2.1 Navier-Stokes equations and their limitations at the nanoscale

Navier-Stokes equations, the backbone of fluid dynamics, face challenges at the nanoscale. As we shrink down, surface forces dominate, continuum assumptions break, and molecular interactions take center stage. These changes demand new approaches to understand and model nanofluidic behavior.

The limitations of Navier-Stokes at the nanoscale are crucial to grasp. From breakdown of continuum assumptions to time and length scale issues, these constraints highlight the need for alternative models. Understanding these limits is key to developing accurate nanofluidic simulations and devices.

Fluid Dynamics at the Nanoscale

Surface Forces and Continuum Breakdown

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• Surface forces dominate over body forces at the nanoscale altering fluid behavior compared to macroscale systems
• Continuum assumption breaks down at the nanoscale necessitating alternative fluid modeling approaches
• Molecular-level interactions become increasingly important influencing fluid properties and flow characteristics
• No-slip boundary conditions may not hold due to increased fluid-surface interactions
  • Slip length becomes a crucial parameter in describing fluid-solid interfaces
  • Molecular dynamics simulations reveal complex boundary behaviors

Thermal Effects and Electric Double Layer

• Brownian motion and thermal fluctuations play a significant role affecting particle movement and fluid behavior
  • Einstein-Smoluchowski relation describes the diffusion coefficient: $D = \frac{k_B T}{6\pi \eta r}$
  • Mean squared displacement of particles follows: $\langle x^2 \rangle = 2Dt$
• Electric double layer (EDL) formation near charged surfaces becomes crucial especially in electrolyte solutions
  • Debye length characterizes EDL thickness: $\lambda_D = \sqrt{\frac{\epsilon_r \epsilon_0 k_B T}{2 N_A e^2 I}}$
  • EDL overlap in narrow channels can lead to ion selectivity and unique electrokinetic phenomena

Limitations of Navier-Stokes for Nanofluidics

Breakdown of Continuum Assumptions

• Navier-Stokes equations assume fluid continuum becoming invalid at nanoscale where discrete molecular effects are significant
• Traditional assumptions of constant fluid properties (viscosity, density) may not hold due to heterogeneous nature of fluids at nanoscale
• Equations do not account for increased importance of surface forces and molecular interactions dominating nanofluidic behavior
  • Surface-to-volume ratio increases dramatically at nanoscale ($\sim 10^6$ m²/m³ for 100 nm channel)
  • Molecular dynamics simulations reveal complex fluid structuring near surfaces

Time and Length Scale Limitations

• Time and length scales associated with nanofluidic phenomena often beyond scope of Navier-Stokes equations' applicability
  • Characteristic length scales: molecular size ($\sim 0.1$ nm) to channel width ($\sim 100$ nm)
  • Relevant time scales: molecular collision time ($\sim 10^{-12}$ s) to diffusion time ($\sim 10^{-6}$ s)
• Equations fail to capture quantum effects relevant at extremely small scales particularly for certain materials or under specific conditions
  • Quantum confinement effects become significant for channel sizes below $\sim 10$ nm
  • Wave nature of particles may need consideration (de Broglie wavelength)

Boundary Conditions and Discreteness

• Slip flow conditions occurring at fluid-solid interfaces in nanochannels not adequately described by classical no-slip boundary conditions
  • Slip length can range from few nanometers to micrometers depending on surface properties
  • Navier slip condition: $u_s = b \frac{\partial u}{\partial y}|_{y=0}$, where $b$ is slip length
• Navier-Stokes equations do not account for discreteness of fluid particles significant when channel dimensions approach molecular sizes
  • Knudsen number ($Kn = \lambda / L$) becomes important parameter
  • Transition from continuum to molecular flow regimes occurs as $Kn$ increases

Intermolecular Forces in Nanochannels

Van der Waals and Electrostatic Interactions

• Van der Waals forces between fluid molecules and channel walls influence fluid dynamics leading to fluid layering near surfaces
  • Lennard-Jones potential describes intermolecular interactions: $V(r) = 4\epsilon [(\frac{\sigma}{r})^{12} - (\frac{\sigma}{r})^6]$
  • Oscillatory density profiles observed near walls in molecular dynamics simulations
• Electrostatic interactions in ionic solutions result in formation of electric double layers affecting fluid flow and transport phenomena
  • Gouy-Chapman model describes EDL structure
  • Zeta potential characterizes surface charge and EDL strength

Hydrogen Bonding and Surface Properties

• Hydrogen bonding in water and polar liquids creates structured fluid layers near surfaces altering viscosity and flow characteristics
  • Water shows increased viscosity and decreased diffusivity in nanoconfinement
  • Hydrogen bond network can extend several molecular layers from surface
• Surface roughness at nanoscale impacts fluid-wall interactions potentially inducing local turbulence or altering flow patterns
  • Wenzel and Cassie-Baxter models describe wetting behavior on rough surfaces
  • Roughness can enhance or diminish slip depending on scale and geometry
• Hydrophobicity or hydrophilicity of channel surfaces leads to slip or no-slip boundary conditions influencing flow velocity profiles
  • Contact angle measurements characterize surface wettability
  • Superhydrophobic surfaces (contact angle > 150°) can achieve large slip lengths

Molecular Effects and Surface Functionalization

• Steric effects due to finite size of fluid molecules become significant in extremely narrow channels leading to size-dependent fluid behavior
  • Entropic exclusion of larger molecules from small pores
  • Size-dependent diffusion coefficients observed in nanopores
• Chemical functionalization of nanochannel surfaces modulates fluid-wall interactions enabling control over fluid transport and separation processes
  • Self-assembled monolayers (SAMs) modify surface properties
  • Stimuli-responsive coatings allow dynamic control of surface interactions

Surface-to-Volume Ratio in Nanofluidics

Enhanced Surface Effects

• Dramatically increased surface-to-volume ratio in nanofluidic systems leads to dominance of surface effects over bulk fluid properties
  • Surface-to-volume ratio scales as $1/L$ for characteristic length $L$
  • For 10 nm channel, over 50% of fluid molecules interact with surfaces
• Enhanced fluid-surface interactions due to high surface-to-volume ratio result in apparent changes in fluid viscosity and density near interfaces
  • Layering of fluid molecules near surfaces observed in molecular dynamics simulations
  • Effective viscosity can increase by orders of magnitude in extreme confinement

Adsorption and Capillary Effects

• Formation of adsorbed layers on channel walls becomes significant potentially altering effective channel dimensions and flow characteristics
  • Langmuir adsorption isotherm describes surface coverage: $\theta = \frac{K c}{1 + K c}$
  • Adsorbed layers can reduce effective channel size by several nanometers
• Capillary forces and surface tension effects become dominant influencing fluid behavior and transport phenomena
  • Young-Laplace equation describes pressure difference across curved interfaces: $\Delta P = \gamma (\frac{1}{R_1} + \frac{1}{R_2})$
  • Capillary filling in nanochannels follows Lucas-Washburn equation: $L^2 = \frac{\gamma R \cos \theta}{2\eta} t$

Energy Dissipation and Confinement Effects

• High surface-to-volume ratio leads to increased energy dissipation in nanofluidic flows affecting efficiency of fluid transport and mixing processes
  • Enhanced viscous dissipation near walls
  • Electrokinetic effects contribute to additional energy dissipation
• Nanoconfinement effects due to proximity of surfaces induce changes in fluid structure potentially altering thermodynamic properties
  • Melting point depression observed for confined fluids
  • Shifts in critical point and phase behavior reported in nanopores
• Enhanced surface area significantly impacts chemical reactions and catalytic processes potentially increasing reaction rates or enabling novel reaction pathways
  • Surface-to-volume ratio affects reaction kinetics and equilibrium
  • Nanoconfinement can stabilize reaction intermediates or alter reaction mechanisms