# Surface tension and excess pressure

• Urmi Roy
In summary, the excess pressure for a liquid-gas interface with radii of curvature is given by ΔP= σ(1/R1 + 1/R2). To determine which radii/lengths to plug into the equation, one must look at the liquid-gas interface and see what radii are on the opposite sides. The curvature is different at different points around the plate for the rectangular plate, and for the bubble the curvatures have the same value, 1/R1.

#### Urmi Roy

Let σ be the surface tension ...
I got to know that the excess pressure for a liquid-gas interface with radii of curvature (see http://en.wikipedia.org/wiki/Surface_tension#Surface_curvature_and_pressure ...the part on Surface curvature and pressure and Young-Laplace equation).

is given by ΔP= σ(1/R1 + 1/R2)

I have 3 cases as shown in my attached figure... in there cases, how do we know which radii/lengths to plug into the above formula?

how do we/on what basis do we make the selection..?

I'm guessing that we just look at the liquid-gas interface and see what radii are on the opposite sides...but that doesn't work for the rectangular plate...

also, especially in regard to the liquid droplet, if we take the inner radius to be one radius, on the other side of the boundary of the droplet, we have an infinite radius!

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As the Wikipedia article states, the quantity in parentheses is known as the "mean curvature" of the surface. 1/R1 and 1/R2 are known as the principal curvatures. For a smooth surface if you look at all curves on the surface that pass through the central point, their curvature will vary. There will be two directions in which the curvature takes an extreme value, and by definition these are the principal curvatures. The two principal directions will be perpendicular to each other.

1) You've correctly identified r and r', only remember that r' is negative.
2) For the rectangular plate the curvature will be different at different points around the plate. Along one of the straight sides, you've identified one principal curvature r' (it's negative). In the other principal direction parallel to the edge the curvature is zero (r is infinite, that's fine)
3) For the bubble, both curvatures have the same value, 1/R1.

I'm a little confused by the diagrams- if the fluid is not pinned to the edges of the plate, the mean curvature is simple to understand- one principal radius is r', and the other is r, and note that r varies with height. This means r' has to vary with height as well: IIRC, if there is no gravity the profile is a catenary, and if gravity is present you get an 'amphora' shape, and the boundary condition is given by the contact angle.

Now, if the fluid is pinned to the plate edges, things change considerably: for the circle plate there's no conceptual issue (except the contact angle can vary), but the rectangular plate *is* much more complicated because of the corners. The radius of curvature there is infinite, so the fluid can't stay pinned there. I'm unaware of a solution to this problem, although the fluid shape is a minimal surface, so there may be a solution out there.

Bill_K said:
1/R1 and 1/R2 are known as the principal curvatures.

For a smooth surface if you look at all curves on the surface that pass through the central point, their curvature will vary.
There will be two directions in which the curvature takes an extreme value, and by definition these are the principal curvatures.

The two principal directions will be perpendicular to each other.

I guess the bolded parts are important to remember...

Bill_K said:
2) For the rectangular plate the curvature will be different at different points around the plate. Along one of the straight sides, you've identified one principal curvature r' (it's negative). In the other principal direction parallel to the edge the curvature is zero (r is infinite, that's fine)

I don't understand the bold parts...why is the radius of curvature in one direction zero? How does the curvature vary around the rectangle?

Also, you said that the curvatures are supposed to be taken perpendicular to one-another...this does not happen here...neither for the circular disc case for that matter...could you elaborate more on this example,please?

Andy Resnick said:
Now, if the fluid is pinned to the plate edges, things change considerably: for the circle plate there's no conceptual issue (except the contact angle can vary), but the rectangular plate *is* much more complicated because of the corners. The radius of curvature there is infinite, so the fluid can't stay pinned there.

How are the radii of curvature at the corners infinite? For that matter, I don't think I understand what happens along the edges either..what about if it were an infinite rectangular plate?

The liquid isn't pinned, I guess...we're just sort of balancing the plate on a film of liquid...much like in a contact lens-eye interface...

Urmi Roy said:
How are the radii of curvature at the corners infinite?

No, the curvature is infinite- the radius is zero (k = 1/r) because the surface is discontinuous at an edge or corner.

Urmi Roy said:
For that matter, I don't think I understand what happens along the edges either..what about if it were an infinite rectangular plate?

The liquid isn't pinned, I guess...we're just sort of balancing the plate on a film of liquid...much like in a contact lens-eye interface...

If the liquid isn't pinned, then the shape of the plate doesn't matter.

Andy Resnick said:
No, the curvature is infinite- the radius is zero (k = 1/r) because the surface is discontinuous at an edge or corner.
If the liquid isn't pinned, then the shape of the plate doesn't matter.

I think I'm having a basic problem in how to decide which radii I'm talking about...please could you also look at post #4...maybe if you answered those questions first, it would be easier for me...

If you look the derivation it would be easy.

## 1. What is surface tension and how does it work?

Surface tension is the force that causes the surface of a liquid to behave like a stretched elastic membrane. It is caused by the cohesive forces between the molecules of the liquid. This results in the surface of the liquid having a higher energy and therefore, behaving as if it has a stretched surface.

## 2. How does surface tension affect the behavior of objects on the surface of a liquid?

Surface tension creates a "skin" on the surface of the liquid, making it more difficult for objects to penetrate or sink into the liquid. This is why small objects, like paper clips, can float on the surface of water. It also causes liquids to form droplets, as the cohesive forces pull the liquid molecules together into a more compact shape.

## 3. What is the relationship between surface tension and temperature?

Generally, as temperature increases, surface tension decreases. This is because higher temperatures cause the molecules to move more rapidly, reducing the cohesive forces between them. However, there are some exceptions to this, such as with water, which has a slightly higher surface tension at higher temperatures due to its unique molecular structure.

## 4. How is excess pressure related to surface tension?

Excess pressure is the difference in pressure between the inside and outside of a curved liquid surface. It is directly related to surface tension, as the surface tension of a liquid creates a pressure imbalance that causes the liquid to curve. This is why small insects, like water striders, can walk on the surface of water without breaking the surface tension.

## 5. How is surface tension measured?

Surface tension is typically measured in units of force per unit length (such as Newtons per meter) or energy per unit area (such as Joules per square meter). It can be measured using various techniques, such as the pendant drop method or the capillary rise method. These techniques involve measuring the force or height of a liquid column to determine its surface tension.