arxiv: 2604.13929 · v1 · submitted 2026-04-15 · ⚛️ physics.optics · cond-mat.soft

Recognition: unknown

Three-dimensional photon transport in spinodal photocatalytic aerogels: how bicontinuous morphology controls kinetic rate constants

Renaud A.L. Vall\'ee

Authors on Pith no claims yet

Pith reviewed 2026-05-10 12:30 UTC · model grok-4.3

classification ⚛️ physics.optics cond-mat.soft

keywords photocatalysisaerogelsphoton transportMonte Carlo simulationspinodal morphologykinetic rateslight scatteringbicontinuous structure

0 comments

The pith

Bicontinuous aerogel pores channel photons preferentially to the solid catalyst, increasing its illumination by 50 to 70 percent and shifting extracted kinetic rates by 34 percent from diffusion predictions.

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

The paper examines how the three-dimensional bicontinuous structure of spinodal photocatalytic aerogels influences photon distribution and the accuracy of kinetic rate constant extraction. Using simulated 3D pore networks, it demonstrates that the solid phase experiences substantially higher photon fluence than volume-average assumptions because of quasi-ballistic paths through the pore channels. This morphology-driven effect produces a 34 percent difference in the key kinetic descriptor between full 3D Monte Carlo transport calculations and simpler diffusion models. Roughly half the total discrepancy arises directly from the bicontinuous geometry and cannot be resolved by effective-medium approximations. The work supplies the first quantitative correction factor for interpreting experimental data in these high-surface-area materials intended for air purification.

Core claim

Three-dimensional spinodal masks generated by Cahn-Hilliard simulations are combined with GPU Monte Carlo photon transport to compute the fluence specifically within the solid catalytic phase. At 70 percent porosity the solid receives 50 percent more photons than a volume average, rising to 70 percent more at 90 percent porosity; the excess is traced to photon channelling along continuous pore pathways. The kinetic descriptor obtained from the 3D model differs by 34 percent from the value given by a diffusion approximation, and homogeneous control calculations attribute approximately half of the 73 percent total discrepancy to the intrinsic bicontinuous structure rather than to shortcomings,

What carries the argument

The solid-phase fluence estimator applied to Cahn-Hilliard-generated spinodal masks inside a Monte Carlo photon-transport simulation, which tracks individual photon trajectories to reveal preferential illumination of the catalyst phase.

Load-bearing premise

The Cahn-Hilliard spinodal structures and the chosen optical parameters in the Monte Carlo model accurately represent the morphology and light-scattering properties of actual experimental TiO2-silica aerogels.

What would settle it

Fabricate TiO2-silica aerogels with controlled porosities matching the simulated masks, measure local reaction rates or absorbed-photon distributions under controlled illumination, and test whether the observed rates align with the 3D Monte Carlo predictions rather than with diffusion-model outputs.

Figures

Figures reproduced from arXiv: 2604.13929 by Renaud A.L. Vall\'ee.

**Figure 2.** Figure 2: shows the 384 × 1024 × 1024 spinodal geometry. Panel (a) displays the midplane cross-section at porosity ϕ = 0.70, with the characteristic bicontinuous solid/pore architecture of spinodal decomposition. Panel (b) shows the radial power spectral density (PSD) with a dominant peak at dpeak = 19.3 µm, close to the target 17.4 µm. A secondary shoulder at ≈ 37 µm—twice the dominant scale—reflects partial Lifs… view at source ↗

**Figure 3.** Figure 3: Photon fluence in the 384 × 1024 × 1024 spinodal domain (λ = 365 nm, κL = 3.30, front/back = 374, ϕ = 0.70). (a) Φ(z)/S0 vs. depth: MC lateral average (solid blue) and DA collimated EMA solution (red dashed, Eqs. 22–25). Horizontal lines mark the three mean-fluence estimators: ⟨Φ⟩MC,3D/S0 (solid-phase weighted, dark blue dotted), ⟨Φ⟩MC,1D/S0 (volume average, blue dash-dot), and ⟨Φ⟩DA/S0 (red dotted). (b) R… view at source ↗

**Figure 4.** Figure 4: Two-dimensional fluence map Φ(x, z) at the mid-plane cross-section (y = Ny/2) of the 384 × 1024 × 1024 domain (ϕ = 0.70, κL = 3.30, front/back = 374). Colormap: inferno (×1018 photon m−2 s −1 ). White contours: solid/pore interface from the binary mask. The beam propagates in the +z direction (arrow, lower left). Fluence decays rapidly from the illuminated face (left), yet isolated high-fluence islands (or… view at source ↗

**Figure 5.** Figure 5: Three-dimensional isosurface rendering of the solid phase of the spinodal aerogel [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗

**Figure 6.** Figure 6: Validation of the collimated-source DA against MC for a homogeneous solid [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗

**Figure 7.** Figure 7: Validation of the collimated-source DA against MC for a homogeneous EMA [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗

**Figure 8.** Figure 8: Three-dimensional isosurface rendering of the solid phase at [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗

**Figure 9.** Figure 9: Photon fluence in the 384 × 1024 × 1024 spinodal domain at ϕ = 0.90 (λ = 365 nm, κL = 1.10, front/back = 30). Same layout as [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗

**Figure 10.** Figure 10: Two-dimensional fluence map Φ(x, z) at the mid-plane cross-section for the spinodal domain at ϕ = 0.90 (κL = 1.10, front/back = 30). Same layout as [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗

read the original abstract

Porous monolithic photocatalysts based on anatase TiO2 in silica aerogels are promising for air purification. Their bicontinuous spinodal architecture offers high surface area and strong light scattering. However, extracting intrinsic kinetic rates requires accurate optical models. Current methods replace the complex 3D pore network with a homogeneous 1D slab, an approximation whose error is unknown for spinodal geometries. We combine 3D spinodal masks from Cahn-Hilliard simulations with GPU Monte Carlo photon transport to quantify this. We introduce a solid-phase fluence estimator that accounts for catalytic site distribution, comparing it to volume averages and diffusion approximations. The solid phase receives 50% more photons than volume averages at porosity 0.70, rising to 70% at 0.90. This preferential illumination stems from quasi-ballistic paths through pore channels, termed photon channelling. The extracted kinetic descriptor differs by 34% between 3D Monte Carlo and diffusion models. Homogeneous controls show that roughly 50% of the total 73% discrepancy is intrinsic to the bicontinuous structure and cannot be fixed by effective medium theories. These results provide the first quantitative correction for kinetic extraction in such photocatalysts and establish design rules linking synthesis coarsening, pore size, and light efficiency.

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.

Desk Editor's Note private letter to a colleague

The paper's main result is that 3D Monte Carlo on Cahn-Hilliard spinodal masks shows 50-70% higher solid-phase fluence than volume averages due to photon channelling, producing a 34% kinetic shift versus diffusion models.

read the letter

The central finding is that bicontinuous spinodal geometry in TiO2-silica aerogels channels photons preferentially to the solid phase, giving 50% excess fluence at 70% porosity and 70% at 90%, which then changes the extracted kinetic descriptor by 34% compared with diffusion approximations. About half of that gap survives even when homogeneous controls are used, so the morphology itself matters beyond what effective-medium fixes can handle. This is new: prior work on these materials relied on 1D slabs or averaged properties, and the combination of Cahn-Hilliard masks with a solid-phase fluence estimator in GPU Monte Carlo is a direct way to quantify the 3D effect. The forward simulation and the homogeneous controls are clean moves that let the authors separate structure from other variables. The numbers are specific to the chosen refractive indices, scattering lengths, and coarsening parameters, however, and the abstract gives no sensitivity sweeps or experimental cross-checks against real aerogel samples. If the mean free path or index contrast is off, the quasi-ballistic paths and the reported percentages can move. That keeps the quantitative claims provisional until those inputs are tested. The work is aimed at groups who synthesize monolithic photocatalysts and need better optical corrections when they back out intrinsic rates from reactor data. Anyone fitting kinetic models to aerogel experiments would get immediate use from the correction factor, even if they later adjust the optical constants for their own material. I would send it to peer review. The simulation framework is reproducible in principle and the question it asks is practical, so referees can usefully press on validation and parameter robustness.

Referee Report

3 major / 2 minor

Summary. The manuscript uses Cahn-Hilliard simulations to generate bicontinuous spinodal masks for TiO2-silica aerogels and GPU-accelerated Monte Carlo photon transport to model 3D light propagation. It introduces a solid-phase fluence estimator that accounts for catalytic site distribution and reports that the solid phase receives 50% more photons than volume averages at 0.70 porosity (rising to 70% at 0.90 porosity) due to quasi-ballistic photon channelling through pore channels. This produces a 34% difference in an extracted kinetic descriptor relative to diffusion approximations; homogeneous controls indicate that roughly half of the total 73% discrepancy is intrinsic to the bicontinuous morphology and cannot be recovered by effective-medium theories.

Significance. If the quantitative corrections hold, the work supplies the first morphology-specific adjustment for kinetic-rate extraction in spinodal photocatalysts and supplies design rules connecting coarsening time, pore size, and light utilization. The direct forward Monte Carlo approach (rather than parameter fitting) and the use of homogeneous controls as an external check are methodological strengths that avoid circularity in the fluence comparison.

major comments (3)

[Abstract and Results] Abstract and Results: the reported 50–70% solid-phase fluence excess and 34% kinetic shift are presented as specific numbers without any sensitivity sweeps on the optical constants (refractive indices, absorption/scattering lengths) or on the Cahn-Hilliard coarsening parameters. Because the mean free path relative to pore size controls the quasi-ballistic channelling, these percentages are not shown to be robust; the central claim that the bicontinuous structure intrinsically accounts for half the discrepancy therefore rests on untested inputs.
[Methods] Methods: the solid-phase fluence estimator and the precise definition of the kinetic descriptor are introduced without an equation or algorithmic description of how photon paths are binned onto the solid voxels or how the 73% total discrepancy is partitioned between structure and effective-medium error. This prevents independent verification of the 50% intrinsic attribution.
[Results] Results: no experimental benchmark or even a comparison to measured transmittance/reflectance of real TiO2-silica aerogels is provided. Without such grounding, it remains unclear whether the Monte Carlo model with the chosen parameters reproduces the actual photon transport regime of the experimental materials.

minor comments (2)

[Introduction] The term 'photon channelling' is used without reference to analogous quasi-ballistic transport concepts already present in the porous-media optics literature.
[Figures] Figure captions should explicitly state the porosity values, optical parameters, and number of photon trajectories used for each panel to allow direct comparison with the reported percentages.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive comments and for recognizing the methodological strengths of the direct Monte Carlo approach and homogeneous controls. We address each major comment point by point below, with planned revisions indicated.

read point-by-point responses

Referee: [Abstract and Results] Abstract and Results: the reported 50–70% solid-phase fluence excess and 34% kinetic shift are presented as specific numbers without any sensitivity sweeps on the optical constants (refractive indices, absorption/scattering lengths) or on the Cahn-Hilliard coarsening parameters. Because the mean free path relative to pore size controls the quasi-ballistic channelling, these percentages are not shown to be robust; the central claim that the bicontinuous structure intrinsically accounts for half the discrepancy therefore rests on untested inputs.

Authors: We agree that the absence of sensitivity sweeps leaves the quantitative robustness untested. In the revised manuscript we will add sweeps over refractive index, absorption/scattering lengths, and Cahn-Hilliard coarsening times, confirming that the 50–70 % fluence excess and 34 % kinetic shift remain qualitatively stable within the physically relevant range for TiO2-silica aerogels. revision: yes
Referee: [Methods] Methods: the solid-phase fluence estimator and the precise definition of the kinetic descriptor are introduced without an equation or algorithmic description of how photon paths are binned onto the solid voxels or how the 73% total discrepancy is partitioned between structure and effective-medium error. This prevents independent verification of the 50% intrinsic attribution.

Authors: We accept that explicit equations and algorithmic details are required for reproducibility. The revised Methods section will include the mathematical definition of the solid-phase fluence estimator, the precise binning algorithm for photon paths onto solid voxels, and the step-by-step procedure used to partition the 73 % discrepancy between bicontinuous morphology and effective-medium error. revision: yes
Referee: [Results] Results: no experimental benchmark or even a comparison to measured transmittance/reflectance of real TiO2-silica aerogels is provided. Without such grounding, it remains unclear whether the Monte Carlo model with the chosen parameters reproduces the actual photon transport regime of the experimental materials.

Authors: The study is computational and quantifies morphology-specific corrections via Monte Carlo transport; the homogeneous controls already provide an internal consistency check that avoids circularity. Direct experimental transmittance/reflectance data for the exact simulated structures are not available to the authors. In revision we will expand the discussion of model assumptions and applicability limits, but cannot add new experimental benchmarks. revision: partial

standing simulated objections not resolved

Direct experimental transmittance or reflectance measurements on real TiO2-silica aerogels matching the simulated structures for model benchmarking.

Circularity Check

0 steps flagged

No circularity: all quantitative claims are direct outputs of forward Monte Carlo simulation on independently generated morphologies

full rationale

The paper generates spinodal masks via Cahn-Hilliard simulation, then runs GPU Monte Carlo photon transport to compute fluence distributions. The solid-phase fluence estimator is a defined post-processing step applied to the simulated photon paths; the reported 50-70% excess, 34% kinetic descriptor difference, and homogeneous-control breakdown are computed results rather than inputs or self-referential definitions. No fitted parameters are relabeled as predictions, no uniqueness theorems or ansatzes are smuggled via self-citation, and the derivation chain does not reduce any central claim to its own inputs by construction. The work is self-contained forward modeling whose outputs can be independently reproduced or falsified by changing the optical parameters or morphology generator.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The work rests on standard simulation techniques whose validity is assumed rather than re-derived. Porosity values are chosen parameters; no data-fitted constants appear in the abstract. The new estimator and channelling concept are descriptive rather than new physical postulates.

free parameters (2)

porosity
Specific values 0.70 and 0.90 are selected to illustrate the trend; these are input choices rather than outputs.
Cahn-Hilliard coarsening parameters
Used to generate the 3D masks but not numerically specified in the abstract.

axioms (2)

domain assumption Cahn-Hilliard equation produces representative spinodal bicontinuous morphologies for the aerogels
Invoked to create the 3D masks that replace real experimental structures.
domain assumption Monte Carlo photon transport with the chosen optical properties correctly models light propagation in the porous medium
Core assumption enabling the fluence calculations and channelling observation.

invented entities (2)

solid-phase fluence estimator no independent evidence
purpose: To weight photon counts by the actual distribution of catalytic sites instead of volume averaging
New computational construct introduced to quantify the preferential illumination effect.
photon channelling no independent evidence
purpose: Descriptive label for quasi-ballistic photon paths through pore channels
Term coined to explain the observed fluence excess; not a new physical entity.

pith-pipeline@v0.9.0 · 5535 in / 1988 out tokens · 33431 ms · 2026-05-10T12:30:14.292826+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 20 canonical work pages

[1]

Hoffmann, S.T

M.R. Hoffmann, S.T. Martin, W. Choi, D.W. Bahnemann, Environmental ap- plications of semiconductor photocatalysis, Chem. Rev. 95 (1995) 69–96.https: //doi.org/10.1021/cr00033a004

work page doi:10.1021/cr00033a004 1995
[2]

Performance and Analysis of the Alchemical Transfer Method for Binding-Free-Energy Predictions of Diverse Ligands

A. Vardon, M.-A. Dourges, ´E. Laurichesse, V. Schmitt, A. Bentaleb, F. Nallet, I. Ly, R. Backov, Mastering syntheses of siliceous hierarchical porous self-standing monoliths through the integration of the sol–gel process, complex fluids, and a plan- etary mixer, Langmuir 41 (2025) 26760–26769.https://doi.org/10.1021/acs. langmuir.5c03237

work page doi:10.1021/acs 2025
[3]

Layan, G

´E. Layan, G. Clermont, I. Ly, F. Nallet, R.A.L. Vall´ ee, et al., Bulk photocatalysis within anatase-TiO 2–SiO2 monoliths for efficient VOC photodegradation, Chem. Commun. 61 (2025) 12345.https://doi.org/10.1039/d5cc02552d

work page doi:10.1039/d5cc02552d 2025
[4]

Ollis, Kinetics of photocatalyzed reactions: Five lessons learned, Front

D.F. Ollis, Kinetics of photocatalyzed reactions: Five lessons learned, Front. Chem. 6 (2018) 378.https://doi.org/10.3389/fchem.2018.00378

work page doi:10.3389/fchem.2018.00378 2018
[5]

Bloh, A holistic approach to model the kinetics of heterogeneous photocatalysis, Front

J.Z. Bloh, A holistic approach to model the kinetics of heterogeneous photocatalysis, Front. Chem. 7 (2019) 128.https://doi.org/10.3389/fchem.2019.00128

work page doi:10.3389/fchem.2019.00128 2019
[6]

Garnett, Colours in metal glasses and in metallic films, Philos

J.C.M. Garnett, Colours in metal glasses and in metallic films, Philos. Trans. R. Soc. A 203 (1904) 385–420.https://doi.org/10.1098/rsta.1904.0024

work page doi:10.1098/rsta.1904.0024 1904
[7]

Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press, New York, 1978

A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press, New York, 1978

1978
[8]

Vall´ ee, R

R.A.L. Vall´ ee, R. Backov, From light diffusion to photocatalytic rates: Compact scal- ing laws for strongly scattering porous slabs, J. Quant. Spectrosc. Radiat. Transfer 352 (2026) 109819.https://doi.org/10.1016/j.jqsrt.2026.109819

work page doi:10.1016/j.jqsrt.2026.109819 2026
[9]

Vynck, R

K. Vynck, R. Pierrat, R. Carminati, L.S. Froufe-P´ erez, F. Scheffold, R. Sapienza, S. Vignolini, J.J. Sa´ enz, Light in correlated disordered media, Rev. Mod. Phys. 95 (2023) 045003.https://doi.org/10.1103/RevModPhys.95.045003 18

work page doi:10.1103/revmodphys.95.045003 2023
[10]

Leyre, E

S. Leyre, E. Coutino-Gonzalez, J.J. Joos, J. Ryckaert, Y. Meuret, D. Poelman, P.F. Smet, G. Durinck, J. Hofkens, G. Deconinck, P. Hanselaer, Absolute determination of photoluminescence quantum efficiency using an integrating sphere setup, Rev. Sci. Instrum. 91 (2020) 013108.https://doi.org/10.1063/1.5136060

work page doi:10.1063/1.5136060 2020
[11]

Leyre, B

S. Leyre, B. Meuris, J.J. Joos, P.F. Smet, G. Deconinck, P. Hanselaer, Determination of the optical properties of turbid media using a single integrating sphere setup, Appl. Opt. 61 (2022) 792–802.https://doi.org/10.1364/AO.445979

work page doi:10.1364/ao.445979 2022
[12]

Brunser, F.L

S.S. Brunser, F.L. Bargardi, R. Libanori, N. Kaufmann, H. Braun, A. Steinfeld, A.R. Studart, Solar-driven redox splitting of CO 2 using 3D-printed hierarchically channelled ceria structures, Adv. Mater. Interfaces 10 (2023) 2300452.https:// doi.org/10.1002/admi.202300452

work page doi:10.1002/admi.202300452 2023
[13]

Q. Liu, K. Pan, Y. Lu, X. Zhang, Data-driven for accelerated design strategy of photocatalytic degradation activity prediction of doped TiO2 photocatalyst, J. Water Process Eng. 49 (2022) 103126.https://doi.org/10.1016/j.jwpe.2022.103126

work page doi:10.1016/j.jwpe.2022.103126 2022
[14]

Cahn, J.E

J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958) 258–267.https://doi.org/10.1063/1.1744102

work page doi:10.1063/1.1744102 1958
[15]

Cahn, Free energy of a nonuniform system

J.W. Cahn, Free energy of a nonuniform system. II. Thermodynamic basis, J. Chem. Phys. 30 (1959) 1121–1124.https://doi.org/10.1063/1.1730145

work page doi:10.1063/1.1730145 1959
[16]

Provatas, K

N. Provatas, K. Elder, Phase-Field Methods in Materials Science and Engineering, Wiley-VCH, Weinheim, 2010.https://doi.org/10.1002/9783527631520

work page doi:10.1002/9783527631520 2010
[17]

Fang, D.A

Q. Fang, D.A. Boas, Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units, Opt. Express 17 (2009) 20178–20190. https://doi.org/10.1364/OE.17.20178

work page doi:10.1364/oe.17.20178 2009
[18]

Geslin, I

P.-A. Geslin, I. McCue, B. Gaskey, J. Erlebacher, A. Karma, Topology-generating interfacial pattern formation during liquid metal dealloying, Nat. Commun. 6 (2015) 8887.https://doi.org/10.1038/ncomms9887

work page doi:10.1038/ncomms9887 2015
[19]

Cahn, On spinodal decomposition, Acta Metall

J.W. Cahn, On spinodal decomposition, Acta Metall. 9 (1961) 795–801.https: //doi.org/10.1016/0001-6160(61)90182-1

work page doi:10.1016/0001-6160(61)90182-1 1961
[20]

Zhu, L.-Q

J. Zhu, L.-Q. Chen, J. Shen, V. Tikare, Coarsening kinetics from a variable-mobility Cahn–Hilliard equation: Application of a semi-implicit Fourier spectral method, Phys. Rev. E 60 (1999) 3564–3572.https://doi.org/10.1103/PhysRevE.60.3564

work page doi:10.1103/physreve.60.3564 1999
[21]

Okuta, Y

R. Okuta, Y. Unno, D. Nishino, S. Hido, C. Loomis, CuPy: A NumPy-compatible library for NVIDIA GPU calculations, Proc. Workshop on Machine Learning Systems (LearningSys) at NeurIPS, 2017

2017
[22]

Lifshitz, V.V

I.M. Lifshitz, V.V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19 (1961) 35–50.https://doi.org/10.1016/ 0022-3697(61)90054-3

1961
[23]

Wagner, Theorie der Alterung von Niederschl¨ agen durch Uml¨ osen (Ostwald- Reifung), Z

C. Wagner, Theorie der Alterung von Niederschl¨ agen durch Uml¨ osen (Ostwald- Reifung), Z. Elektrochem. 65 (1961) 581–591. 19

1961
[24]

Jolivet, C

J.-P. Jolivet, C. Chan´ eac, E. Tronc, Iron oxide chemistry. From molecular clusters to extended solid networks, Chem. Commun. (2004) 481–487

2004
[25]

L. Yu, F. Nina-Paravecino, D. Kaeli, Q. Fang, Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms, J. Biomed. Opt. (2018), 23(1), 010504

2018
[26]

",isreflect = 1. “Shape RMS

R.C. Haskell, L.O. Svaasand, T.-T. Tsay, T.-C. Feng, M.S. McAdams, B.J. Tromberg, Boundary conditions for the diffusion equation in radiative transfer, J. Opt. Soc. Am. A 11 (1994) 2727–2741.https://doi.org/10.1364/josaa.11. 002727 20 Supplementary Information Three-dimensional photon transport in spinodal photocatalytic aerogels: how bicontinuous morphol...

work page doi:10.1364/josaa.11 1994