Controlled Spherulitic Crystal Growth from Salt Mixtures: A Universal Mechanism for Complex Crystal Self-Assembly

Daniel Bonn; Isa Yusuf; Michael Steiger; Noushine Shahidzadeh; Paul Kolpakov; Romane Le Diz\`es Castell; Simon L\'epinay; Tess Heeremans

REVIEW 3 major objections 5 minor 37 references

Controlled Spherulitic Crystal Growth from Salt Mixtures: A Universal Mechanism for Complex Crystal Self-Assembly

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Adding divalent metal sulfates to an evaporating sodium sulfate solution makes sodium sulfate grow as spherulites through a two-step, diffusion-limited nucleation at about 111 Pa·s.

desk verdict Genuinely new observation of Na2SO4 spherulites in divalent sulfate mixtures, but the paper's own growth-slope data contradict its headline 111 Pa·s viscosity mechanism by orders of magnitude. read the letter →

arxiv 2506.01163 v1 pith:Z5HBSMEX submitted 2025-06-01 cond-mat.soft cond-mat.mtrl-sciphysics.chem-phphysics.geo-ph

Tess Heeremans , Simon Lépinay , Romane Le Dizès Castell , Isa Yusuf , Paul Kolpakov , Daniel Bonn , Michael Steiger , Noushine Shahidzadeh This is my paper

classification cond-mat.softcond-mat.mtrl-sciphysics.chem-phphysics.geo-ph

keywords spherulitessodiumsulfatetwo-stepnucleationdiffusion-limitedgrowthsolutionviscositysaltmixturesevaporativecrystallizationmetastablephaseIII

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

Spherulites are ball-shaped, radially branched polycrystalline aggregates found in rocks, kidney stones, and amyloid plaques, yet the conditions that make them form have been hard to pin down. This paper reports a controlled recipe: adding a divalent metal sulfate (Mg, Fe, Cu, or Zn) to an evaporating sodium sulfate solution at room temperature makes sodium sulfate precipitate as spherulites. The authors measure the onset conditions—supersaturation $\beta_{\mathrm{Mg}} \approx 1.4$ and $\beta_{\mathrm{Fe}} > 3.3$—and show that the spherulites grow by a two-step nucleation path in a solution whose viscosity has risen to about 111 Pa·s, with size increasing as the square root of time. They map the molar-fraction windows where spherulites form and show that the final shape can be tuned by evaporation rate. If the mechanism is general, it turns spherulitic morphology from an accident of growth into a tunable outcome of salt-mixture design.

What carries the argument

The mechanism runs on two coupled elements. The first is the viscosity–supersaturation relation of the mixed salt solution: rheometry gives $\eta = 4\times 10^{-5} \exp(10.6\,\beta_{\mathrm{Mg}})$ with $R^2=0.88$, so that as water evaporates the solution becomes exponentially more viscous, reaching about 111 Pa·s at the measured onset supersaturation $\beta_{\mathrm{Mg}}=1.4$. Through the diffusion–viscosity relation $D \sim 1/\eta$, this viscosity surge throttles ionic diffusion and makes growth diffusion-limited, with the observed $L \approx \Delta\sqrt{t}$. The second element is the two-step nucleation pathway: before spherulites appear, the solution produces micrometer-sized droplet-like clusters near the contact line; these clusters then crystallize into phase III nanocrystals whose slight orientational mismatches branch into the spherulitic form. The supersaturation values used to anchor the picture come from ternary solubility phase diagrams computed with a molality-based ion-interaction thermodynamic model.

What would settle it

Directly measure the viscosity of the x_Mg = 0.12 mixed solution at the supersaturation where spherulites first nucleate (around 1.4), and independently measure the sulfate diffusivity in that solution; if the viscosity is not close to 111 Pa·s or the spherulite radius does not follow the square root of time, the diffusion-limited viscosity mechanism is not supported.

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Extended reading notes

Core claim

The central discovery is that sodium sulfate, which normally crystallizes as faceted polyhedra, can be driven to grow as spherulites of metastable phase III by the presence of divalent ions in an evaporating mixed-salt solution. At molar fractions $0.03 < x_{\mathrm{Fe}} < 0.29$ or $0.04 < x_{\mathrm{Mg}} < 0.38$, the solution crosses the phase III solubility limit at high supersaturation, and precipitation proceeds through micrometer-scale precursor clusters—a two-step, nonclassical nucleation—rather than directly through classical nucleation. The growth is then diffusion-limited: the spherulite radius follows $L \approx \Delta\sqrt{t}$ with $\Delta$ between 3.9 and 9.6, and the viscosity of the solution at onset is inferred to be about 111 Pa·s, consistent with a gel-like medium. The spherulites are metastable; under slow evaporation they can sprout bladed crystals that later recrystallize into thenardite, the stable phase V. The paper presents this as a universal mechanism, supported by the same spherulitic growth with Fe, Mg, Cu, and Zn sulfates, and by the absence of spherulites when only monovalent sulfates are mixed.

Load-bearing premise

The central quantitative claim rests on extending a viscosity curve from measurements that stop well before crystals appear, plus a unit-prefactor assumption in the diffusion-to-viscosity conversion; if the viscosity trend changes beyond the measured range, the 111 Pa·s value and the diffusion-limited mechanism are not established.

Editorial extensions

If this is right

Sodium sulfate spherulites form in a predictable compositional window: $0.03 < x_{\mathrm{Fe}} < 0.29$ for iron sulfate and $0.04 < x_{\mathrm{Mg}} < 0.38$ for magnesium sulfate, outside of which the solution makes faceted crystals or a gel-like amorphous state.
Final morphology can be tuned by evaporation rate: fast evaporation gives many small, smooth spherulites, while slow evaporation gives larger, rougher spherulites that can sprout bladed crystals and later recrystallize into thenardite.
Divalent ion identity is not the key variable: magnesium, iron, copper, and zinc sulfates all produce sodium sulfate spherulites, whereas mixing two monovalent sulfate salts does not, so divalence itself is the trigger.
The growth law $L \approx \Delta\sqrt{t}$ places sodium sulfate spherulites in the diffusion-limited class, and the inferred local viscosity varies by a factor of roughly 12 to 70 across an evaporating droplet.
The recipe scales from 1 µL droplets to 10 mL bulk solutions, producing millimeter-sized spherulites under low humidity, so the controlled growth is not limited to microdroplets.

Reading between the lines

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

A testable extension is to apply the same viscosity-based recipe to non-sulfate salts: any monovalent salt whose concentrated solution is made very viscous by a divalent additive should show diffusion-limited branching growth, so the design rule may transfer to carbonates, nitrates, or pharmaceuticals.
The unidentified vibrational-spectroscopy phase of the blade-like crystals offers a concrete follow-up: if it is a new metastable sodium sulfate hydrate or a double salt, mapping its structure would complete the phase-III-to-thenardite transformation path and could explain similar blades on natural gypsum spherulites.
The paper's diffusion-to-viscosity inference assumes a prefactor of unity; a direct measurement of sulfate diffusivity in the concentrated mixed solution would test that assumption and, if the prefactor differs, would revise the inferred 111 Pa·s onset viscosity without necessarily changing the square-root growth law.

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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

Two things you should know about arXiv:2506.01163. First, it reports something genuinely new: sodium sulfate, normally a boring polyhedral crystallizer, forms spherulites when divalent sulfate salts (Mg, Fe, Cu, Zn) are present, and the authors back this up with Raman-confirmed phase III identification, measured supersaturation at onset, and sqrt(t) growth kinetics. That part is solid and worth a look. Second, the paper's headline mechanism—that viscosity rises to ~111 Pa·s and forces diffusion-limited growth—contradicts the paper's own growth-slope data by three orders of magnitude. The stress-test note is correct and it lands right on the load-bearing claim.

The novel pieces are real: first quantitative supersaturation values for spherulite onset in mixed sulfate systems (β_Mg = 1.4 ± 0.3, β_Fe > 3.3), a clean demonstration that monovalent K2SO4 doesn't do it, and a nice morphology map across molar fraction and relative humidity. The Pitzer-based ternary phase diagrams look like standard Steiger-group work, which is good.

The soft spots are not minor. From Fig. 5a the growth slopes are Δ = 3.9–9.6 µm·s^-1/2; the paper sets D = Δ², yielding D = 15–92 µm²/s. Plugging that into their own Stokes-Einstein relation with D_i = 1070 µm²/s gives η/η_i = 12–70, i.e., 12–70 mPa·s. The rheology extrapolation gives 111 Pa·s at the same β = 1.4. That's a factor of 10^3–10^4 gap. A prefactor in R ~ λ√(Dt) could shift D by λ², but λ would need to be 40–100, not O(1). The paper doesn't acknowledge this. Either the growth unit isn't the sulfate ion, or the growth law isn't the simple diffusion-limited one they assume, or the growth is measured at much lower β than the nucleation onset. All three could be fixed with careful wording and a few more experiments, but as written the quantitative mechanism is not supported.

Also, the extrapolation to 111 Pa·s comes from an exponential fit to β ≤ 0.73 with R² = 0.88; that number was already soft. The "universal" framing should be toned down to divalent sulfate salts. Error propagation is missing from the supersaturation numbers.

Who's this for? People working on crystallization, salt damage, and out-of-equilibrium morphology. The empirical discovery is likely reproducible and will be useful even if the mechanism needs revision.

Recommendation: send it to peer review, but with a clear request to reconcile the viscosity estimates and to either justify or remove the D = Δ² step. If I were the editor, I would not desk-reject; the observation is too clean. But the paper in its current form shouldn't be accepted. I wouldn't cite it yet until the numbers are consistent.

Referee Report

3 major / 5 minor

Summary. The manuscript reports that the addition of divalent metal sulfates (Mg2+, Fe2+, Cu2+, Zn2+) to evaporating Na2SO4 solutions induces spherulitic growth of sodium sulfate in the metastable phase III, and it maps the composition ranges where spherulites form in ternary Na2SO4-MSO4-H2O systems. The authors measure the supersaturation at the onset of spherulite nucleation (β_Mg = 1.4 ± 0.3, β_Fe > 3.3), observe diffusion-limited growth (L ∝ √t), and use rheometry to extrapolate a bulk viscosity of about 111 Pa·s at the onset. They propose that divalent ions drive a two-step nucleation process in a highly viscous solution, resulting in diffusion-limited growth and a universal mechanism for spherulite formation. The paper also documents the influence of evaporation rate on the subsequent morphological evolution of the spherulites toward more stable phases.

Significance. The core experimental observation is valuable and appears well documented: spherulitic Na2SO4 (phase III) grows reliably from mixed sulfate solutions containing divalent ions, with Raman-confirmed phase identity and reproducible morphology maps across several cations. The quantification of supersaturation at onset and the growth kinetics is a step beyond prior qualitative reports. If the mechanistic interpretation were quantitatively consistent, the paper would offer a practical route to controlled crystal morphologies. However, the central quantitative support for the proposed high-viscosity, diffusion-limited mechanism is weakened by a large internal inconsistency between the viscosity inferred from the growth-slope data and that extrapolated from the rheology experiments; this needs to be reconciled before the mechanism can be considered established.

major comments (3)

[Growth mechanism driven by viscosity changes (Fig. 5)] The two viscosity estimates presented in this section are mutually inconsistent. The growth-slope data in Fig. 5a yield D = Δ² = 15.2–92.2 µm²/s, which with Eq. (3) and D_i = 1.07×10⁻⁹ m²/s gives η/η_i ≈ 12–70, i.e., η ≈ 12–70 mPa·s. The rheology fit in Fig. 5b, extrapolated to the onset supersaturation β = 1.4, gives η ≈ 111 Pa·s = 1.11×10⁵ mPa·s. The two estimates differ by three to four orders of magnitude at the same nominal supersaturation. If the solution at the growing interface had the extrapolated viscosity, the expected growth slope would be Δ ≈ 0.1 µm/s^1/2, not the measured Δ = 3.9–9.6 µm/s^1/2. The paper uses the 111 Pa·s value as the basis for the diffusion-limited mechanism while also using the growth-derived 12–70 mPa·s to support the same mechanism, without addressing the discrepancy. This must be resolved: either the relevant viscosity for growth is not the bulk viscosity (for example because of local concentration depletion near the growing crystal), or the exponential extrapolation is invalid; the manuscript does not currently provide a consistent picture.
[Growth mechanism driven by viscosity changes (Eq. 3 and text)] The identification of the growth-slope Δ with √D is not justified. For diffusion-limited growth, the radius typically advances as R = λ√(Dt), where λ is a dimensionless prefactor that depends on the supersaturation, the geometry, and the boundary conditions; in general λ is not equal to 1. Setting Δ = √D implicitly assumes λ = 1. The inferred D, and hence the inferred viscosity via Eq. (3), changes by a factor λ². While λ is often order one for simple growth geometries, the manuscript provides no derivation or citation for the relation used. Although λ cannot by itself explain the orders-of-magnitude discrepancy between the growth-derived and rheology-derived viscosities, the unsupported identification leaves the growth-based viscosity estimate without a firm foundation.
[Growth mechanism driven by viscosity changes (Fig. 5b)] The viscosity fit η = 4×10⁻⁵ exp(10.6β) is based on data only up to β = 0.73, and the extrapolation to β = 1.4 involves a factor of about 10³ in viscosity. Exponential fits of this type are typically not robust over such large extrapolations, and the exponent 10.6 carries no reported uncertainty; R² = 0.88 indicates a moderate fit. The paper should either directly measure viscosity closer to β = 1.4 (for example using a composition that remains fluid at that concentration) or provide a quantitative uncertainty bound on the extrapolated value. Without this, the statement that the viscosity 'reaches approximately 111 Pa·s' at the onset of spherulite precipitation is not supported to the precision implied.

minor comments (5)

[Results and Discussion] The text states that spherulites form for 0.04 < x_Mg < 0.38, but the caption of Fig. 4 gives 0.07 < x_Mg < 0.38; the correct range should be stated consistently in the text, figure, and conclusion.
[Growth mechanism driven by viscosity changes] The units in Eq. (3) are inconsistent as written: D_i is given in m²/s while Δ and D are expressed in µm²/s. The paper should state that D_i = 1.07×10⁻⁹ m²/s = 1070 µm²/s explicitly to avoid a factor of 10⁶ confusion in the viscosity ratio.
[Abstract and Conclusion] The abstract and conclusion repeat the 'approximately 111 Pa·s' value without noting that it is an extrapolated value based on a fit to data up to β = 0.73; this qualifier should accompany every occurrence of the number.
[Title and Abstract] The word 'universal' in the title and abstract is stronger than the evidence presented: the experiments cover four divalent cations (Mg, Fe, Cu, Zn) in sulfate mixtures, which I understand as a general family of systems, not necessarily universal across all salt mixtures. A more modest phrasing such as 'general' or 'common' would be more defensible.
[Materials and Methods (Viscosity measurements)] The viscosity was measured at a single shear rate of 900 s⁻¹. Since the solutions are described as approaching a gel-like state, it would be helpful to report a shear-rate sweep at the highest concentrations to exclude shear-thinning effects, which could affect the extrapolation.

Circularity Check

1 steps flagged · score 2.0 of 10

One minor self-referential viscosity estimate from growth slopes; the central experimental claims rest on independent measurements.

self definitional [Results and Discussion, 'Growth mechanism driven by viscosity changes', Fig. 5a and Eq. (3)]
"The slope ∆ of the spherulite growth measurements in Fig. 5a, corresponds to √D (in µm/s1/2), from which we infer that 15.2µm2/s ≤ D ≤ 92.2 µm2/s. By the following relation, the solvent viscosity is estimated: η/ηi = Di/D = kT/(6πηiR) 1/D ... A 12-fold viscosity increase is estimated in the bulk region (spherulite 5), and a 70-fold viscosity increase near the droplet edge (spherulite 1)."

The growth slope Δ is taken from the same L = Δ√t fit that is used to conclude diffusion-limited growth. Defining D = Δ² and then applying Stokes–Einstein (Eq. 3) turns the 'viscosity increase' into η/ηi = Di/Δ², which is a rearrangement of the slope itself rather than an independent measurement. The inset's claimed linear relationship between increased viscosity and inverse diffusivity is therefore an identity by construction. Using this growth-derived viscosity gradient to argue that 'viscosity variations and viscosity gradients drive morphological transitions' is self-referential, because the viscosity gradient is derived from the same growth-rate gradient it is invoked to explain.

full rationale

The paper's central experimental quantities are independently measured: the onset supersaturation (β_Mg = 1.4 ± 0.3, β_Fe > 3.3) comes from microcapillary volume-loss experiments combined with a Pitzer-based solubility model; the spherulite composition comes from Raman microspectroscopy; and the viscosity trend is measured by rheometry up to β = 0.73. None of these are defined in terms of the spherulitic-growth outcome, so there is no fundamental circularity in the main derivation chain. The one self-referential element is the growth-slope-based viscosity estimate in Fig. 5a/Eq. (3), where the inferred viscosity increase is just a rescaling of the observed slope under the diffusion-limited assumption. That estimate is not used as the primary support for the 111 Pa·s claim, but it is presented as independent confirmation of viscosity gradients, which is partly circular. The 111 Pa·s value itself is an extrapolation of an exponential fit well beyond the measured range (β ≤ 0.73 to β = 1.4) and is inconsistent with the growth-derived estimate (roughly 12–70 times water, i.e. tens of mPa·s, versus 111 Pa·s); this is a correctness and robustness concern rather than a circularity, because the extrapolation is not equivalent to its inputs by construction. The two-step nucleation interpretation is supported by direct observation of pre-nucleation clusters and by external literature, not solely by self-citation. Overall, the central experimental findings stand on their own, with only a minor self-referential step in the viscosity-gradient argument.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central mechanism rests on the Pitzer-based phase diagram (which supplies the solubility reference m*_S), on Stokes-Einstein conversion of growth-inferred diffusion coefficients to viscosity, on the equality D = delta^2, and on the transfer of microcapillary supersaturation values to droplet geometry. The only numbers fitted to data are the exponential viscosity coefficients and the Fe ternary Pitzer parameters; no new entities are introduced.

free parameters (3)

Viscosity exponential fit prefactor and exponent = 4e-5 Pa·s and 10.6 (R^2 = 0.88)
Fit to rheometry data on x_Mg = 0.12 solutions up to saturation beta = 0.73; used to extrapolate viscosity to 111 Pa·s at beta = 1.4.
Pitzer ternary interaction parameter psi_Na,Fe,SO4 = 0.00234
Determined from available solubility data for Na2SO4-FeSO4-H2O; used to build the phase diagram that defines the reference solubility m*_S for the supersaturation calculation. theta_Na,Fe set to 0.
Phase III solubility reference m*_S = 4.6 mol/kg at x = 0.12 from model
Used as the denominator in beta = m_S/m*_S; obtained from the Pitzer model, not measured directly in the mixed solution.

assumptions (4)

domain assumption The Pitzer ion-interaction model accurately predicts solubilities, water activities, and densities in the ternary Na2SO4-FeSO4-H2O and Na2SO4-MgSO4-H2O systems at 21°C.
Invoked in Fig. 4 and Methods to construct phase diagrams and define m*_S; relies on literature parameters plus ternary parameters fitted to solubility data.
domain assumption The Stokes-Einstein relation D ~ 1/eta holds for sulfate ion diffusion in the concentrated, partially gelled solution, so D measured from spherulite growth can be converted to solvent viscosity.
Used in Eq. (3) and Fig. 5a inset; the solution at onset is a viscous, non-ideal electrolyte, so the hydrodynamic relation may not hold quantitatively.
domain assumption Spherulite growth L = delta sqrt(t) implies diffusion-limited growth with D = delta^2, i.e., the prefactor linking the parabolic rate constant to the diffusion coefficient is 1.
Stated in Results: 'The slope delta ... corresponds to sqrt(D)'; no prefactor from the moving-boundary solution is included, which can introduce order-unity or larger errors.
domain assumption The supersaturation measured in microcapillaries (0.5 x 0.5 mm^2) represents the onset conditions in the evaporating droplets on glass slides.
Supersaturation at onset was measured in capillaries (Supplementary D) and applied to the droplet experiments; confinement and surface differ.

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Cite this review

Pith. "Pith review of Controlled Spherulitic Crystal Growth from Salt Mixtures: A Universal Mechanism for Complex Crystal Self-Assembly." pith.science (2026). https://pith.science/paper/Z5HBSMEX

@misc{pith2026250601163,
  author       = {Pith},
  title        = {Pith review of: Controlled Spherulitic Crystal Growth from Salt Mixtures: A Universal Mechanism for Complex Crystal Self-Assembly},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z5HBSMEX}},
  note         = {Machine review of arXiv:2506.01163}
}

abstract

Spherulites are complex polycrystalline structures that form through the self-assembly of small aggregated nanocrystals starting from a central point and growing radially outward. Despite their wide prevalence and relevance to fields ranging from geology to medicine, the dynamics of spherulitic crystallization and the conditions required for such growth remain ill-understood. Here, we report on the conditions to induce controlled spherulitic growth of sodium sulfate from evaporating aqueous solutions of sulfate salt mixtures at room temperature. We reveal that introducing divalent metal ions in the solution cause spherulitic growth of sodium sulfate. For the first time, we quantify the supersaturation at the onset of spherulitic growth from salt mixtures and determine the growth kinetics. Our results show that the nonclassical nucleation process induces the growth of sodium sulfate spherulites at high supersaturation in highly viscous solutions. The latter reaches approximately 111 Pa$\cdot$s, triggered by the divalent ions, at the onset of spherulite precipitation leading to a diffusion limited growth. We also show that spherulites, which are metastable structures formed under out-of-equilibrium conditions, can evolve into other shapes when supersaturation decreases as growth continues at different evaporation rates. These findings shed light on the conditions under which spherulites form and offer practical strategies for tuning their morphology.

Figures

Figures reproduced from arXiv: 2506.01163 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** a,b show a precipitated premature and fully developed spherulite at the center of the evaporating droplet. These spherulites are covered by an amorphous (gel-like) state [14]; in Fig. 3c, the internal structure of the precipitated spherulites at the edge of the droplet can be seen: edge-grown spherulite are flattened because of the geometrical constraints imposed by the contact line of the evaporating droplet. Fig… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: illustrates the morphological transitions during spherulitic growth, highlighting the key experimental factors that influence the final morphology. Higher molar ratios x of divalent ions lead to more viscous solutions, which in turn results in densely grown, smooth …

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 13.** Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 14.** Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]

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