]—to the segregation [28] andphase separation [29] within unicellular (bacteria) communities, pattern formation in groups of complex multicellularorganisms [30,31], besides experiments [32–34] and theoretical [10] accounts for artificial active matter undergoing
motility-induced phase separation. Amid such a multidisciplinary environment, the iconic and perhaps the simplest
example of ordering remains being the (kinetic) Ising model[35,36] because of its simple conceptualization and theoreticaland numerical tractability, both of which are accompanied bya fundamental universality that often emerge from nontrivial,
real dynamics (e.g., Refs. [28,37]).
Fascination on phase-ordering kinetics then stems fromits virtual ubiquitousness and complexity that yet bears universal aspects [5–11]. Among these, generically one finds a
dynamic scaling property that guarantees that the domain mosaic becomes statistically time-independent whenever lengths
are measured in the unit of a single, asymptotically emergent, growing scale l (t ). Free to relax in the absence of054103-1©2021 American Physical Society

ALMEIDA AND TAKEUCHIPHYSICAL REVIEW E 104, 054103 (2021)

quenched disorder, geometric frustration and external forces,domains display an algebraic growth l (t ) ∼ t 1/z , where z is
the universal dynamic exponent. For kinetics where the system’s order parameter is not conserved, viz. the Model A
dynamics in the classification by Hohenberg and Halperin[38], one has [39] z = 2 [3,4,40]. Although simulations andexperiments extensively support this dynamic scaling, rigorous derivations have been limited to the one-dimensional
nonconserved kinetic Ising model (1DIM) [41,42] and to thenonconserved O(n → ∞) model [43], where n is the numberof the components of the order parameter. In the case of scalar(n = 1), two-dimensional (2D) continuum Model A systems
quenched from a fully disordered regime to an equivalentzero-temperature (T = 0) state, dynamic scaling with z = 2
is obtained from an exact result that relies on the assumed
convergence to the critical percolation fixed point [44]. A
similar conclusion is reached for quenches performed fromthe critical state [44], for which the cluster size distribution isexactly known [45].
Theoretically suggested [46,47], then observed in models[48] and real systems [49] around the end of the 1970s, timeindependent forms of two-point spatial correlators remain
hitherto exactly known only for specific cases [41–43,50].
These cases do not include the scalar 2D Model A whose
approximate forms were proposed along a remarkable dealof effort made around the 1980s [51,52] and in the 1990s[53–57], where the Ohta-Jasnow-Kawasaki (OJK) theory [51]and the theory of unstable growth (TUG) by Mazenko [52,53]
played the major roles [58]. Tested against discrete [59,60]
and continuum [61,62] models, both OJK and TUG functional
forms generically succeeded [63] in fitting to data. The OJK
form was tested in an notable experiment [64] back to the
1980s, although agreement was mostly confined to the short
intradomain scale ruled by the Porod’s regime [65].
On the Porod’s regime, Bray and Humayun [66] derived
exact formulas for the short-distance limit of the spatial correlator (or the tail of the structure factor)—hereinafter referred
to as the Bray-Humayun amplitude—so that there is no needto rely on approximate theories in this regime. Moreover, because the formulas were derived under minimal assumptions
on the existence of defects, the Bray-Humayun amplitude
shall be valid regardless of the conservation laws of the order parameter and whether the dynamic scaling holds [66].
Despite so, to our knowledge, no experimental measurement
has turned attention to this prediction.
The memory of systems in the coarsening stages is anadditional aspect of interest because corresponding two-timeand infinitely many time statistics may also display scale
invariance [5,67,68], including aging akin to that observedin glassy materials [69]. As revealed by Fisher and Huse[70], the description of two-time correlators requires anadditional nonequilibrium exponent—defined in Sec. IV A,
Eq. (19)—λ(n, d ) [70,71] which is affected by the presence oflong-range order in initial conditions [72,73]. For the Model
A class, exact results for λ are known for the 1DIM [42],the 1D noiseless time-dependent Ginzburg-Landau (TDGL)
equation [74], including for the O(n) model in terms of the
1/n expansion [71]. In the 2D scalar case, Fisher and Huse
(FH) conjectured λFH = 5/4 for the 2DIM after noting that,when viewed as a continuum system, its dynamics shall con-verge to the critical percolation fixed point [70]. By making
a parallel between the scaling law for the correlation length
in the percolation problem [75] and a scaling hypothesis forthe evolution of the fraction of Ising spins that has flipped
an even number of times, the FH conjecture is achieved withthe further assumptions: Spins in the random and unbiased
initial conditions shall be fully time decorrelated to their counterparts in final configurations; for biased initial conditions,the dynamics shall enhance the growth of the biased phase[70]. The Fisher and Huse conjecture λFH = 5/4 was initially
supported by 2DIM [70,72,73] and 2D TDGL simulations[76], but recent studies based on finite-size scaling for the verysame models suggest a value closer to the prediction by TUG[77,78], λTUG ≈ 1.29 [79]. In parallel, the OJK theory predicts a rather different value, λOJK = 1 [80], coinciding withthe lower bound of the inequality λ  d/2 for nonconserved
dynamics [70,81]. The OJK value has not been observed so
far but for 2DIM simulations with strongly long-range interactions [22]. Noting that deviations from the Z2 -symmetricdynamics must be taken into account when analyzing the twotime correlator—otherwise strongly biasing and artificiallychanging its corresponding asymptotic decay—it is important
to elucidate on the experimental support to λFH in Ref. [82] inthe light of dealing with systems that are not Z2 symmetric,
as it happens in experiments.
More fundamentally, experimental support to the dynamic
scaling according to aging hypothesis [69] is surprisingly elusive for the Model A class. As consequence, functional forms
for the two-time correlator given by OJK [80], TUG [79], andthe local scale invariance (LSI) theory by Henkel [83], all lacka test with real dynamics.
Another nontrivial facet of phase-ordering kinetics is encoded in their first-passage statistics [84], specifically in thelocal persistence probability Q(t, t0 ) [67,68,85]. In the language of the Ising model, Q(t, t0 ) is defined as the fractionof spins at time t which has never flipped since a previous
time t0 . By construction, it involves correlation of infinitely
many points. In the thermodynamic limit (L → ∞, followed
by t → ∞ with fixed t/t0 ), Q often decays as Q(t, t0 ) ∼(t/t0 )−θ for large t/t0 with the persistence exponent θ , which
is normally independent of z and λ [67,68]. The associatedbody of knowledge was seeded in the numerical work by
Derrida et al. [86] on the 1D q-state Potts model (1DPM)
and then grew and took shape thanks to 1D-5D IM simulations [87], analytical approaches for the noiseless 1D TDGL
equation [74], reaction-diffusion schemes [88,89], exact solutions for 1DPM [90,91], approximations for simple diffusion(DIF) with random initial conditions (RIC) [92,93], besides
being nurtured by experiments [94–98]. Solvability of Q in
Model A systems is constrained to 1D cases [74,90,91]. In
the scalar sector of 2D Model A, early numerics predictedθ ≈ 0.22 [86,87] while a recent 2DIM simulation suggests
θ = 0.199(2) [99]. This value is rather discernible from theexact θDIF = 3/16 = 0.1875 value derived from the 2D DIFRIC [100] and OJK theory [93], which is also faced as a
candidate value for 2D Model A. The early experiment by
Yurke, Pergellis, Majumdar, and Sire reported θ = 0.19(3)[96], albeit precision does not allow to rule out θDIF ; in thisparticular case, one requires a more accurate outcome. Furthermore, the fractal morphology of clusters formed by local

054103-2

PHASE-ORDERING KINETICS IN THE ALLEN-CAHN …

PHYSICAL REVIEW E 104, 054103 (2021)

persistence spins in the 2D nonconserved Ising model (2DIM)[101–103] calls for substantiation.
A traditional experimental approach to nonconserved kinetics has been the twisted nematic liquid crystal (TNLC)[64,82,96], which can be quenched through a first-ordertransition between the high-temperature fluid phase and thelower-temperature nematic phase. Not forgetting to pay due
tribute to these previous systems that aimed at measuring
dynamic scaling properties, we, however, note that the inexorable slow quench rate implied in the isotropic-nematictransition and the issues of metastability, nucleation and nuclei growth stages, have conspired to blur comparisons withpredictions made from genuinely sudden transitions acrosssecond-order points. It is also highly desirable to seek fora versitle setup in which different sorts of kinetics can betriggered.
Here, we exploit electrically driven regimes in TNLC
layers to induce a virtually instantaneous switch from a spatiotemporally chaotic (disordered-like) state to an equilibrium,
two-phase competing state. Because of the sudden removal ofthe driving, problems related to first-order transitions are negligible (if existent), while thermal effects are finely controlled.
In addition, this setup offers an elegant way to induce different
sorts of kinetics by transiting between nonequilibrium states.
We focus on the simplest transition type that is toward theequilibrium, so that we assess the several aspects on 2D Model
A waiting for experimental elucidation.
We first show well-established Model A features suchas the sharp evidence of dynamic scaling with z = 2. Then
we move to assess the Bray-Humayun amplitude and testOJK and TUG theories in length scales much larger than
the Porod’s regime. Paying attention to the asymmetry inthe twisted phases, we confirm that the two-time correlator
collapses on a master function when rescaled by the dimensionless ratio of its two times, as theoretically assumed by the
aging phenomenology [69], but not experimentally confirmed
hitherto for 2D Model A systems. This master function is thenused to test the OJK, TUG, and LSI theories. We find that a
particular solution of LSI, which fits 2DIM data and assumes
the Fisher-Huse conjecture, gives the superior account for theexperiment. Although our estimate of λ, in the asymptotic
regime, agrees with both the FH and TUG values within theuncertainty, analysis from the correlator form itself rules out
the TUG scenario from a global viewpoint. The local persistence statistics is also investigated; we measure a precise
value of θ that agrees with a recent, numerically estimatedvalue for 2DIM but that declines the suggestion of the analytic
DIF-RIC (OJK) theory. Morphology of persistence clusters isshown as a fractal whose dimension, we remark here to beclose to the golden ratio, is in harmony with some of the 2DIM
simulations available in the literature.
This contribution is organized as follows. Section II introduces the electrohydrodynamic phenomenon of nematic
liquid crystals, the experimental setup and the protocol used totrigger the ordering kinetics. Morphology of TNLC domains
and the shrinking rate of their interfaces are the opening
themes of Sec. III. The sequence quantifies the asymmetry
in twisted phases. Two-point spatial correlators, the growthlaw, dynamic scaling and the Bray-Humayun amplitude are
addressed in Sec. III C. Comparison with OJK and TUG func-

tions closes Sec. III. Time correlation properties are studiedin Sec. IV. Two-time correlators are presented in Sec. IV A;
it covers the measurement of λ, dynamic scaling aspects, andtests of theoretical forms. Local persistence probability andmorphology of persistence clusters are presented in Sec. IV B.
We conclude this contribution in Sec. V.
II. EXPERIMENTAL SETUP AND
ELECTRICAL SWITCHING

We exploit the electrohydrodynamic convection of nematic liquid crystal [104–108] that arises when a thin layer(typically 10 to 100 μm) of nematic liquid crystal—with negative dielectric anisotropy (ε − ε⊥ < 0) [109] and positive
conductance anisotropy (σ − σ⊥ > 0)—is subjected to anelectric voltage of ∼10 V or higher [110]. On increasing V ,one observes a sequence of convective patterns [111–113] andfinally a spatiotemporally chaotic state called the dynamic
scattering mode 2 (DSM2) [114–118]. DSM2 stands out by
the presence of a high density of disclinations; fluctuations of
the director field are short-range correlated in space and time:≈1–3 μm and 10 ms, respectively, for a 50- μm-thick layer[111]. We use DSM2 as a disordered initial condition to study
relaxation toward the equilibrium state.
We built a capacitor of parallel glass plates coated withindium tin oxide. The gap was set by polyester spacers ofthickness 12 μm and enclosed an empty region 16 mm ×
16 mm. Prior to assembly, a polyvinyl alcohol film wascoated over each plate and rubbed to set the direction of thesurface alignment. Rubbing directions were set to be orthogonal between the two plates to realize a TNLC cell [119].
The capacitor was then filled with N-4-methoxybenzylidene4-butylaniline (purity >98.0%, Tokyo Chem. Ind., Japan)
doped with 0.01 wt% of tetrabutylammonium bromide (Tokyo
Chem. Ind., Japan). Director field twisted along the bulk ineither left- or right-handed orientation in the cell [Fig. 1(c)].
The cell was inserted in a handmade thermocontroller[Fig. 1(b)]; the set was placed on the stage of an inverted
microscope (IX73, Olympus, Japan). Green-filtered, circularly polarized light was emitted through the cell; transmitted
images were recorded by a charge-coupled device [Fig. 1(a)].
We monitored an area of dimensions Lx × Ly = 2.9 mm ×
2.2 mm in the sample by a ×4 objective. The whole apparatus
was inserted in a thermally isolated chamber whose inner temperature was kept roughly constant by a thermostat circulator.
Peltier elements in the thermocontroller were connected to abipolar current supply set by a proportional-integral-derivative
feedback using the signal of one of the thermistors located at
1 mm to the cell. During the experiments, we kept the cell temperature at 25 ◦ C with fluctuations smaller than ±1.5 mK according to the reference thermistor. Prior to the experiments,
we also monitored the temperature at different positions near
the cell. Thermal fluctuations of thermistors not used for the
feedback control were at most ±10 mK, whilst the temperature gradient formed across the cell was a few tens of mK.
Ordering kinetics was triggered as follows. We applied
a 100 Hz, 70 V sinusoidal voltage across the cell to generate the DSM2 state. Equilibrium twist alignment of thedirector field was then replaced by disordered and fluctuating
configurations. The cell was kept in DSM2 during 120 s

054103-3

ALMEIDA AND TAKEUCHIPHYSICAL REVIEW E 104, 054103 (2021)

FIG. 1. (a) Schematics of the experimental setup. (b) Sketch of the thermocontroller. Using a personal computer (PC), we convert theresistance of a thermistor to the temperature and determine the current to drive the Peltier elements by a proportional-integral-derivative
operation. (c) Illustration of a TNLC cell in which the director field is represented by coloured ellipses; different colours represent thedistinct macroscopic twist orientations. Dashed line represents a domain boundary (disclination). (d) Part of a real TNLC image with domainsof opposite handedness detected by bright (φ = +1) and dark (−1) shades of gray. A dashed line is drawn (not in scale for the sake ofvisualization) along the domain boundary. Acronyms: halogen lamp (HAL) and charged-coupled camera (CCD). Objects are not to scale inpanels (a)–(c).

(∼104 correlation times of the DSM2 state) before we instantaneously [120] removed the electric field. From thisswitching time t = 0 s onwards, the twisted boundary conditions for the director field induced either left- or right-handed
twist; these twisted alignments were formed after complex
disentangling of disclination loops. We recorded the dynamics at 3 Hz over 2000s with pixel size a ≈ 1.82 μm; 20
independent relaxation histories were collected. Due to thepolarization optics, twists of opposite handedness were distinguished in the images by darker and brighter shades of gray[Fig. 1(d)]. We attributed to each pixel a local state variable
φ(r, t ) that took φ = −1 and +1 at the dark- and bright-graycolor pixels, respectively. Domains of the opposite handedness were bordered by disclination lines [Figs. 1(c) and 1(d)]that appeared as dark lines of thickness ξ ≈ a. Because it was
hard to fully detect disclinations out of dark-gray color pixels,
we regarded disclinations as part of the φ = −1 phase: Thisis a fair approximation because ξ ≈ a and because interest
is in the scaling regime where ξ /l (t ) → 0. Nonetheless, wewarn for specific instances where the binarization effect mightplay some role. Data for t  2.33 s are neglected because thelarge fraction of disclinations blurred rational detection of
domains.

III. SPATIAL SCALING AND PHASE ASYMMETRY

shrinking ones. Around 10 clusters are left at t ≈ 150 s, butordering is only completed a decade later, at a finite-size
equilibration time t ∗ ≈ 1.2×103 s.
To quantify shrinkage of interfaces (disclinations), we
measure the interface density ρ(t ) as the number of neighboring pairs with the opposite handedness multiplied by a/Lx Ly .
Based on the scaling hypothesis [5]:ρ(t ) ∼ t −1/z .

(1)Figure 3 shows that such a decay, dictated by a slope near
1/2, occurs in the experiment for nearly two decades. The−1(t ) =
decay is better quantified by the effective exponent, zeff−d[ln ρ(t )]/d ln t, whose plateau for 6 s  t  60 s indicatesthe core of the algebraic regime (Fig. 3, inset). Time averaging
in this regime, or fitting Eq. (1) to the data of each realization
and averaging, we obtain:0.505(15) (effective exponent)
1/z =
.
(2)
0.498(6)(power-law fit)
The number(s) into parentheses refers to the uncertainty(standard deviation for the effective exponent; standard error
for power-law fit) in the last digit(s). Converting to z: z =1.98(6) (effective exponent) and z = 2.008(22) (power-lawfit). These results are in accurate agreement with the theory
for curvature-driven interface dynamics [3,4,40], a hallmarkof the Model A systems.

A. Shrinkage law

Figure 2 shows TNLC domain configurations in ordering(see also supplemental videos 1 and 2 [121]). Configurations
at t = 2.66 s appear as a random patchwork of hundreds of
domains (or clusters) that progressively coarsen afterwards.
Initial irregular contours of domains become smoother intime. While the total length of boundaries tends to decrease,larger domains increase their area at the expense of smaller

B. Phase asymmetry and domain shrinkage

The twist phases in the cell are not strictly Z2 symmetricbecause the rubbing method inserts a nonvanishing pretilt
angle for the director field at the anchoring surfaces [122]. The
strength of this asymmetry cannot be precisely anticipated
before the manual cell assembly, which in turn inserts a small,≈1◦ , misalignment between the top and bottom surfaces. This

054103-4

PHASE-ORDERING KINETICS IN THE ALLEN-CAHN …

PHYSICAL REVIEW E 104, 054103 (2021)

FIG. 2. Configurations of the ordering of TNLC during a relaxation toward the equilibrium. The dynamics was triggered by a virtually
instantaneous electrical switch (at t = 0 s) from a spatiotemporally chaotic called DSM2 toward the two-phase competing, equilibrium state.
Different shades of gray correspond to twists of opposite handedness in the nematic director field. See also supplemental videos 1 and 2 [121].

misalignment may also affect the symmetry balance between
the phases [123].
Asymmetry in the kinetics is captured by the “magnetization,” M(t ) = 1 − 2A(t ), defined from the area fraction A(t )of the φ = −1 (darker) phase [Fig. 4(a), inset]. Note thatM(t ) contains a contribution from disclinations included in
φ = −1. To remove this effect, we carried out an extra set of
20 independent experimental runs with the cell placed upside
down: By this changing, the brightness of the two competingdomains is inverted, so that the sign of φ was flipped—except
for disclinations which, keeping looking as dark-coloredpixels, remained regarded as φ = −1 after binarization. Measuring the magnetization Mflip (t ) for this sign-flipped data set,
we compute M (t ) = [M(t ) − Mflip (t )]/2 as the disclinationfree M (t ) magnetization [Fig. 4(a), main panel]. The M (t )reveals that the φ = −1 phase is slightly favored in themain data set, albeit asymmetry is small; for instance,|M(t )|, |M (t )|  0.2 for t  60 s. Time average in this interval gives M = −0.17(3), which corresponds to A = 0.58(2).

1/2

-2

-3

10

0.6

1/zeffρ(t) [μm-1]

10

0.5
0.4

-4

10

10

1

10

1

10

t [s]

2

10
t [s]

2

Asymmetry is also seen in the shrinkage rate of domainsof phase φ surrounded by a “sea” of −φ. For curvature-driven
dynamics, the squared radius R2 (t ) of a circular domain (or“bubble”) evolves, from a reference time t0 , as [4]:R2 (t ) − R2 (t0 ) = −2DAC (t − t0 ) + K,

(3)

where DAC is the Allen-Cahn diffusion parameter and K isa constant. We monitored 6 independent, isolated and nearlycircular bubbles; 3 of them made of the φ = −1 favoredphase, and the remaining ones made of φ = +1. Measuring
their area and equalizing π R2 (t ) to define R2 (t ) for eachbubble, we accompany their squared radius in time [Fig. 4(b)].
−Fitting according to Eq. (3) yields DAC= 117.3(17) μm2 s−1+
2 −1
and DAC = 126.0(6) μm s for φ = −1 and +1 bubbles,
respectively. The asymmetry translates in nearly 4% of deviation of each diffusion constant relative to the average,
DAC = 122(4) μm2 s−1 ,(4)where the number in the parentheses was set to cover the−+interval [DAC
, DAC].
We remind that the initial state of the experiment is DSM2,
which may provide an unbiased (or weakly biased) initialcondition because the surface anchoring of the director field is
not preserved [124] and because the bulk director is strongly
disturbed by the high density of disclinations. Thus, the observed asymmetry—including the one in the earlier times not
captured in Fig. 4(a)— is likely developed along with the
kinetics instead of being set by initial conditions.
The usual presence of asymmetry in experiments demands
additional care when comparison is made with predictions
for Z2 -symmetric models. As we shall see, although this asimple task for some quantities such as the spatial correlator (Sec. III C), they are subtle for, e.g., the time correlator(Sec. III C).
C. Growth law and the Bray-Humayun amplitude

3

10

The two-point spatial correlator,

FIG. 3. Density of the interface length in time. Error bars indicate one standard deviation. Dashed black line is a guide for eyeswith the slope −1/2 expected for Model A dynamics. Inset shows
the time series of the effective exponent, 1/zeff = −d[ln ρ(t )]/d ln t,which fluctuates around 1/2 (dashed black line) within the core of
the power-law decay, ρ(t ) ∼ t −1/2 , for 6 s  t  60 s.

Cs (r, t ) =φ(r , t )φ(r + r, t ) r e ,(5)where ... r denotes the spatial average and ... e the ensemble
average, is shown as function of r = |r| in Fig. 5(a). The larger
the t, the slower the Cs decay because of the domain growing
size l (t ): Here set it as the length at which Cs (r, t ) = 0.2 for|r| = l (t ). The inset of Fig. 5(a) displays the growth law with

054103-5

ALMEIDA AND TAKEUCHIPHYSICAL REVIEW E 104, 054103 (2021)

0

(a)

(b)

0

φ = −1
φ = +1R2(t) - R2(t0) [μm2]

-0.4
0

-0.6

-0.2

M(t)

M’(t)

-0.2

-0.4
-0.6

-1×10

-2×10

4

4

-0.8

-0.8

1

2

10

10

3

10

10

t [s]

1

2

10
t [s]

0

3

10

25

50
t - t0 [s]

75

100

FIG. 4. (a) “Magnetization” of the TNLC system in time with no contribution of disclinations (free from binarization effects; see text). The
inset shows the bare magnetization, M(t ); the guide line (red) marks the averaged value M ≈ −0.17 for 2.66 s  t  60 s. In both the panels,
error bars indicate one standard deviation. (b) Evolution of the square radius R2 (t ) (converted from the area) of approximately circular TNLC
domains of phase φ surrounded by a “sea” of the opposite phase. Curves with different patterns and shades of cyan circles (magenta squares)
are independent realizations for bubbles made of the slightly favored (unfavored) φ = −1 (+1) phase. The insets show snapshots of a φ = −1
bubble at two different times, where the red dashed lines is a guide for a circle of radius R(t ) at each time.

z = 2,l (t )

Bt

1/2(ξ  l (t )  Lsys ),(6)with a nonuniversal amplitude B discussed below.
A time-independent scaling function F (·),
Cs (r, t )F (x) [x = |r|/l (t )],Cs (r, t )1 − αx(for x  1),

(7)

(9)

We also remind that Bray and Humayun [66] exactly derived Cs (r, t ) at r  l (t ), for n  d:
Cs (r, t )1 + A(d, n)ρEuc (t )r n

[ξ  r  l (t )],(−n/2) (d/2) 2 [(n + 1)/2],[(n + d )/2] (n/2)

Cs (r, t )

(12)

(11)

for odd n, where (·) denotes the gamma function. For oursystem, d = 2 and n = 1, so that A(2, 1) = −4/π .
To assess Eq. (10) in the experiment, we convert ρEuc toρ in the Manhattan metric [125] because this metric (theManhattan one) was used in Fig. 3. Thus, ρEuc = LEuc /Lx Ly
is replaced by ρ = LMan /Lx Ly , where LEuc (LMan ) is the

1 − CA∗ rt −1/2r[ξ  r  l (t )],= 1 − A∗ BC
l (t )(13)

and we identify α = A∗ BC.
From the plateau in the plot of ρ(t )t 1/2 versus t[Fig. 5(d)], observed for 6 s  t  60 s, we read C =0.0381(11) s1/2 μm−1 . Similarly from (1 − Cs )t 1/2 /r versus
r in Fig. 5(e), the plateau interval 0  r  0.5l (t ) ≈ 57 μm
for t = 21.33s—which was used in the measurement of α in
Eq. (9)—gives CA∗ = 0.0377(14) μm−1 , so that we find:A∗ = 0.99(7)(10)

where ρEuc is the interface density measured with the Euclidian metric; A(d, n) is the universal Bray-Humayun (BH)
prefactor [66]:A(d, n) = π (n/2)−1[ξ  r  l (t )],with the BH amplitude A = 1.
We test such prediction as follows. Defining C as the prefactor of ρ(t ) Ct −1/2 (Fig. 3), and recalling B/l (t ) t −1/2
[Eq. (6)], we insert these in Eq. (12) to obtain:(8)

is emphasized in the inset of Fig. 5(b). The angular coefficientα, estimated from the convergence of (1 − Cs )/x for x → 0 at
a late time [t = 21.33 s; see Fig. 5(c)], reads:
α = 0.92(9).

1 − A∗ ρ(t )r∗is defined when dynamic scaling holds, as it does in theexperiment within the accuracy of δCs (x, t ) ≈ 0.07 (standard
deviation) for t  21 s and x  3 [see Fig. 5(b)]. Moreover,the Porod’s law [65],Cs (r, t )interface length in the Euclidean (Manhattan) ruler. Notingthat LMan /LEuc = 4/π holds for isotropic space [66], Eq. (10)becomes(14)

in striking accordance with the BH prediction A∗ = 1 [66].
By passing, from α = A∗ BC = 0.92(9) in Eq. (9) and CA∗ =0.0377(14) μm−1 above, one has B = 24(3) μm s−1/2 : avalue that describes very well the experimental growth law[Fig. 5(a), inset].
D. Experimental test of Gaussian theories

In scalar 2D Model A, the order parameter field tends tochange discontinuously in the scaling limit when interfaces
are crossed. Because such discontinuity is hard to deal
with analytically, both OJK and TUG theories replace theorder parameter by an auxiliary field that smoothly varies

054103-6

PHASE-ORDERING KINETICS IN THE ALLEN-CAHN …

(b)l(t) [μm]10

Cs(r,t)0.8

31(24 μm s-1/2) t1/2
1/2

10

0.8

2

0.6
10

1

10

t [s]

2

2.66 s5.33 s10.66 s21.33 s

0.4

1

Cs(r,t)1

Cs(r,t)(a)

PHYSICAL REVIEW E 104, 054103 (2021)

0.25

0.6

0

0.2

0.4

0.6

0.8

r / l(t)

0.2

0
0

100

50

0
0

150

1

(c)

(e)

α = 0.92(9)13CA* = 0.0377(14) μm-1s1/2

0.2

0.4

x

0.6

0.8

1

(d)
C = 0.0381(11) μm-1s1/2

0.04

6s

0.03

(1-Cs) t1/2 / r [μm-1s1/2]

0.04

0.8
0

2
x = r / l(t)r [μm](1-Cs) / x

0.5

0.4

0.2

ρt1/2 [μm-1s1/2]

0.75

0.03

60 s

10

100

0

t [s]

20

40

60

r [μm]

FIG. 5. (a) Spatial two-point correlator Cs (r, t ) [Eq. (5)] versus r = |r| at different times. Data were obtained by rotational average[averaging values of Cs (r, t ) at different r with the same r = |r|], data binning, and ensemble average. Error bars indicate one standard
deviation. Dashed line indicates Cs (r, t ) = 0.2. Inset shows the growth of l (t ), where dashed red line shows l (t ) = Bt 1/2 with B = 24 μm s−1/2
estimated below Eq. (14). (b) Dynamic scaling hypothesis is confirmed from the time independence of Cs (r, t ) versus x = r/l (t ). Gray dashedline represents the Porod law, Cs (r, t ) 1 − αx, with α = 0.92(9) estimated from (c). Inset of (b) is a zoom in on the region x  1 forthe main plot. (c) Estimation of α by reading (1 − Cs )/x for x → 0; dashed line indicates the mean value, 0.92, of the ordinates in therange x  0.5. (d) Plateau of the rescaled interface density ρt 1/2 for 6 s  t  60 s; dashed line indicates the mean value 0.0381 μm−1 s1/2 .
(e) Plot of (1 − Cs )t 1/2 /r versus r from which the amplitude CA∗ is estimated in the limit r → 0. Dashed line indicates the average valueCA∗ = 0.0377 μm−1 s1/2 . Legends in (b), (c), and (e) are the same as that in (a).

in the whole space. OJK does so by describing interfaces
as a collection of positions satisfying u(r, t ) = 0, whereu(r, t ) is an auxiliary field such that φ(r, t ) = sgn[u(r, t )][5,51]. Starting from the Allen-Cahn interface motion [4]and relying on approximating ∇ j u∇ k u/|∇u|2 ≈ δ jk /d withj, k = 1, 2, . . . , d, OJK reaches at the diffusion equation foru(r, t ). Assuming u(r, 0) as a Gaussian random field [126],
the form of F (·) in Eq. (7) [5,51] is [127]:
FOJK (r/t

1/2




−r 2
2
−1
exp
,
) = sinπ
8DOJKtwith DOJK = DAC /2 for d = 2 [51].

(15)

We test Eq. (15) in the experiment, free from fit, using DOJK = DAC /2 = 61(2) μm2 s−1 after Eq. (4). Because
Cs (r, t ) → M(t )2 for |r| → ∞ and M(t ) = 0 in the experiment [Fig. 4(a), inset], but M = 0 for OJK, comparison isrestricted to Cs (r, t )  M(t )2 . Figure 6(a) compares the OJK
form with TNLC data in the region Cs (r, t )  M(t )2 ≈ 0.04
for t = 42.66 s, and 0.02 for shorter times. We find noteworthy agreement not only in the initial linear decay (Porod’s
regime) but also in the interdomain length scale beyond it.
Agreement is better for larger times. We also may look at
the TNLC growth law, l (t ) Bt 1/2 with B = 24(3) μm s−1/2(Sec.
III C), and compare it with the OJK outcome, l (t ) =√−8DOJK ln[sin(π /10)]t 1/2 from FOJK [l (t )/t 1/2 ] = 0.2 in

054103-7

ALMEIDA AND TAKEUCHI

(b)

1

0.8

DOJK = 61 μm2s-1

0.6

Cs(r,t)

1
0.8

0.4

Cs(r,t)

0.8
0.6

1

2.66 s5.33 s10.66 s21.33 s42.66 sCs(r,t)(a)

PHYSICAL REVIEW E 104, 054103 (2021)

0
1

0

100

200

10

r / t1/2 [μm s-1/2]

0.4
21.33 s42.66 sOJK
TUG (112 μm2s-1)
TUG (104 μm2s-1)

0
0

0

0.4
0.2

0.2

0.2

0.6

10

20
30
r / t1/2 [μm s-1/2]

r [μm]

40

50

FIG. 6. Two-point spatial correlator Cs (r, t ), Eq. (5), in the experiment (symbols) compared with the forms predicted by the OJK theory and
the TUG own to Mazenko. The TNLC data for t  21.33 s are identical to those in Figs. 5(a) and 5(b), but the number of data points is reduced
for the sake of visibility. Error bars indicate one standard deviation. (a) Cs (r, t ) for the experiment (symbols) at several times and the OJK
theoretical form (dashed lines), Eq. (15), with DOJK = 61 μm2 s−1 obtained from the relation with the Allen-Cahn constant: DOJK = DAC /2.
The dash-dotted brown line indicates Cs = 0.04 whose value stems from the observation that Cs (r, t ) → M 2 (t ) for |r| → ∞ and M 2 (t ) = 0 in
the experiment, while M = 0 for OJK. Thus, the comparison shall be constrained to Cs  M 2 (t ), which reads M 2 ≈ 0.04 for t = 42.66 s andsmaller for shorter times. (b) Rescaled Cs (r, t ) in the experiment compared with both OJK (dashed black line) and TUG (dash-dotted blue andsolid turquoise lines) functions. The two TUG curves were obtained with different values of DTUG as indicated in the legend. The inset shows
the same data in the main plot, but the horizontal axis is displayed in logarithmic scale.

Eq. (15); this comparison yields DOJK = 64(18) μm2 s−1 incompatibility with DAC /2 = 61(2) μm2 s−1 .
Now we turn attention to TUG. Developed from the ddimensional TDGL equation, TUG introduces an auxiliary
field m(r, t ) interpreted as the shortest distance between r
and the nearby interface [128]. Assuming m(r, t ) as a Gaussian field, TUG predicts that F (·) in Eq. (7) is given by
F (r/t 1/2 ) = FTUG (g), which satisfies the following differential equation [52,53]:dFTUG (g)d 2 FTUG (g)d −1+ gμ(d )
+
dg2
g
dg


π
FTUG (g) = 0.
(16)
+ tan
2
with FTUG (0) = 1, g = r/(4DTUGt )1/2 , and DTUG is a constant. The value μ(d ) is μ(2) ≈ 1.104 [52,53] for the case ofinterest here [129].
To test TUG form, we numerically integrated Eq. (16)by a Euler-Heuns discretization with a uniform step δg =10−6 , in d = 2, and boundary conditions FTUG (0) = 1,
dFTUG (g)|(g=0) = −(2/π )1/2 . After, to determine DTUG , wedgrely on the Porod’s regime g  1 of the TUG correlator[53,54]:
1/2

r
1
.
(17)
FTUG (g) 1 −DTUG 2π (d − 1)
t 1/2
1/2
Relating Eq. (17) with Cs (r, t ) 1 − (α/B)(r/t)√
2[from Eqs. (6) and (8)], we find DTUG = (B/α 2π ) =1/2π (CA∗ )2 = 112(8) μm2 s−1 .

Figure 6(b) compares the experimental data with FTUG (g)for DTUG = 112 μm2 s−1 (the most probable value), as wellas with FOJK (r/t 1/2 ) for DOJK = 61 μm2 s−1 . For the sakeof visibility, we show only data for t = 21.33 s and 42.66 s
because they are in the core of the z = 2 scaling regime (seeFig. 3). Figure 6(b) suggests that FOJK (r/t 1/2 ) describes theexperiment data better than the TUG counterpart; however,by adjusting the value of DTUG within the uncertainty, specifically by setting DTUG = 104 μm2 s−1 , the form FTUG (g) can
also be made reasonably close to the data [Fig. 6(b), inset].
Thus, both the OJK and TUG theories are good descriptors(within the statistical accuracy) for the TNLC correlator andthis includes the description of length scales larger than thosein the Porod’s regime. Unlike DOJK , DTUG does not seemto have a trivial connection with DAC , albeit it can also be
reliably evaluated by fitting the experimental correlator to the
TUG Porod’s regime, Eq. (17), as we have done. It is overall
noteworthy that OJK performs remarkably well without such
fitting.
IV. TIME STATISTICS
A. Two-time autocorrelation

The two-time autocorrelator,
Ct (t, t0 ) =φ(r, t )φ(r, t0 ) r e(t  t0 ),(18)

is anticipated [130] to decay in the coarsening stages (for mostof the cases [131]) as [5]:054103-8

Ct (t, t0 )H (t/t0 ),H (y) ∼ y−λ/z(y  1),(19)

PHASE-ORDERING KINETICS IN THE ALLEN-CAHN …

PHYSICAL REVIEW E 104, 054103 (2021)

for large t0 and t, where H (y) is a universal scaling functionand λ is the autocorrelation exponent introduced by FH [70].
As described in Sec. I, 2D Model A theories suggest: λFH =5/4 [70], λOJK = 1 [80], and λTUG ≈ 1.2887 [79].
The scaling function H (y) in the OJK model is [5,80]:

d/4
4y
2
−1
HOJK (y) = sin.
(20)π(1 + y)2
For TUG, the function HTUG (g, τ (y)), analogously
defined from the two-point spatiotemporal correlatorφ(r + r, t )φ(r , t0 ) r e , with g = r/(4DTUGt )1/2 andτ (y) = (ln y)/4, satisfies the differential equation [79]:∂HTUG [g, τ (y)]1
= ∇g2 HTUG + 2g · ∇ gHTUG
∂τ
μ


π
1
HTUG .
+ tan
μ
2

(21)

This equation can be integrated numerically from the initialcondition HTUG (g, 0), which is just the vectorial analog of
FTUG (g) that we also numerically obtained in Eq. (16). Here,
we are interested in HTUG (y) = HTUG (0, τ (y)).
The LSI theory is based on the hypothesis that correlators transform covariantly under the conformal group (thatis, under space-time scale transformations set by the dynamic
exponent z) [83]. For 2D scalar Model A, LSI theory yields[132]:


y+1
λ/2
−λ
HLSI (y) = y (y − 1) 
,
(22)
y−1
with



1 3−2λ1+(λ − 1) − γ (λ − 1, E3 q)
(q) = E1 qq1γ (λ, E3 q) + E2 E31−λ q−λ γ (λ, E3 q)+
E3 q
e−E3 q − 1 + 2 F1 (λ − 1, 1; 3 − λ; −1/q), (23)+q1−λwhere E1 , E2 , and E3 are constants, is the gamma function,
γ is the lower incomplete gamma function, and 2 F1 is thehypergeometric function. Constants were determined from
2DIM simulations after a quench from the paramagnetic state
to the lower-temperature T = 0 state, with λ = λFH ; they read[133]: E1 = −0.601, E2 = 3.94, and E3 = 0.517. Hereafter,
HLSI (y) with this set is referred to as the LSI-2DIM correlator.
Before comparing experiment with theories, it is crucial
to subtract effects from a nonvanishing M(t ) in the kinetics,
otherwise, the asymptotic decay of the two-time correlator
in Eq. (18) becomes dependent on which φ = ±1 phaseequilibrates the system. Evaluating the magnetization M̂(t ) =φ(r, t ) r and the bare correlator Ĉt (t, t0 ) = φ(r, t )φ(r, t0 ) r
for each realization, we define a modified correlator Ct (t, t0 ):Ĉt (t, t0 ) − M̂(t )M̂(t0 )Ct (t, t0 ) =.
1 − M̂(t )M̂(t0 )e(24)This correlator satisfies Ct (t0 , t0 ) = 1 and Ct (t → ∞, t0 )→0,
since Ĉt (t → ∞, t0 ) → M̂(t )M̂(t0 ).

Note that Ct (t, t0 ) ≈ Ct (t, t0 ) as long as Ĉt (t, t0 ) M̂(t )M̂(t0 ), which is roughly ≈0.03 for t, t0  60 s for instance.
Figure 7(a) displays Ct (t, t0 ) in the experiment for t0 =2.66 s, 5.33 s, and 10.66 s against t (inset) and y = t/t0 (mainpanel) in the double logarithmic scales. From the collapse of
the three data sets shown in the main panel, we confirm the
dynamic scaling hypothesis as prescribed in Eq. (19). Those
data are then compared with the asymptotic power law decay
Ct (t, t0 ) ∼ (t/t0 )−λ/z with λ/z = 1/2 (OJK), = 5/8 (FH), and≈0.644 (TUG). After the initial nonalgebraic region, the data
at intermediate times 2  y  10 tend to decay with a OJKlike value prior to their crossover to a faster, asymptotic decay.
Using t0 = 2.66 s for which the asymptotic power law is mostprominent, and z = 2 shown in Eq. (2), we find1.03(5)(2  y  10)
,
(25)
λ=
1.28(11) (15  y  75)
whose values for the asymptotic TNLC decay are compatible with both the FH conjecture and TUG theory within
uncertainty (uncertainties are evaluated from the effective exponents).
Figure 7(b) compares Ct (t, t0 ) in the experiment with the
OJK [Eq. (20)], TUG [Eq. (21)], and LSI-2DIM [Eq. (23)]functions.
OJK outcome is well above the TNLC data, a result similarly seen in tests with 2DIM simulations [133]. The TUG
form, which to our knowledge has not been confronted withsimulations or experiments, exhibits pronounced deviations
from the experiment for y  25 while it approaches to thecollapsed correlator from below. This suggests that neither
OJK or TUG theory may perform well to describe the universal autocorrelator form in scalar, 2D Model A dynamics.
Interestingly, the experimental correlator is best accounted forby the LSI-2DIM theory, except for y  2 since LSI solutions
cannot describe such regime [83,133]. The nice agreementwith LSI-2DIM can also be appreciated in the log × linear
plot shown in the inset of Fig. 7(b).
In an overall view, individual theories cannot account simultaneously for all aspects (or regimes) of the experimental
two-time autocorrelator. Indeed, although both the FH conjecture and TUG theory are good descriptors for the slope of
the power-law decay, the autocorrelator functional form itself
is captured by neither TUG nor OJK. Instead, this form isoverall best accounted for by the LSI-2DIM correlator, except
for the y  2 regime; note that LSI-2DIM incorporates the FH
assumption, which is also in harmony with our experimental
observations.
B. Local persistence

The local persistence probability Q(t, t0 ) is the probability
that sgn[φ(r, t˜)] does not change during t˜ ∈ [t0 , t]. It typicallyshows nontrivial algebraic decay [67,68],
Q(t, t0 ) ∼ (t/t0 )−θ(t  t0 ),(26)

with the persistence exponent θ , as we can here directly appreciate in the experiment for several values of t0 [Fig. 8(a)].
The plot Q(t, t0 ) versus y = t/t0 yields an excellent
data collapse within δQ ≈ 0.04 (standard deviation) for

054103-9

10

(b)

0

t0 = 2.66 s
5.33 s10.66 s
FH
OJK
TUG

-1

10

10

0

t0 = 2.66 s
5.33 s10.66 s
LSI-2dIM
OJK
TUG

1
0.8
0.6
0.4
0.2

0.6

0
0
10

0.4

y

10

1

-1

0.2
10

-2

10

10

1
0.8

C’t (t, t0)

10

C’t (t, t0)

C’t (t, t0)(a)

PHYSICAL REVIEW E 104, 054103 (2021)

C’t (t,t0)

ALMEIDA AND TAKEUCHI

1

0

10

2

10

t [s]

3

1

10
y = t / t0

10

0
0

2

10

20

30
y = t / t040

50

FIG. 7. (a) Two-time autocorrelators Ct (t, t0 ) [Eq. (24)] for the TNLC experiment (symbols) versus t (inset), and y = t/t0 (main panel)according to the dynamic scaling hypothesis. The solid orange, dashed black, and dash-dotted blue lines are guides to the eyes whose slopes,−λ/z, are set by the Fisher-Huse (−5/8), OJK (−1/2), and the theory of unstable growth (TUG; ≈ −0.644) predictions, respectively. (b) SameTNLC data compared with forms for the following theoretical predictions: HLSI (y) [solid orange line; Eq. (23) with E1 = −0.601, E2 = 3.94,
E3 = 0.517, and λ = λFH , which is a solution of the LSI theory that describes the 2DIM correlator (see text)], HOJK (y) [dashed black line;
Eq. (20) derived from the OJK theory], and HTUG (y) [dashed-dotted blue line; Eq. (21) derived from the TUG theory]. Inset displays the samedata, but the horizontal axis is displayed in logarithmic scale. In both the panels, error bars indicate one standard deviation.

y  102 [Fig. 8(b)]. The decay is quantified by the exponent, θeff (t, t0 ) = −d[ln Q(t, t0 )]/d (ln t ), shown in the inset
of Fig. 8(b) for t0 = 2.66 s and 5.33 s. Both of the θeff (t, t0 )curves fluctuate around their mean values for 8  y  50.
Averaging θeff data in this interval gives 0.1938(14) and
0.1976(27) for t0 = 2.66 s and 5.33 s, respectively. We also
evaluate θ by applying a least-squares fit individually toQ(t, t0 ) data of each realization, constrained to the interval
8  y  50, and then ensemble averaging; the results are:
θ = 0.1941(26) and 0.199(4) for t0 = 2.66 s and 5.33 s. Integrating these results, we report
0.196(3)θ=
0.196(4)(effective exponent),(power-law fit)

(27)

where uncertainties include (sum up) both the uncertainty
from each estimate and the deviation between the two estimates for t0 = 2.66 s and 5.33 s. We remark that these results
do not significantly change when the interval 8 < y < 60 s/t0
is used, since it corresponds to the interval t  60 s predominantly used in Sec. III.
Results in Eq. (27) agree with systematic 2DIM outcomes [99] [θ ≈ 0.195 and θ = 0.199(2) for free and periodic
boundary conditions, respectively], but they significantly differ from θDIF = 3/16 = 0.1875 [100] and, hence, from the
OJK theory as well [93]. This may suggest, based on a concrete experience, the former as the true θ for 2D Model A
systems.
To probe morphological aspects of persistence, we define
an index χ (r, t ) such that χ = 1 if sgn[φ(r, t˜)] does notchange during t˜ ∈ [t0 , t], with t0 hereafter pinned at t0 =2.66 s, and χ = 0 otherwise. The index χ (r, t ) can onlychange from 1 to 0; such change happens only when aninterface crosses the position r for the first time since t0 .
Snapshots of {χ (r, t )} are presented in Fig. 9; they reveal
how a single χ = 1 cluster is progressively fragmented because of the interface motion in real space. Surviving clusters
develop ragged edges in both the internal and external contours resembling fractal objects (see Supplemental Videos 3
and 4 [121]). The fractal dimension d f of this morphology isestimated by combining the gliding-box method [134] and themethod in Ref. [135]. Here a square of side ζ glides over each{χ (r, t )} image while counting the number of χ = 1 pixels
inside the ith box, gi , with i = 1, 2, . . . , n(r); n(r) is the total
number of distinct boxes that contain at least one χ = 1 pixel.
The mean counting is G(ζ , t ) = gi n e , where ... n is theaverage over n(r) boxes glided on a snapshot at t. For fractal
objects [135,136]:
G(ζ , t ) ∼ ζ d f(a  ζ  Lsys ).

(28)

In the experiment, G exhibits a crossover from fractal to
Euclidian length scales at a certain correlation length ζ ∗ (t )[Fig. 10(a)]. The fractal regime is progressively built in timesince it is confined to emerge for a  ζ  ζ ∗ (t ), with a growing length, ζ ∗ (t ), that starts from ζ ∗ (t0 ) = 0. This progressivebuilding is captured by d f −eff (ζ , t ) = d (ln G(ζ , t ))/d (ln ζ )[Fig. 10(b)] which shows a concave dependence with ζ andforms a plateau around a t-dependent minimum. For ζ ζ ∗ (t ), d f −eff crossovers to d = 2. Finite-size and finite-time
effects preclude observation of clear t-independent plateau forζ  ζ ∗ ; however, from the plateau at the minimum of d f −eff
for large times, t = 1365 s, we roughly read d f ≈ 1.65(3)[Fig. 10(b)]. The number in the parentheses indicate one standard deviation.

054103-10

PHASE-ORDERING KINETICS IN THE ALLEN-CAHN …

(a)

PHYSICAL REVIEW E 104, 054103 (2021)

where η is a universal exponent. At the crossover length,Cχ ∼ (ζ ∗ )−η ∼ t −θ . Thus, if ζ ∗ (t ) ∼ t 1/z holds [138], then
η = zθ . Because G(ζ , t ) ∼ box Cχ (r, t )d d r [135], where theintegration is carried out over a box of side ζ centered at
r = 0, integration after Eq. (30) yields: d−η[a  ζ  ζ ∗ (t )]ζG(ζ , t ) ∼ −θ d
.
(31)
t ζ
[ζ ∗ (t )  r  Lsys ]

1

0.5

19

0.

Q(t,t0)t0 = 2.66 s
5.33 s10.66 s21.33 s6Comparing with Eq. (28), we identify:d f = d − θ z,

0.25

1

2

10
(b)

and, consequently, zθ  d.
Before using Eq. (32) to evaluate d f , we test Eq. (30) in theexperiment. As shown in Fig. 10(c), we detect a nearly timeindependent decay Cχ (r, t ) ∼ r −η (arguably from t = 43 s)
within a region 20 μm  r  ζ ∗ (t ) that extends in time. The
exponent ηeff (r) = −d[ln Cχ (r, t )]/d (ln r) measured at t =1365.33 s provides us with η = 0.39(4) after averaging datafor 38 μm  r  150 μm [see left inset of Fig. 10(c)]. Notethat this value is consistent with η = 0.428(7) obtained for
persistent 2DIM clusters [102]. We also observe Cχ (r, t ) ∼t −θ for r  ζ ∗ in the right inset of Fig. 10(c), which displaysCχ (r, t ) at |r| = 500 μm. Finally, the assumption ζ ∗ (t ) ∼ t 1/2
is verified by the collapse of d f −eff (ζ , t ) [Fig. 10(d)] andCχ (r, t ) [Fig. 10(e)] when plotted versus ζ /t 1/2 and r/t 1/2 ,respectively.
Deviations from collapsed forms for small lengths are due
to effects of microscopic scales, such as a and ξ .
Now we safely rely on Eq. (32) to evaluate d f . Using θ in
Eq. (27) and z in Eq. (2), we find
1.613(14) (effective exponent)
,
(33)
df =
1.609(9)(power-law fit)

3

10
t [s]

10

1

0.5
θeff (t,t0)

Q(t,t0)θ = 0.196

0.21

t = 2.66 s
5.33 s

0.2
0.19
0.18
10

0.25 0
10

1

10

y

2

1

10
y = t / t02

10

FIG. 8. (a) Local persistence probability Q(t, t0 ) for the ordering
kinetics of TNLC, for several reference times t0 . Error bars indicate
one standard deviation. Dashed line is a guide to eyes whose absolutevalue of its slope is indicated in the plot. (b) Collapse of Q whenplotted versus y = t/t0 and Eq. (26) with θ = 0.196. Legend ofsymbols as shown in (a). Inset shows the effective exponent θeff (t, t0 )for t0 = 2.66 s and 5.33 s data. Dashed line indicates the mean value,
0.196, for the 8  y  50 interval.

Although t = 1365 s is far from the time interval whereM(t ) is constant [Fig. 4(a), inset], properties of persistence
clusters at long times are reminiscent of those in the early
dynamics: This is revealed by the inner structures of persistence domains in the panel at t = 1365 s of Fig. 9, which aremostly frozen and already present since t = 43 s. Note alsothat the rough estimate 1.65(3) is consistent with the interval
d f = 1.58–1.62 reported for persistence 2DIM clusters [102].
To evaluate d f from shorter times, for which |M(t )| is smalland z = 2 is clearly identified, we appeal to the correlator Cχ ,Cχ (r, t ) =χ (r , t )χ (r + r , t ) r e,
χ (r , t ) r ewhich behaves as [102,135,137]: −η[r  ζ ∗ (t )]rCχ ∼ −θ,[r  ζ ∗ (t )]
t

(32)

(29)

(30)

where the notes in the parentheses indicate the methods used
to evaluate 1/z and θ in Eqs. (2) and (27), respectively. Finalestimates are slightly lower than the rough estimate d f =1.65(3) (as may be expected). More important, they nicely
agree with the interval, d f = 1.58–1.62, numerically extracted
from the 2DIM model in Ref. [102], while all of them deny a
proposed d f = 1.53(2) from Ising simulations in Ref. [103].
Altogether, experimental results in this section provide firm
evidence that the statistics of the local persistence probability
correspond to a universal feature of 2D Model A systems.
In addition, one can realize that the presence of a weakphase asymmetry has negligible, or very small, effect on thosequantities.
V. CONCLUSIONS

We have studied the nonconserved phase-ordering kineticsof 2D TNLC as a model experimental system to elucidate universal aspects in the Allen-Cahn (Model A) universality class.
Exploiting electrically driven hydrodynamical regimes ofliquid crystals, we triggered a genuinely sudden transition between a disordered-like, spatiotemporally chaotic state, called
DSM2, toward a two-phase competing, equilibrium one. The
sudden removal of the driving avoids the inexorable slow
quench rate of thermally induced transitions, besides it avoids
the issues of metastability, nucleation, and nuclei growth,
implied in first-order transitions. Because spatiotemporal

054103-11

ALMEIDA AND TAKEUCHIPHYSICAL REVIEW E 104, 054103 (2021)

FIG. 9. Configurations of the local persistence index χ (r, t ) (see text) computed in the ordering of TNLC for a reference time t0 = 2.66 s.
The black and white pixels correspond to χ = 0 (nonpersistence) and 1 (persistence), respectively. See also Supplemental Videos 3 and 4[121].

(a)

(b)

6

10

t = 5.33 s
21.33 s85.33 s1365.33 s

2

1.9

4

G(ζ,t)

df-eff (ζ,t)

2

10

t = 5.33 s
10.66 s21.33 s42.66 s85.33 s170.66 s
341.33 s
682.66 s1365.33 s

1.8

2

10

5

1.6

1.7

0

10

0

1

10

10

2

1.6

3

10
ζ [μm]

2

3

10
ζ [μm]

10(d) 2.1

0

2

Cχ(r,t)

1.9
1.8
1.7

0.3

ζ / t1/2 [μm s-1/2]

9(e) 1.6
4
Cχ(r,t) at |r| = 500 μm
1

0.45

0.19

6

0.5

0.3

-1

Cχ(r,t) tθ [sθ]ηeff(r)

10

1

10

10

df-eff (ζ,t)

(c)

10

r [μm]

0.15
10

1

10

1

10

2

t [s]

0.25
10

2

10
r [μm]

1

10

2

3

Cχtθ = 1.20 sθ
0.3

9

2
1
r / t1/2 [μm s-1/2]

10

10

0

1

10

2

10

FIG. 10. Statistics of the local persistence in the ordering kinetics of TNLC phases. In all the panels, dash-dotted and dashed lines are guideto eyes; the values of their slopes or ordinates are indicated in the respective plots. (a) Mean counting G(ζ , t ) of χ = 1 (persistence) pixels
found inside a square of size ζ that glides over the configurations of the persistence index χ (r, t ) (reference time t0 = 2.66 s). (b) Effective
dimension d f −eff (ζ , t ) for the morphology of persistence clusters as a function of ζ . (c) Covariance Cχ (r, t ) [Eq. (29)] of the index χ versus
r = |r|. Data were obtained by rotational average (averaging over different r with the same r = |r|), data binning, and ensemble average. The
legend of the symbols is given in (b). Left inset depicts the effective exponent ηeff (r) related to a curve, t = 1365.33 s, in the main panel. Right
inset shows Cχ (r, t ) in time for a length (r = 500 μm) much larger than the crossover length ζ ∗ (t ). [(d) and (e)] d f −eff (ζ , t ) and Cχ (r, t ) versus
ζ /t 1/2 and r/t 1/2 , respectively, following the dynamic scaling hypothesis. Legends in (c), (d), and (e) are as that shown in (b).
054103-12

PHASE-ORDERING KINETICS IN THE ALLEN-CAHN …

PHYSICAL REVIEW E 104, 054103 (2021)

TABLE I. Universal quantities measured in the TNLC experiment
Quantity

Experiment (this work)

1/z
A∗
λ

0.505(15), 0.498(6)a0.99(7)1.03(5) for short times (2  y  10)
1.28(11) for late times (15  y  75)

Fig. 3, Eq. (2)Eq. (14)
Fig. 7(a), Eq. (25)θ

0.196(3), 0.196(4)bFig. 8, Eq. (27)df

1.65(3), 1.613(14), 1.609(9)dFig. 10

TheoryReferences

1/2
1
1
5/4 = 1.25
≈1.2887≈0.195, 0.199(2)c
3/16 = 0.1875
1.58–1.62
1.53(2)

Exact predictions [3,4,40]
BH prediction [66]
OJK [80]
FH [70]TUG [79]
2DIM simulations [99]DIF-RIC [100] and OJK [93]2DIM simulations [102]2DIM simulations [103]Both the estimates were obtained from the decay of the interface density, ρ(t ) ∼ t 1/z . The time averaged effective exponent and the least-square
fit of the power law give 0.505(15) and 0.498(6), respectively. Please see Eq. (2) and text for detail.
b
The estimate θ = 0.196(3) was obtained from the time average of the effective exponent; 0.196(4) from the least-squares fit of the power law.
See Eq. (27) and text for detail.
c
The two values θ ≈ 0.195 and 0.199(2), from Ref. [99], were obtained from simulations with free and periodic boundary conditions,
respectively.
d
The estimate d f = 1.65(3) was obtained directly from Eq. (28) by using data at t = 1365 s, which is however far from the interval where M(t )is constant and far from the core of the z = 2 scaling regime. The other values d f = 1.613(14) and 1.609(9) are our final estimates obtained
through the scaling relation (32). Please see Eq. (33) and text for detail.
acorrelations of the director field are negligible in the
DSM2 state [111], DSM2 essentially generates random initial conditions. When electrically switched (t = 0 s) toward
equilibrium, the ensuing competition between two possible
conformations for the director field, in left- and righthanded twists (here noted φ = ±1), resembles that between
Ising spins when quenched from the higher-temperature(T  Tc , Tc is the critical temperature) regime to the lowertemperature (T  Tc ) phase (see Fig. 2). Nonetheless, theexperiment develops a weakly asymmetry likely along its
dynamics: This is revealed by the “magnetization” |M | 
0.2 = 0, for t  60 s [Fig. 4(a)], and by the shrinking rateof φ = ±1 spherical domains (“bubbles”) [Fig. 4(b)]. Noting the curvature-driven dynamics of disclinations (interfaces)[Eq. (3); Fig. 4(b)], the asymmetry between the twisted
phases also manifests in the Allen-Cahn diffusion coefficients,
which take slightly different values [4% deviation from themean DAC = 122(4) μm2 s−1 ] for φ = ±1. As we discussed
throughout the study, this weak asymmetry—that is usual inexperiments [96,139]—must be taken into account when one
aims at testing theories for Z2 -symmetric models.
After observing that the experiment firmly shows wellknown Model A features—such as the dynamic exponentz = 2 [see Fig. 3; Eq. (2)] and dynamic scaling [Fig. 5(b)]—we move forward to comprehensively elucidate aspects thatremain either controversial or not experimentally assessed yet
for 2D Model A. Main results are summarized in Tables I
and II.

From the spatial correlator at the Porod’s regime, we measured the universal amplitude A∗ (see Table I) that confirms
the long-standing Bray-Humayun theory [66].
Beyond the Porod’s regime, the correlator form is well
captured by both the OJK theory [51] and the TUG own
to Mazenko [52,53] [Fig. 6(b)]. Such conclusion emerges
from comparisons done with no free parameters: the constant
DOJK was found from its direct relation with DAC [51], which
we measured independently from the “bubble” experiments;
DTUG was determined from the functional form in the Porod’s
regime. We have seen that although the agreement between
experiment and theory is more pronounced for OJK (considering the most probable values for DOJK and DTUG ), TUG
also may account for the TNLC correlator if the lower limit
for DTUG , set by its uncertainty, is used [Fig. 6(b)]. To ourknowledge, this is the first experimental evidence [140] thattheoretical spatial correlators, such as OJK and TUG, describe
experimental data in the nontrivial scale beyond the Porod’s
regime, for a curvature-driven (z = 2) system.
Regarding the time correlations, we measured the two-timecorrelator after proposing its due modification to account for
the weak asymmetry in the kinetics [Eq. (24)]. These correlators, Ct (t, t0 ), collapse on a master curve when rescaledby y = t/t0 , thus offering evidence of dynamic scaling hypothesis for 2D Model A systems as demanded by aging
phenomenology [69]; see Fig. 7. The exponent λ [Eq. (19)]displays a crossover from an initial OJK-like value to an
asymptotic one which is compatible with both the FH

TABLE II. Universal scaling functions measured in the TNLC experiment and compared with theoretical predictions.
Scaling function

Experimental result

Spatial correlator F (g)Temporal correlator H (y)Fig. 6
Fig. 7(b)Conclusion
Consistent with both OJK and TUG predictions
Best described by LSI-2DIM prediction for y  2054103-13

ALMEIDA AND TAKEUCHIPHYSICAL REVIEW E 104, 054103 (2021)

conjecture and the TUG prediction (values in Table I): This
reinforces the difficult matter, witnessed by simulations, ofdistinguishing between the FH and TUG predictions. Light isshed by comparisons with the functional form itself, which
elucidate that neither OJK or TUG theory can account for
the experimental, 2D Model A, correlation function. Instead,the LSI theory own to Henkel [83], that describes the 2DIM
correlator along with the FH conjecture [133], provides thebest description for the experiment in the regime, t/t0  2,that LSI deals with [Fig. 7(b)]. These results might indicate
support to the FH conjecture from a global perspective, in
harmony with many numerical studies [22,70,72,73,76,133].
The agreement with LSI also raises up the question whether
the paradigmatic 2D Model A systems may display conformal
invariance in some stage of their kinetics [83].
We have also assessed the local persistence statistics.
Thanks to the accuracy of our data, we measure a local persistence exponent θ (Fig. 8; Table I) in accordance with therecent estimates from 2DIM [99], but in significant disagreement with the candidate 2D Model A value obtained fromthe diffusion equation [100] and the OJK theory [93]. This
experimentally based observation thus points out the former
as the genuine value in the 2D Model A class. Additionally,
we have observed that the mosaic of persistence clusters,
defined by the persistence index χ (r, t ), develops a fractal
morphology up to the scale of a correlation length ζ ∗ (t ) ∼ t 1/2(see Fig. 9 and Fig. 10). The fractal dimension d f ≈ 1.61 ofsuch morphology is measured experimentally (Table I) andsubstantiated as universal. Correlations among persistence
clusters respect the dynamic scaling hypothesis [Fig. 10(e)].
Interesting√to note that d f ≈ 1.61 is close to the golden ratio,
φ = (1 + 5)/2 = 1.618 . . . , although no present argument
exists for conjecturing d f = φ, so that one could obtain anexact value for θ . This may be a matter of theoretical andnumerical exploration.

Having a versatile experimental model system in the AllenCahn (Model A) class opens novel paths for experimental
investigation, too. The many interesting electroconvective patterns in the liquid crystals [111–113], for instance, can beused to assess unusual sorts of ordering kinetics by transiting
between nonequilibrium states, with the advantage of the genuinely sudden electrical switching. Moreover, the independent
control on the temperature allows one to tune thermal effects
by adjusting the distance from the nematic-isotropic transition
point [110]. Such control can be particularly important for experimentally assessing the fundamental theoretical hypothesis
that, from the viewpoint of renormalization group [4,5,44], thebath temperature merely factorizes nonuniversal amplitudes,
while it preserves universal features—scaling exponents andcorrelation functions—for a subcritical dynamics. Anotherappealing direction is suggested by the numerically observed[44] self-tuning of 2D Model A systems to the critical percolation fixed point [141–143], which theoretically has boosted
derivations of exact solutions for the distributions of hullareas [44,139], perimeters [144], domain crossing probabilities [145], besides proposing a generalization of the dynamic
scaling hypothesis [146,147]. Access to response functions[73,83] is an additional and attractive avenue that calls for,and it is in principle amenable to, experimentation.

[1] I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitationfrom supersaturated solid solutions, J. Phys. Chem. Sol. 19, 35(1961).
[2] C. Wagner, Theorie der alterung von niederschlägen durch
umlösen (ostwald-reifung), Z. Elektrochem. 65, 581 (1961).
[3] I. M. Lifshitz, Kinetics of ordering during second-order phasetransitions, Sov. Phys. JETP 15, 939 (1962).
[4] S. M. Allen and J. W. Cahn, A microscopic theory for
antiphase boundary motion and its application to antiphasedomain coarsening, Acta Metall. 27, 1085 (1979).
[5] A. J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 51,
481 (2002).
[6] A. Onuki, Phase Transition Dynamics (Cambridge University
Press, Cambridge, UK, 2002).
[7] S. Puri, Kinetics of Phase Transitions, 1st ed. (CRC Press,
Boca Raton, FL, 2009).
[8] P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of
Statistical Physics (Cambridge University Press, Cambridge,
UK, 2010).
[9] L. F. Cugliandolo, Coarsening phenomena, Comp. Rend. Phys.
16, 257 (2015).

[10] M. E. Cates and J. Tailleur, Motility-induced phase separation,
Ann. Rev. Condens. Matter Phys. 6, 219 (2015).
[11] J. Berry, C. P. Brangwynne, and M. Haataja, Physical principles of intracellular organization via active and passive phasetransitions, Rep. Prog. Phys. 81, 046601 (2018).
[12] D. Bouttes, E. Gouillart, E. Boller, D. Dalmas, and D.
Vandembroucq, Fragmentation and Limits to Dynamical Scaling in Viscous Coarsening: An Interrupted in Situ X-Ray
Tomographic Study, Phys. Rev. Lett. 112, 245701 (2014).
[13] P.-A. Geslin, I. McCue, B. Gaskey, J. Erlebacher, and A.
Karma, Topology-generating interfacial pattern formation during liquid metal dealloying, Nat. Commun. 6, 8887 (2015).
[14] M. Tateno and H. Tanaka, Power-law coarsening in networkforming phase separation governed by mechanical relaxation,Nat. Commun. 12, 912 (2021).
[15] J. Lambert, R. Mokso, I. Cantat, P. Cloetens, J. A. Glazier, F.
Graner, and R. Delannay, Coarsening Foams Robustly Reacha Self-Similar Growth Regime, Phys. Rev. Lett. 104, 248304(2010).
[16] M. Castro, R. Cuerno, M. M. García-Hernández, and L.
Vázquez, Pattern-Wavelength Coarsening from Topological

ACKNOWLEDGMENTS

The authors acknowledge enlightening discussions with J.
J. Arenzon, A. Azevedo-Lopes, F. Corberi, L. F. Cugliandolo,
Y. T. Fukai, S. Majumdar, M. Mucciconi, M. Picco, and T.
Ohta. This work was supported by Japan Society for the
Promotion of Science (JSPS). R.A.L.A. was funded by JSPS
KAKENHI Grant No. JP16J06923. K.A.T. was funded by
JSPS KAKENHI Grants No. JP25103004, No. JP16H04033,
No. JP19H05800, and No. JP20H00128.

054103-14

PHASE-ORDERING KINETICS IN THE ALLEN-CAHN …

PHYSICAL REVIEW E 104, 054103 (2021)

Dynamics in Silicon Nanofoams, Phys. Rev. Lett. 112, 094103(2014).
[17] C. Misbah, O. Pierre-Louis, and Y. Saito, Crystal surfaces inand out of equilibrium: A modern view, Rev. Mod. Phys. 82,
981 (2010).
[18] J. J. Arenzon, L. F. Cugliandolo, and M. Picco, Slicing thethree-dimensional ising model: Critical equilibrium and coarsening dynamics, Phys. Rev. E 91, 032142 (2015).
[19] P. Bhandari, V. Malik, and S. Puri, Logarithmic coarsening in
the coulomb glass, Phys. Rev. E 99, 052113 (2019).
[20] A. de Azevedo-Lopes, A. R. de la Rocha, P. M. C. de Oliveira,and J. J. Arenzon, Dynamical cluster size heterogeneity,
Phys. Rev. E 101, 012108 (2020).
[21] F. Corberi, E. Lippiello, and P. Politi, Quasideterministic
dynamics, memory effects, and lack of self-averaging in
the relaxation of quenched ferromagnets, Phys. Rev. E 102,020102(R) (2020).
[22] H. Christiansen, S. Majumder, M. Henkel, and W. Janke,
Aging in the Long-Range Ising Model, Phys. Rev. Lett. 125,
180601 (2020).
[23] M. P. O. Loureiro, J. J. Arenzon, L. F. Cugliandolo, and A.
Sicilia, Curvature-driven coarsening in the two-dimensional
Potts model, Phys. Rev. E 81, 021129 (2010).
[24] M. P. Loureiro, J. J. Arenzon, and L. F. Cugliandolo, Geometrical properties of the potts model during the coarsening
regime, Phys. Rev. E 85, 021135 (2012).
[25] S. Redner, Reality-inspired voter models: A mini-review,Comp. Rend. Phys. 20, 275 (2019).
[26] A. A. Hyman, C. A. Weber, and F. Jülicher, Liquid-liquid
phase separation in biology, Ann. Rev. Cell Dev. Biol. 30, 39(2014).
[27] S. Alberti, A. Gladfelter, and T. Mittag, Considerations andchallenges in studying liquid-liquid phase separation andbiomolecular condensates, Cell 176, 419 (2019).
[28] L. McNally, E. Bernardy, J. Thomas, A. Kalziqi, J. Pentz,
S. P. Brown, B. K. Hammer, P. J. Yunker, and W. C. Ratcliff,
Killing by Type VI secretion drives genetic phase separationand correlates with increased cooperation, Nat. Commun. 8,14371 (2017).
[29] G. Liu, A. Patch, F. Bahar, D. Yllanes, R. D. Welch, M. C.
Marchetti, S. Thutupalli, and J. W. Shaevitz, Self-Driven Phase
Transitions Drive Myxococcus Xanthus Fruiting Body Formation, Phys. Rev. Lett. 122, 248102 (2019).
[30] Q.-X. Liu, A. Doelman, V. Rottschäfer, M. de Jager, P. M. J.
Herman, M. Rietkerk, and J. van de Koppel, Phase separation explains a new class of self-organized spatial patterns inecological systems, Proc. Natl. Acad. Sci. USA 110, 11905(2013).
[31] E. Demir, Y. I. Yaman, M. Basaran, and A. Kocabas, Dynamics of pattern formation and emergence of swarming in
Caenorhabditis elegans, eLife 9, e52781 (2020).
[32] D. Geyer, D. Martin, J. Tailleur, and D. Bartolo, Freezing
a Flock: Motility-Induced Phase Separation in Polar Active
Liquids, Phys. Rev. X 9, 031043 (2019).
[33] M. N. van der Linden, L. C. Alexander, D. G. A. L. Aarts, and
O. Dauchot, Interrupted Motility Induced Phase Separationin Aligning Active Colloids, Phys. Rev. Lett. 123, 098001(2019).

[34] J. Zhang, R. Alert, J. Yan, N. S. Wingreen, and S. Granick,
Active phase separation by turning towards regions of higher
density, Nat. Phys. 17, 961 (2021).
[35] R. J. Glauber, Time-dependent statistics of the ising model,
J. Math. Phys. 4, 294 (1963).
[36] K. Kawasaki, Diffusion constants near the critical point for
time-dependent ising models. I, Phys. Rev. 145, 224 (1966).
[37] H. Wioland, F. G. Woodhouse, J. Dunkel, and R. E. Goldstein,
Ferromagnetic and antiferromagnetic order in bacterial vortex
lattices, Nat. Phys. 12, 341 (2016).
[38] P. C. Hohenberg and B. I. Halperin, Theory of dynamic criticalphenomena, Rev. Mod. Phys. 49, 435 (1977).
[39] Exceptions to this law are the nonconserved one-dimensional
XY model, which yields z = 4 [50], and the two-dimensional
voter model for which clusters have no typical scale √inthe sense of l (t ), although the largest scale grows as ∼ t[148,149].
[40] A. J. Bray and A. D. Rutenberg, Growth laws for phase ordering, Phys. Rev. E 49, R27 (1994).
[41] J. G. Amar and F. Family, Diffusion annihilation in one dimension and kinetics of the ising model at zero temperature,
Phys. Rev. A 41, 3258 (1990).
[42] A. J. Bray, Universal scaling function for domain growth in the
Glauber-Ising chain, J. Phys. A: Math. Gen. 23, L67 (1990).
[43] A. Coniglio and M. Zannetti, Multiscaling in growth kinetics,
Europhys. Lett. 10, 575 (1989).
[44] J. J. Arenzon, A. J. Bray, L. F. Cugliandolo, and A. Sicilia,
Exact Results for Curvature-Driven Coarsening in Two Dimensions, Phys. Rev. Lett. 98, 145701 (2007).
[45] J. Cardy and R. M. Ziff, Exact results for the universal area
distribution of clusters in percolation, ising, and potts models,J. Stat. Phys. 110, 1 (2003).
[46] K. Binder et al., Phys. Rev. Lett. 33, 1006 (1974); Phys. Rev.
B 15, 4425 (1977); Z. Phys. B 30, 183 (1978).
[47] H. Furukawa, Structure Functions of Quenched OffCritical Binary Mixtures and Penormalizations of Mobilities,
Phys. Rev. Lett. 43, 136 (1979).
[48] J. Marro, J. L. Lebowitz, and M. H. Kalos, Computer Simulation of the Time Evolution of a Quenched Model Alloy in the
Nucleation Region, Phys. Rev. Lett. 43, 282 (1979).
[49] Y. C. Chou and W. I. Goldburg, Angular distribution of lightscattered from critically quenched liquid mixtures, Phys. Rev.
A 23, 858 (1981).
[50] T. J. Newman, A. J. Bray, and M. A. Moore, Growth of order invector spin systems and self-organized criticality, Phys. Rev.
B 42, 4514 (1990).
[51] T. Ohta, D. Jasnow, and K. Kawasaki, Universal Scaling in
the Motion of Random Interfaces, Phys. Rev. Lett. 49, 1223
(1982).
[52] G. F. Mazenko, Theory of Unstable Thermodynamic Systems,
Phys. Rev. Lett. 63, 1605 (1989).
[53] G. F. Mazenko, Theory of unstable growth, Phys. Rev. B 42,4487 (1990).
[54] G. F. Mazenko, Universal features in growth kinetics: Someexperimental tests, Phys. Rev. B 43, 8204 (1991).
[55] A. J. Bray, Topological defects, correlation functions, andpower-law tails in phase-ordering kinetics, Phys. Rev. E 47,
228 (1993).

054103-15

ALMEIDA AND TAKEUCHIPHYSICAL REVIEW E 104, 054103 (2021)[56] G. F. Mazenko, Post-Gaussian approximations in phase ordering kinetics, Phys. Rev. E 49, 3717 (1994).
[57] G. F. Mazenko, Perturbation expansion in phase-ordering kinetics: I. Scalar order parameter, Phys. Rev. E 58, 1543 (1998).
[58] Other theories such as the Kawasaki-Yalabik-Gunton (KYG)
theory [150], the non-Gaussian theories by Mazenko et al.
[56,151], and the perturbative ones [55,57], generically recover either OJK or TUG in the leading order. The KYG
theory recovers the OJK function for a particular value of OJK
diffusion constant [5].
[59] K. Humayun and A. J. Bray, Scaling functions in phaseordering dynamics: A comparison of theory and simulations,Phys. Rev. B 46, 10594 (1992).
[60] R. E. Blundell, A. J. Bray, and S. Sattler, Absolute test fortheories of phase-ordering dynamics, Phys. Rev. E 48, 2476
(1993).
[61] A. J. Bray and K. Humayun, Towards a systematic calculationof the scaling functions for the ordering kinetics of nonconserved fields, Phys. Rev. E 48, R1609 (1993).
[62] G. Brown, P. A. Rikvold, and M. Grant, Universality andscaling for the structure factor in dynamic order-disorder transitions, Phys. Rev. E 58, 5501 (1998).
[63] However, Ref. [60] pointed out that Gaussian theories such
as OJK and TUG do not describe, simultaneously, data of
the correlator of the order parameter field and those of thesquared field. Limitations of Gaussian approximations werealso discussed in Refs. [152,153].
[64] H. Orihara and Y. Ishibashi, Dynamics of disclinations intwisted nematics quenched below the clearing point, J. Phys.
Soc. Jpn. 55, 2151 (1986).
[65] G. Porod, Kolloid-Zeitschrift 124, 83 (1951); 125, 51 (1952).
[66] A. J. Bray and K. Humayun, Universal amplitudes of powerlaw tails in the asymptotic structure factor of systems with
topological defects, Phys. Rev. E 47, R9 (1993).
[67] S. N. Majumdar, Persistence in nonequilibrium systems,
Curr. Sci. 77, 370 (1999).
[68] A. J. Bray, S. N. Majumdar, and G. Schehr, Persistence andfirst-passage properties in nonequilibrium systems, Adv. Phys.
62, 225 (2013).
[69] M. Henkel, Non-Equilibrium Phase Transitions Volume 2:
Ageing and Dynamical Scaling Far from Equilibrium (Springer
Science & Business Media, New York, 2010).
[70] D. S. Fisher and D. A. Huse, Equilibrium behavior of the spinglass ordered phase, Phys. Rev. B 38, 386 (1988).
[71] T. J. Newman and A. J. Bray, J. Phys. A: Math. Gen. 23, L279
(1990); 23, 4491 (1990).
[72] K. Humayun and A. J. Bray, Non-equilibrium dynamics of theising model for t less-than/equal-toTc, J. Phys. A: Math. Gen.
24, 1915 (1991).
[73] F. Corberi and R. Villavicencio-Sanchez, Role of initial stateand final quench temperature on aging properties in phaseordering kinetics, Phys. Rev. E 93, 052105 (2016).
[74] A. J. Bray and B. Derrida, Exact exponent λ of the autocorrelation function for a soluble model of coarsening, Phys. Rev.
E 51, R1633 (1995).
[75] Fisher and Huse assumed that the correlation length, ξ p , behaves for random percolation at the continuum d = 2, in thevicinity to the critical point pc , as ξ p = |p − pc |ν=4/3 , where pis the model parameter, as this was the scaling indicated to FH

by several previous independent results that were summarized
in the Table VI, p. 261, of Ref. [154].
[76] G. Brown, P. A. Rikvold, M. Sutton, and M. Grant,
Speckle from phase-ordering systems, Phys. Rev. E 56, 6601
(1997).
[77] J. Midya, S. Majumder, and S. K. Das, Aging in ferromagnetic
ordering: Full decay and finite-size scaling of autocorrelation,
J. Phys.: Condens. Matter 26, 452202 (2014).
[78] N. Vadakkayil, S. Chakraborty, and S. K. Das, Finite-size
scaling study of aging during coarsening in non-conserved
ising model: The case of zero temperature quench, J. Chem.
Phys. 150, 054702 (2019).
[79] F. Liu and G. F. Mazenko, Nonequilibrium autocorrelations inphase-ordering dynamics, Phys. Rev. B 44, 9185 (1991).
[80] C. Yeung and D. Jasnow, Two-time correlations, selfaveraging, and an analytically solvable model of phaseordering dynamics, Phys. Rev. B 42, 10523 (1990).
[81] C. Yeung, M. Rao, and R. C. Desai, Bounds on the decay of
the autocorrelation in phase ordering dynamics, Phys. Rev. E
53, 3073 (1996).
[82] N. Mason, A. N. Pargellis, and B. Yurke, Scaling Behavior of
Two-Time Correlations in a Twisted Nematic Liquid Crystal,Phys. Rev. Lett. 70, 190 (1993).
[83] M. Henkel, Phenomenology of local scale invariance: From
conformal invariance to dynamical scaling, Nucl. Phys. B 641,
405 (2002).
[84] S. Redner, A Guide to First-Passage Processes (Cambridge
University Press, Cambridge, UK, 2001).
[85] F. Aurzada and T. Simon, Persistence probabilities and exponents, in Lévy Matters V: Functionals of Lévy Processes(Springer International Publishing, Cham, 2015), pp. 183–224.
[86] B. Derrida, A. J. Bray, and C. Godreche, Non-trivial exponents
in the zero temperature dynamics of the 1d ising and pottsmodels, J. Phys. A: Math. Gen. 27, L357 (1994).
[87] D. Stauffer, Ising spinodal decomposition at T = O in one to
five dimensions, J. Phys. A: Math. Gen. 27, 5029 (1994).
[88] P. L. Krapivsky, E. Ben-Naim, and S. Redner, Kinetics ofheterogeneous single-species annihilation, Phys. Rev. E 50,
2474 (1994).
[89] J. Cardy, Proportion of unaffected sites in a reaction-diffusion
process, J. Phys. A: Math. Gen. 28, L19 (1995).
[90] B. Derrida, V. Hakim, and V. Pasquier, Exact First-Passage
Exponents of 1D Domain Growth: Relation to a ReactionDiffusion Model, Phys. Rev. Lett. 75, 751 (1995).
[91] B. Derrida, V. Hakim, and V. Pasquier, Exact exponent for thenumber of persistent spins in the zero-temperature dynamics
of the one-dimensional potts model, J. Stat. Phys. 85, 763(1996).
[92] S. N. Majumdar, C. Sire, A. J. Bray, and S. J. Cornell, Nontrivial Exponent for Simple Diffusion, Phys. Rev. Lett. 77, 2867
(1996).
[93] B. Derrida, V. Hakim, and R. Zeitak, Persistent Spins in the
Linear Diffusion Approximation of Phase Ordering and Zerosof Stationary Gaussian Processes, Phys. Rev. Lett. 77, 2871
(1996).
[94] M. Marcos-Martin, D. Beysens, J. Bouchaud, C. Godrèche,and I. Yekutieli, Self-diffusion and ‘visited’ surface in thedroplet condensation problem (breath figures), Physica A:
Stat. Mech. App. 214, 396 (1995).

054103-16

PHASE-ORDERING KINETICS IN THE ALLEN-CAHN …

PHYSICAL REVIEW E 104, 054103 (2021)[95] W. Y. Tam, R. Zeitak, K. Y. Szeto, and J. Stavans, First-Passage
Exponent in Two-Dimensional Soap Froth, Phys. Rev. Lett.
78, 1588 (1997).
[96] B. Yurke, A. N. Pargellis, S. N. Majumdar, and C. Sire, Experimental measurement of the persistence exponent of the planarising model, Phys. Rev. E 56, R40 (1997).
[97] G. P. Wong, R. W. Mair, R. L. Walsworth, and D. G. Cory,
Measurement of Persistence in 1D Diffusion, Phys. Rev. Lett.
86, 4156 (2001).
[98] J. Soriano, I. Braslavsky, D. Xu, O. Krichevsky, and J. Stavans,
Universality of Persistence Exponents in Two-Dimensional
Ostwald Ripening, Phys. Rev. Lett. 103, 226101 (2009).
[99] T. Blanchard, L. F. Cugliandolo, and M. Picco, Persistence inthe two dimensional ferromagnetic ising model, J. Stat. Mech.
(2014) P12021.
[100] M. Poplavskyi and G. Schehr, Exact Persistence Exponent
for the 2d-Diffusion Equation and Related Kac Polynomials,
Phys. Rev. Lett. 121, 150601 (2018).
[101] G. Manoj and P. Ray, Persistence in higher dimensions: A
finite size scaling study, Phys. Rev. E 62, 7755 (2000).
[102] S. Jain and H. Flynn, Scaling and persistence in the twodimensional ising model, J. Phys. A: Math. Gen. 33, 8383
(2000).
[103] S. Chakraborty and S. K. Das, Fractality in persistence decay
and domain growth during ferromagnetic ordering: Dependence upon initial correlation, Phys. Rev. E 93, 032139(2016).
[104] R. Williams, Domains in liquid crystals, J. Chem. Phys. 39,384 (1963).
[105] E. F. Carr, Influence of electric fields on the molecular alignment in the liquid crystal p-(anisalamino)-phenyl acetate,
Mol. Cryst. 7, 253 (1969).
[106] W. Helfrich, Conduction-induced alignment of nematic liquid
crystals: Basic model and stability considerations, J. Chem.
Phys. 51, 4092 (1969).
[107] E. Dubois-Violette, P. de Gennes, and O. Parodi, Hydrodynamic instabilities of nematic liquid crystals under a. c. electricfields, J. Phys. Fr. 32, 305 (1971).
[108] R. Ribotta, A. Joets, and L. Lei, Oblique Roll Instability in an
Electroconvective Anisotropic Fluid, Phys. Rev. Lett. 56, 1595
(1986).
[109] By convention, quantities with  and ⊥ subscripts indicate
the component parallel and perpendicular to the director field,
respectively.
[110] P.-G. De Gennes and J. Prost, The Physics of Liquid Crystals(Oxford University Press, Oxford, UK, 1993), Vol. 83.
[111] A. Joets and R. Ribotta, Hydrodynamic transitions to chaosin the convection of an anisotropic fluid, J. Phys. Fr. 47, 595(1986).
[112] A. Joets, X. Yang, and R. Ribotta, Singularities in the transitions to chaos of a convective anisotropic fluid, Physica D 23,
235 (1986).
[113] S. Kai and W. Zimmermann, Pattern dynamics in the electrohydrodynamics of nematic liquid crystals, Prog. Theor. Phys.
Suppl. 99, 458 (1989).
[114] There also exists a regime called dynamic scattering mode 1(DSM1) that is observed at lower applied voltages. DSM1 ischaracterized by the absence of sustained elongated disclinations and by the presence of anisotropy along the direction of
the underlying roll structure [155–157].

[115] G. Heilmeier, L. Zanoni, and L. Barton, Dynamic scattering:
A new electrooptic effect in certain classes of nematic liquid
crystals, Proc. IEEE 56, 1162 (1968).
[116] A. Sussman, Secondary hydrodynamic structure in dynamic
scattering, Appl. Phys. Lett. 21, 269 (1972).
[117] J. Nehring and M. Petty, The formation of threads in the
dynamic scattering mode of nematic liquid crystals, Phys. Lett.
A 40, 307 (1972).
[118] R. Chang, “secondary” dynamic scattering in negative nematic
liquid crystal films, J. Appl. Phys. 44, 1885 (1973).
[119] C. Mauguin, Sur les cristaux liquides de M. Lehmann,
Bull. Soc. Fr. Miner. 34, 71 (1911).
[120] The time taken for this removal of the electric field was much
shorter than 10 ms, which was the period of the alternating
field and also the order of correlation time of the DSM2 state[111].
[121] See
Supplemental
Materialat
http://link.aps.org/
supplemental/10.1103/PhysRevE.104.054103 for videos
1 and 2 that show the twisted domain configurations inordering, for videos 3 and 4 that display the evolution of thelocal persistence index, and for a very detailed text description
about the videos.
[122] M. E. Becker, R. A. Kilian, B. B. Kosmowski, and D. A.
Mlynski, Alignment properties of rubbed polymer surfaces,
Mol. Cryst. Liq. Cryst. 132, 167 (1986).
[123] T. Nagaya, H. Orihara, and Y. Ishibashi, Collective motion ofan assembly of disclinations in the non-orthogonally twisted
nematics quenched below the clearing point, J. Phys. Soc. Jpn.
56, 3086 (1987).
[124] V. S. U. Fazio and L. Komitov, Alignment transition in anematic liquid crystal due to field-induced breaking of anchoring, Europhys. Lett. 46, 38 (1999).
[125] For the two-dimensional square lattice, the minimum distanceδs between two arbitrary points described by Cartesian coordinates (x1 , y1 ) and (x2 , y2 ) is, in the Manhattan metric, givenby δs = δx + δy, with δx = |x2 − x1 | and δy = |y2 − y1 |. Notethe
 difference from the Euclidean metric, for which δs =(δx)2 + (δy)2 .
[126] Note that OJK’s assumption of a Gaussian field was recently
shown to be a reasonable approximation in d = 2 by numericalsimulations of the TDGL equation [153].
[127] It was shown that the nonlinear terms neglected in the OJK theory by the approximation ∇ j u∇ k u/|∇u|2 ≈ δ jk /d contributeto a correction to Eq. (15) in the order of O(1/d 2 ) for large d[158].
[128] More specifically, the relation between the continuous orderparameter field φ and the auxiliary field m is obtained by
2 φ(m)= dVdφ(φ) , where V (φ) is a symmetric doublerequiring d dm
2well potential, with the boundary conditions limm→±∞ φ(m) =±1 and φ(0) = 0. This is the equilibrium condition for the
TDGL equation, with m playing the role of the coordinate
normal to the interface [5].
[129] Equation (16) admits two linearly independent solutions forg  1, one with a Gaussian decay and the other with an algebraic one. The general solution is a linear combination of both:ππg2 ) + F1 g− 2μ , where F0 and
FTUG (g  1) = F0 g(d− 2μ ) exp( −μ2F1 are constants [53]. The power-law solution in TUG shouldbe neglected to guarantee the integrability of the associatedpower-spectrum. The value of μ(d ) in TUG is then given insuch a way that FTUG (g  1) yields a solution with F1 = 0.

054103-17

ALMEIDA AND TAKEUCHIPHYSICAL REVIEW E 104, 054103 (2021)[130] H. Furukawa, Multi-time scaling for phase separation, J. Phys.
Soc. Jpn. 58, 216 (1989).
[131] An exception is the 1dXY model whose Ct (t/t0 ) decay isexponential [50].
[132] A. Picone and M. Henkel, Local scale-invariance and ageing
in noisy systems, Nucl. Phys. B 688, 217 (2004).
[133] M. Henkel, A. Picone, and M. Pleimling, Two-time autocorrelation function in phase-ordering kinetics from local scale
invariance, Europhys. Lett. 68, 191 (2004).
[134] C. Allain and M. Cloitre, Characterizing the lacunarity of
random and deterministic fractal sets, Phys. Rev. A 44, 3552
(1991).
[135] G. Manoj and P. Ray, Scaling and fractal formation in persistence, J. Phys. A: Math. Gen. 33, L109 (2000).
[136] B. B. Mandelbrot and B. B. Mandelbrot, The Fractal Geometryof Nature (W. H. Freeman, New York, 1982), Vol. 1.
[137] D. H. Zanette, Distribution of persistent sites in diffusing systems, Phys. Rev. E 55, 2462 (1997).
[138] It has been shown by Bray and O’Donoghue [159] that, for
1D Potts models, the general law for the characteristic persistence length is ζ ∗ ∼ t max(1/z,θ ) . It turns out that for all integerq  3 one has θ > 1/z [90,91], so that the relation ζ ∗ ∼ t 1/2
breaks down. For the TNLC experiment, however, ζ ∗ ∼ t 1/2 isconfirmed in Fig. 10(d).
[139] A. Sicilia, J. J. Arenzon, I. Dierking, A. J. Bray, L. F.
Cugliandolo, J. Martínez-Perdiguero, I. Alonso, and I. C.
Pintre, Experimental Test of Curvature-Driven Dynamics in
the Phase Ordering of a Two Dimensional Liquid Crystal,Phys. Rev. Lett. 101, 197801 (2008).
[140] Interestingly, OJK-like form was also seen to describe the spatial correlations of polycrystals formed by colloids and whose
overall kinetics do not follow z = 2 [160]. It is also the firsttime that the TUG two-point spatial correlator [see Eq. (16)] is
confronted with an experiment after Professor Mazenko called
for experimental tests in Ref. [54].
[141] D. Stauffer, Scaling theory of percolation clusters, Phys. Rep.
54, 1 (1979).
[142] N. Araújo, P. Grassberger, B. Kahng, K. Schrenk, and R.
Ziff, Recent advances and open challenges in percolation,
Eur. Phys. J. Spec. Top. 223, 2307 (2014).
[143] A. A. Saberi, Recent advances in percolation theory and its
applications, Phys. Rep. 578, 1 (2015).
[144] A. Sicilia, J. J. Arenzon, A. J. Bray, and L. F. Cugliandolo, Domain growth morphology in curvature-driven two-dimensionalcoarsening, Phys. Rev. E 76, 061116 (2007).

[145] J. Olejarz, P. L. Krapivsky, and S. Redner, Fate of 2D Kinetic
Ferromagnets and Critical Percolation Crossing Probabilities,
Phys. Rev. Lett. 109, 195702 (2012).
[146] T. Blanchard, F. Corberi, L. F. Cugliandolo, and M. Picco,
How soon after a zero-temperature quench is the fate of theising model sealed? Europhys. Lett. 106, 66001 (2014).
[147] F. Corberi, L. F. Cugliandolo, F. Insalata, and M. Picco, Coarsening and percolation in a disordered ferromagnet, Phys. Rev.
E 95, 022101 (2017).
[148] J. T. Cox and D. Griffeath, Diffusive clustering in the twodimensional voter model, Ann. Prob. 14, 347 (1986).
[149] M. Scheucher and H. Spohn, A soluble kinetic model forspinodal decomposition, J. Stat. Phys. 53, 279 (1988).
[150] K. Kawasaki, M. C. Yalabik, and J. D. Gunton, Growth of fluctuations in quenched time-dependent ginzburg-landau model
systems, Phys. Rev. A 17, 455 (1978).
[151] R. A. Wickham and G. F. Mazenko, Phase ordering of the O(2)model in the post-Gaussian approximation, Phys. Rev. E 55,
2300 (1997).
[152] C. Yeung, Y. Oono, and A. Shinozaki, Possibilities and limitations of Gaussian-closure approximations for phase-ordering
dynamics, Phys. Rev. E 49, 2693 (1994).
[153] C. Yeung, Direct test of the Gaussian auxiliary field ansatz
in nonconserved order parameter phase ordering dynamics,
Phys. Rev. E 97, 062107 (2018).
[154] F. Y. Wu, The potts model, Rev. Mod. Phys. 54, 235 (1982).
[155] T. Krupkowski and W. Ruszkiewicz, Evidence for anisotropic
expansion of secondary dynamic scattering, Mol. Cryst. Liq.
Cryst. 49, 47 (1978).
[156] T. Nagaya, T. Takeda, and H. Orihara, Dynamic image analysis for dynamic scattering modes in a nematic liquid crystal,J. Phys. Soc. Jpn. 68, 3848 (1999).
[157] G. Strangi, C. Versace, N. Scaramuzza, D. E. Lucchetta, V.
Carbone, and R. Bartolino, Photopolarimetric characterizationof the transition between two turbulent states in a nematic
liquid crystal film, Phys. Rev. E 59, 5523 (1999).
[158] C. L. Emmott, Perturbative corrections to the Ohta-JasnowKawasaki theory of phase-ordering dynamics, Phys. Rev. E 58,
5508 (1998).
[159] A. J. Bray and S. J. O’Donoghue, Unusual dynamical scaling
in the spatial distribution of persistent sites in one-dimensional
potts models, Phys. Rev. E 62, 3366 (2000).
[160] F. A. Lavergne, D. G. A. L. Aarts, and R. P. A. Dullens,Anomalous Grain Growth in a Polycrystalline Monolayer ofColloidal Hard Spheres, Phys. Rev. X 7, 041064 (2017).

054103-18