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Module 6: Laplace Pressure and Young Laplace Equation

Lecture 6: Laplace Pressure and Young Laplace Equation

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6.0 Laplace Pressure and Young Laplace Equation

So far our basic Fluid Dynamics knowledge has taught us that for a fluid surface which is

stagnant and is in mechanical equilibrium, the pressure on the two sides of the surface is equal.

For example, if we consider a tank full of water exposed to atmosphere, then at the liquid – air

interface the pressure on both sides of the interface is Patm. At any point at a depth h from the

interface the absolute pressure would be Patm + ρgh (where ρ is the density of the fluid), and this

situation is referred to as “the pressure distribution is purely hydrostatic”. The implicit

assumption that goes in when we consider the above example is that the liquid – air interface is

perfectly flat. However what happens in case the interface is NOT flat, or has a curvature? We

will discuss those cases in this chapter and introduce the very important concept of Laplace

Pressure. In case you are wondering why or where can a liquid – air interface be flat, then let me

remind you that in your senior school you were taught about “Parallax Error” and was also

cautioned how to take the correct reading, when measuring the rise of a liquid column through a

narrow capillary tube.

The discontinuity in pressure across a non planar interface was discovered by Thomas Young

(1805) and Pierre-Simon Laplace (1806). A spherical interface is a special case of a general non

planar interface. Any general curved surface at any point can be identified in terms of two local

radius of curvature (R1 and R2) orthogonal to each other at that point. For a spherical interface

the two orthogonal radius of curvature are same, and that makes calculations simpler

(R1=R2=R)! For a spherical interface, the law expresses the pressure discontinuity at a given

point (∆P) of an interface as

∆P = 2γ/R (6.1)

As already discussed in earlier, in a non condensed liquid phase, the curvature is caused due to

surface tension and consequently results in a pressure imbalance. Due to inward resultant or

effective attraction, the molecules reorganize themselves and this re organization is based on

energy minimization. This is also the phenomenon that forms a meniscus and the pressure

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difference on the either sides of meniscus leads to the development of a net normal force, which

is balanced by surface.

We now look at how the expression given in equation 6.1 can be obtained mathematically. Let us

consider a liquid bubble of radius R, shown in figure 6.1. Lets say the pressure inside and outside

the bubble is P// and P/ respectively. Now lets consider that the size of the bubble expands from

R to R + dR. The work necessary for the expansion W is given as

Figure 6.1: Schematic of Liquid Drop under consideration

(6.2)

Now, (6.3)

Therefore, (6.4)

As a manifestation of increase of radius, the surface energy of the bubble increases. Consequent

to this, Es (energy of interaction of a pair of molecules at surface) changes by dEs.

(6.5)

Therefore, combining equations (6.4) and (6.5), we get

(6.6)

Which eventually gives (6.7)

From equation (6.7), we can say that ∆P is positive and hence the P// ie the pressure in the

concave side is higher than P/.

6.1 Expression of Young Laplace Equation for a non spherical Curved Surface

Consider a non spherical curved surface of area xy. Any point in 3-D space needs to be defined

by 3 co-ordinates and any surface in 3-D phase can be represented by 2 curvatures.

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Figure 6.2: View of a segment of the expanded spherical surface along with projections along X and Y axis.

On supplying some energy, the change in surface area, dA can be written as

(6.8)

Change in surface energy, (6.9)

Work done, (6.10)

Now, assuming that the vales of the angles of the curved surface, θ and α are very small, the two

parts of the curved surface can be treated as similar triangles and hence from figure it can be seen

that (6.11)

and (6.12)

Therefore, substituting equations (6.11) and (6.12) in equation (6.10) we have

(6.13)

Now comparing equation (6.9) and (6.13), we get

(6.14)

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This is the expression for Young-Laplace equation for a n arbitrarily curved surface with at any

point where the two orthogonal Radii of curvature are R1 and R2 respectively. Further, R1 and

R2 may not be uniquely defined but their combination is unique and represents the curvature of

the surface. Further, if ΔP is known, then the curvature of the surface can be determined at

equilibrium using Young-Laplace equation. However, the problem is, for any arbitrary curved

surface, at every point R1 and R2 may not be known. Thus, the following formulation of the

same equation for an axi symmetric surface becomes important.

Figure 6.3: Some examples of Axi-symmetric surfaces: (A) Cone; (B) Cylinder; (C) Cylindrical Surface; (D) Converging Nozzle.

6.2 Generalized Expression for Young-Laplace Equation for an Axi-Symmetric Surface

An axi-symmetric surface is a type of a surface that can be generated by rotating a line segment

or a curve about an axis by 360º in space. For example, if a line segment is rotated around

another vertical line, one would get a cone. In case the line segment that is rotated is parallel to

the line around which it is rotated, then one would get a cylinder. Similarly, if one rotates an arc,

one would get a curved surface. Some examples of axi-symmetric surfaces can be seen in figure

6.3. We will soon realize that some additional simplification is possible in the expression of

Laplace pressure for an axi-symmetric liquid surface.

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To develop the formalism for the Laplace pressure on a curved axi symmetric surface we first

look at the Helmholtz free energy of the surface. It is the thermodynamic potential that measures

the useful work obtainable from a closed thermodynamic system at a constant temperature and

volume. It is minimized at equilibrium. It is the maximum amount of work extractable from a

thermodynamic process at constant volume and temperature. If F represents Helmholtz free

energy, U is internal energy of the system, S is the entropy and γj gives the surface tension, then

F is given as

(6.15)

From the first law of thermodynamics

(6.16)

Σμixi gives the Gibbs free energy (G) which represents the useful or process initiating work

obtainable for an isothermal or isobaric thermodynamic system. It is the measure of maximum

non-expansion work that can be extracted from a closed reversible system. As S and V are

extensive variables,

(6.17)

From equations 6.15, 6.16 and 6.17 we get

(6.18)

Figure 6.4: Construction of elemental area on an axi symmetric surface

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Consider an axi-symmetic surface as shown in figure created by rotating any curved surface

about an axis. From x→0 to x→R gives the extent of the phase in x direction. If z=f(x) is known,

then we know the equation of the surface and F will represent the energy of the surface as a

function of shape. In particular, the intention is to find out a specific shape that corresponds to

the optima of energy which will also correspond to the shape of the surface at equilibrium.

For a two component system

(6.19)

Total volume, V = V’ + V” (6.20)

Therefore,

(6.21)

As we want to look at the variation of energy (F) as V” is changed, with total volume (V)

remaining constant.

(6.22)

Now, writing in terms of the shape of the interface gives

The area of an axi-symmetric surface and volume can be given as

(6.23)

(6.24)

For small values of dz and dx, dl can be assumed to be a straight line which results in the above

equations.

Now, putting equations in equation, we have

(6.25)

d

dx

d

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As the surface is axi-symmetric, we have Z = f(x). However, we would like to optimize F. F is to

be optimized with respect to x. However, F is NOT an explicit function of x

Rather F is of the form (6.26)

This is what is known as a functional. Evaluation of the optimum of a functional falls in the

branch of mathematics known as ‘Calculus of Variations’

The necessary condition of for optima for a functional F is (6.27)

We will not discuss any more details about calculus of variation and would progress based on the

above expression.

For our system, (6.28)

Now we evaluate the individual consistent terms:-

(6.29)

Therefore, we get

(6.30)

We look at the last two terms of the above expression

=

=

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Substituting these equations in the condition of optima and rearranging, we have

(6.31)

This is the general form of Young-Laplace equation for an axi-symmetric surface.

The equation is analogous to (6.32)

Which is a much simpler looking equation. However, in space it is not possible to determine R1

and R2 for an arbitrarily curved surface, and therefore though the expression is simple, obtaining

ΔP at every point of a non planar surface/interface cannot be dome directly. In contrast, as

already mentioned, for an axi-symmetric surface it is indeed possible to express Z = f(x) and

therefore one can analytically evaluate Zx and Zxx and therefore ΔP can be calculated.

A question to ask at this point is “is there a physical significance of the two terms”. Of course we

understand that one of the terms corresponds to 1/R1 and the other one corresponds to 1/R2. Can

one refer to any of the term as 1/R1 and 1/R2?

However, it turns out that the choice of R1 and R2 are based on convention. Typically the 1/R1

term refers to the in plane curvature and 1/R2 is the out of plane curvature.

It turns out that the general equation for axi-symmetric surface we have

(6.33)

The utility of this equation lies in the fact that if the equation of the line/ arc is known, which has

lead to the creation of the surface; then the Laplace pressure can be calculated. This is in fact

achievable. In fact, though equation 6.32 looks simple, as we have already said, it is often not

possible to measure R1 and R2 in real situation.