Thermodynamic deviations of the mechanical equilibrium conditions for fluid surfaces: Young's and Laplace's equations

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Third entry by Kelly Miller, AP225 Fall 2011

Information

Title: "Thermodynamic derivations of the mechanical equilibrium conditions for fluid surfaces: Young's and Laplace's equations"[1]

Author: P. Roura

Journal: American Journal of Physics


Keywords

Laplace's Law, Young's equation, surface energy, surface tension

Introduction

The fundamental laws governing the mechanical equilibrium of solid-fluid systems are Laplace's Law (which describes the pressure drop across an interface) and Young's equation for the contact angle. Laplace's Law and Young's equation were established in 1805 and 1806 respectively. At the time of their conception, the laws were supported from a purely mechanical approach. Subsequently, in 1880, Gibbs showed, thermodynamically, that these laws were necessary conditions for the equilibrium of heterogeneous systems. The authors of this paper revisit Gibbs' derivation and simplify it to make it more accessible to the undergraduate level student. The authors also derive Young's and Laplace's equations using energy balance on a 'local' volume element located a the surface. They argue that their derivations have several advantages compared to the more traditional 'global' approaches which minimize the energy at constant entropy or the Helmholtz free energy at constant total volume. These advantages are: the derivations are simpler, they allow for the analysis of nonequilibrium situations, and they allow a natural identification of the surface energy with the surface tension of the liquid-vapor interface.


Overview of Laplace's and Young's Equations

The relationship between the hydrostatic pressure (P) and the acceleration due to gravity (g) at any point in the bulk of a fluid at rest is:

Equation 1.png where p=density of the fluid and the force of gravity is exerted downward along the z axis


However, additional if we are dealing with the surface of a fluid (as opposed to the bulk) additional surface forces need to be considered for the system to be in mechanical equilibrium. Figure 1 shows that the pressure on a thin fluid element must be balanced by surface forces acting on its contour.

Figure 1.png

The surface forces Equation 2.png arise from a surface tension Equation 4.png

Equation 3.png

Laplace Derived the second general equation for the equilibrium of fluids in 1806:

Equation 5.png

This equation relates the pressure discontinuity at the surface with the surface curvature. Basically, the larger the curvature (smaller radii), the larger the difference in pressure between the liquid and vapor.

Young derived the following equation which explains the equilibrium contact angle <math>\Theta</math>:


Young.png

The mechanical equilibrium of an element of volume, located along the contact line between the liquid-vapor (LV) surface and a solid S can be explained by the action of two surface tensions of the fluid-solid surfaces Gammasv.png and Gammasl.png, which equilibrate Equation 4.png

Figure 2.png

"Local" Thermodynamic Derivations

Typically, thermodynamically, the two equations above are derived using the condition of minimum energy (Gibbs' approach). The authors derive Young's Equation and Laplace's equation from a different approach; using the condition of energy balance during a reversible displacement of a volume element located at an interface. Gibb's approach involves looking at the energy of a system from a global perspective, whereas the approach proposed by the authors involves analyzing local energy variations of a volume element. In equilibrium, any displacement of the surfaces is reversible and, the work done by external forces on a volume element (<math>\delta w</math>) will increase the Helmholtz free energy (F) (at constant temperature). This equilibrium condition in the local approach is that the free energy of any volume element changes when surfaces move (in contrast with the free energy of the entire system, which remains constant.

<math>\delta F = \delta w</math>

Laplace's Equation

To derive Laplace's equation using this 'local' approach, the authors consider a thin film of fluid L that contains a portion of the LV surface (see figure 5). Any displacement of the entire LV surface will correspond to a particular value <math>\delta t</math>

External forces will cause a deformation that can be described by the magnitude <math>\delta t</math>

Screen shot 2011-10-02 at 11.49.14 PM.png


The free energy changes due mostly to the change of the LV surface area and somewhat due to variations of the bulk of the thin element of fluid L


Eq 25.png

If we assume a thin film, the O(h) becomes negligible compared to the surface term and so the equation becomes:

Eqn 26.png

<math>\Gamma _{S}</math> is the contour of the surface element.

To calculate external work, we know that the LV surface has stress on it characterized by the surface tension <math>\gamma _{LV}</math> which acts perpendicularly to the contour of the SL surface. The hydrostatic pressure on either side of this surface will be different.

Eqn 27.png


When the system is in equilibrium, <math>\delta F= \delta w</math>

The terms proportional to <math>\delta N</math> provide Laplace's equation

Young's Equation

To derive Young's Equation, the authors analyze boundary conditions for the same volume element chosen for the above discussion.


Fig 6.png

The volume element has boundary surfaces parallel to the solid surface (see above diagram).

When the contact line recedes by <math>\delta T</math> the liquid will be deformed near the SL surface, whereas far away from the contact line, it will not be deformed.

The forces that do the work during this process are:

the hydrostatic pressure gravity SL surface tension

The variation in energy can be expressed as:

Eqn 29.png (Equation A)


The work can be calculated by considering the component of the surface force perpendicular and tangent to the solid surface.

Using the same thin film approximations as previously,

Eqn 33.png (Equation B)

Equilibrium condition <math>\delta F= \delta w</math> means that the terms proportional to <math>\delta T</math> in Equations A and B (above) must be the same, which yields:

Eqn 34.png (Equation C)


<math>\gamma _{LV} = \sigma _{LV}</math> substituting this equality into Equation C yields Young's Equation

Conclusion

The authors derive Young's and Laplace's equations using energy balance on a 'local' volume element located a the surface. They argue that their derivations are simpler and less constrained than the more traditional 'global' approaches (Gibbs) which minimize the energy at constant entropy or the Helmholtz free energy at constant total volume.


References

"Thermodynamic derivations of the mechanical equilibrium conditions for fluid surfaces: Young's and Laplace's equations," by P. Roura [Am. J. Phys. 73 (12), 1139–1147 (2005)]