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arxiv: 2605.01820 · v1 · submitted 2026-05-03 · ❄️ cond-mat.soft · physics.flu-dyn

Recognition: unknown

Hindered transport of spherical particles in cylindrical pores: The role of structural heterogeneity in rejection-permeability trade-offs

Debanik Bhattacharjee , Yaniv Edery , Guy Z. Ramon
Authors on Pith no claims yet

Pith reviewed 2026-05-09 16:46 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn
keywords hindered transportmembrane separationstructural heterogeneityrejection-permeability trade-offPeclet numberspherical particlescylindrical poressteric exclusion
0
0 comments X

The pith

Dual heterogeneity in particle and pore sizes shifts the rejection-permeability trade-off toward higher permeability at fixed rejection.

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

The paper introduces a steric hindered-transport framework that resolves variability in both particle sizes and pore sizes for spherical particles moving through cylindrical pores. It finds that the average rejection rises with the particle-to-pore size ratio and with the Peclet number, and that flow advection strengthens exclusion by up to 20 percent at intermediate ratios. When the two heterogeneities are coupled, the spread of local flow conditions widens, producing more variable and sometimes anomalous rejection patterns while moving the overall trade-off curve to allow higher fluid throughput for any given rejection level. A reader would care because real membranes always contain size variations, so the results indicate how that variability can be turned into a controllable design feature rather than treated only as a drawback.

Core claim

In a steric hindered-transport framework for spherical particles in cylindrical pores that explicitly resolves both single and coupled dual heterogeneity in size distributions, the ensemble-averaged rejection increases with the particle-pore aspect ratio λ and with the Péclet number Pe, while advection enhances steric exclusion by up to ∼20% at intermediate λ. Dual heterogeneity broadens the distribution of effective Pe, increases the variability and incidence of anomalous rejection trends, and systematically shifts the rejection-permeability trade-off toward higher permeability at fixed rejection.

What carries the argument

The steric hindered-transport framework that computes local advection-diffusion balances for each pore-particle pair and then performs ensemble averaging over coupled distributions of particle and pore sizes.

Load-bearing premise

The steric hindered-transport framework and ensemble averaging over size distributions fully capture the local advection-diffusion balance in real heterogeneous media without additional particle-pore interactions or non-spherical effects.

What would settle it

Direct measurement of rejection and permeability in membranes fabricated with independently controlled distributions of pore diameters and particle diameters, then comparison against the model's predictions for the same dual-heterogeneity statistics.

Figures

Figures reproduced from arXiv: 2605.01820 by Debanik Bhattacharjee, Guy Z. Ramon, Yaniv Edery.

Figure 1
Figure 1. Figure 1: (a) Schematic of single-pass transport of spherical particles through a cylindrical view at source ↗
Figure 2
Figure 2. Figure 2: Rejection (χ) as a function of the aspect ratio (λ = r/R) for single-pass transport of spherical particles through homogeneous cylindrical pores at different Péclet numbers (P e), spanning diffusion- to convection-dominated regimes. Here, λ is prescribed directly as the ratio of a single representative particle radius r to pore radius R (i.e., a monodisperse suspension and a given membrane pore size), and … view at source ↗
Figure 3
Figure 3. Figure 3: Single heterogeneity ensemble construction. Top: two representative particle-pore size distributions drawn from the 104 independently sampled configurations. In each configuration, particle radii are heterogeneous, r ∼ N (¯r, σr) with σr = 10 nm and r¯ scanned from 10 to 1000 nm (navy histogram), while pore radii are effectively monodisperse, R ∼ N (500 nm, 0 nm) (magenta spike). Bottom: histogram of λ¯ ov… view at source ↗
Figure 4
Figure 4. Figure 4: Single-heterogeneity transport response. (a) Configuration-averaged re￾jection χ¯ versus configuration-averaged aspect ratio λ¯ across 104 independently sampled particle-pore configurations, (b) Ensemble histograms of P e (top) and χ¯ (bottom); bars are colored by the same P e intervals to connect transport regime to rejection outcome. Particles are polydisperse with r ∼ N (¯r, σr), where r¯ is scanned fro… view at source ↗
Figure 5
Figure 5. Figure 5: Dual heterogeneity ensemble construction: Top: two representative particle-pore size distributions drawn from the 104 independently sampled configurations (labels “1” and “10000” denote two illustrative indices from the full ensemble). In each configuration, both particle radii and pore radii are heterogeneous: r ∼ N (¯r, σr) (navy) and R ∼ N (R, σ ¯ R) (magenta), with σr = σR = 10 nm; the means r¯ and R¯ … view at source ↗
Figure 6
Figure 6. Figure 6: Dual-heterogeneity transport response. (a) Configuration-averaged rejec￾tion χ¯ as a function of the configuration-averaged aspect ratio λ¯ for 104 independently sam￾pled particle-pore configurations. Each point is colored by the corresponding configuration￾averaged Péclet number P e, with 0 ≤ P e ≤ 5.35. (b) Ensemble distributions of P e (top) and χ¯ (bottom). Histogram bins are colored using the same P e… view at source ↗
Figure 7
Figure 7. Figure 7: Effect of dual heterogeneity on permeability at matched rejection levels. (a) Mean permeability ratio,  P i P dual P i i P single i  , plotted versus applied pressure drop ∆p = 10−3 , 10−2 , 10−1 Pa for five narrow rejection windows defined by the mean rejection χ: W1 [0.49, 0.51], W2 [0.59, 0.61], W3 [0.69, 0.71], W4 [0.79, 0.81], and W5 [0.89, 0.91] (colors). “Single” heterogeneity denotes heterogeneou… view at source ↗
read the original abstract

Membrane separations rely on balancing rejection and permeability. Extensive work has clarified how pore structure and operating conditions control each quantity in idealized or weakly heterogeneous systems. However, it remains unclear how this trade-off emerges in strongly heterogeneous media, where coupled distributions of pore and particle sizes shape the local balance between advection and diffusion and generate substantial variability in performance among distribution realizations. Here we present a steric hindered-transport framework for spherical particles in cylindrical pores that explicitly resolves both single and coupled dual heterogeneity in size distributions. We show that the ensemble-averaged rejection increases with the particle-pore aspect ratio $\lambda$ and with the P\'eclet number $Pe$, while advection enhances steric exclusion by up to $\sim$20\% at intermediate $\lambda$. Dual heterogeneity broadens the distribution of effective $Pe$, increases the variability and incidence of anomalous rejection trends, while systematically shifting the rejection-permeability trade-off toward higher permeability at fixed rejection. These results suggest that controlled heterogeneity can serve as a design lever to expand the attainable operating space for simultaneous high selectivity and high throughput.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript develops a steric hindered-transport framework for spherical particles in cylindrical pores that incorporates single and dual heterogeneity via distributions of particle and pore sizes. It computes ensemble-averaged rejection and permeability by resolving local advection-diffusion balance through size-based partitioning and Péclet number, reporting that rejection rises with aspect ratio λ and Pe, that advection enhances steric exclusion by up to ~20% at intermediate λ, and that dual heterogeneity broadens the effective-Pe distribution, increases anomalous-rejection variability, and shifts the rejection-permeability trade-off toward higher permeability at fixed rejection.

Significance. If the central results hold, the work demonstrates that controlled heterogeneity can expand the attainable membrane operating space beyond uniform-pore limits, providing a quantitative design lever for simultaneous selectivity and throughput. The explicit ensemble treatment of coupled size distributions and the resulting variability among realizations constitute a clear advance over idealized single-pore analyses.

major comments (3)
  1. [§2] §2 (Model formulation): The steric-only framework excludes hydrodynamic hindrance factors (e.g., Bungay-Brenner corrections) whose λ-dependence can couple to the same pore- and particle-size distributions used for the ensemble averages. Because the reported broadening of effective Pe and the systematic trade-off shift are generated entirely from these averages, the omission constitutes a load-bearing assumption whose quantitative impact on the claimed 20% enhancement and permeability shift must be bounded or shown to be negligible.
  2. [§4] §4 (Results, ensemble averages): The abstract states quantitative outcomes (20% advection enhancement, systematic shift) without accompanying error bars, sensitivity to distribution moments, or comparison against limiting cases (monodisperse, zero-Pe). These omissions make it impossible to assess whether the reported trends are robust or artifacts of particular distribution choices.
  3. [§3] §3 (Numerical implementation): No validation against known analytic limits (e.g., uniform pore, pure diffusion, or single-particle trajectory) or against existing hindered-transport literature is presented. Without such benchmarks, the reliability of the ensemble statistics that underpin the dual-heterogeneity claims cannot be verified.
minor comments (2)
  1. Notation for the local and effective Péclet numbers is introduced without a clear distinction in the text; a single equation or table defining both would improve readability.
  2. Figure captions should explicitly state the number of distribution realizations used for each ensemble average and the sampling method.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify areas where additional validation, sensitivity analysis, and discussion of assumptions will strengthen the manuscript. We will revise accordingly, as outlined in the point-by-point responses below.

read point-by-point responses

  1. Referee: §2 (Model formulation): The steric-only framework excludes hydrodynamic hindrance factors (e.g., Bungay-Brenner corrections) whose λ-dependence can couple to the same pore- and particle-size distributions used for the ensemble averages. Because the reported broadening of effective Pe and the systematic trade-off shift are generated entirely from these averages, the omission constitutes a load-bearing assumption whose quantitative impact on the claimed 20% enhancement and permeability shift must be bounded or shown to be negligible.

    Authors: We acknowledge that the steric-only framework is a deliberate modeling choice to isolate heterogeneity effects on partitioning and local Pe. Full hydrodynamic corrections are position-dependent within each pore and would require additional radial integration, substantially complicating the ensemble treatment. In the revision we will add a limitations subsection that (i) states the assumption explicitly, (ii) cites literature values for uniform-pore hydrodynamic factors, and (iii) provides order-of-magnitude bounds showing that the heterogeneity-induced broadening of effective Pe and the direction of the trade-off shift remain qualitatively robust even when hydrodynamic hindrance is approximated. Full quantitative incorporation lies beyond the present scope. revision: partial

  2. Referee: §4 (Results, ensemble averages): The abstract states quantitative outcomes (20% advection enhancement, systematic shift) without accompanying error bars, sensitivity to distribution moments, or comparison against limiting cases (monodisperse, zero-Pe). These omissions make it impossible to assess whether the reported trends are robust or artifacts of particular distribution choices.

    Authors: We agree that quantitative claims require supporting robustness checks. The revised manuscript will include: (i) error bars or shaded regions showing standard deviation across 200–500 independent realizations of each distribution, (ii) additional figures or panels varying the variance and shape parameters of the log-normal pore- and particle-size distributions, and (iii) explicit comparisons to the monodisperse limit and the pure-diffusion (Pe = 0) case. These additions will confirm that the reported ~20 % enhancement and permeability shift are consistent features rather than distribution-specific artifacts. revision: yes

  3. Referee: §3 (Numerical implementation): No validation against known analytic limits (e.g., uniform pore, pure diffusion, or single-particle trajectory) or against existing hindered-transport literature is presented. Without such benchmarks, the reliability of the ensemble statistics that underpin the dual-heterogeneity claims cannot be verified.

    Authors: We accept that explicit benchmarks were omitted from the initial submission. The numerical procedure samples size distributions and evaluates the exact analytic transmission probability for each realization under combined steric and advection-diffusion transport. In the revision we will add a dedicated validation subsection that recovers: (i) the classic steric rejection formula R = 1 − (1 − λ)^2 for uniform pores at Pe = 0, (ii) the single-pore hindered-transport curves from the literature (e.g., Deen 1987), and (iii) convergence of ensemble averages with increasing sample size. These checks will precede the heterogeneity results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies stated transport equations to input distributions

full rationale

The paper introduces a steric hindered-transport model based on advection-diffusion balance with size-based partitioning and local Peclet number, then computes ensemble averages over single and dual heterogeneity distributions. The reported trends (broadening of effective Pe, anomalous rejection variability, and shift in rejection-permeability trade-off) are generated outputs of these calculations rather than inputs or fitted parameters. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain; the framework remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract supplies insufficient detail to enumerate all free parameters or axioms; the framework rests on standard advection-diffusion transport with steric exclusion and ensemble averaging over size distributions.

axioms (2)
  • domain assumption Particle transport is governed by steric hindrance combined with advection and diffusion
    Core premise of the hindered-transport framework stated in the abstract
  • domain assumption Ensemble averages over independent or coupled size distributions represent effective media behavior
    Required to obtain the reported rejection and permeability trends

pith-pipeline@v0.9.0 · 5499 in / 1289 out tokens · 47742 ms · 2026-05-09T16:46:42.732520+00:00 · methodology

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