Surface Tension - Lung Alveoli

Question

So, the way I understand this is as follows :

The alveoli (pretend they're bubbles) have diameters of the order of microns implying a massive pressure required to inflate them by the Young-Laplace equation.

$p_{in}-p_{out}=\frac{2\gamma}{r}$

However, the presence of pulmonary surfactant molecules (lets just pretend they're like detergents molecules in washing liquid) can effectively reduce the surface tension at the unexpanded alveoli and hence allow easy inflation.

Now this bit I don't understand :

As the alveoli expand the distance between the individual surfactant molecules on the alveoli increases and hence the surface tension rises again therefore decreasing the rate of expansion.

What is the mathematical connection between surface tension and separation between surfactant molecules ? How can I rationalise the statement in bold ?

Answer 1

Suppose the surface tension of a pure interface (without surfactants) is $\gamma_0$. Now adding the surfactants lowers the surface tension to $\gamma(a) = \gamma_0 - \Pi(a)$, where $a$ is the area per surfactant (or the inverse of the surface density; one can use either, but $a$ is more customary). The term $\Pi(a)$ is called the surface pressure precisely because it acts like a two-dimensional pressure. Let me elaborate.

Taking this view, then, the pure interface wants to contract, get smaller, to minimize the size of the interface. As the surfactants thus become ever closer to one another, they start repelling and create a force to the opposite direction. It is this force that is called the surface pressure.

In fact if you were to push "too many" surfactants to the interface very quickly (this assumption of very quicky is important, for otherwise the surfactants would jump off the interface), the surface tension would become negative and the surface would then tend to bend, or change its shape, so that the size of the surface would actually grow. This happens a lot in certain classes of systems (say, lipid bilayers, which are more or less connected to the monolayers that you are talking about when you talk about lung alveoli).

As to how to model $\Pi(a)$, the simplest models might be that of ideal gas, or van der Waals equations of state. Some more elaborate, but still quite elementary methods would be some lattice models (a la Langmuir adsorption model), and at the more advanced end you have density functional theory and fundamental measure theory and the stuff that branches off of them. The literature on the subject is huge.