Surface tension and capillarity are critical phenomena in fluid mechanics, chemistry, biology, and engineering. These properties govern the behavior of fluids at interfaces, in small tubes, and near solid surfaces. Understanding these phenomena is essential for explaining droplet formation, liquid rise in capillaries, detergency, and many industrial and biological processes.
This post provides a detailed exploration of surface tension, capillarity, their principles, mathematical formulations, examples, and applications.
1. Introduction to Surface Tension
Surface tension (γ) is the property of a liquid surface that causes it to behave as if it were a stretched elastic sheet.
- Caused by cohesive forces between molecules within the liquid
- Molecules at the surface experience unbalanced inward forces, leading to minimum surface area
Definition:
Surface tension is the force acting along a line of unit length on the surface of a liquid, perpendicular to the line.
Formula: γ=FL\gamma = \frac{F}{L}γ=LF
Where:
- γ\gammaγ = surface tension (N/m)
- FFF = force acting along the surface (N)
- LLL = length over which force acts (m)
SI Unit: N/m (Newton per meter)
Typical Values:
- Water at 25°C: γ≈0.0728 N/m\gamma \approx 0.0728 \, N/mγ≈0.0728N/m
- Mercury: γ≈0.485 N/m\gamma \approx 0.485 \, N/mγ≈0.485N/m
2. Molecular Basis of Surface Tension
- Cohesive Forces: Molecules within the bulk attract equally in all directions → no net force
- Surface Molecules: Experience net inward cohesive force, creating tension at the interface
- Minimization of Energy: Liquid surface reduces area to minimize potential energy
Implication: Droplets tend to form spherical shapes, which have minimum surface area for a given volume.
3. Work Done in Increasing Surface Area
- Increasing surface area requires work against surface tension
- For a liquid surface of area AAA:
W=γ ΔAW = \gamma \, \Delta AW=γΔA
Where WWW = work done, ΔA\Delta AΔA = change in surface area
Example: Soap bubbles expand because surface tension acts to minimize area.
4. Measurement of Surface Tension
4.1 Capillary Rise Method
- Liquid rises in a thin tube due to surface tension and adhesion
- Height hhh given by:
h=2γcosθρgrh = \frac{2 \gamma \cos \theta}{\rho g r}h=ρgr2γcosθ
Where:
- rrr = radius of capillary
- θ\thetaθ = contact angle
- ρ\rhoρ = liquid density
4.2 Drop Weight or Drop Volume Method
- Surface tension calculated from weight of droplet:
γ=mg2πr\gamma = \frac{mg}{2 \pi r}γ=2πrmg
Where mmm = mass of drop, rrr = radius of tube
4.3 Maximum Bubble Pressure Method
- Pressure inside bubble related to surface tension:
ΔP=2γR\Delta P = \frac{2 \gamma}{R}ΔP=R2γ
Where RRR = radius of bubble
5. Capillarity (Capillary Action)
Definition: Capillarity is the rise or fall of a liquid in a narrow tube (capillary) due to surface tension.
Mechanism:
- Adhesion between liquid and tube
- Cohesion within the liquid
- Gravity acting downward
Capillary rise equation: h=2γcosθρgrh = \frac{2 \gamma \cos \theta}{\rho g r}h=ρgr2γcosθ
- hhh = height of liquid rise or depression
- rrr = radius of capillary
- θ\thetaθ = contact angle
- γ\gammaγ = surface tension
- ρ\rhoρ = density of liquid
Observation:
- Wetting liquids (θ<90°\theta < 90°θ<90°) → rise in capillary
- Non-wetting liquids (θ>90°\theta > 90°θ>90°) → depression in capillary
6. Contact Angle
- Angle formed between liquid surface and solid surface at contact line
- Determines wetting behavior:
- θ<90°\theta < 90°θ<90° → liquid spreads, adhesive forces dominate → rise
- θ>90°\theta > 90°θ>90° → liquid contracts, cohesive forces dominate → depression
Applications: Adhesives, paints, coatings, inkjet printing
7. Pressure Difference Across Curved Surfaces
- Curved liquid surfaces create pressure difference due to surface tension:
ΔP=γ(1R1+1R2)\Delta P = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2} \right)ΔP=γ(R11+R21)
Where R1R_1R1 and R2R_2R2 = principal radii of curvature
- Example: Soap bubbles, droplets, meniscus
7.1 Soap Bubble
- Both inner and outer surfaces contribute:
ΔP=4γr\Delta P = \frac{4 \gamma}{r}ΔP=r4γ
Where rrr = radius of bubble
7.2 Liquid Droplet
ΔP=2γr\Delta P = \frac{2 \gamma}{r}ΔP=r2γ
- Explains shape and stability of droplets
8. Meniscus Formation
- Liquid forms concave or convex curve at container wall due to adhesion and cohesion
- Concave meniscus: Water in glass (θ<90°\theta < 90°θ<90°)
- Convex meniscus: Mercury in glass (θ>90°\theta > 90°θ>90°)
Implication: Accurate volume measurement in laboratory glassware requires reading bottom of meniscus for concave, top for convex.
9. Factors Affecting Surface Tension
- Temperature: Increases → reduces surface tension
- Impurities / Surfactants: Lower surface tension (soap, detergent)
- Liquid type: Polar liquids (water) → higher γ, non-polar liquids (oil) → lower γ
- Pressure: Minor effect on surface tension
10. Applications of Surface Tension
10.1 Droplet Formation
- Raindrops, inkjet printing, and sprays rely on surface tension to maintain shape
- High γ → stable droplets, less fragmentation
10.2 Capillary Rise in Plants
- Water rises in xylem tubes via capillarity
- Supports transport of water from roots to leaves
10.3 Detergency and Cleaning
- Surfactants reduce surface tension, allowing liquids to wet surfaces and remove dirt
- Soap molecules align at surface → decrease γ → improve cleaning
10.4 Floating of Light Objects
- Small insects (water striders) float due to surface tension supporting their weight
- Engineering applications: microfluidics
10.5 Meniscus Reading in Laboratory
- Accurate volume measurement in pipettes, burettes, and graduated cylinders
- Consider meniscus curvature to avoid error
10.6 Foam and Bubbles
- Bubble formation controlled by surface tension of liquid film
- Applications: detergents, carbonated drinks, fire-fighting foams
11. Applications of Capillarity
11.1 Paper Chromatography
- Capillary action moves liquid solvent along stationary phase
- Separates mixture components based on solubility
11.2 Oil Lamps and Wicks
- Capillary rise draws liquid fuel upward in wick → continuous combustion
11.3 Soil Moisture Movement
- Water moves upward in soil pores via capillarity
- Essential for agriculture and irrigation
11.4 Thin Layer Liquid Films
- Capillary forces create thin films in coatings, printing, and microfluidics
11.5 Inkjet Printers
- Capillary action ensures precise ink flow into printheads
12. Capillary Phenomena in Nature
- Water transport in plants: Roots → stem → leaves
- Absorption in sponges and towels
- Movement of fluids in small blood vessels
- Soil capillarity: Influences water retention and plant growth
13. Mathematical Examples
Example 1: Capillary Rise
- Water (γ=0.0728 N/m\gamma = 0.0728 \, N/mγ=0.0728N/m), θ=0°\theta = 0°θ=0°, radius r=0.5 mmr = 0.5 \, mmr=0.5mm, density ρ=1000 kg/m3\rho = 1000 \, kg/m^3ρ=1000kg/m3, g=9.81g = 9.81g=9.81
h=2γcosθρgr=2⋅0.07281000⋅9.81⋅0.0005≈29.6 mmh = \frac{2 \gamma \cos \theta}{\rho g r} = \frac{2 \cdot 0.0728}{1000 \cdot 9.81 \cdot 0.0005} \approx 29.6 \, mmh=ρgr2γcosθ=1000⋅9.81⋅0.00052⋅0.0728≈29.6mm
- Water rises ~3 cm in capillary
Example 2: Pressure Difference Across a Bubble
- Soap bubble radius r=2 mmr = 2 \, mmr=2mm, γ=0.03 N/m\gamma = 0.03 \, N/mγ=0.03N/m
ΔP=4γr=4⋅0.030.002=60 Pa\Delta P = \frac{4 \gamma}{r} = \frac{4 \cdot 0.03}{0.002} = 60 \, PaΔP=r4γ=0.0024⋅0.03=60Pa
- Explains bubble stability and tendency to expand
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