1.3 Surface tension

Surface tension is a crucial property of liquids that causes their surfaces to behave like elastic sheets. It arises from the imbalance of cohesive forces between molecules at the liquid surface, resulting in various phenomena like capillary action and meniscus formation.

Understanding surface tension is essential in fluid dynamics, as it affects fluid behavior in narrow spaces and at interfaces. This knowledge has applications in microfluidics, inkjet printing, respiratory physiology, and even insect locomotion on water surfaces.

Fundamentals of surface tension

• Surface tension is a property of liquids that causes their surfaces to behave like elastic sheets due to the cohesive forces between liquid molecules
• Plays a crucial role in various phenomena such as capillary action, meniscus formation, and the behavior of soap films and bubbles

Molecular origin of surface tension

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• Arises from the imbalance of cohesive forces experienced by molecules at the liquid surface compared to those in the bulk
• Molecules at the surface have fewer neighboring molecules to interact with, resulting in a net inward force
• This inward force minimizes the surface area of the liquid, causing it to behave like an elastic sheet

Cohesive vs adhesive forces

• Cohesive forces are the attractive forces between molecules of the same substance (liquid-liquid interactions)
• Adhesive forces are the attractive forces between molecules of different substances (liquid-solid or liquid-gas interactions)
• The balance between cohesive and adhesive forces determines the wetting behavior of liquids on surfaces

Surface tension coefficient

• Denoted by the Greek letter $\gamma$ and expressed in units of force per unit length (N/m) or energy per unit area (J/m²)
• Represents the force required to create a new unit area of the liquid surface or the energy required to increase the surface area by a unit amount
• Typical values range from 20-80 mN/m for most liquids at room temperature (water: 72 mN/m, ethanol: 22 mN/m)

Factors affecting surface tension

• Temperature: Surface tension decreases with increasing temperature due to increased thermal motion of molecules and reduced cohesive forces
• Solutes: Dissolved substances can either increase (surfactants) or decrease (inorganic salts) the surface tension depending on their effect on cohesive forces
• Electric fields: Applied electric fields can modify the surface tension by inducing charge redistribution at the liquid surface
• Surface contamination: Impurities or contaminants at the liquid surface can significantly alter the surface tension

Capillary action

• The ability of liquids to flow through narrow spaces without the assistance of, or even in opposition to, external forces like gravity
• Plays a vital role in the transport of fluids in plants, wicking of liquids in porous materials, and the operation of microfluidic devices

Capillary rise in tubes

• When a liquid is placed in a narrow tube, it can spontaneously rise (or fall) to a certain height due to the interplay between surface tension and gravity
• The height of the liquid column depends on the liquid's surface tension, the tube's radius, and the contact angle between the liquid and the tube wall
• For a wetting liquid (contact angle < 90°), the liquid rises in the tube; for a non-wetting liquid (contact angle > 90°), the liquid is depressed

Contact angle and wettability

• The contact angle $\theta$ is the angle formed between the liquid-solid interface and the liquid-vapor interface at the three-phase contact line
• Determines the wettability of a surface: complete wetting ($\theta = 0°$), partial wetting ($0° < \theta < 90°$), or non-wetting ($\theta > 90°$)
• Affects capillary rise, droplet shape, and the ease with which a liquid spreads on a surface (water on glass: $\theta \approx 0°$, mercury on glass: $\theta \approx 140°$)

Jurin's law

• Relates the height of the liquid column in a capillary tube to the tube's radius, the liquid's surface tension, and the contact angle
• Mathematically expressed as: $h = \frac{2\gamma \cos\theta}{\rho g r}$, where $h$ is the height of the liquid column, $\gamma$ 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 tube radius
• Predicts that narrower tubes result in higher liquid rise (or fall) for a given liquid and contact angle

Capillary pressure

• The pressure difference across a curved liquid-vapor interface caused by surface tension
• Responsible for the flow of liquids through porous media and the stability of liquid bridges and menisci
• Given by the Young-Laplace equation: $\Delta P = \gamma (\frac{1}{R_1} + \frac{1}{R_2})$, where $\Delta P$ is the capillary pressure, $\gamma$ is the surface tension, and $R_1$ and $R_2$ are the principal radii of curvature of the interface
• Positive for concave menisci (liquid rises) and negative for convex menisci (liquid is depressed)

Meniscus shapes

• The curved liquid-vapor interface formed when a liquid is in contact with a solid surface
• Determined by the balance between surface tension and gravity, as well as the wetting properties of the liquid on the solid surface

Concave vs convex menisci

• Concave menisci occur for wetting liquids (contact angle < 90°) and are characterized by a liquid surface that curves upward near the solid surface
• Convex menisci occur for non-wetting liquids (contact angle > 90°) and are characterized by a liquid surface that curves downward near the solid surface
• The shape of the meniscus affects the capillary pressure and the liquid's behavior in confined spaces (water in glass tubes: concave meniscus, mercury in glass tubes: convex meniscus)

Radius of curvature

• A measure of the curvature of the liquid-vapor interface at any point
• Determined by the principal radii of curvature, $R_1$ and $R_2$, which are the radii of the circles that best fit the surface in two perpendicular planes
• Smaller radii of curvature correspond to more highly curved interfaces and larger capillary pressures

Laplace pressure

• The pressure difference across a curved liquid-vapor interface, as described by the Young-Laplace equation
• Proportional to the surface tension and the curvature of the interface (inverse of the radii of curvature)
• Responsible for the pressure difference between the inside and outside of bubbles, droplets, and menisci
• Explains the stability of foams and emulsions, as well as the pressure drop in capillary flows

Surface tension effects

• Surface tension gives rise to various fascinating phenomena in nature and technology, showcasing the unique properties of liquid surfaces and interfaces

Soap films and bubbles

• Soap molecules (surfactants) reduce the surface tension of water, allowing the formation of stable thin films and bubbles
• Soap films consist of a thin layer of liquid bounded by two air-liquid interfaces, with the soap molecules oriented such that their hydrophilic heads face the water and their hydrophobic tails face the air
• Bubbles are spherical shells of soap film enclosing a volume of air, with the internal pressure being higher than the external pressure due to the Laplace pressure

Marangoni effect

• A mass transfer phenomenon driven by surface tension gradients along a liquid-fluid interface
• Occurs when there are local variations in surface tension caused by temperature, concentration, or surfactant distribution
• Induces fluid flow from regions of low surface tension to regions of high surface tension, leading to the formation of convection cells and interfacial instabilities
• Plays a role in various processes, such as the spreading of oil spills, the drying of paint films, and the behavior of tear films in the eye

Tears of wine phenomenon

• The formation of droplets (tears) that run down the inside of a glass after swirling wine
• Caused by the Marangoni effect, which is driven by the surface tension gradient between the wine and the evaporating alcohol
• As the alcohol evaporates from the thin film of wine on the glass, it creates a region of higher surface tension, causing the liquid to be pulled up the glass in the form of tears
• The tears eventually become too heavy and fall back down into the bulk of the wine, creating a self-sustaining cycle

Capillary waves

• Small ripples or waves that propagate along the surface of a liquid due to the restoring force of surface tension
• Distinct from gravity waves, which are driven by the restoring force of gravity and have longer wavelengths
• Capillary waves have short wavelengths (typically less than a few millimeters) and are rapidly damped by viscous forces
• Play a role in the scattering of light from liquid surfaces, the formation of capillary bridges, and the atomization of liquids in sprays and jets

Measurement techniques

• Various experimental methods have been developed to measure the surface tension of liquids, each with its own advantages and limitations

Capillary rise method

• Based on measuring the height of the liquid column that rises in a narrow capillary tube due to surface tension
• The surface tension is calculated using Jurin's law, which relates the height to the tube radius, the liquid density, and the contact angle
• Simple and inexpensive, but requires precise measurement of the tube radius and the contact angle
• Suitable for liquids with moderate to high surface tension values (water, organic solvents)

Wilhelmy plate method

• Involves measuring the force exerted on a thin plate (usually made of platinum or glass) that is partially immersed in the liquid
• The surface tension is calculated from the force, the plate perimeter, and the contact angle between the liquid and the plate
• Provides a direct and accurate measurement of surface tension, but requires a sensitive force sensor and a clean, well-defined plate surface
• Widely used for studying the dynamic surface tension of surfactant solutions and the wetting behavior of liquids on solid surfaces

Du Noüy ring method

• Uses a thin wire ring (usually made of platinum or platinum-iridium alloy) that is pulled through the liquid-air interface
• The maximum force required to detach the ring from the interface is measured and related to the surface tension using a correction factor that accounts for the ring geometry and the liquid density
• Provides a quick and reliable measurement of surface tension, but is sensitive to the cleanliness of the ring and the alignment of the ring with the interface
• Commonly employed in industrial settings for quality control and product development

Pendant drop method

• Based on analyzing the shape of a liquid droplet hanging from a needle tip
• The droplet shape is determined by the balance between surface tension and gravity, and is described by the Young-Laplace equation
• The surface tension is obtained by fitting the theoretical drop shape to the observed profile using numerical methods
• Provides a non-invasive and accurate measurement of surface tension, especially for liquids with low surface tension values or in situations where the liquid volume is limited
• Requires a high-resolution imaging system and a stable droplet formation setup

Applications of surface tension

• Surface tension plays a crucial role in various fields, from biological systems to industrial processes and advanced technologies

Microfluidics and lab-on-a-chip devices

• Surface tension is a dominant force at the microscale, enabling the control and manipulation of fluids in microchannels and microchambers
• Capillary forces can be used to passively pump liquids, form droplets, and create stable interfaces in microfluidic devices
• Surface tension-driven flows can be used for mixing, sorting, and separating particles or cells in lab-on-a-chip platforms
• Understanding and engineering surface properties is crucial for the design and operation of microfluidic systems (electrowetting, surface patterning)

Inkjet printing technology

• Surface tension plays a critical role in the formation, stability, and deposition of ink droplets in inkjet printing
• The ink's surface tension affects the droplet size, shape, and spreading behavior on the substrate, which in turn influence the print quality and resolution
• Surfactants are often added to the ink to modify its surface tension and improve the printing performance
• Controlling the surface tension is essential for achieving reliable and high-quality prints on various substrates (paper, plastic, textiles)

Lung surfactants and respiratory disorders

• Lung surfactants are complex mixtures of lipids and proteins that reduce the surface tension of the alveolar fluid, preventing the collapse of the alveoli during exhalation
• Insufficient or dysfunctional lung surfactants can lead to respiratory disorders, such as neonatal respiratory distress syndrome (NRDS) and acute respiratory distress syndrome (ARDS)
• Surfactant replacement therapy involves administering exogenous surfactants to improve lung compliance and gas exchange in affected individuals
• Understanding the surface tension properties of lung surfactants is crucial for developing effective therapies and diagnostic tools for respiratory disorders

Insect locomotion on water surfaces

• Many insects, such as water striders, can walk or jump on water surfaces without sinking due to the interplay between surface tension and their specialized leg structures
• The insect's weight is supported by the surface tension force acting on its hydrophobic legs, creating dimples on the water surface
• The legs are covered with microscopic hairs (setae) that further increase the water repellency and the contact area with the water surface
• Insects can also use surface tension to propel themselves by creating ripples or vortices on the water surface (water strider leg rowing, whirligig beetle gyration)
• Studying insect locomotion on water can inspire the design of water-walking robots and novel propulsion mechanisms for aquatic applications