Bubble pressure

Diffusion and adsorption coefficients

In reality the surface of a liquid is not a distinct separation line, but rather a thin layer. Within this layer, which is only a few molecules thick, the properties of the solution such as density or concentration are not homogeneous, but depend on the height position within the layer. The surfactant concentration increases as the proximity to the gas phase increases.

In order to be able to describe surface processes mathematically, Gibbs defined an interface boundary without any height. If this model line is placed at the level of the highest surfactant concentration then the inhomogeneous region below the line can be called the “subsurface”. This model allows the thermodynamic differentiation of two processes: diffusion and adsorption. Diffusion describes the molecular movements from the homogeneous solution to the subsurface, adsorption the movement from the subsurface to the surface. These processes take place at different rates; these rates influence the total rate of the reduction in surface tension.

The total rate of the alteration in surface tension can, depending on surface age, concentration, molecular and solvent characteristics, be influenced more strongly by the diffusion rate or more strongly by the adsorption rate. The slower of the two processes dominates the total rate; this is the so-called “rate determining step”.

The diffusion and adsorption coefficients are important quantities for the movement rates during surface formation; these quantities are independent of the concentration and the temperature of the surfactant solution and can therefore be regarded as being characteristics of a surfactant-solvent system.
In order to calculate the two coefficients, measuring curves of the surface tension as a function of surface age are evaluated at various surfactant concentrations. These evaluations are based on the models of Joos & Rillaerts for the diffusions coefficient and Ward & Tordai for the adsorption coefficient.

Diffusion coefficient according to Joos and Rillaerts

According to Joos and Rillaerts, in the region of the diffusion-related reduction of the surface tension the following relationship applies:

$\gamma_t = \gamma_o - 2 RTc  (\frac {D_s t}{\pi})^{0,5}$

where:

$\gamma_t$ = surface tension at surface aget
$\gamma_o$ = surface tension of pure solvent
$R$ = universal gas constant
$T$ = absolute temperature
$c$ = surfactant concentration
$D$ =diffusion coefficient

In the evaluation in LabDesk the value $D_s$ is calculated for each individual measured value of a curve and plotted against $1/c^2  t$ .

In this way the concentration is used as a standard, i.e. the measuring curves for different concentrations overlap, at least in the low concentration range.

One region of the measuring curves should form a plateau, and the required diffusion coefficient is obtained from this plateau value for $D_s$ or from the mean value for the plateaux at different concentrations.

The rate in the higher concentration and surface age range is no longer determined by diffusion; accordingly the value for $D_s$ shows no plateau here.

Adsorption coefficient according to Ward and Tordai

As concentration increases and at older surface ages the movement toward the surface is no longer determined by diffusion, but the movement of the molecules from the subsurface to the surface (adsorption) is decisive for the history of the surface tension. According to Ward & Tordai the following relationship applies to this region:

$\gamma_t = \gamma_{eq} + \frac {RT \tau^2}{2}\ \frac {D_l t}{\Pi})^{o,5}$

where:

$\gamma_t$ = surface tension at surface age t
$\gamma_{eq}$ = surface tension in dynamic equilibrium
$R$ = universal gas constant
$T$ = absolute temperature
$\tau$ = excess concentration
$c$ = surfactant concentration
$D_l$ =adsorption coefficient

As with the diffusion coefficient, when the concentration is used as the standard (plotted against $1/c^2 t$ ) the measuring curves whose plateaux are to be evaluated agree to a large extent. The excess concentration G describes the difference between the amount of substance adsorbed at the surface and the concentration of the substance in the solution. This quantity cannot be accessed directly in a bubble pressure measurement, but must be determined by making CMC measurements. These can be carried out using a KRÜSS Tensiometer and the CMC Add-in of the LabDesk tensiometer software.