Surface tension:

Capillary pressure, scaling, wetting

John W. M. Bush

Department of Mathematics MIT

18.357: Lecture 3

Surface tension:

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σ =forcelength

analogous to a negative surface pressure

P = F/A

! gradients in surface tension necessarily drive surface motion

! measure the force required to withdraw a plate from a free surface

A simple way to measure surface tension

NOTES

r · n̂ =1

R1+

1

R2

Curvature

where R1 , R2 are the principal radii of curvature

n̂

R1

R2

Curvature:

Which way does the air go?

r · n̂ =1

R1+

1

R2for a sphere= 2/R

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R

Who cares about surface tension?

Capillary pressures in biology

1 mm

Ostwald ripening

The scaling of surface tension

Fundamental Concept

The laws of Nature cannot depend on arbitrarily chosen system of units.A system is most succinctly described in terms of dimensionless variables.

DIMENSIONAL ANALYSIS

Deduction of Dimensionless groups: Buckingham’s Theorem

For a system with M physical variables (e.g. density, speed, length, viscosity)describable in terms of N fundamental units (e.g. mass, length, time, temperature),there are M - N dimensionless groups that govern the system.

E.g. Translation of a sphere

Physical variables:

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U , a,ν , ρ ,D ⇒

Fundamental units: M , L , T

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⇒M = 5N = 3

Ua

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ρ ,νD

M - N = 2 dimensionless groups:

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Cd =DρU 2 , Re =

U aν

System uniquely determined by a single relation:

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Cd = F(Re)

The scaling of surface tension g

U

water

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ρ ,νvacuum

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Bo =ρ ga2

σ=

GRAVITYCURVATURE

= Bond number

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We =ρU 2aσ

=INERTIA

CURVATURE= Weber number

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Ca =ρνUσ

=VISCOSITYCURVATURE

= Capillary number

a

Note: is dominant relative to gravity when

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σ

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Bo < 1

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a < σρg⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

= ℓ c = capillary length ~ 2mm for air-wateri.e.

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σ

When is surface tension important relative to gravity?

• when curvature pressures are large relative to hydrostatic:

a ρ

σgBo ! 1

Bo > 1

i.e. for drops small relative to the capillary length:

Bond number:

2 mm for air-watera < lc =

(

σ

ρg

)1/2

∼

Bo =ρga

σ/a=

ρga2

σ< 1

(σ = 70 dynes/cm)

Surface tension dominates the world of insects - and of microfluidics.

Falling rain drops

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a < ℓ c = σ /ρg ≈ 2mm

If a drop is small relative tothe capillary length

maintains it against the destabilizing influence ofaerodynamic stresses.

,

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ρU 2a2 ~ M g = 43 π a

3 ρ g

Small drops

Force balance:

Fall speed:

Drop integrity requires:

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σ

U ∼

√

ρ ga

ρa

a

ρaU2∼ ρga < σ/a

Drops larger than the capillarylength

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a > ℓ c ≈ 2mm

,

break up under the influence ofaerodynamic stresses.

The break-up yields drops with size of order:

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ℓ c ≈ 2mm

Big drops

Puddles

Wetting

Who cares about wetting?

The world’s smallest lizard: the Brazilian Pygmy Gecko

Partial wetting

Total wetting

Water on glass