There has been a significant amount of theoretical work on dynamic compression at the molecular level, considering not only biological surfactants but also other phospholipids, liquid crystals, etc. In this case, the theoretical question is, how do the properties of the individual molecules affect their phases and orientation at various pressures? (c.f. Kaganer et al., 1999, Film Structure online) There has also been some work on the high-level properties of thin surfactant-covered films. The tools come from fluid dynamics, and questions include where surfactant will pool and where the film will break. (c.f. Ida and Miksis, parts I and II, 1998, online)

A small group of articles look for a theory behind the measurements of biological surfactants made in the lab. These assume that surface tension is (by definition) entirely controlled by the amount of surfactant at the interface.

Basic Scenario:

Getting surfactant to the interface is a two-step process. First, during diffusion, surfactant particles move from the bulk solution to just below the surface.

Next, during adsorption, particles adhere to the interface. Particles may also desorb into the solution.

Diffusion and adsorption occur at different rates. An important question in building a model is which limits the accumulation of surfactant. (Horn and Davis, 1975, binder)

Once the concentration of surfactant at the interface has been found, we can determine the surface tension through an "equation of state" that relates the two quantities. (Otis et al., 1994, binder)

Horn and Davis, 1975

Horn and Davis considered the factors that might create apparent hysteresis in the surface tension of an oscillating spherical shell. They built four models: diffusion-limited, adsorption- and desorption- limited, bulk viscosity controlled, and surface viscosity controlled. They concluded that bulk viscosity was unlikely to contribute to apparent hysteresis, unless the substance was near the viscosity of pitch. However, surface viscosity could have significant effects. (Horn and Davis, 1975, binder)

The Horn and Davis model is limited because it only considered near-equilibrium surfactant behavior, and made no provision for over-compression or the squeeze-out of surfactant material from the interface. (Otis et al., 1994, binder)

Otis et al., 1994

The Otis group modeled the surfactant behavior of a pulsating bubble system. They simplified the problem by assuming that a spherical bubble of air was oscillating in an infinite solution of surfactant.

In this model, two different types of surfactant concentration are important. is the two-dimensional concentration at the interface, measured in g/ cm², while C is the concentration in the bulk, measured in g/ cm³.

Two major constants delineate regimes of surfactant behavior. is the highest possible surface concentration under equilibrium conditions, as bulk concentration approaches infinity. is the greatest surface concentration possible as the interface is dynamically compressed. This corresponds to the lowest attainable surface tension.

There are three different regimes of adsorption behavior:

I.

In this regime, we assume that adsorption observes standard Langmuir behavior:

where , the amount of surfactant in the interface (g), k₁ is an adsorption coefficient, and k₂ is a desorption coefficient. We may rewrite this equation in terms of gamma as:

II.

Here we assume that the monolayer is insoluble: the surface concentration will change only because of changes in the area. Thus, we can write the concentration in the compression part of the cycle as

where A^* is the area at which is obtained. We may rewrite this as

III.

If the surface area is reduced past the point of minimum surface tension, overcompression results, and surfactant is squeezed into the bulk phase. We assume that the concentration remains constant:

Once we know concentration as a function of time, we can find the surface tension using an equation of state. Otis et al. use an equation of state that consists of two line segments with slopes m₁ and m₂:

If we have values for k₁ and k₂, we can find m₁ from equilibrium measurements, using

Since m₂ should be constant across experiments, we may determine it by fitting a line to experimental data.

In implementing their model, Otis et al. vary k₁ and k₂ iteratively until they have an equation of state and a hysteresis loop that match experimental data. This model has very good qualitative results: the theoretical loops have similar shapes to the experimental ones. Moreover, the model shows that hysteresis is possible as a direct consequence of surfactant loss, and "islands" of surfactant in a reexpanding fluid are not a necessary assumption. However, the quantitative results are not reliable; the uncertainty in k₁ and k₂ is about 25%. (Otis et al., 1994, binder)

Extensions

Morris et al., 2001

This group, part of the original Otis group, extended the Otis model to include the effects of diffusion. Including diffusion means that the bulk concentration C becomes time and space dependent. This adjustment to the model allows it to model the transient effects for a newly oscillating bubble more closely. It also predicts a layer of high surfactant concentration immediately below the interface, which may be related to experimentally observed collapse structures. (Morris et al., 2001, binder)

Krueger and Gaver, 2000

This group altered the Otis model by explicitly including a secondary collapse layer. Their results also showed an increased agreement with experiment. (Krueger and Gaver, 2000, online)

Questions about the Models

Can surface viscosity be ignored, as Otis et al. claim?

What about mechanical respreading mechanisms, such as folds "unzipping," or layers unstacking?

How does a monolayer's elasticity affect respreading properties?

How does the equation of state vary between experiments? (i.e., what factors affect and ?)

Can we determine k₁ and k₂ experimentally for systems capable of fast oscillations?