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How water-droplet/dew stick to spider's web? What keeps them there?

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The water droplets on a spider's web are an example of dew. Cold air holds less water than warm air, so if the air has absorbed a lot of water during the day, as it cools overnight the water condenses on anything that can provide a nucleus.

There is an energy barrier to the formation of water droplets from water vapour, so the droplets generally form where there is something to act as a nucleus. This Wikipedia article describes the role of nuclei in formation of clouds, but the same principle applies to condensation onto a spider's web. There's nothing special about a spiders web: dew forms on just about anything. It's just prettier on a spider's web so we notice it more.

The reason you get droplets is due to a phenomenon called Rayleigh Instability. This causes a smooth film of water on a fibre to break up into droplets.

The droplets stick to the fibres of the web due to capillary forces.

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