Determination of the mass transport parameters in thin membranes by phase-sensitive photoacoustics in the optically transparent and mixed regimes

Pawel Rochowski

arxiv: 2604.05647 · v1 · submitted 2026-04-07 · ⚛️ physics.app-ph

Determination of the mass transport parameters in thin membranes by phase-sensitive photoacoustics in the optically transparent and mixed regimes

Pawel Rochowski This is my paper

Pith reviewed 2026-05-10 19:07 UTC · model grok-4.3

classification ⚛️ physics.app-ph

keywords photoacousticsmass transportthin membranesFickian diffusionoptical regimesphase-sensitiveGreen's function

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The pith

Phase-sensitive photoacoustics determines mass transport parameters in thin membranes under optically transparent and mixed regimes.

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

The paper investigates how a sample's absorption properties affect the quantification of slow mass transport processes in thin membranes when the material is optically transparent or semi-opaque. It applies the Green's function approach to model the photoacoustic response, with heat distributions assumed to track the evolving mass concentration profiles from Fickian diffusion. The separation of fast photoacoustic timescales from slow transport timescales supports this linkage. Experimental data on pigment transport into a thin porous membrane is used to examine and validate the approach.

Core claim

The ability to quantify mass transport parameters in thin membranes via phase-sensitive photoacoustics holds in optically transparent and mixed regimes. The framework uses the Green's function approach as formalized by Mandelis. Heat distributions are assumed to follow the temporal mass concentration profiles predicted by a Fickian diffusion process because of the separation of photoacoustic and mass transport timescales. The approach is examined and validated with experimental data on pigment transport into a thin porous membrane.

What carries the argument

Green's function approach that models photoacoustic signals from absorption properties tied to Fickian mass concentration profiles over time.

If this is right

Mass transport parameters can be extracted from phase data even when the membrane absorbs little probe light.
The method extends reliably to semi-opaque conditions with partial absorption.
Experimental results on pigment diffusion confirm that transport coefficients remain accessible in these optical regimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Pairing the optical signals with electrical circuit models could enable simultaneous tracking of multiple species without changing the core assumptions.
The timescale separation may allow similar parameter extraction in other thin-layer diffusion systems if the optical conditions match the modeled cases.

Load-bearing premise

Heat distributions follow the instantaneous mass concentration profiles because photoacoustic processes occur much faster than mass transport.

What would settle it

If the measured phase-sensitive photoacoustic signals during pigment transport experiments deviate systematically from the curves predicted by the Fickian diffusion model, the link between heat and mass profiles would not hold.

read the original abstract

The research complements and extends the scope of the work reported in the article "LED-based multibeam photoacoustics combined with electrical circuit-based modeling for the analysis of multispecies mass transport through thin membranes" (arXiv:2602.20902). In particular, this work investigates the impact of the sample's absorption properties on the ability to quantify slow mass transport processes in thin membranes under optically transparent and semi-opaque conditions. The theoretical framework is based on the Green's function approach as formalized by Mandelis. Owing to the separation of photoacoustic and mass transport timescales, the heat distributions are assumed to follow the temporal mass concentration profiles predicted by a Fickian diffusion process. The approach is further examined and validated using experimental data on pigment transport into a thin porous membrane. Detailed experimental results are provided in the referenced paper.

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

This is a narrow technical extension of the author's prior photoacoustics work that adds absorption-regime analysis but depends almost entirely on the earlier paper for its data and validation.

read the letter

This paper extends the same author's earlier arXiv:2602.20902 by examining how sample absorption affects photoacoustic extraction of mass-transport parameters in thin membranes. It covers the optically transparent case and the mixed semi-opaque regime, using a Green's function solution under the assumption that photoacoustic timescales are fast enough for the temperature field to follow the instantaneous Fickian concentration profile at each moment. Experimental checks use pigment transport into a porous membrane, with the detailed results placed in the referenced prior work. The approach is straightforward and fits the practical needs of applied membrane studies where light absorption varies. The timescale-separation step simplifies the forward model in a way that matches standard photoacoustic practice, and the focus on absorption effects fills a small gap left by the first paper. The central modeling choice is reasonable when the separation holds, and the Fickian assumption is stated clearly. The main weakness is that this manuscript supplies almost no equations, error analysis, or raw data of its own. Validation reduces to the earlier paper, so any reader must consult both to judge whether the phase-lag inversion actually recovers the transport coefficients reliably. The stress-test point about circularity is fair: because the same forward model is used both to generate predictions and to fit the measurements, agreement does not independently confirm the Fickian or timescale assumptions. Non-Fickian effects such as adsorption in the porous matrix would propagate directly into the extracted parameters without being flagged. The paper is therefore best read as a short technical note rather than a standalone result. It is aimed at researchers already working on optical or photoacoustic methods for diffusion in thin films and membranes. Those readers will see incremental value in the absorption-regime discussion, but the work does not shift broader understanding of mass transport or photoacoustics. It deserves peer review because the underlying technique is reproducible in principle and the topic has clear applied interest, provided the authors are asked to include enough self-contained modeling and a clearer summary of the prior validation so the claims can be checked without the companion paper.

Referee Report

2 major / 1 minor

Summary. The paper extends the author's prior work on LED-based multibeam photoacoustics for multispecies mass transport through thin membranes by examining how sample absorption properties affect quantification of slow transport processes under optically transparent and semi-opaque (mixed) conditions. It employs a Green's function framework (following Mandelis) under the assumption that photoacoustic and mass-transport timescales are separated, allowing heat distributions to follow instantaneous Fickian mass-concentration profiles; the approach is stated to be validated by experimental pigment-transport data into a thin porous membrane, with all detailed results referred to the prior preprint.

Significance. If the timescale-separation and Fickian assumptions prove robust, the work would usefully broaden phase-sensitive photoacoustics to a wider range of membrane optical regimes, supporting non-invasive extraction of mass-transport parameters. The reliance on an established Green's-function formalism is a strength, but the absence of standalone experimental data or independent tests of the core assumptions here reduces the immediate incremental impact.

major comments (2)

[Abstract] Abstract: the statement that 'the approach is further examined and validated using experimental data' is undercut because all detailed results and parameter fits are deferred to the referenced prior paper (arXiv:2602.20902); the present manuscript therefore supplies no independent data or error analysis with which to assess the claimed validation.
[Theoretical framework] Theoretical framework (Green's function + Fickian assumption): the load-bearing claim that heat distributions follow the instantaneous Fickian c(x,t) profiles rests on timescale separation, yet no quantitative bound on the separation (or sensitivity analysis for the pigment/porous-membrane case) is supplied; any violation would directly bias the predicted phase lag in the mixed regime.

minor comments (1)

[Abstract] The citation to Mandelis' Green's-function formalism should include the specific reference(s) rather than the author's name alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and indicate planned revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the statement that 'the approach is further examined and validated using experimental data' is undercut because all detailed results and parameter fits are deferred to the referenced prior paper (arXiv:2602.20902); the present manuscript therefore supplies no independent data or error analysis with which to assess the claimed validation.

Authors: We agree the abstract phrasing risks implying standalone validation. This manuscript develops the theoretical extension of phase-sensitive photoacoustics to transparent and mixed optical regimes via the Green's function formalism, while the full experimental validation (pigment transport into a porous membrane, parameter fits, and error analysis) is contained in the companion preprint arXiv:2602.20902. We will revise the abstract to state explicitly that validation details appear in the referenced work and that the present paper focuses on the impact of absorption properties under the timescale-separation assumption. revision: yes
Referee: [Theoretical framework] Theoretical framework (Green's function + Fickian assumption): the load-bearing claim that heat distributions follow the instantaneous Fickian c(x,t) profiles rests on timescale separation, yet no quantitative bound on the separation (or sensitivity analysis for the pigment/porous-membrane case) is supplied; any violation would directly bias the predicted phase lag in the mixed regime.

Authors: The model relies on clear separation between fast photoacoustic generation (microsecond scale) and slow Fickian mass transport (minutes to hours for the pigment/porous-membrane system). We will add a quantitative estimate of the timescale ratio together with a brief sensitivity analysis showing that phase-lag predictions remain robust within the expected range of diffusion coefficients and membrane thicknesses for this experimental case. revision: yes

Circularity Check

1 steps flagged

Validation and core modeling reduce to self-cited prior paper by same author

specific steps

self citation load bearing [Abstract]
"The approach is further examined and validated using experimental data on pigment transport into a thin porous membrane. Detailed experimental results are provided in the referenced paper."

The manuscript claims to examine and validate the photoacoustic method for extracting mass transport parameters, but explicitly defers the detailed experimental results and (per the reader's context) the core modeling to the author's prior arXiv:2602.20902. The present paper's central validation therefore reduces to that self-cited work by construction.

full rationale

The paper's theoretical framework invokes the Green's function solution with the explicit assumption that heat distributions follow instantaneous Fickian concentration profiles due to timescale separation. However, the abstract states that the approach is validated using experimental data whose detailed results are provided only in the referenced prior paper (arXiv:2602.20902) by the same author. This makes the load-bearing validation step and the parameter extraction claims dependent on the self-cited work rather than independently derived or tested within the present manuscript. The central claim of determining mass transport parameters therefore reduces to the earlier paper's data and model fits.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on timescale separation allowing heat to track mass concentration, the Green's function formalism from Mandelis, and Fickian diffusion; experimental validation details are external to this manuscript.

axioms (2)

domain assumption Heat distributions follow the temporal mass concentration profiles predicted by a Fickian diffusion process due to separation of photoacoustic and mass transport timescales.
Explicitly stated in the abstract as the basis for the theoretical framework.
standard math Green's function approach as formalized by Mandelis applies to the photoacoustic signal generation in the membrane.
Cited as the theoretical foundation in the abstract.

pith-pipeline@v0.9.0 · 5441 in / 1413 out tokens · 75764 ms · 2026-05-10T19:07:33.224831+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Owing to the separation of photoacoustic and mass transport timescales, the heat distributions are assumed to follow the temporal mass concentration profiles predicted by a Fickian diffusion process.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

Introduction The work demonstrates the potential of the phase-sensitive photoacoustic technique for measuring the dynamics of pigment transport through thin membranes. As will be shown, investigating the evolution of the signal phase over a broad range of sample transmissivities enables direct and accurate determination of transport constants, even when r...

work page
[2]

For the PA signal generation we summarize the Green’s function approach formalized in works of Mandelis, McGahan and Cole [2–4]

PA signal generation for arbitrary pigment distributions in a sample. For the PA signal generation we summarize the Green’s function approach formalized in works of Mandelis, McGahan and Cole [2–4]. For the sample region under modulated light excitation 𝑒𝑗𝜔𝑡 the heat transport equation takes the form: 𝑑2Θ 𝑑𝑥2 − 𝜎𝑠 2 Θ = − 𝐻(𝑥) 𝑘𝑠 , (1) where 𝜎𝑠 2 = 𝑗𝜔/𝛼𝑠 ...

work page
[3]

As such, the adaptation of the model to the mass transport studies requires specifying an appropriate distribution of the heat sources 𝐻(𝑥) and calculating the integral given by eq

The heat source distribution, the mass transport and the PA phase The PA response is in general is proportional to the surface temperature Θ(0). As such, the adaptation of the model to the mass transport studies requires specifying an appropriate distribution of the heat sources 𝐻(𝑥) and calculating the integral given by eq. 4. Now we are in position to m...

work page
[4]

Qualitatively the measurable phase difference exhibits an exponential form: 𝜙(𝑡0) − 𝜙eq ≡ Δ𝜙(𝑡0) = ∑ 𝐴𝑛 ∞ 𝑛=0 𝑒 −𝑡0 𝜏𝑛 (11) By considering a timescale such as 𝑡0 ≫ 𝜏1 the 𝑛 = 0 dominates and the temporal phase evolution can be approximated by: Δ𝜙(𝑡0) ≈ 𝐴𝑒−𝑡0/𝜏0 = 𝐴𝑒−𝜋2𝐷 4𝑙2 𝑡0 (12)

work page
[5]

center of mass

Experimental validation and simulations For the validation we take the results presented in [6]. In particular, we consider a 1D model pigment (dithranol) transport from a solid Vaseline formulation (backing) into a thin l = 15µm dodecanol/collodion membrane sample (dodecanol represents a permeable phase). The data were collected for the PA sampling depth...

work page 1960
[6]

centre of mass

The dual-frequency test In the case of the transient system at least two process affects the signal phase, the temporal pigment distribution and the optical saturation. From the Rosencwaig -Gersho model it is expected, that for the 𝛿(0) thermal excitations (optically opaque regime) the difference between signal phases recorded at two distinct modulation f...

work page
[7]

centre of mass

The combined signal-phase analysis The analysis presented so far has shown that the signal phase is highly sensitive to the position of the pigment “centre of mass” within the membrane, while varying only weakly for samples whose transmission spans a very broad range of values. On the other hand, the signal curvature encodes both the evolution of the pigm...

work page
[8]

LED- based multibeam photoacoustics combined with electrical circuit -based modelling for the analysis of multispecies mass transport through thin membranes

Summary The aim of this study was to investigate the possibility of determining pigment diffusion parameters based on the analysis of phase evolution and the photoacoustic signal of a sample in response to mass sorption from a semi-infinite medium into a thin membrane. The theoretical approach relied on the Green's function formalism, assuming a pseudo -s...

work page
[9]

Rosencwaig, A

A. Rosencwaig, A. Gersho, Theory of the photoacoustic effect with solids, J. Appl. Phys. 47 (1976) 64–69. doi:10.1063/1.328255

work page doi:10.1063/1.328255 1976
[10]

Mandelis, Diffusion-Wave Fields, Springer New York, New York, NY , 2001

A. Mandelis, Diffusion-Wave Fields, Springer New York, New York, NY , 2001. doi:10.1007/978-1-4757-3548-2

work page doi:10.1007/978-1-4757-3548-2 2001
[11]

McGahan, K.D

W.A. McGahan, K.D. Cole, Solutions of the heat conduction equation in multilayers for photothermal deflection experiments, J. Appl. Phys. 72 (1992) 1362–1373. doi:10.1063/1.351747

work page doi:10.1063/1.351747 1992
[12]

Ab Initio Transport Coefficients of Gaseous Hydrogen

K.D. Cole, Analysis of Photothermal Characterization of Layered Materials - Design of Optimal Experiments, Int. J. Thermophys. 25 (2004) 1567–1584. doi:10.1007/s10765- 004-5759-4

work page doi:10.1007/s10765- 2004
[13]

Crank, The Mathematics of Diffusion, Second Edi, Clarendon Press, Oxford, 1975

J. Crank, The Mathematics of Diffusion, Second Edi, Clarendon Press, Oxford, 1975

work page 1975
[14]

Rochowski, LED-based multibeam photoacoustics combined with electrical circuit- based modeling for the analysis of multispecies mass transport through thin membranes, (2026)

P. Rochowski, LED-based multibeam photoacoustics combined with electrical circuit- based modeling for the analysis of multispecies mass transport through thin membranes, (2026). doi:10.48550/arXiv.2602.20902

work page internal anchor Pith review doi:10.48550/arxiv.2602.20902 2026

[1] [1]

Introduction The work demonstrates the potential of the phase-sensitive photoacoustic technique for measuring the dynamics of pigment transport through thin membranes. As will be shown, investigating the evolution of the signal phase over a broad range of sample transmissivities enables direct and accurate determination of transport constants, even when r...

work page

[2] [2]

For the PA signal generation we summarize the Green’s function approach formalized in works of Mandelis, McGahan and Cole [2–4]

PA signal generation for arbitrary pigment distributions in a sample. For the PA signal generation we summarize the Green’s function approach formalized in works of Mandelis, McGahan and Cole [2–4]. For the sample region under modulated light excitation 𝑒𝑗𝜔𝑡 the heat transport equation takes the form: 𝑑2Θ 𝑑𝑥2 − 𝜎𝑠 2 Θ = − 𝐻(𝑥) 𝑘𝑠 , (1) where 𝜎𝑠 2 = 𝑗𝜔/𝛼𝑠 ...

work page

[3] [3]

As such, the adaptation of the model to the mass transport studies requires specifying an appropriate distribution of the heat sources 𝐻(𝑥) and calculating the integral given by eq

The heat source distribution, the mass transport and the PA phase The PA response is in general is proportional to the surface temperature Θ(0). As such, the adaptation of the model to the mass transport studies requires specifying an appropriate distribution of the heat sources 𝐻(𝑥) and calculating the integral given by eq. 4. Now we are in position to m...

work page

[4] [4]

Qualitatively the measurable phase difference exhibits an exponential form: 𝜙(𝑡0) − 𝜙eq ≡ Δ𝜙(𝑡0) = ∑ 𝐴𝑛 ∞ 𝑛=0 𝑒 −𝑡0 𝜏𝑛 (11) By considering a timescale such as 𝑡0 ≫ 𝜏1 the 𝑛 = 0 dominates and the temporal phase evolution can be approximated by: Δ𝜙(𝑡0) ≈ 𝐴𝑒−𝑡0/𝜏0 = 𝐴𝑒−𝜋2𝐷 4𝑙2 𝑡0 (12)

work page

[5] [5]

center of mass

Experimental validation and simulations For the validation we take the results presented in [6]. In particular, we consider a 1D model pigment (dithranol) transport from a solid Vaseline formulation (backing) into a thin l = 15µm dodecanol/collodion membrane sample (dodecanol represents a permeable phase). The data were collected for the PA sampling depth...

work page 1960

[6] [6]

centre of mass

The dual-frequency test In the case of the transient system at least two process affects the signal phase, the temporal pigment distribution and the optical saturation. From the Rosencwaig -Gersho model it is expected, that for the 𝛿(0) thermal excitations (optically opaque regime) the difference between signal phases recorded at two distinct modulation f...

work page

[7] [7]

centre of mass

The combined signal-phase analysis The analysis presented so far has shown that the signal phase is highly sensitive to the position of the pigment “centre of mass” within the membrane, while varying only weakly for samples whose transmission spans a very broad range of values. On the other hand, the signal curvature encodes both the evolution of the pigm...

work page

[8] [8]

LED- based multibeam photoacoustics combined with electrical circuit -based modelling for the analysis of multispecies mass transport through thin membranes

Summary The aim of this study was to investigate the possibility of determining pigment diffusion parameters based on the analysis of phase evolution and the photoacoustic signal of a sample in response to mass sorption from a semi-infinite medium into a thin membrane. The theoretical approach relied on the Green's function formalism, assuming a pseudo -s...

work page

[9] [9]

Rosencwaig, A

A. Rosencwaig, A. Gersho, Theory of the photoacoustic effect with solids, J. Appl. Phys. 47 (1976) 64–69. doi:10.1063/1.328255

work page doi:10.1063/1.328255 1976

[10] [10]

Mandelis, Diffusion-Wave Fields, Springer New York, New York, NY , 2001

A. Mandelis, Diffusion-Wave Fields, Springer New York, New York, NY , 2001. doi:10.1007/978-1-4757-3548-2

work page doi:10.1007/978-1-4757-3548-2 2001

[11] [11]

McGahan, K.D

W.A. McGahan, K.D. Cole, Solutions of the heat conduction equation in multilayers for photothermal deflection experiments, J. Appl. Phys. 72 (1992) 1362–1373. doi:10.1063/1.351747

work page doi:10.1063/1.351747 1992

[12] [12]

Ab Initio Transport Coefficients of Gaseous Hydrogen

K.D. Cole, Analysis of Photothermal Characterization of Layered Materials - Design of Optimal Experiments, Int. J. Thermophys. 25 (2004) 1567–1584. doi:10.1007/s10765- 004-5759-4

work page doi:10.1007/s10765- 2004

[13] [13]

Crank, The Mathematics of Diffusion, Second Edi, Clarendon Press, Oxford, 1975

J. Crank, The Mathematics of Diffusion, Second Edi, Clarendon Press, Oxford, 1975

work page 1975

[14] [14]

Rochowski, LED-based multibeam photoacoustics combined with electrical circuit- based modeling for the analysis of multispecies mass transport through thin membranes, (2026)

P. Rochowski, LED-based multibeam photoacoustics combined with electrical circuit- based modeling for the analysis of multispecies mass transport through thin membranes, (2026). doi:10.48550/arXiv.2602.20902

work page internal anchor Pith review doi:10.48550/arxiv.2602.20902 2026