💧Multiphase Flow Modeling Unit 3 – Interfacial Phenomena & Surface Tension

Key Concepts and Definitions

• Interfacial phenomena encompasses interactions and processes occurring at the boundary between two phases (liquid-liquid, gas-liquid, solid-liquid)
• Surface tension ($\gamma$) represents the energy required to increase the surface area of a liquid by a unit amount
  • Measured in units of force per unit length (N/m) or energy per unit area (J/m²)
• Capillarity describes the ability of a liquid to flow in narrow spaces without the assistance of external forces
• Wetting refers to the ability of a liquid to maintain contact with a solid surface
  • Determined by the balance of adhesive and cohesive forces
• Contact angle ($\theta$) quantifies the wettability of a solid surface by a liquid
  • Measured as the angle formed between the liquid-solid interface and the liquid-vapor interface
• Interfacial forces include surface tension, disjoining pressure, and capillary pressure
  • Play a crucial role in the behavior and stability of multiphase systems
• Adsorption involves the accumulation of molecules or atoms at an interface
  • Can modify interfacial properties and affect system behavior

Physical Principles of Interfacial Phenomena

• Interfacial phenomena arise from the imbalance of molecular forces at the interface between two phases
• Molecules at the interface experience unequal attractive forces compared to those in the bulk phase
  • Leads to an excess energy at the interface, known as surface free energy
• Minimization of surface free energy drives the system towards a state of minimum surface area
• Surface tension originates from the cohesive forces among liquid molecules
  • Stronger cohesive forces result in higher surface tension values
• Interfacial tension exists between two immiscible liquids or a liquid and a gas
  • Governs the shape and stability of droplets and bubbles
• Laplace pressure ($\Delta P$) describes the pressure difference across a curved interface
  • Given by the Young-Laplace equation: $\Delta P = \gamma (\frac{1}{R_1} + \frac{1}{R_2})$, where $R_1$ and $R_2$ are the principal radii of curvature
• Marangoni effect refers to the mass transfer along an interface due to surface tension gradients
  • Caused by temperature, concentration, or surfactant gradients

Surface Tension: Causes and Effects

• Surface tension arises from the imbalance of molecular forces at the liquid-vapor interface
• Cohesive forces between liquid molecules are responsible for surface tension
  • Stronger intermolecular forces (hydrogen bonding, dipole-dipole interactions) lead to higher surface tension
• Surface tension minimizes the surface area of a liquid to achieve a state of lowest energy
  • Causes liquids to form spherical droplets or menisci in capillaries
• Factors affecting surface tension include temperature, pressure, and the presence of surfactants
  • Increasing temperature reduces surface tension by weakening intermolecular forces
  • Surfactants lower surface tension by adsorbing at the interface and reducing the imbalance of forces
• Surface tension influences the formation and stability of emulsions, foams, and thin films
• Capillary rise or depression in narrow tubes is a consequence of surface tension
  • Height of rise or depth of depression depends on the liquid's surface tension and the tube's radius
• Marangoni convection occurs due to surface tension gradients
  • Drives the movement of liquids from regions of low surface tension to high surface tension

Wetting and Contact Angles

• Wetting refers to the ability of a liquid to spread on a solid surface
• Contact angle ($\theta$) quantifies the wettability of a solid surface by a liquid
  • Measured as the angle between the liquid-solid interface and the liquid-vapor interface at the three-phase contact line
• Young's equation relates the contact angle to interfacial tensions: $\gamma_{SV} = \gamma_{SL} + \gamma_{LV} \cos \theta$
  • $\gamma_{SV}$, $\gamma_{SL}$, and $\gamma_{LV}$ represent the solid-vapor, solid-liquid, and liquid-vapor interfacial tensions, respectively
• Wetting behavior is classified based on the contact angle:
  • Complete wetting: $\theta = 0°$, liquid spreads completely on the surface
  • Partial wetting: $0° < \theta < 90°$, liquid partially spreads on the surface
  • Non-wetting: $\theta > 90°$, liquid tends to form droplets on the surface
• Wettability is influenced by surface roughness, chemical heterogeneity, and surface energy
  • Roughness can enhance or reduce wettability depending on the surface chemistry
  • Chemical heterogeneity leads to contact angle hysteresis and pinning effects
• Superhydrophobic surfaces exhibit extreme water repellency with contact angles greater than 150°
  • Achieved by combining surface roughness with low surface energy materials
• Wetting transitions can occur due to changes in temperature, pressure, or surface chemistry
  • Examples include the transition from Cassie-Baxter state (droplet sitting on surface asperities) to Wenzel state (droplet penetrating surface asperities)

Capillarity and Its Applications

• Capillarity describes the ability of a liquid to flow in narrow spaces without external forces
• Capillary action is driven by the interplay between surface tension and adhesive forces
  • Liquid rises in a capillary tube when adhesive forces (liquid-solid) dominate cohesive forces (liquid-liquid)
• Capillary rise height ($h$) is given by Jurin's law: $h = \frac{2\gamma \cos \theta}{\rho g r}$
  • $\gamma$ is the surface tension, $\theta$ is the contact angle, $\rho$ is the liquid density, $g$ is the acceleration due to gravity, and $r$ is the capillary radius
• Capillary pressure ($P_c$) is the pressure difference across a curved liquid-vapor interface
  • Described by the Young-Laplace equation: $P_c = \frac{2\gamma \cos \theta}{r}$
• Capillarity plays a crucial role in various applications:
  • Wicking in porous media (paper, textiles, soils)
  • Ink delivery in printing processes
  • Oil recovery in petroleum engineering
  • Microfluidic devices for fluid transport and manipulation
• Capillary fingering occurs when a less viscous fluid displaces a more viscous fluid in a porous medium
  • Leads to the formation of finger-like patterns and affects fluid displacement efficiency
• Capillary condensation describes the condensation of vapor in confined spaces (pores, cracks) below the saturation vapor pressure
  • Governed by the Kelvin equation: $\ln \frac{P}{P_0} = \frac{2\gamma V_m \cos \theta}{rRT}$, where $P$ is the vapor pressure, $P_0$ is the saturation vapor pressure, $V_m$ is the molar volume, $R$ is the gas constant, and $T$ is the temperature

Interfacial Forces in Multiphase Systems

• Interfacial forces play a crucial role in the behavior and stability of multiphase systems
• Surface tension forces arise from the imbalance of molecular forces at the interface
  • Act tangentially to the interface and minimize the surface area
• Disjoining pressure is the pressure difference between a thin film and the bulk phase
  • Originates from intermolecular forces (van der Waals, electrostatic, steric) acting across the film
  • Contributes to the stability and thickness of thin films, such as soap films and liquid coatings
• Capillary pressure is the pressure difference across a curved liquid-vapor interface
  • Determined by the Young-Laplace equation: $P_c = \gamma (\frac{1}{R_1} + \frac{1}{R_2})$
  • Influences the shape and stability of droplets, bubbles, and menisci
• Marangoni stresses arise from surface tension gradients along an interface
  • Caused by temperature, concentration, or surfactant gradients
  • Drive the movement of liquids from regions of low surface tension to high surface tension
• Interfacial shear stresses develop when there is a relative motion between two immiscible fluids
  • Affect the flow behavior and mixing in multiphase systems
• Interfacial forces influence the stability and rheology of emulsions, foams, and suspensions
  • Determine the droplet/bubble size distribution, coalescence, and phase separation
• Interfacial instabilities (Rayleigh-Taylor, Kelvin-Helmholtz) can occur due to the interplay of interfacial forces and fluid properties
  • Lead to the breakup of interfaces and the formation of complex flow patterns

Measurement Techniques and Experimental Methods

• Various techniques are employed to measure interfacial properties and study interfacial phenomena
• Pendant drop method measures surface or interfacial tension by analyzing the shape of a suspended droplet
  • Droplet shape is determined by the balance between surface tension and gravitational forces
• Wilhelmy plate method measures surface tension by measuring the force exerted on a vertically suspended plate partially immersed in a liquid
  • Surface tension is calculated from the measured force and the plate's perimeter
• Du Noüy ring method measures surface or interfacial tension using a wire ring pulled through the interface
  • Maximum force required to detach the ring from the interface is related to the surface tension
• Sessile drop method measures the contact angle of a liquid droplet on a solid surface
  • Droplet profile is analyzed using image processing techniques to determine the contact angle
• Capillary rise method measures surface tension by measuring the height of liquid rise in a capillary tube
  • Surface tension is calculated using Jurin's law: $h = \frac{2\gamma \cos \theta}{\rho g r}$
• Spinning drop tensiometry measures ultralow interfacial tensions between two immiscible liquids
  • Droplet elongation in a rotating capillary is related to the interfacial tension
• Langmuir-Blodgett trough studies the behavior of monolayers at the air-water interface
  • Measures surface pressure-area isotherms and provides insights into molecular packing and phase transitions
• Brewster angle microscopy visualizes monolayers and thin films at interfaces without the need for labeling
  • Based on the principle of Brewster angle, where p-polarized light is not reflected from the interface

Modeling Approaches for Interfacial Phenomena

• Modeling interfacial phenomena is essential for understanding and predicting the behavior of multiphase systems
• Continuum models treat the interface as a sharp boundary separating two bulk phases
  • Governed by conservation equations and boundary conditions at the interface
  • Examples include the Volume of Fluid (VOF) method and the Level Set method
• Sharp interface models explicitly track the interface and resolve its dynamics
  • Suitable for problems with well-defined interfaces and moderate interface deformations
  • Examples include the Front Tracking method and the Immersed Boundary method
• Diffuse interface models represent the interface as a thin region with a smooth variation of properties
  • Capture interface dynamics through the evolution of an order parameter (phase-field variable)
  • Examples include the Phase-Field method and the Cahn-Hilliard equation
• Molecular dynamics simulations model the behavior of individual molecules or particles
  • Provide insights into interfacial phenomena at the molecular scale
  • Suitable for studying adsorption, wetting, and interfacial structure
• Lattice Boltzmann methods (LBM) simulate fluid flow and interfacial dynamics using a discrete kinetic approach
  • Capture interfacial physics through appropriate boundary conditions and force models
• Smoothed particle hydrodynamics (SPH) is a meshless Lagrangian method for modeling fluid flow and interfacial phenomena
  • Particles carry fluid properties and interact through smoothing kernels
• Hybrid models combine different modeling approaches to leverage their respective strengths
  • Examples include coupled VOF-LBM, Phase-Field-SPH, and Molecular Dynamics-Continuum methods
• Model validation and comparison with experimental data are crucial for assessing the accuracy and predictive capability of interfacial models