the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 13 Jul 2026
Determination of water diffusion coefficients in aerosols based on characteristic time analysis
Abstract. The water diffusion coefficient in atmospheric particles is a key parameter characterising particle-phase water transport and is essential for understanding aerosol phase state, phase transitions, and multiphase chemical processes. In this study, we develop a characteristic-time-based method for directly measuring the water diffusion coefficient in an individual droplet. The progress of water diffusion is represented by the relative abundance of H2O and D2O within the droplet, which is retrieved from Raman spectra and quantified here as the D2O fraction. The characteristic time is derived as the time at which the measured D2O fraction reaches the value predicted by the aqueous-phase diffusion model. Since the characteristic time derived from this model depends only on the water diffusion coefficient and droplet radius, the water diffusion coefficient can be calculated directly from the measured characteristic time and droplet radius. Using this method, water diffusion coefficients in sucrose droplets at 30–45 % RH were determined to be 1×10-16 to 1×10-14 m2s-1. The water diffusion coefficient showed a clear RH dependence, with lower coefficients observed at lower RH. These results are consistent with previous experimental measurements, supporting the reliability of our method. A key advantage of this method is that it does not require tracking the complete H2O/D2O exchange process, as measurements are only needed until the characteristic time, thereby shortening the experimental observation time and making the method particularly suitable for diffusion measurements in highly viscous aerosol particles. This method applies to both spherical droplets and non-spherical diffusion systems, which allows it to be adapted to different experimental platforms. It therefore provides a basis for understanding mass transport in aerosols and related atmospheric chemical processes.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-3646', Anonymous Referee #1, 27 Jul 2026
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AC1: 'Reply on RC1', Shuqi Guo, 14 Sep 2026
We systematically reviewed the relevant literature and supplemented the manuscript with additional model discussions and theoretical analyses to address and improve every issue raised by the reviewer. We provide detailed point-by-point responses below, including full derivations, supplementary analyses, and additional figures and tables in the Supplementary Material.
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AC1: 'Reply on RC1', Shuqi Guo, 14 Sep 2026
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RC2: 'Comment on egusphere-2026-3646', Anonymous Referee #2, 31 Jul 2026
This manuscript presents a characteristic-time-based method for determining water diffusion coefficients in individual sucrose droplets using Raman H2O/D2O isotope-exchange measurements. The approach is potentially useful because measurements of very slow diffusion processes can require long experimental observation periods. The theoretical framework appears internally consistent under the assumptions adopted, and the measured diffusion coefficients are broadly comparable with previous sucrose measurements.
However, several methodological assumptions have not been sufficiently validated. Please see my comments below.
1. The principal advantage claimed for the proposed method is that it can determine Dw without observing the complete isotope-exchange process. However, the manuscript does not compare the characteristic-time method with conventional full-curve fitting using the same experimental datasets. The authors should apply both methods to the available experiments and compare the resulting diffusion coefficients, uncertainties, and required observation times. Agreement with previous literature alone is not sufficient to validate the new retrieval method. Alternatively, the authors should explain well why such experiments on full-curve fitting has not been done, or why the current methodology is enough to support your statements.
2. Section 5.1 primarily describes similarities and differences between the present results and previous studies, but does not sufficiently explain what these comparisons imply for the reliability of the proposed method. The discussion should focus more directly on the authors’ results, methodological assumptions, and limitations. Claims regarding non-spherical systems, different experimental platforms, low temperatures, and RH conditions below the measured range should be moderated because these applications were not demonstrated in the present study.
3. Figure 4 includes a curve labelled “Fit,” but the fitting method and fitted parameters are not explained. It is also unclear whether the reported Dw was derived from this fit or from the characteristic-time method. The authors should define the fit and use it to compare the two retrieval approaches where possible. The reduction in observation time should also be quantified rather than described only qualitatively.
Minor comments:
1. Lines 30 to 34 move rapidly from hindered molecular diffusion to particle size distributions, gas-particle partitioning, chemical kinetics, ice nucleation, human health, air quality, and climate. The mechanistic links should be explained more clearly, or the broader claims should be narrowed.
2. Line 44, the sentence would read more naturally as “Price et al. (2014) subsequently developed...”.
3. RH is defined earlier in the Introduction. The use of “RH” and “relative humidity” should be made more consistent throughout the manuscript. The authors used "relative humidity" throughout the manuscript in many places.
4. The final paragraph of the Introduction should be revised to distinguish the study objective, the main finding, and potential future applications. The statement that the results are “consistent with previous studies” is too general.
5. Please standardise the use of “temperature – humidity,” “temperature-humidity,” “O – D,” “O–D,” “O – H,” and “O–H.” “Temperature and humidity probe” would be clearer.
6. Numerical and unit formatting should be standardised.
7. Please specify whether the upstream or downstream RH probe was used to determine the stabilisation time.
8. At Line 126, “Where” should be changed to “where.”
9. The explanation following the captions of Figs. 3 and 4 repeats information already provided in the captions. The paragraph should be shortened and focused on additional interpretation.
9. The Fig. 5 legend appears to use “Davis et al.” This should be corrected to “Davies and Wilson.” Similarly, “Davies et al.” is not appropriate for a publication with two authors.
10. There is a missing space before the citation at approximately Line 320, “concentration(Zobrist et al....”.
11. The statement that differences among previous studies may reflect “experimental platforms, instrumental performance, and environmental control” is too general. Specific mechanisms should be discussed.
12. The statement that the data are “well described” by the Vignes-type parameterization should be supported by quantitative fit statistics.
13. The figures resolution are somehow low, please increase the resolution (such as 600 dpi or 900 dpi).
Citation: https://doi.org/10.5194/egusphere-2026-3646-RC2 -
AC2: 'Reply on RC2', Shuqi Guo, 23 Sep 2026
We thank the reviewer for these detailed comments. We expanded the manuscript to address the methodological comparison, key model assumptions, and the proposed approach's range of applicability. First, we applied the characteristic-time (CT) method and the conventional full-curve fitting (FCF) method to the same experimental datasets and directly compared the resulting Dw values, including uncertainties from random fluctuations in the experimental data. The comparison shows that the two methods retrieve Dw values that follow a near 1:1 relationship. CT yields absolute Dw values comparable to those from FCF and reproduces the relative variation in Dw across experimental conditions. The half-widths of the corresponding 95% confidence intervals are also similar for both methods, indicating that CT retains retrieval stability comparable to FCF while requiring less observation duration.
Second, we quantified the reduction in observation duration achieved by CT. At present, no generally accepted criterion is available for defining the minimum observation duration required for FCF. We therefore progressively increased the length of the data included in the fit and examined the convergence of both the retrieved Dw and its associated uncertainty. Based on this, we determined the minimum observation duration sufficient for stable FCF retrievals. In our experiments, CT reduced the required observation duration by approximately 54.0%–73.8% relative to FCF, with a mean reduction of approximately 65.7%. These results provide a quantitative assessment of the reduction in experimental record achieved by CT.
In addition, we further examined two key assumptions underlying the model. For the initial concentration profile, we clarified the mathematical and physical basis for adopting a quadratic approximation, while noting that the experiment did not directly measure the full radial concentration distribution and that the quadratic profile therefore remains an approximation. For the concentration-averaging treatment, we further clarified that the choice between radial and volume averaging should be dictated by the measurement's spatial sampling characteristics. In response to a related comment from another reviewer, we also examined the potential influence of a small laser-beam offset from the droplet centre on the retrieval. Because the optical tweezers provide stable three-dimensional confinement of the trapped droplet, substantial displacement relative to the beam axis is constrained under stable trapping conditions. We nevertheless considered an offset corresponding to 20% of the droplet radius as a sensitivity test. Even at this comparatively large offset, the retrieved Dw changed only slightly, further supporting the validity of the radial-averaging approximation under the present experimental conditions.
Based on these analyses, we revised the discussion to focus more directly on this study's experimental results, the assumptions underlying the method, and their associated limitations. We also revised the discussion about potential applications to non-spherical systems and other experimental platforms. The detailed procedures and results are provided in the Supplement PDF.
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AC2: 'Reply on RC2', Shuqi Guo, 23 Sep 2026
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RC3: 'Comment on egusphere-2026-3646', Anonymous Referee #3, 10 Aug 2026
Guo et al. present a characteristic time-based analytical framework to constrain water diffusion coefficients (Dw) in viscous droplets. By leveraging eigenfunction solutions to Fick’s second law, the authors demonstrate that the isotope fraction at characteristic time (τ) scales linearly with the initial spatial distribution. The method provides a promising route to shorten experimental runtimes in complex conditions. The theoretical derivation is generally sound and clearly presented. The manuscript is suitable for publication after addressing the following questions.
Major comments:
- Defining the precise onset of RH stabilization is critical, as it directly fixes the time origin for calculating τ and the derived Dw. Currently, the choice of five characteristic times (5ω) for RH stabilization appears empirical. The authors should provide a clear physical mechanism or appropriate literature references to justify this criterion, or adopt a more rigorous mathematical definition for the stabilization point. Moreover, using the term "characteristic time" for both the experimental RH response parameter (ω) and the isotope diffusion timescale (τ) introduces unnecessary confusion.
- The authors should include, for at least a subset of the measured droplets, a side-by-side comparison of Dw values obtained via the new characteristic-time method and conventional fitting of the entire D2O-fraction time series. Such comparison would provide a more direct and convincing validation of the proposed method, rather than relying solely on qualitative agreement with literature values obtained under different conditions and platforms.
- The Introduction and Discussion repeatedly emphasize that the method is particularly advantageous for slow-diffusion (low-RH, high-viscosity) systems where conventional isotope-tracer measurements become impractical. However, the RH range investigated here (30–45%) does not extend into this regime. I recommend that the authors extend the measurements to lower RH (or lower temperature) to directly demonstrate the method's capability under conditions where conventional approaches fail.
Minor suggestions:
- In Figure 3, the x-axis should be extended to show data prior to the D2O flow switch to illustrate the initial perturbation dynamics.
- In Figure 5, "Davis et al." should be corrected to "Davies and Wilson".
Citation: https://doi.org/10.5194/egusphere-2026-3646-RC3 -
AC3: 'Reply on RC3', Shuqi Guo, 24 Sep 2026
We thank the reviewer for the detailed and constructive comments. In response, we clarified the rationale and criteria for determining the experimental starting point and added relevant references to support this approach. We also directly compared Dw values obtained using the conventional full-curve-fitting (FCF) method with those obtained using the proposed characteristic-time (CT) method. In addition, we revised relevant statements throughout the manuscript to more clearly distinguish conclusions directly supported by the experimental data obtained under the conditions investigated in this study from the method's potential advantages under conditions not yet experimentally examined. Because several of these additions involve equations and figures, we provide detailed point-by-point responses and the corresponding supplementary analyses in the attached PDF.
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RC4: 'Comment on egusphere-2026-3646', Anonymous Referee #4, 17 Aug 2026
The manuscript Determination of water diffusion coefficients in aerosols based on characteristic time analysis develops a characteristic-time-based method for directly measuring the water diffusion coefficient in an individual droplet. The method is useful, but several points require clarification.
- In line 57, what does “these two experimental systems” refer to? Similarly, which method does “measurements” refer to in line 59?
- Why does the combination of optical tweezers and Raman isotope tracing show a relatively short equilibration time? What process does “equilibration” describe here?
- Is the diffusion process temperature-dependent?
- At RH < 30%, the results obtained using the characteristic-time-based method differ considerably from those of Zobrist et al. Which results are more reliable?
Citation: https://doi.org/10.5194/egusphere-2026-3646-RC4 -
AC4: 'Reply on RC4', Shuqi Guo, 24 Sep 2026
We sincerely thank the reviewer for the detailed and constructive comments. In response, we clarified several statements in the manuscript that were previously insufficiently defined and expanded the discussion of discrepancies among the reported data. Because these revisions involve figures, references, and detailed manuscript changes, we provide point-by-point responses to each comment in the supplementary PDF.
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This manuscript presents an innovative, time-saving methodology for directly measuring the water diffusion coefficient within single levitated aerosol droplets. Understanding internal mass transport limitations in highly viscous or semisolid atmospheric particles is an active area of atmospheric chemistry, and this study addresses a major bottleneck: the prohibitive experimental observation timescales required by traditional full-exchange Raman isotope tracer methods.
By framing the calculation around a mathematically derived characteristic diffusion time rather than a complete fit of the exchange curve, the authors demonstrate that can be successfully retrieved using only the initial segment of the diffusion profile. This offers a major technical advantage for probing highly viscous states at low relative humidity (RH) or low temperatures where diffusion is exceptionally slow. The manuscript is well-structured, mathematically rigorous, and the results for sucrose droplets correlate nicely with literature trends. I recommend this manuscript for publication after addressing the following specific points.
Specific Comments