How Does Surface Energy Affect Thin Fluid Films?

[member 428835]

I'm studying thin fluid films, and the text writes free surface energy of a film (puddle) over domain $(0,X)$ can be expressed as $$E=\int_0^X \left[\frac{h_x^2}{2}+\omega(h)+G\frac{h^2}{2} \right]\, dx$$ where $X$ is a length that the thin film (puddle) rests on, $h$ is the height of the film (puddle), $G$ is a gravity term (0 for no gravity and 1 for gravity), and $\omega(h)$ is energy density due to van der waals forces. Firstly, can someone explain the derivation of this integral to me?

However, that's not my main question. This integral is subject to the constraint $$A = \int_0^X h\, dx$$ where $A$ is area. To minimize the energy subject to the constraint yields the funcitonal $$F = \int_0^X \left[\frac{h_x^2}{2}+\omega(h)+G\frac{h^2}{2} -p(\bar{h})h\right]\, dx$$ where $p$ is a Lagrange multiplier. Here's where I'm stuck: the author proceeds by saying "neglecting the energy in the narrow transition regions at the extremes of the plateau region of thickness $h_0$ and length $L$ (the puddle length, not confused with domain $(0,X)$), the energy per unit length can be calculated as $$g(h_0) = \omega(h_0)+G\frac{h_0^2}{2} -p(\bar{h})h_0.$$
Why is this the energy per unit length when we said $E$ was the free energy?

 

[eljose79]

As a scientist studying thin fluid films, allow me to explain the derivation of this integral and address your main question.

The derivation of the integral can be understood by considering the forces acting on the thin film. The first term in the integral, $\frac{h_x^2}{2}$, represents the kinetic energy of the film due to its motion. The second term, $\omega(h)$, represents the energy density due to van der Waals forces, which are attractive forces between molecules at close distances. The third term, $G\frac{h^2}{2}$, represents the potential energy of the film due to gravity. By integrating over the length $X$, we are considering the total energy of the film over the entire domain.

Now, onto your main question. The author is neglecting the energy in the narrow transition regions at the extremes of the plateau region, which means that they are assuming that the energy in those regions is small compared to the rest of the film. This assumption allows them to simplify the integral and calculate the energy per unit length, which is represented by the function $g(h_0)$. This is a common approach in physics and engineering, where we often make simplifying assumptions to make calculations easier.

Furthermore, the energy per unit length, $g(h_0)$, is equivalent to the free energy, $E$, since energy per unit length is just the energy divided by the length. Therefore, the author is still considering the free energy of the film, but they are using a simplified expression for it. This is a common technique in physics, where we often use simplified expressions to make calculations easier while still capturing the important physical concepts.

I hope this explanation helps to clarify the derivation and the reasoning behind the author's approach. If you have any further questions, please don't hesitate to ask. As scientists, it is important to have a clear understanding of the concepts and methods we use in our research.