Problem Statement
Define the spreading coefficient $S = \sigma_{SG}-\sigma_{SL}-\sigma_{LG}$ and state the conditions for spreading, partial wetting, and non-wetting.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: The spreading coefficient determines whether a liquid droplet spreads on a solid surface:
Step 2 — Apply the relevant physical law or equation: $$S = \sigma_{SG} – \sigma_{SL} – \sigma_{LG}$$
Step 3 — Solve algebraically for the unknown:
- $S > 0$: Complete wetting — liquid spreads indefinitely, forming a thin film.
- $S = 0$: Critical (neutral) wetting — Young’s equation with $\theta=0°$.
- $S < 0$: Partial wetting — droplet forms with contact angle $\theta > 0°$ (Young’s equation: $\cos\theta = (\sigma_{SG}-\sigma_{SL})/\sigma_{LG} = 1 + S/\sigma_{LG}$).
Step 4 — Substitute numerical values with units: Example: Water on clean glass: $S>0$ (complete wetting). Water on waxed surface: $S<0$ (beads up, $\theta\approx110°$). Mercury on glass: $S\ll0$ ($\theta\approx140°$).
Worked Calculation
$$S = \sigma_{SG} – \sigma_{SL} – \sigma_{LG}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{S = \sigma_{SG} – \sigma_{SL} – \sigma_{LG}}$$
The spreading coefficient determines whether a liquid droplet spreads on a solid surface:
$$S = \sigma_{SG} – \sigma_{SL} – \sigma_{LG}$$
- $S > 0$: Complete wetting — liquid spreads indefinitely, forming a thin film.
- $S = 0$: Critical (neutral) wetting — Young’s equation with $\theta=0°$.
- $S < 0$: Partial wetting — droplet forms with contact angle $\theta > 0°$ (Young’s equation: $\cos\theta = (\sigma_{SG}-\sigma_{SL})/\sigma_{LG} = 1 + S/\sigma_{LG}$).
Example: Water on clean glass: $S>0$ (complete wetting). Water on waxed surface: $S<0$ (beads up, $\theta\approx110°$). Mercury on glass: $S\ll0$ ($\theta\approx140°$).
Answer
$$\boxed{S = \sigma_{SG} – \sigma_{SL} – \sigma_{LG}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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