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Coarsening Kinetics in Active Model B+: Macroscale and Microscale Phase Separation

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Pith's one-line read Adding a rotational current to the minimal active Model B slows droplet growth from t^(1/3) to t^(1/4) in the macroscale regime, and in the reverse-Ostwald regime arrests coarsening in a hexagonal crystal of monodisperse droplets.

desk verdict Solid numerical study of AMB+ coarsening, but the asymptotic t^{1/4} claim is supported by effective exponents that haven't plateaued. read the letter →

arxiv 2506.14548 v2 pith:2GJZ4Z3X submitted 2025-06-17 cond-mat.soft

Pradeep Kumar Yadav , Shradha Mishra , Sanjay Puri This is my paper · ORCID
classification cond-mat.soft
keywords activematterModelB+coarseningkineticsphaseseparationOstwaldripeningmicrophasesurfacediffusionLifshitz-Slyozovlaw
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper investigates how the two additional activity terms in Active Model B+ (AMB+) alter the kinetics of phase separation in a conserved order parameter. The central result is that the rotational current term of strength $\xi$ reshapes the mass-transport current into closed loops along domain walls, so that material travels around droplet surfaces rather than along straight bulk paths. In the forward-Ostwald region (e.g., $\lambda=0$, $\xi<0$), this changes the asymptotic growth law from the Lifshitz–Slyozov $L(t) \sim t^{1/3}$ to $L(t) \sim t^{1/4}$, with a crossover time scaling as $\xi^{-2}$. In the reverse-Ostwald region ($\lambda<0$, $\xi<0$), coarsening stops: the system reaches a steady state of monodisperse droplets arranged on a hexagonal lattice, with saturation size $L_s(\lambda,\xi)$ that appears to scale as $(-\xi)^{-2/3}$. These results matter because they show that a minimal nonequilibrium field theory can produce both slower coarsening and stable microphase-separated patterns without long-range interactions.

What carries the argument

The central object is the dimensionless AMB+ equation $\partial\psi/\partial t = -\nabla\cdot\mathbf{J}$ with $\mathbf{J} = -\nabla(-\psi + \psi^3 - \nabla^2\psi + \lambda|\nabla\psi|^2) + \xi\nabla^2\psi\nabla\psi$, where $\psi$ is the conserved order parameter. The $\lambda$-term is a rotation-free nonequilibrium current, the $\xi$-term a rotational current; neither can be written as the gradient of a free energy. The paper shows that this current forms closed loops on domain walls, and that the loop circumference scales with the droplet size $L$; mass is transported along these loops, effectively renormalizing time from $t$ to $\tau \sim t/L$ and converting a diffusive $L \sim \tau^{1/3}$ into $L \sim t^{1/4}$. The second load-bearing ingredient is the static one-dimensional kink solution of the model, which gives the nonequilibrium chemical potential $\mu_s = 4\alpha/15$ with $\alpha = \lambda - \xi/2$, and the coexisting phase values $\psi_1 = 1 + \mu_s/2$, $\psi_2 = -1 + \mu_s/2$; this asymmetry fixes which phase forms droplets in MPS and enters the conservation-law relation $a = \sqrt{2\pi(\psi_1-\psi_2)/(\sqrt{3}|\psi_2|)} L_s$ used to connect the hexagonal lattice spacing to the droplet size in $\mu$PS.

What would settle it

Extend the $\lambda=0$, $\xi=-2$ MPS simulation to a $2048^2$ lattice and $t=10^7$ and measure the running local exponent $1/z_{\rm eff}$; if it rises back toward $1/3$, the apparent $t^{1/4}$ asymptotic regime is a transient. Similarly, run the $\mu$PS case $\lambda=-4$, $\xi=-1$ on a $1024^2$ box and track $L(t)$ past $t=10^4$: if $L(t)$ keeps growing or the hexagonal ordering anneals away, the claimed steady-state droplet crystal is a finite-size artifact.

Watch

Extended reading notes

Core claim

The paper establishes that AMB+—the scalar conserved-order-parameter theory with a zero-curl activity term of strength $\lambda$ and a rotational activity term of strength $\xi$—has two distinct kinetic regimes after a quench at critical composition. In the forward-Ostwald regime (for instance $\lambda=0$, $\xi<0$), the current becomes sharply peaked at the interfaces and organizes into alternating clockwise and anticlockwise loops separated by nodal points. Mass transfer from small to large droplets occurs only by moving along these surface loops, whose length grows with droplet size, so the effective transport is slowed; the domain size crosses over from the Model B Lifshitz–Slyozov law $L(t) \sim t^{1/3}$ at early times to an asymptotic $L(t) \sim t^{1/4}$, with crossover time $t_c \sim \xi^{-2}$. In the reverse-Ostwald regime ($\lambda<0$, $\xi<0$), the current loops between droplets cancel, so ripening is reversed and the system reaches a steady state: monodisperse droplets on a hexagonal lattice, with saturation size $L_s(\lambda,\xi)$ consistent with $L_s \sim (-\xi)^{-2/3}$ over the studied parameter range, and a lattice spacing set by the conservation law. The paper also shows that the correlation function obeys dynamical scaling in both regimes, that the early-time scaling function coincides with the Model B (superuniversal) function, and that at late times in MPS the scaling function becomes parameter-dependent because the two coexisting phases are asymmetric.

Load-bearing premise

The analysis assumes that the two coexisting phase compositions taken from a one-dimensional static interface also describe the saturated droplet and background phases in the two-dimensional patterns, although in the microphase-separated state the paper notes the small droplets do not reach those compositions.

Editorial extensions

If this is right

  • In the forward-Ostwald region, the asymptotic growth law of AMB+ is $L(t) \sim t^{1/4}$, not the $t^{1/3}$ of Model B, even at critical composition, because the morphology becomes droplet-like and transport is surface-diffusion mediated.
  • The crossover from $t^{1/3}$ to $t^{1/4}$ occurs at a time $t_c$ that decreases with increasing $|\xi|$, consistent with $t_c \sim \xi^{-2}$, so stronger rotational activity reaches the slow-growth regime sooner.
  • In the reverse-Ostwald region, coarsening terminates in a steady-state hexagonal crystal of monodisperse droplets; the saturation size $L_s$ decreases with $|\xi|$ ($L_s \sim (-\xi)^{-2/3}$ in the studied range) and the lattice spacing is tied to $L_s$ by the conservation law.
  • Both MPS and $\mu$PS exhibit dynamical scaling of the correlation function: the early-time MPS scaling function is the superuniversal Model B form, while the late-time MPS function depends on $\lambda$ and $\xi$; the $\mu$PS scaling function is universal and shows oscillations from the crystalline order.
  • Porod's law holds in both regimes, confirming that the interfaces are sharp even though the transport is nonequilibrium.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension not pursued in the paper would be to measure how the current-loop length grows with $L$ in MPS and to check whether $L(t)$ collapses when time is rescaled by $t_c(\xi)\sim \xi^{-2}$, which would sharpen the surface-diffusion mechanism into a quantitative scaling prediction.
  • Because Eq. (20) is used in a regime where the bulk phases do not saturate to the kink values, measuring the actual droplet and background compositions in $\mu$PS would quantify the error in the predicted lattice spacing and may explain the scatter in $L_s(-\xi)$.
  • Restricting to critical composition means off-critical quenches are unexplored; by the model's $\psi\to-\psi$ symmetry, off-critical initial conditions could select different droplet-size scaling or different steady-state patterns in $\mu$PS.
  • If the $t^{1/4}$ law is exact rather than an effective exponent, it implies that mass transport is controlled by the interfaces, not the bulk; introducing a small but nonzero bulk mobility should eventually restore $t^{1/3}$ at extremely long times, a prediction that could be tested numerically.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript reports Euler-discretized numerical solutions of Active Model B+ (AMB+) for critical quenches in two dimensions. For parameters assigned to forward Ostwald ripening (macroscale phase separation, MPS), it claims a crossover from Lifshitz-Slyozov growth L(t) ~ t^(1/3) to an asymptotic slower law L(t) ~ t^(1/4), with crossover time tc ~ |xi|^(-2), and it attributes the slowdown to circulating interfacial current loops. For parameters assigned to reverse Ostwald ripening (microscale phase separation, muPS), it claims that the system saturates to a hexagonal crystal of monodisperse droplets with saturation length Ls satisfying Ls ~ (-xi)^(-2/3), and it derives a conservation-law relation between the hexagonal lattice spacing and Ls. The paper also reports dynamical scaling of the correlation function in both regimes, with a superuniversal early-time form and a parameter-dependent late-time form in MPS, and a crystalline oscillatory scaling function in muPS.

Significance. If the MPS exponent and the muPS saturation scaling are correct, the paper is a valuable extension of phase-separation kinetics to a minimal active field theory, showing that the rotational active current can change the growth law and produce a steady-state microphase-separated crystal. The numerical protocol is standard and clearly specified (N=512, dx=dy=0.5, dt=0.01, averages over 50 runs for the main data sets), and the use of effective exponents, correlation functions, and current-field visualization is transparent. The authors are also honest about limitations, explicitly noting the restricted parameter range for the Ls data and the lack of a predictive theory for Ls. However, the central quantitative claims are empirical and the asymptotic regime is not fully demonstrated: the effective exponent for the MPS case has not plateaued by the largest simulation time, and the Ls scaling is fitted over a narrow range. The significance is therefore conditional on the additional numerical evidence discussed below.

major comments (4)
  1. [Section III.A, Figs. 2(b)-2(e)] The claimed asymptotic growth law L(t) ~ t^(1/4) is not established by the presented data. For xi < 0, the inverse effective exponent 1/z_eff initially follows 1/3, then dips below 1/4, and only approaches 1/4 from below at the largest times; there is no plateau by t = 10^6. Since the asymptotic value is inferred from a non-monotonic transient, the paper should either extend the simulations to longer times (with a check for finite-size effects on the N = 512 lattice) or use an alternative analysis, for example windowed log-log slopes with an extrapolation procedure, that demonstrably converges to a plateau. Without this, the central MPS claim remains an interpretation of the final decade of data rather than a measured asymptotic exponent.
  2. [Section III.A, paragraph after Fig. 5] The loop-transport mechanism is a post hoc heuristic: the assumption of ballistic transport along current loops, encoded in tau ~ t/L, is introduced after observing the 1/4 fit, and the step 'for diffusive transport L ~ tau^(1/3), yielding L ~ t^(1/4)' is not derived from the AMB+ equation. The manuscript should provide a direct numerical test of this mechanism, for instance by measuring the current-loop length and traversal time, or by checking whether the data collapse onto L ~ tau^(1/3) when t is renormalized as tau ~ t/L. As written, the mechanism does not independently support the fitted exponent.
  3. [Section III.B, Eqs. (19)-(20) and Fig. 9] Equation (20) uses the one-dimensional static-kink saturation values psi1 and psi2 from Eq. (12), but the manuscript itself notes in Section III.B and Fig. 9 that the droplet and background phases in the two-dimensional muPS state do not saturate to psi1 and psi2 because the droplets are small. This is exactly the regime in which Eq. (20) is used to connect the hexagonal lattice spacing to Ls. Please quantify the error by measuring the actual plateau values inside the droplets and in the background, and state whether Eq. (20) remains quantitatively accurate, or restrict the claim to cases where the phases are closer to saturation. Relatedly, the steady-state nature of the muPS morphology is only shown up to t = 10^4; longer runs are needed to rule out a slow coarsening process at later times.
  4. [Section III.B, Fig. 10(b)] The claimed scaling Ls ~ (-xi)^(-2/3) is fitted over a very narrow range of xi (roughly from -1 to -2 for each lambda), and the paper acknowledges that the accessible range is limited by finite-size effects and numerical stability. With only about a factor of 2 in xi, a power-law fit is not strongly discriminative; alternative functional forms, such as exponential or crossover forms, would also fit the data. To make this a load-bearing quantitative claim, the authors should broaden the parameter range (for example with adaptive mesh or larger systems) or provide a theoretical derivation of the exponent. The non-monotonic Ls vs. -lambda data in Fig. 10(c) are presented without a quantitative model, which further limits the predictive content of the muPS results.
minor comments (5)
  1. [Fig. 3] The x-axis label '124 xi' appears to be a typographical artifact; it should be '-xi' or '|xi|' to match the text.
  2. [Section II, Eq. (7)] The rescaling that removes the coefficients of the Ginzburg-Landau free energy and defines the dimensionless lambda and xi is not shown explicitly; stating the rescaling factors would make the parameter values fully reproducible.
  3. [Section III.A, after Fig. 5(c)] The sentence 'The total number of vortices is even due to the periodic domain, which enforces zero net vorticity' is imprecise: periodic boundary conditions enforce zero total vorticity, not an even number of vortices. Please rephrase to avoid a topological statement that is not justified.
  4. [Section III.A, paragraph after Fig. 5] When citing Refs. [37,38] for the surface-diffusion growth law, the text should explicitly state that those works derive L ~ t^(1/4) in d = 2 for conserved order-parameter dynamics with surface diffusion, so that the analogy is concrete.
  5. [General] A data/code availability statement would strengthen the paper's reproducibility; the numerical scheme is standard, but the exact central-difference discretization of the xi-term (a mixed second-order derivative) is not fully specified, and providing the code or a detailed discretization formula would remove ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are numerical measurements and fits with acknowledged limitations; self-citations are not load-bearing.

full rationale

I walked the claimed derivation chain. The paper does not claim a first-principles derivation of the t^{1/4} law; it reports numerical measurements of L(t), defines z_eff, and states that the late-time data are 'consistent with L(t) ∼ t^{1/4}' (Sec. III.A). The loop-transport argument and citations [37,38] supply an interpretive mechanism, not a fit relabeled as a prediction. Likewise, t_c is defined operationally as the first crossing of 1/z_eff = 1/4 and then plotted against ξ; this is a data-analysis statement, with the admitted limitation that 'the range of ξ is limited'. The μPS L_s scaling is explicitly a fit ('The data appear to be consistent with the power law L_s ∼ (−ξ)^{−2/3}'), not a derived prediction, and Eqs. (19)-(20) form an area-balance identity given the assumed saturated ψ_1, ψ_2; the paper itself cautions that in μPS 'the bulk domains do not saturate to the values ψ_1 and ψ_2 due to the small droplet sizes.' The perturbative kink results (Eqs. 10-12) are cited from prior work, but they are parameter-free expansions with stated small-α assumptions that do not contain the paper's coarsening or scaling results; citing them is therefore independent support, not a circular reduction. There are self-citations to the authors' earlier AMB study [28] and to surface-diffusion work [37,38], but the present numerical data for AMB+ stand independently, and no result in the paper is forced by definition or by a self-citation chain. I find no significant circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central quantitative results (theta=1/4, tc~xi^(-2), Ls~(-xi)^(-2/3)) are fitted from simulation data, and the explanatory mechanism rests on a heuristic ballistic-loop assumption. No new physical entities are postulated; the model and phase diagram are imported from prior work.

free parameters (3)
  • Asymptotic growth exponent theta (MPS) = ~ 1/4
    Extracted from the late-time slope of L(t) vs. t (Fig. 2a); the central claim of t^(1/4) growth rests on this fit over times up to 10^6, where 1/zeff shows a transient dip below 1/4 before recovering.
  • Crossover time exponent = tc ~ xi^(-2)
    Power-law fit to tc vs. xi in Fig. 3 over xi = 1 to 4; the paper calls it approximate power law behavior.
  • Saturation length exponent (microPS) = Ls ~ (-xi)^(-2/3)
    Best-fit slope in Fig. 10(b) for lambda = -2.5, -3, -4 and xi = 1 to 2, i.e., a narrow parameter window; the lambda-dependence in Fig. 10(c) is non-monotonic and not fitted.
assumptions (5)
  • domain assumption The AMB+ equation (Eq. 7) with the two activity terms lambda|grad psi|^2 and xi grad^2 psi grad psi is the minimal continuum model for active phase separation kinetics.
    Taken from Tjhung et al. [26]; the paper adopts this as the model under study without re-deriving the truncation.
  • domain assumption Zero noise is adequate: thermal noise is asymptotically irrelevant for domain growth in AMB+.
    Stated in Sec. 2, citing [29,30] for Model B; assumed to carry over to AMB+ without new argument.
  • standard math Perturbative static kink result mu_s(alpha) = 4 alpha / 15 + O(alpha^2) (Eq. 10), from the authors' prior work [28], is valid for the alpha values used.
    Standard asymptotic expansion of the kink equation in Sec. 2; accepted from the cited reference.
  • ad hoc to paper Mass transport along interfacial current loops is ballistic, so traversing a loop of length ~ L renormalizes the effective time as tau ~ t/L.
    Introduced in Sec. III.A to rationalize the observed 1/4 exponent; no microscopic justification is given for ballistic rather than diffusive loop transport.
  • domain assumption The equilibrium-like volume-fraction balance using a hexagonal unit cell (Eq. 19) correctly counts droplet and background areas.
    Used to derive Eq. (20) for the lattice spacing a; the prefactor 3 pi Ls^2 per cell is inconsistent with the honeycomb lattice having two droplets per cell, and the bulk fields do not reach the saturated values psi1, psi2.

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Pith. "Pith review of Coarsening Kinetics in Active Model B+: Macroscale and Microscale Phase Separation." pith.science (2026). https://pith.science/paper/2GJZ4Z3X

@misc{pith2026250614548,
  author       = {Pith},
  title        = {Pith review of: Coarsening Kinetics in Active Model B+: Macroscale and Microscale Phase Separation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2GJZ4Z3X}},
  note         = {Machine review of arXiv:2506.14548}
}
abstract

We perform a comprehensive numerical investigation of the coarsening kinetics of active Brownian particles modeled by the {\it Active Model B+} (AMB+). This model was introduced by Tjhung et al. [Phys. Rev. X {\bf 8}, 031080 (2018)] and is a generalization of Model B for a conserved order parameter, with two additional activity terms. These terms correspond to rotation-free current (of strength $\lambda$) and rotational current (of strength $\xi$). We find that the presence of rotational current $(\xi \neq 0)$ significantly affects growth kinetics. Depending on the parameter values, AMB+ exhibits either {\it macroscale phase separation} (MPS) or {\it microscale phase separation} ($\mu$PS). We present detailed results for the kinetics of MPS and $\mu$PS in AMB+ with critical composition.

Figures

Figures reproduced from arXiv: 2506.14548 by the authors.

Figure 1
Figure 1. FIG. 1. Evolution snapshots of AMB+ at [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Log-log plot of domain size [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Log-log plot of the crossover time [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (First row): Snapshots of the current magnitude [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Orientation angle [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Correlation function [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) The top row shows snapshots of [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Plot of [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (a) The top frame shows a snapshot of [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. (a) Log-log plot of [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Plot of [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.