Kinetics of Salt Creeping on a Free Surface: From Nucleation to Saturation

1. Problem Statement

The paper addresses the phenomenon of "creeping salt," where salt crystals spontaneously grow upward along partially immersed vertical surfaces (e.g., glass rods) against gravity. While this process is known to cause structural degradation in coastal infrastructure (corrosion of concrete and metals) and has applications in solar desalination, its macroscopic kinetics and microscopic mechanisms remain only partially understood.

Specifically, the authors aim to:

• Characterize the temporal evolution of the salt deposit's height and thickness from nucleation to saturation.
• Identify the governing kinetic regimes.
• Develop a unified theoretical model linking macroscopic growth to microscopic crystal structure under varying environmental conditions (humidity and temperature).

2. Methodology

Experimental Setup

• Apparatus: A glass rod (or flat microscope slide for microscopic observation) was partially immersed in a saturated NaCl solution within a temperature-controlled, ventilated tank.
• Control: Relative Humidity (RH) was controlled by injecting dry air; temperature was monitored. Experiments were conducted at room temperature with RH levels of 25%, 26%, 30%, and 32%.
• Data Collection:
  • Macroscopic: A camera captured images every 90 seconds to track the height and width of the crystalline layer. A scale monitored the weight of the rod every minute.
  • Microscopic: Optical microscopy was used on glass slides to analyze crystal size distribution.
  • Evaporation Rate: Measured independently via mass loss of a brine beaker ($J \approx 0.56 \times 10^{-4} \text{ kg m}^{-2} \text{ s}^{-1}$).

Theoretical Modeling

The authors derived an analytical model based on mass conservation at the crystal-air interface.

• Key Assumptions: Cylindrical symmetry, constant crystal volume ($\langle V \rangle$), and a fixed ratio of NaCl to water molecules ($N_e = 9$) in the saturated solution.
• Flux Model: The water evaporation flux $\Phi$ is modeled as dependent on height ($z$) and crystal thickness ($e$):
  $$\Phi(z, t) = \Phi_0 e^{-z/h_\infty} e^{-\xi e(z,t)}$$
  Where $h_\infty$ represents the characteristic height limited by gravity (capillary rise limits), and $\xi$ accounts for the hindrance of radial flow due to crystal accretion.
• Governing Equation: The evolution of the deposit is described by integrating the flux over the height of the crystal, leading to three distinct asymptotic regimes.

Numerical Simulations

• Approach: A 2D cellular automaton was used to simulate mesoscopic crystal growth on a grid.
• Mechanisms: The simulation updated water configuration (wetting/capillary advance), local salt concentration (using Ostwald-Freundlich relation for curvature effects), and stochastic crystallization based on Boltzmann probabilities.
• Goal: To reproduce crystal size distributions and test the influence of temperature and evaporation rates on morphology.

3. Key Contributions

1. Identification of Three Kinetic Regimes: The study successfully unifies the description of salt creeping into three distinct phases:
   • Regime 1 (Nucleation/Initial): Exponential growth of height with constant thickness.
   • Regime 2 (Linear Propagation): Linear growth of height while thickness remains constant.
   • Regime 3 (Saturation): Height saturates (plateaus) while the deposit thickens logarithmically.
2. Unified Analytical Model: The derivation of equations (Eq. 5, 7, and 9) that mathematically describe the transition between these regimes based on evaporation flux, gravity, and capillary constraints.
3. Micro-Macro Link: The integration of numerical simulations with experimental data to explain how microscopic parameters (crystal size, capillary spacing) dictate macroscopic outcomes (growth rate, final height).

4. Results

Macroscopic Dynamics

• Height Evolution:
  • Initially, the height grows exponentially ($h(t) \propto e^{t/\tau_1}$).
  • This transitions to a linear regime ($h(t) \propto t$) with a slope $a_2 \approx 0.8 \text{ cm/h}$ (experimental) vs. $3 \text{ cm/h}$ (theoretical). The discrepancy is attributed to fluctuations in evaporation rates and experimental noise.
  • Finally, the height saturates as gravity prevents further capillary rise ($h \to h_\infty \approx 0.1 \text{ m}$).
• Thickness Evolution:
  • Once height saturation occurs, the width (thickness) of the deposit increases logarithmically over time ($e(z,t) \propto \ln(t)$).
• Environmental Dependence:
  • Humidity: Primarily affects the kinetics (speed of growth) but not the final crystal size distribution.
  • Temperature: Affects both kinetics and the morphology. Higher temperatures lead to smaller average crystal sizes and steeper growth fronts.

Microscopic Observations & Simulation

• Crystal Size: Experimental microscopy showed an average crystal size of $\approx 3.1 \mu\text{m}$. Simulations with a grid spacing of $0.1 \mu\text{m}$ yielded an average of $3.9 \mu\text{m}$, showing satisfactory agreement.
• Temperature Effect: Simulations revealed that increasing temperature reduces the prevalence of large crystals (shifting the 84th percentile down by $>1 \mu\text{m}$).
• Mechanism: Smaller crystals create smaller capillary spaces. According to Jurin's law, this allows water to rise higher (increasing max height) but increases flow resistance ($\xi$), restricting lateral spreading (keeping width small).

Dead Ends & Limitations

• Mass Measurement: Attempts to model the mass evolution of the rod failed to fit the exponential growth model of Regime 1. The data fit better with a diffusion-limited aggregation model ($m(t) \propto e^{t/\tau} - 1$), suggesting the initial conditions are more complex than the macroscopic model assumed.
• OCT Imaging: Optical Coherence Tomography failed to visualize the main capillary between the rod and the crystal due to light diffusion caused by the thick crystal layer. X-ray imaging is proposed as a future alternative.

5. Significance and Conclusion

This work provides a comprehensive framework for understanding salt creeping, moving beyond qualitative descriptions to a quantitative kinetic model.

• Scientific Impact: It resolves the debate on the mechanism of upward propagation by validating a model where capillary rise and evaporation-driven crystallization compete with gravity.
• Practical Application: The findings are crucial for predicting the durability of coastal structures and optimizing solar desalination systems.
• Future Directions: The authors suggest that future research should focus on the temperature dependence of the phenomenon (rather than just humidity) and employ continuous mass monitoring or X-ray imaging to better resolve the transition between kinetic regimes and the internal capillary structure.

In summary, the paper successfully bridges the gap between microscopic crystal physics and macroscopic structural growth, offering a predictive tool for salt creep dynamics from nucleation to saturation.