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REVIEW 3 major objections 4 minor 44 references

A hybrid neural network predicts spatial-ALD coverage in milliseconds and reveals that kinetic-inversion precision is set by parameter degeneracy, not fitting power.

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2026-08-04 01:00 UTC pith:GXYJYVPR

load-bearing objection Careful, self-critical hybrid-surrogate plus identifiability paper on a synthetic SALD benchmark; the analytic slope law is a real increment, but the diagnostic's statistical calibration is weaker than the abstract claims and there is no experimental validation. the 3 major comments →

arxiv 2608.00212 v1 pith:GXYJYVPR submitted 2026-07-31 cs.LG physics.comp-ph

A Physics-Chemistry-Informed Neural Network (PCINN) for Real-Time Spatial-ALD Coverage Prediction and Reliable Kinetics Inversion

Ning Hu , Chang Liu , Yunlei Jiang , Yuan Dong This is my paper
classification cs.LG physics.comp-ph
keywords physics-informed neural networkspatial atomic layer depositionsurface coverage predictionkinetic inversionparameter identifiabilityprofile likelihoodArrhenius degeneracyhybrid gray-box modeling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that a physics-chemistry-informed neural network (PCINN) can serve as a real-time surrogate for spatial atomic layer deposition (SALD) coverage prediction while keeping kinetic parameter inversion reliable. The central claim is that the precision boundary of the inversion is controlled by the degeneracy structure of the parameter space, not by the network's fitting power. It derives an analytic slope law for the multi-temperature degeneracy between the Arrhenius prefactor and adsorption energy — dE_ads/dlog10ν = k_B T_eff ln10 — that is nearly invariant under any mismatch preserving a single Arrhenius process and shifts only when a second thermally activated process appears. If correct, PCINN gives millisecond coverage predictions with R²_log ≈ 0.998 from only 30 training cases, and the identifiability limits of hybrid inversion can be characterized analytically and tested statistically.

Core claim

PCINN divides the inverse problem between a small neural network that learns the operating-condition to near-wall concentration closure and a hard-coded Langmuir–Arrhenius chemistry layer integrated along the substrate trajectory, compressing the data-driven freedom to a single scalar. The central discovery is that at a single temperature the adsorption rate k_ads is not separately identifiable (only the product k_ads·c_wall is), and across multiple temperatures the prefactor ν and adsorption energy E_ads remain bound along a weakly identifiable degeneracy valley whose slope is predicted analytically as k_B T_eff ln10 — about 0.065 eV/decade for the 300–360 K window. The slope persists under

What carries the argument

The central object is the PCINN hybrid architecture: a tiny MLP physics branch that maps (v_sub, U_curtain, T) to an effective near-wall concentration C*_s, coupled to a hard-coded, trainable Langmuir kinetics layer that integrates coverage along the substrate trajectory. The key identity is the analytic degeneracy slope dE_ads/dlog10ν = k_B T_eff ln10, derived directly from the Arrhenius form with T_eff the harmonic mean of the sampled temperatures; this slope fixes the geometry of the ν–E_ads valley and doubles as a reliability diagnostic.

Load-bearing premise

The entire identifiability boundary and slope diagnostic rest on the premise that the real SALD surface chemistry is well described by the same single-site Langmuir–Arrhenius kinetics used to generate and invert the data; if coverage-dependent barriers, adsorption-side nonlinearity, site heterogeneity, or temperature-dependent transport are present in reality, the reported parameters and the slope threshold could shift.

What would settle it

Measure the multi-temperature degeneracy slope from real spatial-ALD coverage data (or from a well-characterized surface with known two-site heterogeneity) using the same PCINN profile-likelihood pipeline; if the empirical slope stays within the single-Arrhenius band µ±1.64σ while an independent spectroscopic measure confirms site heterogeneity, the slope diagnostic's specificity fails. Conversely, a clean single-site surface whose measured slope departs from k_B T_eff ln10 beyond the threshold would falsify the law.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Coverage can be predicted in about 7 ms per query, roughly 5×10^4 times faster than a high-fidelity CFD solve, with test R²_log ≈ 0.998 from only 30 training cases, enabling real-time operating-window scans and control-loop deployment.
  • At a single temperature, k_ads is not separately identifiable; only the product k_ads·c_wall is constrained. At multiple temperatures ν and E_ads remain bound along a weak valley, and E_ads is recovered to within 0.3% only when ν is conditioned on.
  • A measured degeneracy slope departing from k_B T_eff ln10 is a falsifiable flag for unmodelled site heterogeneity or another thermally activated process, even when the surrogate fit remains excellent.
  • The embedded physics provides extrapolation gains only along the structurally known residence-time axis, not along the data-driven transport axis, delineating precisely where the learned closure does and does not help.
  • The identifiability conclusions and the slope diagnostic persist under moderate model mismatch, including desorption-side coverage dependence, adsorption-side nonlinearity, and non-Fickian transport, with the valley flattening and conditional E_ads degrading by only about 1%.

Where Pith is reading between the lines

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

  • The slope law is likely to transfer to any single-channel activated-rate inversion beyond ALD surface kinetics, offering a design tool: the harmonic-mean temperature window determines how easily a prefactor and activation energy can be separated, and widening that window directly tightens the inferable interval.
  • A testable extension is to apply the slope diagnostic to real SALD thickness data: if multi-temperature coverage measurements from a production reactor give a slope within the single-Arrhenius band, that supports a single-site Langmuir model; a departure would indicate hidden site heterogeneity even if the surrogate fit is excellent.
  • The single-scalar bottleneck suggests a natural route toward field-level prediction — replacing the trajectory-averaged scalar with full spatial coverage fields via a neural operator — while retaining the same per-parameter identifiability analysis on each output dimension.
  • The paper's own boundary analysis implies that a temperature-dependent transport mismatch (e.g. diffusivity or viscosity varying with temperature) would shift the degeneracy slope just as a second Arrhenius process would, so the diagnostic may also catch thermal-transport errors — an inference the authors explicitly leave to future work.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes PCINN, a hybrid physics-chemistry-informed neural network for spatial ALD coverage prediction. A small MLP learns only the operating-condition-to-effective-near-wall-concentration closure, while a hard-coded Langmuir-Arrhenius chemistry layer integrates coverage along the substrate trajectory. From 30 CFD-generated training cases the surrogate achieves test R^2_log ≈ 0.998 and millisecond inference. The paper's main methodological contribution is an identifiability analysis (Fisher information, profile likelihood, initial-value drift) showing that k_ads is not separately identifiable at a single temperature and that, across multiple temperatures, ν and E_ads are bound along a degeneracy valley whose slope is derived analytically as k_B T_eff ln10. This slope is proposed as a diagnostic flag for unmodelled site heterogeneity, supported by a seven-chemistry mismatch matrix. The authors explicitly frame the study as a simulation-based concept verification with known Langmuir-Arrhenius ground truth and no experimental validation.

Significance. If the claims are sustained, the paper offers a useful interpretable surrogate for SALD and, more importantly, a transferable identifiability workflow for hybrid neural-ODE models. The analytic slope law is an elegant and nontrivial result: it connects the geometry of the ν–E_ads degeneracy to the temperature sampling window and is robust to the form of a known prefactor. The paper is unusually honest: it discloses the saturated-end 13–36% bias, the non-convergence of the near-wall auxiliary concentration, the parking-artifact of free-inversion point estimates, and the circularity of simulation-based validation. The LOOCV, prior-ablation, bootstrap, and coarse-grid checks strengthen confidence in the surrogate and in the basic identifiability conclusions. However, the central operational claim — the statistically calibrated false-positive rate of the slope diagnostic — is not supported by the evidence presented, and the abstract overstates the diagnostic's universality.

major comments (3)
  1. [§7.3, Table 7, Fig. 10] The statistical calibration of the slope diagnostic is not established. The null distribution is a convenience sample of seven hand-picked single-Arrhenius cases, not draws from a well-defined ensemble of plausible mismatches. The Gaussian tail probability (p≈1.5×10^-5) and the 5%/1% thresholds (μ+1.64σ, μ+2.33σ) assume normality and a known σ. With n=7, the sampling error in σ is large (SE(σ)≈0.0007), and a 95% tolerance bound for the 95th percentile is roughly μ+3.4σ≈0.0736 eV/decade, only ~0.002 below the dual-site signal 0.0755; for the 1% false-positive rate the bound can exceed the signal. The set of null cases also mixes exact-model and mismatched single-process cases, and the selection/exclusion of rows (e.g., excluding Temkin β=4 but including Freundlich n=0.5 with degraded R²=0.86–0.99) is ad hoc. The paper should either define a proper null ensemble and report nonparametric to
  2. [Abstract, §1.3, §7.3, Fig. 11a] The abstract and contribution statement claim the degeneracy slope 'shifts only when a second thermally activated process is introduced.' This is contradicted by the paper's own energy-split sweep: dual-site models with ΔE=0.04 and ΔE=0.148 eV give slopes of 0.0663 and 0.0673, both inside the single-process band (0.063–0.070). The body text later correctly states the slope is 'specific but not universally sensitive' and that absence of a slope excursion does not exclude heterogeneity. The abstract and Section 1.3 should be reworded to present the slope as a one-sided flag — a departure implies heterogeneity, but non-departure does not imply its absence — and the bi-conditional language should be removed.
  3. [§1.3, §8.2, Abstract] The entire quantitative validation is generated by simulation from the same Langmuir-Arrhenius kinetic form that is used for inversion. The paper discloses this clearly and positions the work as a concept verification, which is commendable. Nevertheless, the title and abstract's 'reliable kinetics inversion' overstate what is demonstrated: the reliability is established only within a simulated model world, with a mismatch matrix covering a limited selection of kinetic/transport perturbations and no experimental data or experimental uncertainty model. The authors should temper the reliability language in the title/abstract, or explicitly add a qualifier such as 'in simulation' to the reliability claim. This is not a request for new experiments, but for a scope-bound statement consistent with the evidence.
minor comments (4)
  1. [§4.5, Table 1] The auxiliary near-wall concentration ⟨c⟩/C0 used to anchor the learned closure does not converge under mesh refinement (0.0148→0.0123→0.0206 across m_f=2,4,8), yet the production mesh is m_f=2. The paper's argument that the identifiable E_ads is read from the temperature slope and is mesh-stable is plausible, and the coarse-grid experiment supports it. Still, the non-converged supervision is a source of uncertainty in the learned C*_s and effective k_ads; this should be acknowledged more directly as an uncertainty, not only as a benign artifact.
  2. [§6.1 vs §5.4] The abstract reports test R^2_log=0.998, while Section 6.1 gives 0.9975±0.0005 over 8 seeds. The former is an appropriate single-run highlight, but the abstract should note it is one representative run or a rounded summary.
  3. [§7.3, Table 7] The text says strong single-process mismatch (Temkin β=4, Freundlich n=0.5) causes R^2_log to fall to 0.69–0.86, but Table 7 lists Freundlich n=0.5 as 0.86–0.99. The range is inconsistent; please clarify which value corresponds to which condition and whether the reference cluster in Fig. 10 includes Freundlich n=0.5 despite its degraded fit.
  4. [§6.7] The LOOCV R^2_raw=0.974 and maximum log error 0.187 are mentioned; providing the corresponding worst-case condition (which appears to be v_sub=1.2) as a table or explicit text would help readers locate the saturated-corner bias.

Circularity Check

0 steps flagged

No significant circularity: the analytic degeneracy slope is derived from the Arrhenius law, and the synthetic-data inversion is explicitly framed as a self-consistency check rather than a hidden prediction.

full rationale

The paper's central derivations do not reduce to their inputs. The slope law dE_ads/dlog10 nu = k_B T_eff ln10 (Eq. 6, Section 7.2) is obtained by direct differentiation of the Arrhenius relation under the stated assumption that data constrain k_des(T0); it depends only on the sampled temperature set and the exponential activation form, not on fitted parameter values or on the neural network. The empirical slope 0.0647 is compared with this analytic prediction 0.0652, which is a legitimate model-derived check rather than a self-fulfilling fit. The surrogate accuracy claim (R2_log = 0.998 from 30 training cases) is a standard train/test evaluation on CFD-generated labels; although the chemistry layer is hard-coded from the same governing equations that generated the data, the operating-condition-to-near-wall-concentration closure is learned from data, so the accuracy claim is an empirical interpolation result. The closest-to-circular element is that the simulated data are generated from the same Langmuir-Arrhenius form used for inversion. The paper explicitly and repeatedly acknowledges this: Section 1.3 states 'the data are generated by high-fidelity simulation from known Langmuir–Arrhenius ground truth and inverted with the same kinetic form,' and Section 8.2 states 'the data are synthetic measurements without real process data.' The study is therefore framed as a self-consistency and identifiability-boundary verification, not as discovery of real kinetic parameters. The mismatch matrix (Section 7.3) tests the slope diagnostic under independently generated Temkin, Freundlich, and dual-site chemistries; these are forward simulations designed to probe whether the diagnostic responds as predicted, not fitted quantities renamed as predictions. The paper also explicitly avoids ML-based data augmentation because it 'would form a circular validation loop and add no independent information' (Section 7.4), demonstrating awareness of the circularity hazard. No load-bearing self-citations or imported uniqueness theorems were found. The main weakness is that the claimed false-positive rate (~1.5e-5) is calibrated from only seven hand-picked null models with no defined ensemble, and the energy-split sweep shows the slope is not universally sensitive. That is a statistical validity concern, not a circularity concern. Overall, the derivation chain is self-contained and the paper's claims are appropriately scoped.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 1 invented entities

The central claims rest primarily on the assumed kinetic form and the single-scalar closure. No new physical entities are introduced; the learned C*_s is a latent closure variable. The free parameters are simulation truth, loss weights, noise, and temperature window.

free parameters (4)
  • Ground-truth kinetic constants (E_ads=0.774 eV, log10 ν=13, k_ads=0.01 m/s, k_des=1 s^-1 at 300 K) = chosen simulation truth
    Hand-selected inputs for the synthetic benchmark. The paper argues the identifiability geometry only translates with the truth, but the absolute numbers and the DFT-plausibility reference (0.5–1.1 eV) are not fitted to real data.
  • Loss weights w_C, w_m, w_p = 0.1, 0.1, 0.01
    Hand-set in Eq. (5). Only w_p (k_des prior) is ablated; w_C and w_m affect the learned C*_s and could influence the inversion, especially the k_ads·c_wall degeneracy.
  • Label noise σ = 5% relative noise on train/val labels
    Chosen noise level; noise robustness is tested only via added noise in Section 7.4, not against real experimental uncertainty.
  • Multi-temperature sampling window {300,320,340,360} K = T_eff = 328.5 K
    The slope law and the 0.6-decade interval are direct functions of this chosen window; the authors state a wider window would lose saturated anchoring cases.
axioms (7)
  • domain assumption Single-site Langmuir–Arrhenius surface kinetics: J_net = k_ads c_wall(1−θ) − k_des Γ_s θ, k_des=ν e^{-E_ads/k_B T}.
    Section 3 Eq. (2). Used both to generate synthetic data and as the hard-coded chemistry layer; the mismatch matrix tests deviations but real surface chemistry is not validated.
  • ad hoc to paper Two-segment trajectory closure: uniform effective concentration C*_s·C0 across the A-zone and c_wall=0 downstream.
    Section 5.3 Eq. (4). This is the surrogate's main modeling approximation and is stated to cause the 13–36% saturated-end bias; it is load-bearing for the coverage prediction accuracy.
  • ad hoc to paper Effective near-wall concentration is a single scalar function of operating conditions.
    Section 5.2. Compresses all transport modulation into one learned scalar; the paper shows this is the expressivity limit at high coverage.
  • domain assumption Steady 2-D incompressible laminar flow; temperature enters only through k_des (all transport coefficients T-independent).
    Sections 3 and 4.2. Verified for the simulation via identical wall shear rates across temperatures, but real heated SALD gaps have T-dependent diffusivity/viscosity, which the paper flags as untested.
  • domain assumption Observed data consist of substrate-averaged coverage θ̄_A only, with i.i.d. log-normal noise (σ=5%) on training labels.
    Section 4.7 and Section 5.4. The identifiability conclusions are conditional on this observation model; spatially resolved profiles would break some degeneracies.
  • standard math Fisher information computed as Gauss–Newton with σ^{-2} weights and profile likelihood using χ^2_1 likelihood-ratio threshold.
    Section 6.5 and Section 7.2. Standard statistical tools; assumptions of Gaussian noise and independent errors are not tested against real data.
  • standard math Multi-temperature degeneracy direction is weighted by 1/T, giving T_eff = harmonic mean.
    Section 7.2. The least-squares derivation is not shown; the paper asserts the weighting and confirms empirically (0.0647 vs 0.0652). Included as an unproved but standard algebraic step.
invented entities (1)
  • Learned effective near-wall concentration scale C*_s no independent evidence
    purpose: A single scalar output of the physics-branch MLP that represents the operating-condition-dependent effective precursor concentration feeding the hard-coded chemistry layer.
    Not a physically measured quantity; it is anchored in magnitude to the COMSOL surface-averaged near-wall concentration (about 5.5× higher) and justified by the leading-edge concentration, but it is a latent closure variable with no direct experimental handle. Flagging it keeps the modeling invention explicit.

pith-pipeline@v1.3.0-alltime-deepseek · 26855 in / 21438 out tokens · 204098 ms · 2026-08-04T01:00:15.766826+00:00 · methodology

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read the original abstract

Spatial atomic layer deposition (SALD) is a leading atmospheric-pressure, high-throughput route to industrial ALD, but design and control are limited by the cost of predicting surface coverage: high-fidelity CFD is far too slow for operating-window scans, while analytic models miss transport modulation such as the gas curtain. We present a physics-chemistry-informed neural network (PCINN), a hybrid surrogate with CFD-level accuracy at real-time speed: a query returns coverage in about 7 ms, roughly 5x10^4 times faster than a CFD solve, reaching a test R^2_log = 0.998 (leave-one-out R^2_raw = 0.974) from only 30 training cases spanning four orders of magnitude in coverage. The architecture is not a black box: a small network learns only the operating-condition to near-wall concentration closure, while the known surface kinetics is a hard-coded, trainable chemistry layer integrated along the substrate trajectory. This single-scalar bottleneck keeps it accurate under sparse data, interpretable and invertible. We add a full identifiability analysis (Fisher information, profile likelihood). The adsorption energy E_ads and desorption rate k_des are robustly identifiable; k_ads is not separately identifiable at a single temperature (only k_ads*c_wall is). Across four temperatures the prefactor nu and E_ads bind along a weakly identifiable degeneracy valley of slope 0.065 eV/decade, derived analytically as k_B T_eff ln(10) and turned into a reliability diagnostic: a seven-chemistry mismatch matrix shows it is invariant under any single-Arrhenius mismatch and shifts only when a second thermally activated process appears, so a slope departure flags unmodelled site heterogeneity. Data come from simulation with known ground truth inverted by the same kinetic form, so the study verifies pipeline self-consistency and the identifiability boundary, not real parameters.

Figures

Figures reproduced from arXiv: 2608.00212 by Chang Liu, Ning Hu, Yuan Dong, Yunlei Jiang.

Figure 1
Figure 1. Figure 1: PCINN architecture. A physics branch (small MLP) maps the operating condition (v ∗ ,U ∗ [,T ∗ ]) to an effective near-wall concentration C ∗ s ; a hard-coded chemistry layer with trainable kinetics integrates coverage along the substrate trajectory in two segments (A-zone exposure and downstream desorption) to yield θ¯ A. The data-driven freedom is compressed to a single scalar. 5.3. Chemistry branch and t… view at source ↗
Figure 2
Figure 2. Figure 2: Model-form bias versus coverage. (a) Raw relative prediction bias(θˆ −θ)/θ: a systematic under-prediction of 13–36% appears only at the saturated (high-coverage) end, the expressivity limit of the single-scalar near-wall closure. (b) The same residuals in log space log10(θˆ/θ) stay within the σ = 5% label-noise band across four decades of coverage (RMS 0.09); the bias is therefore real but log-compressed, … view at source ↗
Figure 3
Figure 3. Figure 3: Surrogate accuracy (representative run). Predicted versus true θ¯ A across about four orders of magnitude on the clean test set; PCINN reaches test R 2 log = 0.998 using only 30 training cases [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sample efficiency (primary caliber θ¯ A, two-segment). PCINN attains R 2 log ≈ 0.92 already at n = 6 training cases with small seed variance, whereas a structure-free MLP needs ∼ 30 cases to catch up and has an order-of￾magnitude larger variance at the sparse end. yields an exit coverage, as noted.) We stress that this is not a fair accuracy contest and we do not use it to inflate PCINN: the plug-flow mode… view at source ↗
Figure 5
Figure 5. Figure 5: Analytic baseline (Yanguas-Gil–Elam plug-flow self-limiting model). Only the slow-substrate/low-curtain saturated corner falls within 2× of the analytic value; the median over-prediction is 1.55 decades (up to 3.94 at high curtain), delineating the regime where resolved transport (PCINN) is required. Not separately identifiable. kads: the data constrain only the adsorption-flux product kadscwall, so kads i… view at source ↗
Figure 6
Figure 6. Figure 6: Extrapolation to held-out operating regions (8 seeds, mean±s.d.; dots are individual seeds). Along the substrate velocity vsub (residence time, carried by the chemistry branch) PCINN stays robustly positive (R 2 log = 0.90± 0.13, all seeds > 0) while the MLP is unstable; along the curtain velocity Ucurtain (the data-driven transport axis) both fail and are indistinguishable. The physics constraint helps on… view at source ↗
Figure 7
Figure 7. Figure 7: Single-temperature identifiability. Top: Fisher confidence ellipses—once the concentration scale is freed, the likelihood is nearly flat along the kadscwall-invariant direction. Bottom: profile likelihood—the kads profile has a clear minimum offset from the truth by ∼ 7× (minimum off-true), whereas the Eads profile minimum sits at the truth [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Deepened structural analysis. (a) Eigenvalue spectrum of the closure-free Fisher matrix—λ3 is an exact structural zero at the double-precision floor (|λ3|/λ1 ≈ 2 × 10−16). (b) Singular-value spectrum of the prediction￾sensitivity matrix ∂ fi/∂ βj , mirroring the hierarchy. (c) Sloppy-manifold visualisation—moving ±10% along the null direction leaves the model output essentially unchanged, while the same st… view at source ↗
Figure 9
Figure 9. Figure 9: Multi-temperature ν–Eads degeneracy valley. (a) Re-optimized Eads versus fixed log10 ν: the solutions slide along one degeneracy line with the ground truth on it, and conditioning on νtrue recovers Eads ≈ 0.776 eV. (b) Profile likelihood in log10 ν—a shallow one-sided valley; the shaded band is the heuristic “2× floor” region log10 ν ∈ [12.5,13.5], which corresponds to a very high (∼ 99.99%) confidence; th… view at source ↗
Figure 10
Figure 10. Figure 10: Statistical validation of the slope diagnostic. (a) The six single-Arrhenius generating chemistries (plus the non-Fickian transport mismatch) cluster at µ = 0.0655±0.0024 eV/decade; the dual-site (double-Arrhenius) signal at 0.0755 lies +4.2σ above the cluster (false-positive probability ∼ 1.5×10−5 ), well clear of the α = 5% (µ +1.64σ) and α = 1% (µ +2.33σ) flag thresholds. (b) Noise robustness: the sing… view at source ↗
Figure 11
Figure 11. Figure 11: Energy-split calibration and multi-signal detection. (a) Degeneracy slope versus the dual-site energy split ∆E: the excursion above the single-process band (µ ±σ) peaks at the intermediate split (∆E = 0.083, slope-flagged) and returns into the band at the small (0.04) and large (0.148) splits—a detection window predicted by the Teff theory. (b) Complementary signal: the profile-misfit floor. The large spl… view at source ↗
Figure 12
Figure 12. Figure 12: Degeneracy valley under model mismatch (profile likelihood in log10 ν). Across the seven-chemistry matrix the valley and its slope persist; the conditional Eads at νtrue degrades by at most a few percent for single￾Arrhenius mismatches, and only the dual-site (double-Arrhenius) case shifts the slope ( [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗

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