Physics and Mathematics
Angle of Contact
1. Statement of the Concept
The angle of contact is the angle formed between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid.
It determines whether a liquid will wet or not wet a solid surface.
2. Explanation and Mathematical Derivation
When a liquid is in contact with a solid, molecular forces act between their molecules:
- Cohesive force: Force between molecules of the same substance (liquid–liquid).
- Adhesive force: Force between molecules of different substances (liquid–solid).
At the contact point between a liquid and solid:
- If adhesive forces > cohesive forces → liquid spreads (wetting).
- If cohesive forces > adhesive forces → liquid does not spread (non-wetting).
Defining the Angle of Contact [θ]
- Consider a small quantity of liquid in contact with a solid surface and air.
- The tangent to the liquid surface at the point of contact makes an angle θ with the solid surface, measured inside the liquid.
Mathematical Relation
At equilibrium, the horizontal components of surface tensions are balanced:
[T_{SL} + T_{LA}\cosθ] [= T_{SA}]
Where,
- [T_{SL}] = surface tension between solid and liquid
- [T_{LA}] = surface tension between liquid and air
- [T_{SA}] = surface tension between solid and air
Thus,
[\cosθ] [= \dfrac{T_{SA} – T_{SL}}{T_{LA}}]
3. Dimensions and Units
- Surface Tension (T): [ML⁰T⁻²]
- Angle of Contact (θ): Dimensionless (radians)
4. Key Features
- Depends on the nature of the liquid and solid in contact.
- Affected by surface impurities and temperature.
- Determines the shape of a meniscus:
- Acute (θ < 90°) → Concave meniscus (water in glass).
- Obtuse (θ > 90°) → Convex meniscus (mercury in glass).
- Helps explain capillary action and droplet shape.
5. Important Formulas to Remember
| Quantity | Expression | Remarks |
|---|---|---|
| Surface tension balance | [T_{SA}] [= T_{SL} + T_{LA}\cosθ] | At equilibrium |
| Relation for cosθ | [\cosθ] [= \dfrac{T_{SA} – T_{SL}}{T_{LA}}] | Used to find θ |
| Wetting condition | [T_{SA} > T_{SL}] | θ < 90° |
| Non-wetting condition | [T_{SA} < T_{SL}] | θ > 90° |
6. Conceptual Questions with Solutions
1. What does a smaller angle of contact indicate?
Stronger adhesive forces; the liquid wets the surface (e.g., water on glass).
2. Why does mercury form a convex meniscus in glass?
Because cohesive forces (Hg–Hg) are greater than adhesive forces (Hg–glass), leading to θ > 90°.
3. How does temperature affect the angle of contact?
Increasing temperature decreases surface tension, which reduces the angle of contact.
4. Can angle of contact be greater than 180°?
No, θ always lies between 0° and 180°.
5. What is the angle of contact for a perfectly wetting liquid?
For a perfectly wetting liquid, θ = 0°.
6. What happens when TSA = TSL?
Then cosθ = 0 ⇒ θ = 90°, meaning no capillary rise or fall.
7. Why do raindrops on waxed leaves appear spherical?
Because cohesive forces are greater than adhesive forces, producing a large angle of contact and minimal surface area.
8. Why does water spread on glass but not on paraffin?
Adhesive > cohesive for glass (wetting), but cohesive > adhesive for paraffin (non-wetting).
9. Is the angle of contact the same for all liquids on a given solid?
No, θ depends on the specific interactions between the liquid and the solid.
10. What is the significance of measuring θ experimentally?
It helps determine interfacial energies between solid, liquid, and air interfaces.
11. Why is θ measured inside the liquid?
Because surface tension acts tangentially along the surface of the liquid.
12. Can impurities change θ?
Yes, surface impurities alter interfacial tensions, thus changing θ.
13. Why does soap reduce the angle of contact between water and glass?
Soap reduces the surface tension of water, which decreases θ and increases spreading.
14. What is the typical angle of contact between pure water and clean glass?
Approximately 0°, indicating complete wetting.
15. Why does an oil drop spread on the surface of water?
Because adhesive interaction between oil and water exceeds oil–oil cohesion.
7. FAQ / Common Misconceptions
1. Is the angle of contact the same for the top and bottom of a meniscus?
No, local curvature varies, but the equilibrium angle at the contact line is the same.
2. Does a higher θ mean higher surface tension?
Not necessarily; θ depends on the ratio of interfacial tensions, not their absolute values.
3. Does surface roughness affect the angle of contact?
Yes, roughness and contamination can significantly alter the observed θ.
4. If θ = 90°, will capillary rise occur?
No, when θ = 90°, there is no net rise or fall in the liquid level.
5. Can θ change over time?
Yes, if the liquid or solid surface changes (oxidation, contamination, etc.).
6. Does gravity affect the angle of contact?
No, θ is a microscopic property determined by intermolecular forces, not gravity.
7. Are θ and meniscus curvature directly proportional?
No, they are related through surface tension equilibrium, not direct proportionality.
8. Is the angle of contact defined for solids and gases?
No, it’s defined at the interface of a solid, liquid, and gas only.
9. If θ decreases, what happens to wettability?
Wettability increases as θ decreases.
10. Why does adding detergent to water make it spread more?
Detergent reduces the liquid–air surface tension, lowering θ and enhancing wetting.
8. Practice Questions (with Step-by-Step Solutions)
Q1. If [T_{SA} = 0.070 N/m], [T_{SL} = 0.020 N/m], and [T_{LA} = 0.072 N/m], find θ.
Solution:
[\cosθ] [= \dfrac{T_{SA} – T_{SL}}{T_{LA}}] [= \dfrac{0.070 – 0.020}{0.072}] [= 0.694]
[θ = \cos^{-1}(0.694) ≈ 46°]
So, the liquid wets the solid (θ < 90°).
Q2. For mercury on glass, [T_{SA} = 0.51 N/m], [T_{SL} = 0.56 N/m], [T_{LA} = 0.48 N/m]. Find θ.
[\cosθ] [= \dfrac{0.51 – 0.56}{0.48}] [= -0.104] [\Rightarrow θ = 96°]
Hence, mercury does not wet the glass.
Q3. If θ = 0°, what can you conclude about [T_{SA}] and [T_{SL}]?
[\cosθ] [= 1] [= \dfrac{T_{SA} – T_{SL}}{T_{LA}}] [\Rightarrow T_{SA} – T_{SL}] [= T_{LA}]
Thus, perfectly wetting condition.
Q4. Determine θ when [T_{SA} = T_{SL}].
[\cosθ = 0] [\Rightarrow θ = 90°]
Hence, no capillary rise or fall.
Q5. Find θ if [T_{SA} – T_{SL} = 0.5T_{LA}].
[\cosθ = 0.5] [\Rightarrow θ = 60°]
Liquid wets the solid moderately.
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