A combined diffusion/rate equation model to describe charge generation in phase-separated donor-acceptor blends

Atul Shukla; Dieter Neher; Phillip Teschner

REVIEW 3 major objections 5 minor 7 references

A combined diffusion/rate equation model to describe charge generation in phase-separated donor-acceptor blends

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper builds a model that couples exciton diffusion to Marcus-theory charge transfer at the interface and shows that, in low-driving-force organic solar cells, assuming instant quenching at the donor-acceptor boundary makes…

desk verdict Worth engaging for the analytic diffusion-plus-Marcus model; the PM6:Y6 domain-size estimate rests on a load-bearing 20% offset subtraction that needs more scrutiny. read the letter →

arxiv 2411.08812 v1 pith:ZLK4FRNI submitted 2024-11-13 cond-mat.mtrl-sci

Phillip Teschner , Atul Shukla , Dieter Neher This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords organicsolarcellsexcitondiffusionMarcustheorychargetransferstatestransientabsorptionspectroscopydomainsizeestimationnon-fullereneacceptorsbulkheterojunction

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 tries to establish a quantitative account of free charge generation in organic solar cells that treats exciton diffusion and interfacial charge transfer as one coupled process rather than two separate steps. The authors derive an analytical steady-state expression for the charge generation efficiency from a diffusion equation joined to Marcus-theory rate equations at the donor-acceptor interface, and show that the intrinsic exciton lifetime, not just the diffusion length, controls whether a vanishing driving force can still yield high efficiency. Their dynamic analysis leads to the paper's central methodological claim: when the energy offset driving exciton dissociation is small, the measured time for charge generation is much longer than the pure diffusion time, so conventional domain-size estimates that assume instantaneous quenching at the interface are only valid for high driving forces above about $0.2$ eV. Applied to transient absorption data on PM6:Y6, the model reproduces the observed slow hole-transfer rise and yields a fitted Y6 domain radius of $12.47\pm 0.71$ nm, i.e. roughly a $25$ nm domain diameter, together with a fast CT-state dissociation rate of $1.12\pm 0.44$ ps$^{-1}$.

What carries the argument

The central object is a reaction-diffusion system on an acceptor domain $\Omega\subset\mathbb{R}^3$: a diffusion equation $\partial_t n_{\mathrm{LE}}=D\Delta n_{\mathrm{LE}}+G_{\mathrm{LE}}-R_{\mathrm{LE}}$ for the singlet exciton density, coupled to a surface density of charge-transfer states $n_{\mathrm{CT}}$ through a Robin-type boundary condition $-D\nabla n_{\mathrm{LE}}\cdot\nu=\kappa_{\mathrm{diss,LE}}n_{\mathrm{LE}}-k_{\mathrm{ref}}n_{\mathrm{CT}}$ on the donor-acceptor interface, with $n_{\mathrm{CT}}$ evolving by the rate equation $\partial_t n_{\mathrm{CT}}=\kappa_{\mathrm{diss,LE}}n_{\mathrm{LE}}-(k_{\mathrm{ref}}+k_{\mathrm{f,CT}}+k_{\mathrm{diss,CT}})n_{\mathrm{CT}}$. The carrying identity is the closed-form steady-state efficiency, which depends only on the ratios $L_D/R$ and $\kappa_{\mathrm{diss,LE}}/(R\,k_{\mathrm{f,LE}})$ together with the reformation probability $P_{\mathrm{ref}}$ and the CT dissociation probability $P_{\mathrm{diss,CT}}$; this compact reduction is what lets the authors isolate the intrinsic lifetime as the parameter that controls low-offset performance. A second analytic object, the mean survival time of an exciton in a sphere, provides the diffusion time scale used to show that finite interfacial quenching, not pure diffusion, sets the measured rise times.

What would settle it

Measure the Y6 domain size in the same PM6:Y6 blend with an exciton-dynamics-independent structural probe, such as grazing-incidence small-angle X-ray scattering or resonant soft X-ray scattering; if the scattering-derived domain diameter disagrees with the fitted $25$ nm beyond the stated uncertainty, the TAS-based estimate and its underlying single-spherical-domain, fixed-Marcus-parameter assumptions are falsified.

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Extended reading notes

Core claim

The central discovery is that the efficiency and dynamics of free charge generation in phase-separated donor-acceptor blends are governed by the interplay between exciton diffusion to the interface and the finite, Marcus-limited rates of exciton dissociation, CT reformation, and CT dissociation. In the steady state, the charge generation efficiency reduces to an explicit function of only four parameter combinations: the diffusion length over the domain radius, a velocity ratio $\kappa_{\mathrm{diss,LE}}/(R\,k_{\mathrm{f,LE}})$, the reformation probability, and the CT dissociation probability. The model shows that a long intrinsic exciton lifetime raises efficiency at low offset by permitting multiple dissociation attempts, whereas diffusion length alone cannot. Dynamically, for low offsets the ground-state-bleach rise time is considerably longer than the mean diffusion time because weak quenching prevents a concentration gradient from forming; hence interpreting long experimental rise times of roughly $100$ ps as pure diffusion overestimates the diffusion-limited time and invalidates instant-quench domain sizing unless $\Delta E_{\mathrm{LE-CT}}>0.2$ eV. Fitting the model to PM6:Y6 transient absorption data gives $R=12.47\pm 0.71$ nm and $k_{\mathrm{diss,CT}}=1.12\pm 0.44$ ps$^{-1}$.

Load-bearing premise

The fitted $25$ nm Y6 domain size rests on the assumption that the measured PM6:Y6 transient absorption can be cleanly separated into a constant roughly $20\%$ ultrafast component from disordered Y6 and a remainder governed by one spherical aggregated Y6 domain with a single set of Marcus parameters, including the hand-set $1$ nm CT localization thickness and literature values for $|H_{\mathrm{DA}}|$ and $\lambda$.

Editorial extensions

If this is right

Domain-size estimates that assume instant quenching at the donor-acceptor interface are only reliable when the driving force exceeds about $0.2$ eV; below that, fitted sizes will be distorted because slow dissociation itself extends the measured rise time.
Long intrinsic exciton lifetimes, not just long diffusion lengths, can push charge generation efficiency close to unity at zero or slightly negative offsets, making lifetime a design lever for low-voltage-loss solar cells.
Standard rate-equation models that fold diffusion into a fixed reduction factor reproduce efficiency trends but cannot match the explicit domain-size dependence of the present model.
For PM6:Y6, the model attributes the roughly $100$ ps hole-transfer completion to combined diffusion and dissociation rather than diffusion alone, and places the Y6 domain diameter near $25$ nm with rapid CT dissociation on the order of $1$ ps.

Reading between the lines

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

A direct test would be to extract the domain radius from time-resolved photoluminescence survival times using the mean-survival-time expression in the supplement and compare it with the TAS-derived $25$ nm on the same film; agreement would validate the model, while disagreement would reveal which observable is more sensitive to the disordered-phase assumption.
The strong correlation between $\kappa_{\mathrm{diss,LE}}$ and $R$ implies that independent morphology measurements or independently determined Marcus parameters are needed to make the domain-size estimate unique, since multiple (rate, radius) pairs can fit the same transient.
If the roughly $20\%$ ultrafast component is indeed from disordered Y6, then changing the excitation wavelength or probing a sub-ensemble should shift the fitted aggregated-domain radius only within the stated uncertainty; a large shift would indicate that the constant-offset subtraction is not clean.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core contribution here is real: a clean analytic coupling of 3D exciton diffusion to Marcus-theory interfacial rates, producing a closed-form steady-state efficiency expression (eq. 20). The derivation is consistent, the limiting cases behave, and the comparison with the reduction-factor rate model shows why the common 0.1 prefactor is only a rough stand-in for finite quenching. The MFPT analysis in Sec. 4 also supports the qualitative conclusion that low-offset systems are not diffusion-limited in the naive sense. For an OPV audience, that is a useful framework, and the time-domain interpretation of TAS rise times is a step forward over purely phenomenological rate models.

The soft spots are concentrated in the experimental application. The PM6:Y6 fit subtracts about 20% of the GSB as a constant offset and excludes early EA data; if that disordered-phase component is not constant or overlaps with aggregated-domain kinetics, the recovered R = 12.47 nm and k_diss,CT = 1.12 ps^-1 are biased. The authors themselves note the strong correlation between R and kappa_diss,LE, so the headline 25 nm diameter is contingent on that parameter and on hand-set d, |H_DA|, and lambda from prior work. The single-sphere assumption is a simplification, though probably acceptable as an effective-radius estimate. I do not see a fatal flaw in the model itself, but the abstract's 'demonstrated' is too strong: the fit is a two-parameter model applied to data that already required the offset subtraction to look like the model's prediction.

This paper deserves a serious referee. The analytic part is solid and the qualitative physics is likely robust; the experimental section needs revision, ideally with code and data, a sensitivity analysis over kappa_diss,LE and d, and a more careful justification of the offset treatment. I would cite the analytic model if I worked on charge generation modeling, and I would bring it to a reading group focused on OPV kinetics. My recommendation: send it to review, but expect major revision on the fitting and uncertainty quantification.

Referee Report

3 major / 5 minor

Summary. This paper develops a deterministic continuum model that couples exciton diffusion in a donor or acceptor domain to interfacial rate equations describing CT state formation, reformation, geminate recombination, and dissociation to free charges. The authors derive an analytical steady-state charge generation efficiency (Eqs. 17-20), analyze characteristic timescales from a dynamic formulation, and fit the model to transient absorption (TAS) data of a PM6:Y6 blend with the Y6 domain radius R and the CT dissociation rate k_diss,CT as the only free parameters. The fit yields R = 12.47 ± 0.71 nm (diameter ≈ 25 nm) and k_diss,CT = 1.12 ± 0.44 ps^-1. The central qualitative conclusion is that the common assumption of instantaneous exciton quenching at the donor-acceptor interface is only valid for high driving forces, and that the intrinsic exciton lifetime is a key parameter enabling efficient generation at low offsets.

Significance. The model is a useful conceptual advance: it combines a tractable diffusion problem with Marcus-theory rates at the interface, and the steady-state expression (Eq. 20) is analytic and internally consistent. The qualitative finding that finite interfacial quenching, not pure diffusion, controls the charge-generation timescale in low-offset systems is well supported by the mean-survival-time analysis in Sec. 4 and by the illustrative simulations. If the quantitative TAS-based domain-size estimate can be made robust, the paper would also provide an experimentally applicable method for estimating domain sizes from charge-generation dynamics. The manuscript is likely to be of interest to the organic photovoltaics community.

major comments (3)

[Sec. 5, Fig. 4b] The constant 20% offset subtraction is internally inconsistent with the authors' own interpretation of the late-time GSB decay. The model predicts a GSB that starts at zero, while the measured GSB exhibits an instantaneous component that the authors subtract as a time-independent offset and attribute to disordered Y6 domains. However, the same paragraph states that the observed decline in the GSB after ~200 ps is caused by geminate recombination of CT states in these disordered Y6 phases. If those CT states decay, their GSB contribution is time-dependent, so a constant offset subtraction cannot remove the instantaneous component while leaving the decaying component from the same phase. Since the fitted values of R and k_diss,CT depend directly on the amplitude and shape of the normalized rising signal, this data treatment is load-bearing for the quantitative claim. The authors should model the disordered-phase contribution with explicit time-dependent kinetics, or at minimum provide a sensitivity analysis showing that the fitted R and k_diss,CT are stable under alternative, physically motivated treatments of the offset.
[Sec. 5, Fig. 4c] The exclusion of EA data before 0.3 ps is not justified. The EA signal is proportional to the CS population, and its early rise provides direct constraints on k_diss,CT; truncating the early data can bias both the fitted value and the reported uncertainty. The authors should state the reason for the cutoff (e.g., coherent artifact or pump scatter) and test the sensitivity of R and k_diss,CT to the choice of truncation threshold.
[Sec. 5, Eq. (28)] The reported uncertainties on R and k_diss,CT are only the statistical uncertainties of the least-squares fit. The authors acknowledge in the text that kappa_diss,LE and R are strongly correlated when fitted simultaneously, and kappa_diss,LE itself depends on hand-set parameters (d = 1 nm), adopted parameters (|H_DA|, lambda), and the assumed Delta_E_LE-CT. The claim of a 25 nm domain diameter therefore rests on a chain of assumptions whose uncertainty is not propagated. A sensitivity analysis varying d over a reasonable range and using the literature spread of lambda and |H_DA| should be provided before this quantitative result can be considered established.

minor comments (5)

[Sec. 5 vs. Materials and Methods] The main text states that the excitation wavelength is around 800 nm to selectively excite Y6, while the Materials and Methods section describes pump pulses at 720 nm. These statements should be reconciled.
[Supplementary Note SN2, Eq. (S11)] The equation for dn_CT/dt contains a plus sign before the decay terms (kref + kf,CT + kdiss,CT)nCT, whereas the correct equation in the main text, Eq. (11), has a minus sign. This typo should be corrected.
[Sec. 5, Eq. (27)] The factor '(ratio + 1)' in the expression for n_LE,0 is not defined explicitly; the relationship between the PM6:Y6 mixing ratio and the volume fraction of Y6 should be clarified so that the initial density calculation is reproducible.
[Supplementary Note SN6] The sentence 'The final or the second (suitable if k−1 f,LE≫⟨t⟩f) equation of table ST1 can then be used to estimate R' is difficult to parse; please rephrase to clearly identify which limiting case corresponds to which equation.
[Fig. 3 caption] The phrase 'The respective solid line shows' should be 'The solid lines show' for consistency with the plural entries in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the model derivation is self-contained and the PM6:Y6 analysis is an explicit parameter fit, not a disguised prediction.

full rationale

The paper's central derivation is self-contained: the diffusion/rate-equation system (Eqs. 1-6) is defined independently, and the steady-state efficiency expression (Eq. 20), characteristic-time analysis, and mean survival-time formulas in SN6 are obtained by explicit analytic solution of that system. No equation in the derivation is equivalent to its own input by construction. The Marcus-theory rates (Eq. 22) are standard external inputs, and the paper states clearly which parameters are adopted from prior literature, including |H_DA| from ref. 19. The application to PM6:Y6 TAS data is presented as a fit: 'We fit the model to the experimental data using a non-linear least squared fitting method' (Sec. 5), with R and k_diss,CT as free parameters. Because the domain size and CT dissociation rate are estimated by fitting, they are not claimed as independent predictions, so the 'fitted input called prediction' pattern does not apply. The subtraction of the ~20% ultrafast GSB offset and the single-sphere assumption are modeling/data-treatment choices that could bias the fitted values, but they do not make the derivation circular; the paper even warns about the strong correlation between kappa_diss,LE and R. Self-citations, mainly ref. 19 for the electronic coupling and the reference rate model, are used as external parameter sources and for comparison, but the qualitative conclusion that finite interfacial quenching matters for low-offset systems follows from the model's structure and the MFPT analysis rather than from the cited values. No self-citation chain or uniqueness theorem is invoked to force the choice of model. Therefore no circular step is exhibited, and the appropriate score is 0.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The model rests on standard diffusion and Marcus-theory assumptions plus several externally chosen parameters. The application adds a post-hoc separation of aggregated and non-aggregated phases. No new physical entities are postulated.

free parameters (8)

d (CT localization zone thickness) = 1 nm
Hand-set to 1 nm; enters kappa_diss,LE = k_diss,LE * d and controls the interfacial dissociation velocity.
lambda (reorganization energy) = 0.45 eV
Chosen as intermediate between reported values of 0.33 eV and 0.55 eV; used in the Marcus rates of equation (22).
|H_DA| (electronic coupling) = 0.01 eV
Taken from Pranav et al. [19], a paper with overlapping authors; not independently measured here.
k_f,LE (singlet decay rate) = 1 ns^-1
Estimated within the range of literature values; a central parameter for the model's conclusions about exciton lifetime.
k_f,CT (CT decay rate) = 1 ns^-1
Assumed equal to the singlet decay rate; acknowledged as less important in the parameter hierarchy.
Delta_E_LE-CT (energy offset) = 0.11 eV (PM6:Y6)
Taken from Sun et al. [18], a paper with overlapping authors.
R (Y6 domain radius, fitted) = 12.47 +/- 0.71 nm
Fitted to the TAS GSB and EA kinetics; corresponds to a domain diameter of about 25 nm.
k_diss,CT (CT dissociation rate, fitted) = 1.12 +/- 0.44 ps^-1
Fitted to the TAS kinetics together with R.

assumptions (7)

domain assumption Exciton hopping above 150 K is an isotropic diffusion process with constant diffusivity D (equation 1).
Invoked in Section 2 for all temperatures of interest in organic solar cells.
domain assumption Dissociation and reformation at the donor-acceptor interface obey the linear Robin boundary condition (equation 2) with a CT surface density.
Justified by Collins-Kimball and de Falco; the surface density and kappa_diss,LE = k_diss,LE * d are assumed.
domain assumption Charge transfer rates are given by semi-classical Marcus theory with a single reorganization energy and electronic coupling (equation 22).
Used throughout Sections 3-5; the parameters are externally chosen rather than derived.
domain assumption Direct generation into CT states and non-geminate recombination are negligible (G_CT = 0).
Stated in Section 2 as reasonable under one sun and low fluence conditions.
domain assumption The BHJ is represented by a single spherical acceptor domain fully surrounded by donor, with uniform exciton generation.
Stated in Section 2; reduces the problem to one radial dimension.
domain assumption The sample domain is an average over many local configurations, so a deterministic continuum density is valid.
Invoked in Section 2 following Erban and Chapman [25].
ad hoc to paper The ultrafast GSB component from non-aggregated Y6 regions can be subtracted as a constant offset, leaving only aggregated-domain dynamics.
Introduced in Section 5 to reconcile the data with the slower Marcus rate; no independent evidence for this separation is provided.

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Cite this review

Pith. "Pith review of A combined diffusion/rate equation model to describe charge generation in phase-separated donor-acceptor blends." pith.science (2026). https://pith.science/paper/ZLK4FRNI

@misc{pith2026241108812,
  author       = {Pith},
  title        = {Pith review of: A combined diffusion/rate equation model to describe charge generation in phase-separated donor-acceptor blends},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZLK4FRNI}},
  note         = {Machine review of arXiv:2411.08812}
}

read the original abstract

The power conversion efficiency (PCE) of organic solar cells (OSCs) has been largely improved by the introduction of novel non-fullerene acceptors (NFAs). Further improvements in PCE require a more comprehensive understanding of the free charge generation process. Recently, the small PCE of donor-acceptor blends with low offsets between the relevant frontier orbitals was attributed to inefficient exciton dissociation. However, another source of photocurrent loss is the competition between exciton diffusion and decay, which is particularly relevant for bilayers or bulk heterojunction blends with phase separated morphology. Here, we present an analytical model that combines exciton diffusion with a set of rate equations based on Marcus theory of charge transfer. An expression for the charge generation efficiency is derived from the steady-state solution of the model. Thereby, the intrinsic exciton lifetime is identified as a pivotal parameter to facilitate efficient charge generation in spite of a vanishing driving force for exciton dissociation. The dynamic formulation of the model is used to elucidate the characteristic time scales of charge generation. It is found that for low-offset systems, the pure diffusive times are considerably shorter than those associated with charge generation. It can therefore be concluded that when estimating domain sizes via exciton diffusion measurements, the assumption that excitons are instantaneously quenched at the donor-acceptor interface is only valid when a high driving force for exciton dissociation is present. The model is applied to the transient absorption dynamics of a PM6:Y6 blend. It is demonstrated that the charge generation dynamics are determined by the interplay between exciton diffusion and hole transfer kinetics, with an estimated Y6 domain size of 25nm, while interfacial charge transfer (CT) states separate rapidly into free charges.

Figures

Figures reproduced from arXiv: 2411.08812 by the authors.

**Figure 1.** a) General geometry of the model. Ω denotes the set attributed to the acceptor domain. Γd (red) describes the subset of ∂Ω that enables exciton dissociation while Γr (dark grey) models a reflecting boundary. ν denotes the outward unit normal vector on ∂Ω. The complementary donor domain, although not explicitly included in our model, is indicated by the blue shaded area. b) Schematic illustration of the combined char… view at source ↗

**Figure 2.** The charge generation efficiency ηgen,CS as a function of the energetic offset between the local exciton and the interfacial CT state ∆ELE−CT using Marcus theory. kf,CT = 1 1000 ps−1 , kdiss,CT = 0.1 ps−1 , d = 1 nm, λ = 0.45 eV, |HDA| = 0.01 eV and T = 293.15K. a) The domain size radius R is varied from 10 nm (dark) to 50 nm (light) in steps of 10 nm while the diffusion length LD ≈ 31.6 nm (kf,LE = 1 1000 ps−1 , D … view at source ↗

**Figure 3.** kf,LE = kf,CT = 1 1000 ps−1 , kdiss,CT = 0.1 ps−1 , D = 10−2 cm2 s −1 , d = 1 nm, λ = 0.45 eV, |HDA| = 0.01 eV and T = 293.15K. a) The temporal evolution of the sum of CT and CS normalised to the number of initially generated excitons for different driving forces in a TAS-like simulation. The energetic offset ∆ELE−CT is varied from 0 eV (dark) to 0.25 eV (light) in steps of 0.05 eV. The respective solid line shows t… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: TAS spectra and kinetics of a BHJ PM6:Y6 blend. a) Smoothed spectrum for different delay times. The GSB of PM6 and the EA signal are indicated by the respective arrows. b) The kinetics of the GSB of PM6 obtained by averaging the signal between λp = 580 nm and λp = 590 …

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Pith tools

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