Self-similarity in creeping salt crystallization

M. Prat; N. Shahidzadeh; R.J. Wijnhorst

REVIEW 3 major objections 4 minor 1 cited by

Self-similarity in creeping salt crystallization

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Table-salt crusts are self-similar hierarchical porous media whose final height is set by the initial salt mass, not by capillary-viscous competition.

desk verdict The halving rule for NaCl efflorescence is a real, useful observation; the mass-limited height claim is plausible but rests on an untested no-subflorescence assumption. read the letter →

arxiv 2508.18779 v1 pith:NBFCSPAA submitted 2025-08-26 cond-mat.soft physics.app-ph

R.J. Wijnhorst , M. Prat , N. Shahidzadeh This is my paper

classification cond-mat.softphysics.app-ph

keywords saltefflorescenceself-similarityfractaldimensionNaClcrystallizationhierarchicalporousmediacapillary-viscouscompetitionmassbalancecreeping

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

The paper sets out to show that cauliflower-shaped salt crusts formed during evaporation are not random piles but self-similar hierarchical porous media. At each inward layer, the cubic crystals and the pores between them are about half the size of the previous layer, and box-counting gives fractal dimensions of about 1.71 in 2D and 2.56 in 3D, close to those of diffusion-limited aggregation (random particle sticking) and of cauliflower or broccoli. The paper's quantitative claim is that the final height of a salt deposit is set by the initial mass of salt available, through a simple mass balance, rather than by the capillary-viscous pressure competition considered in previous models. If this is right, the height of salt crusts on stones could be predicted from salt supply and geometry alone, without needing microscopic flow parameters.

What carries the argument

The load-bearing object is the halving hierarchy: in layer i, cubic crystals have side di = di−1/2 and slit pores have aperture ai = ai−1/2. This scaling lets the model sum viscous pressure drops across layers using the cubic-law permeability ki = 2^i a_i^3/(12 d1), compare them with the Young-Laplace capillary pressure Pce(n) = 2^n cosθ γ/a1, and then compute the mass-balance height through the efflorescence porosity εe.

What would settle it

Run the same sandstone experiment to completion, then dissolve or section the stone and measure the salt left inside; if a substantial fraction of the initial salt remains as subflorescence, Eq. (6) overpredicts the crust height. Alternatively, vary the initial salt mass per unit surface area and check that final crust height follows the linear relation of Eq. (6) independent of evaporation rate.

Watch

Extended reading notes

Core claim

NaCl efflorescence and creep pillars are self-similar porous media: large cubic crystals form first, leaving pores in which smaller crystals nucleate, and so on. Feret diameters show crystal and pore sizes roughly halve per layer, pores about a quarter of cube size (~20% porosity). Box-counting gives fractal dimensions 1.71 ± 0.03 (2D) and ~2.56 (3D). Summing viscous pressure drops via the cubic law and comparing with Young-Laplace capillary pressure shows capillary-viscous competition is not the height limit. Equating initial dissolved salt with crust salt gives he = ρlC0ε(V/A)/(ρcr(1−εe)+ρlCsatεe), matching observed heights.

Load-bearing premise

The height prediction assumes every gram of salt that started dissolved in the porous stone ends up in the external crust, with none precipitating inside the stone or being lost elsewhere.

Editorial extensions

If this is right

The final efflorescence height should scale with the initial salt inventory per unit surface area, so adding more salt solution produces proportionally taller crusts.
Changing the evaporation rate should not change the ultimate height, because the model finds capillary-viscous competition is not the limiting factor.
The halving rule supplies a recursive structural description: each layer's crystal and pore sizes are determined by the previous layer, so a layer-by-layer 3D model can reproduce the cauliflower morphology.
The outer skin, with the smallest pores, generates the highest capillary suction, keeping the crust wet and sustaining the self-amplifying precipitation.
Fractal dimensions near 1.7 (2D) and 2.5 (3D) imply a large internal surface area, which amplifies evaporation and the creeping process.

Reading between the lines

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

An untested consequence of the mass-balance claim is that crust height should be linear in C0 V/A and insensitive to evaporation rate; a dedicated matrix of experiments varying those two factors separately would confirm or refute mass control.
The halving rule suggests a simple generative algorithm: recursively place cubes of half size into each pore, then compare predicted pore-size distributions directly with SEM data beyond the three layers shown.
Because the measured fractal dimension sits close to diffusion-limited aggregation values, ion diffusion through the finest pore layer may control the precipitation rate; measuring concentration profiles just beneath the crust would test this.
In conservation practice, if height is mass-limited, salt-damage risk maps could be built from salt inventory and porosity instead of evaporation-rate models.

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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

You should know two things about this paper. First, the direct observation that crystal and pore sizes halve from layer to layer in NaCl efflorescence is new, quantitative, and backed by clear SEM images. Second, the height model quietly assumes every gram of salt from the initial solution ends up in the external crust. That assumption is untested and could overestimate heights by a factor of 1/(1-f) if a fraction f remains inside the stone. The authors cite their own earlier work on subflorescence (ref. 19) but never rule it out here.

What's genuinely good: the layered hierarchy in Figs. 2 and 3 is convincing; the halving rule (d_i = d_{i-1}/2, a_i = a_{i-1}/2) goes beyond earlier qualitative "cauliflower" descriptions. The fractal dimensions, while a routine box-counting application, support the claim and match DLA expectations. The mass-balance height formula (Eq. 6) is a simple, zero-free-parameter estimate that captures the order of magnitude of observed heights, though porosity is measured from the same samples so it's not fully independent.

The soft spots are real but not fatal to the core observation. The capillary-viscous calculation covers only n=1 to 6, with h1=0.1 mm, alpha=2, and parameters from the same system. That's enough to say "not limiting in these experiments," but the abstract's broader claim that height is "primarily determined" by initial salt mass is overgeneralized. The subflorescence issue is the more load-bearing gap: if salt precipitates inside the sandstone, Eq. (5) overcounts the mass in the crust and the predicted height is too high. The paper reports no residual salt measurement, so the mass-balance conclusion isn't established even for their own data. Also, the 3D fractal dimension error bar (±0.0013) looks implausibly tight for box-counting on a z-stack; the analysis details are missing and should be supplied.

Who gets value: researchers in salt weathering, efflorescence, and porous-media drying. The self-similarity observation will likely hold up and is worth citing; the height model is a testable hypothesis rather than a settled result.

My recommendation: send it to peer review. The core experimental finding is solid and useful, the model is addressable, and the missing subflorescence check is a concrete request that a referee can make. With those revisions, this would be a respectable contribution.

Referee Report

3 major / 4 minor

Summary. The manuscript reports SEM and micro-CT observations of NaCl efflorescence grown both from bulk brine and from brine-saturated sandstone cones. It claims that the deposits are hierarchical and self-similar: cubic crystal sizes and pore apertures approximately halve from layer to layer, giving fractal dimensions D2D ≈ 1.71 and D3D ≈ 2.56. On this basis, the authors construct a layered slit-pore model to argue that capillary–viscous competition does not set the maximum efflorescence height. They instead equate the initial dissolved salt mass in the stone (Eq. 4) to the salt contained in the external crust (Eq. 5), yielding the height formula Eq. (6), and report order-of-magnitude agreement with their measured heights.

Significance. If the central claim holds, the paper would shift the understanding of efflorescence growth from a capillary-transport-limited process to a source-limited one, with practical implications for salt damage and natural salt structures. The imaging evidence for a self-similar halving cascade is direct and appears robust across the presented experiments; the mass-balance height formula contains no fitted parameter for the height itself, which is a strength. However, the quantitative height claim currently rests on an unstated assumption about the fate of all dissolved salt and on an idealized model of the capillary-viscous limit, so the degree of support is lower than the abstract suggests.

major comments (3)

[Eqs. (4)–(6), 'To investigate the hypothesis…'] The mass balance equates the salt initially dissolved in the sandstone, ms = ρl C0 ε V, to the salt present only in the external efflorescence, ms = A he(ρcr(1−εe)+ρl Csat εe). This assumes zero subflorescence and no dissolved salt left in the stone at the time of measurement. For the 1.7 M NaCl experiments on 30%-porosity sandstone, subflorescence is a known possibility (the authors' own ref. 19 concerns subflorescence formation), yet no residual-salt measurement is reported. If a fraction f of the initial salt remains in the stone, Eq. (6) overestimates he by 1/(1−f). The order-of-magnitude agreement in Fig. 5d therefore does not discriminate between the mass-balance mechanism and an upper bound. Please measure or estimate the residual salt in the sandstone, or explicitly bound f and discuss the sensitivity of the central claim to it.
[Eq. (3) and Fig. 5c] The conclusion that capillary-viscous competition does not limit height relies on an idealized representation: each layer is a set of parallel slit pores of aperture ai, with permeability from the cubic law and with prescribed ratios di=di−1/2, ai=ai−1/2, α=2, and h1=0.1 mm. No sensitivity analysis is given, and the retention-curve calculation below Eq. (2) uses porosity ε=0.28 and grain size d=100 μm, inconsistent with the experimental values reported earlier (ε=30%, average pore diameter 30 μm). Because the viscous term in Eq. (3) grows roughly as 4n while the available capillary pressure grows as 2n, the ratio Pc(he)/Pcmax is sensitive to these choices. Please provide a sensitivity analysis or direct measurements of permeability/capillary pressure for the observed hierarchy, and reconcile the substrate parameters.
[Fig. 5d and Abstract] The validation of Eq. (6) is based on a single initial concentration (1.7 M) and a single sandstone-cone geometry, and the agreement is described only as 'order of magnitude.' To support the abstract's claim that the ultimate height is 'primarily determined by the initial mass of salt,' the paper should either vary C0, V/A, or the total salt mass and test the predicted scaling, or explicitly restrict the claim to the conditions studied. As written, the data do not establish the general source-limited mechanism independently of the subflorescence assumption.

minor comments (4)

[Line following 'From Darcy's law…'] The formula for permeability appears garbled in the text: 'ki = 2i a3 i 12d1' should be displayed as k_i = 2^i a_i^3/(12 d_1). Please use display math and define d_1 and a_1 before Eq. (1)–(3).
[Fig. 2d caption] The caption reads 'average Feret diameter of the pores per layer and the diameter of the pores per layer'; the second quantity is presumably the crystal/cube size. Please clarify the wording.
[Experimental section] Minor typos: 'Péclets' should be 'Péclet'; 'self-organized strucutre' in the introduction should be 'structure'; and the panel callouts for Fig. 1(c,d) should be checked for consistency.
[Eq. (5)] If the final efflorescence height is measured after complete drying, the term ρl Csat εe (dissolved salt in the efflorescence pores) should be omitted or justified; if measured during growth, the timing should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the self-similar hierarchy is empirical, the capillary-pressure check uses independent parameters, and the height model is a mass balance compared with—not fitted to—measured heights.

full rationale

The claimed derivation chain is not circular. The self-similar halving relationship (Fig. 2d: di = di−1/2 and ai = ai−1/2) is an empirical SEM/Feret observation, not an input re-derived as an output. The hierarchical pressure model (Eq. 3) substitutes these observed scalings into standard Darcy/cubic-law/Young-Laplace equations and uses retention-curve parameters (Eq. 2) from Hidri et al.; although M. Prat is a co-author of that paper, the retention-curve parameters (Sc=0.1, n=10, Pcref from porosity/grain size) are independent of the present target quantity (efflorescence height) and are not fitted to it, so this is legitimate independent support rather than a load-bearing self-citation. The height formula (Eqs. 4-6) is a salt-conservation balance: it equates the initial dissolved salt to the salt held in the external crust, with no free parameter fitted to the observed heights; the prediction is tested against the order of magnitude of measured heights. The principal weakness is the unstated assumption that all salt ends up in the external efflorescence and none remains as subflorescence in the sandstone (Eq. 5 has no subflorescence term); if false, Eq. (6) would overestimate heights. This is a real correctness/assumption concern, but it is not circularity: the mass balance is not equivalent to its inputs by construction, and the inputs (C0, ε, V, A, εe, ρcr, Csat) are measured independently of he. Self-citations in the paper (e.g., refs. 7, 13, 17-19, 29) are used as background or for independent parameters, not as an unverified uniqueness argument, so they do not raise the circularity score.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The model rests on four principal inputs: a factor-of-two halving rule measured from the same samples (α, h1, d1, a1), a measured efflorescence porosity, the cubic law for slit permeability, and a retention curve with parameters borrowed from earlier fitted studies. The mass-balance height formula has no fitted constants beyond the measured porosity and salt concentration, giving it independent grounding. No new physical entities are introduced.

free parameters (4)

α (layer thickness growth factor) = 2
Chosen based on SEM observations that each subsequent layer is about twice as thick as the previous one; used in Eq. (3) and Fig. 5c. Not fitted to the final height.
h1 (height of first layer) = 0.1 mm
Set from experimental images as about 10 level-1 cube sizes; used in Eq. (3).
d1 and a1 (largest cube size and largest pore aperture) = not stated in text
Required for the quantitative evaluation of Eq. (3) and Fig. 5c; values are apparently inferred from SEM measurements but are not reported in the paper.
εe (efflorescence porosity) = ~0.2
Measured from Feret diameters as pore size roughly one quarter of cube size; used in the mass-balance height formula, Eq. (6).

assumptions (4)

standard math Darcy's law applies to steady flow through the hierarchical efflorescence
Used to derive the viscous pressure drop Δpi over each level in the model.
domain assumption Cubic law gives the permeability of each level's slit pores
The expression ki = 2^i a_i^3 / (12 d1) follows the fracture-flow cubic law of Zimmerman & Bodvarsson, assuming parallel-plate slit apertures; ignores 3D pore connectivity and tortuosity.
domain assumption Retention curve of Hidri et al. describes capillary pressure in the underlying sandstone
Eq. (2) with n=10, Sc=0.1, and Pcref from prior literature (refs. 29-31); these parameters were fitted in the cited studies, not in the present paper, but are imported without validation on the current sandstone.
domain assumption All initial salt mass is incorporated into the external efflorescence
Eqs. (4)-(6) equate the initial dissolved salt mass to the salt present only in the crust; this ignores subflorescence precipitation inside the porous stone and any other salt loss.

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Cite this review

Pith. "Pith review of Self-similarity in creeping salt crystallization." pith.science (2026). https://pith.science/paper/NBFCSPAA

@misc{pith2026250818779,
  author       = {Pith},
  title        = {Pith review of: Self-similarity in creeping salt crystallization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NBFCSPAA}},
  note         = {Machine review of arXiv:2508.18779}
}

read the original abstract

The self-amplifying creeping of salts can produce striking macroscopic structures, such as desert roses in arid regions and salt pillars near saline lakes. While these formations are visually remarkable, salt crystallization, often seen as efflorescence on surfaces, also poses significant challenges for cultural heritage conservation, materials science, and soil management. In this study, we investigate the mechanisms underlying self-organized crystallization within efflorescence deposits. Our findings reveal that these porous salt deposits exhibit pronounced self-similarity, with the crystallization process recurring at multiple length scales. This results in smaller replicas of the overall structure nested within larger ones, creating fractal geometries similar to those found in cauliflower and broccoli. By performing controlled evaporation experiments and microscale analysis using advanced imaging techniques combined with fractal dimension analysis, we uncover the hierarchical and size-controlled precipitation of cubic microcrystals within the porous efflorescence. Furthermore, we develop a hierarchical growth model demonstrating that the ultimate height of the macroscopic salt deposit is primarily determined by the initial mass of salt, rather than by the interplay of capillary and viscous forces when salt solution flows within the porous salt structure.

Figures

Figures reproduced from arXiv: 2508.18779 by the authors.

**Figure 2.** a) top view of the surface of the sandstone , early stage of efflorescence deposit of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** a) Scanning electron microscopy image of an emergent NaCl creep structure (pillar [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Micro-CT image of a cauliflower shaped NaCl crystal formed by creep showing the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: a) Model of one level hi of NaCl (halite) with slit pores of aperture ai . b) Hierarchical model with three levels of cubes with diameters di in each layer n for 3 layers. c) Ratio of capillary pressure at maximum height he to the maximum capillary pressure for n leve…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Hygroscopic hysteresis drives intermittent salt creeping
cond-mat.soft 2026-07 conditional novelty 7.0 of 10

Hysteresis between deliquescence and efflorescence is sufficient to turn steady confined evaporation of salt solutions into intermittent salt creeping via episodic imbibition.

Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

G.; L \"a uchli, A

Pitman, M. G.; L \"a uchli, A. Global impact of salinity and agricultural ecosystems. Salinity: environment-plants-molecules 2002, 3, 20

work page 2002

[2] [2]

Salt weathering in deserts

Cooke, R. Salt weathering in deserts. Proceedings of the Geologists' Association 1981, 92, 1--16

work page 1981

[3] [3]

Damage in porous media due to salt crystallization

Shahidzadeh-Bonn, N.; Desarnaud, J.; Bertrand, F.; Chateau, X.; Bonn, D. Damage in porous media due to salt crystallization. Physical Review E 2010, 81, 066110

work page 2010

[4] [4]

Washburn, E. R. The creeping of solutions. The Journal of Physical Chemistry 2002, 31, 1246--1248

work page 2002

[5] [5]

Creeping-film phenomenon of potassium chloride solution

Huang, B.-J.; Huang, J.-C. Creeping-film phenomenon of potassium chloride solution. Nature 1976, 261, 36--38

work page 1976

[6] [6]

Hird, R.; Bolton, M. D. Migration of sodium chloride in dry porous materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 2016, 472, 20150710

work page 2016

[7] [7]

Salt creeping as a self-amplifying crystallization process

Qazi, M.; Salim, H.; Doorman, C.; Jambon-Puillet, E.; Shahidzadeh, N. Salt creeping as a self-amplifying crystallization process. Science advances 2019, 5, eaax1853

work page 2019

[8] [8]

J.; Bonn, D.; Shahidzadeh, N

Qazi, M. J.; Bonn, D.; Shahidzadeh, N. Drying of salt solutions from porous media: Effect of surfactants. Transport in Porous Media 2019, 128, 881--894

work page 2019