Theoretical Analysis of Marangoni Flow and its Potential Role in Cactus Spine Water Transport#

2.1 Driving Force: Surface Tension Gradient (∇s​γ)#

The fundamental driving force for Marangoni flow is the existence of a gradient in surface tension (γ) along a fluid interface. Fluid is driven from regions of lower γ to regions of higher γ.13

FMarangoni​∝∇s​γ

This gradient can originate from:

• Thermal Gradients (∇s​T): As γ is typically temperature-dependent (∂γ/∂T<0 for most liquids) 5: $\nabla_s \gamma_T = \frac{\partial \gamma}{\partial T} \nabla_s T$
• Concentration Gradients (∇s​C): Solutes or surfactants alter surface tension (∂γ/∂C=0) 4: $\nabla_s \gamma_C = \frac{\partial \gamma}{\partial C} \nabla_s C$

2.2 Tangential Stress Balance (Boundary Condition)#

At the interface, the tangential Marangoni stress must be balanced by the jump in viscous shear stress across the interface. For a liquid (1)-gas (2) interface, assuming negligible gas-phase stress, this boundary condition links the surface physics to the bulk fluid dynamics (u1​ is liquid velocity, μ1​ is liquid viscosity, t is a tangent vector) 4:

(μ1​(∇u1​+(∇u1​)T)⋅n)⋅t≈(∇s​γ)⋅t

where n is the unit normal vector pointing out of phase 1. This condition dictates that a non-zero ∇s​γ necessitates a velocity gradient (i.e., flow) adjacent to the interface.

2.3 Dimensionless Marangoni Number (Ma)#

The relative strength of Marangoni convection versus viscous dissipation and diffusion is characterized by the Marangoni number (Ma).17

• Thermal Marangoni Number (MaT​): $Ma_T = \frac{|\frac{\partial \gamma}{\partial T}| \Delta T L}{\mu \alpha}$ (L: characteristic length, ΔT: characteristic temperature difference, μ: dynamic viscosity, α: thermal diffusivity).17
• Concentration Marangoni Number (MaC​): $Ma_C = \frac{|\frac{\partial \gamma}{\partial C}| \Delta C L}{\mu D}$ (ΔC: characteristic concentration difference, D: mass diffusivity).17

High Ma values (Ma≫1) signify that Marangoni-driven convection dominates over diffusive transport.17

3.1 Laplace Pressure Gradient (∇PL​)#

The spine’s conical geometry (Rtip​<Rbase​) creates a curvature gradient. According to the Young-Laplace equation, the pressure jump across the curved droplet interface (ΔPL​=γκ, where κ is mean curvature) is higher at the tip (smaller radius, higher κ) than at the base.6

ΔPL,tip​>ΔPL,base​⟹∇PL​=0

This gradient drives flow from high pressure (tip) to low pressure (base): FLaplace​∝−∇PL​.6 Microgrooves can further modulate local curvature and enhance this effect.6

3.2 Wettability / Surface Energy Gradient (∇γeff​)#

Spines often possess a wettability gradient, typically transitioning from more hydrophobic/rougher at the tip (lower effective surface energy, γeff​) to more hydrophilic/smoother at the base (higher γeff​).6 This can arise from gradients in micro/nano-structure (roughness) or surface chemistry.10 Droplets move towards regions of higher wettability (lower contact angle θc​) to minimize the system’s free energy.11

FWettability​∝∇(γLG​cosθc​)∝∇γeff​

4.1 Theoretical Feasibility#

Marangoni flow on a water droplet residing on a cactus spine is theoretically plausible.20 Evaporation from the droplet surface, potentially non-uniform (e.g., enhanced at the contact line or apex under solar radiation), could establish:

• Thermal gradients (∇T): Due to differential evaporative cooling.20
• Concentration gradients (∇C): If dissolved impurities or atmospheric deposits are present and become concentrated during evaporation.

Such gradients would induce a ∇s​γ across the droplet’s air-water interface, potentially driving internal flows or influencing contact line dynamics.20

4.2 Assessment of Significance#

Despite theoretical feasibility, the contribution of the Marangoni effect to the net directional transport along the spine appears secondary compared to the structure-induced gradients:

• Dominance of Structural Gradients: The vast majority of experimental and theoretical studies on cactus water harvesting emphasize the Laplace pressure and wettability gradients as the primary and sufficient driving forces derived directly from the spine’s geometry and surface morphology.6 Biomimetic designs focus on replicating these structural features.6
• Lack of Direct Evidence: There is limited direct experimental evidence or strong theoretical argument supporting a dominant role for Marangoni flow in the specific context of tip-to-base transport on cactus spines. While mentioned as a general mechanism for droplet motion 20, its quantitative contribution relative to the powerful structural gradients in this system is not established.
• Robustness and Reliability: Laplace and wettability gradients are inherent to the spine’s physical structure and relatively stable. In contrast, Marangoni flows depend on potentially transient and environment-dependent factors (evaporation rates, temperature fields, water purity) acting on the droplet itself.20 Evolutionary pressure likely favored the more robust structure-based mechanisms for a critical function like water acquisition.