Review of Otis et al. Model

We would like to consider some extensions of the Otis et al. model of surfactant adsorption behavior. We begin by recalling the major features of the model. We assume that a spherical air bubble is floating in an infinite bath of surfactant solution, and that the surface tension of the bubble is determined by the amount of surfactant at the air-water interface. Two important constants are , the greatest possible interfacial concentration under equilibrium conditions, and , the maximum interfacial concentration under dynamic compression. corresponds to the lowest achievable surface tension. An equation of state relates the concentration at the surface and the surface tension:

We have drawn the equation of state in terms of the ratio between surface concentration and greatest equilibrium surface concentration .

Surfactant adsorption behavior can be described by three different regimes. (Otis et al., 1994, binder)

I.

In this regime, surfactant is free to adsorb to or desorb from the surface:

where A is area, k₁ is an adsorption constant, k₂ is a desorption constant, and C(0, t) represents the bulk surfactant concentration near the interface at time t. (Morris et al., 2001, binder)

II.

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

where A^* is the area at which is obtained.

III.

We assume that when the surface area is reduced past the point of minimum surface tension, overcompression will result, squeezing the surfactant into the bulk phase. Thus, the amount of surfactant in the interface changes, but its concentration remains constant:

(Otis et al., 1994, binder)

Adding Diffusion

We would like to include the effects of diffusion in our model. For convenience, we define a coordinate system with 0 at the surface of the bubble:

We have x = r - R(t), where r is the distance from the center of the bubble to the point of interest, and R(t) is the radius of the bubble as a function of time.

We describe the diffusion process with a form of the convective-diffusion equation:

Here C(x,t) represents the surfactant concentration in the bulk at a given place and time, D is a diffusion coefficient (in cm²/s), and v_rel is the velocity of the bulk fluid relative to the bubble's surface. v_rel is described by the following equation:

Finally, we need a boundary condition at the interface, to provide for concentration of surfactant:

(Morris et al., 2001, binder)

Parameter Values

The model requires the following parameters:

• and , the greatest possible equilibrium concentration and the corresponding surface tension
• and , the maximum possible surface concentration, only achievable under dynamic compression, and the corresponding minimum possible surface tension
• m₁ and m₂, the absolute values of the slopes of the two lines in the equation of state
• k₁ and k₂, adsorption and desorption coefficients from Regime I
• D, the diffusion coefficient

The values of and m₁ depend on other parameters; the remaining values must be determined from experiment. (Morris et al., 2001, binder)

The following are values for commercially available whole lung surfactant, such as Curosurf. Note that dyn/cm and mN/m have identical numerical values.

can be determined by using the lowest surface tension obtained for equilibrium measurements with a high bulk surfactant concentration. Sample values are 25 dyn/cm (Morris et al., 2001, binder) and 22.2 dyn/cm (Ingenito et al., 1999, binder).

has been estimated as 3 * 10^-7 g/cm² in adsorption studies. (Morris et al., 2001, binder)

can be measured directly. 1 dyn/cm is a reasonable estimate. (Morris et al., 2001, binder; Ingenito et al., 1999, binder)

may be calculated from other parameters using the following equation:

= (1 + ( - )/ m₂)

With the Morris et al. parameters, this gives us 3.72 * 10^-7 g/cm².

m₁ is easy to calculate. Since 0 grams of surfactant at the interface corresponds to the surface tension of pure water, in the dimensionless version of the equation of state we have

= (70 dyn/cm - 25 dyn/cm)/ (1-0) using the Morris et al. figures

= 45 dyn/cm

(Morris et al., 2001, binder)

Using the chain rule, we may write:

where A and are any pair of values found in regime II. Sample values include 140 dyn/cm (Ingenito et al., 1999, binder) and 100 dyn/cm (Morris et al., 2001, binder).

k₁ and k₂ are found by varying these parameters until the theoretical isotherm shows a good match to experimental values. In practice, the model is not sensitive to the desorption constant k₂ at higher bulk concentrations, so both Morris et al. and Ingenito et al. hold k₁/k₂ constant at 1.2 * 10⁵ ml/g. Values for k₁ include 10⁵ cm³/g/min (Otis et al., 1994, binder) and 6 * 10⁵ ml/g/min (Morris et al., 2001, binder; Ingenito et al., 1999, binder). Values for k₂ are 1 * 1/min (Otis et al., 1994, binder) and 5 * 1/min (Morris et al., 2001, binder; Ingenito et al., 1999, binder).

Like k₁ and k₂, D may be found by fitting theoretical curves to experimental data. An appropriate value of D for measurements made in the pulsating bubble surfactometer (PBS) is 1 * 10^-6 cm²/s. This value is similar to the diffusion coefficient of DPPC molecules.

Fitting theoretical curves to data from the captive bubble surfactometer (CBS) leads to very different values of k₁ and D: k₁ becomes .07 * 10⁵ ml/g/min, while D changes to 1 * 10^-9 cm²/s. Morris et al. suggest that the differences arise because the bubble in the CBS was compressed further and because the experiment began with the bubble at maximum rather than minimum volume. This could lead to changes in the organization of the surfactant such as the formation of micelles. (Morris et al., 2001, binder)

Questions

• Is an infinite amount of bulk solution an appropriate approximation to the dynamics of the lung? (Limited solution may affect the behavior of the pendant drop; c.f. Jyoti et al., 1997, Droplets binder)
• How would a non-spherical bubble geometry affect the model?
• How can we deal with the effect of other factors on surface tension? (For example, in the CBS the pressure change affects the volume of the bubble, changing its surface area and therefore the surface tension; but changing pressure should also alter the bubble's surface tension directly.)