MSE2: Modelling ionic-surfactant-driven flows

Project completed April 30, 2011

Background

Reducing the surface tension of liquids has a broad spectrum of applications in industrial contexts and is also found in many biological systems. Industrially, surfactants aid in the mechanical clean-up processes of liquid pollutants and in oil recovery by mobilising residual oil. As a biological example, surfactants are essential to lung function.

Soluble surfactants are chemical species dissolved in a liquid that are distinguished by their preferential adsorption at gas-liquid interfaces rather than in the bulk of the liquid. Once adsorbed, they reduce the local surface energy.

Much of the behaviour of surfactants when in solution is not well understood. Researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) have developed mathematical models uncovering new insights into the mechanism of the breakdown of non-ionic surfactant aggregates (also called micelles) following a system dilution, and the adsorption of ionic surfactants at an air-liquid interface (see Figure 1).

Techniques and Challenges

Micellar breakdown kinetics: When dissolved in solution, surfactant molecules tend to exist as monomers (individual molecules). However, as the concentration of surfactant molecules increases, the monomers begin to combine. Once a critical surfactant concentration is reached, termed the critical micelle concentration (CMC), it becomes favourable for aggregates or micelles (combinations of monomers) to form. These aggregates adopt a distribution of sizes, but most exist at the size known as the large optimal aggregation number. 

Following dilution, these aggregates break down until the monomer is replenished to the CMC. It is generally assumed that this replenishment occurs via the stepwise release of individual monomers from aggregates. 

This process is captured mathematically by the Becker-Döring theory – an infinite dimensional system of coupled ordinary differential equations, which do not lend themselves to simple analysis. 

In this project, a significantly simplified continuum model that replicates the Becker-Döring system was derived by exploiting the large optimal aggregation number. The resulting system was simplified further by utilising additional characteristics of the aggregate size distribution. This led to explicit expressions for the re-equilibration behaviour, which was shown to be composed of a fast early release of monomers from aggregates, described by a single ordinary differential equation, followed by a much longer relaxation to equilibrium that is captured by a metastable analysis.

Adsorption kinetics: Upon dissolution in water, ionic surfactants dissociate into charged surfactant ions and oppositely charged counterions. Previous theories assume entire dissociation into respective ions, which lead to an unfeasibly large surface potential and subsurface counterion concentration. While this issue is commonly circumvented by assuming that some of the counterions may recombine with the adsorbed surfactant ions to form a Stern layer directly beneath the surface, such a resolution is physically unsatisfactory as there is no physical justification for why recombination should be restricted to near the interface. 

In this project, the assumption of surface recombination was relaxed to allow recombination of surfactant ions and counterions also to occur within the bulk of the liquid, rather than only at the surface, in order to generate a physically robust mathematical model. A natural consequence of this relaxation was the imposition of an additional constraint that enforces the principal of microscopic reversibility, which requires that, for a system of many reactions to be in equilibrium, all reactions must be individually in equilibrium. This led to natural restrictions on system parameter choices. 

Results

The results of this work on micellar breakdown kinetics highlighted a key failure in the present Becker-Döring theory for the description of surfactants whose typical concentration of smaller aggregates is much lower than the concentration of optimal aggregates. Such systems include the polyoxyethylene glycol alkyl ether surfactants. It was shown that re-equilibration through the traditional stepwise monomer loss route occurs over unfeasibly long timescales. 

A new theory and accompanying mathematical model for micelle breakdown was proposed, which allows for the merging of aggregates to produce large unstable aggregates called super micelles. It was shown that these large aggregates subsequently quickly break down via traditional stepwise monomer release to replenish the concentration of monomer over a timescale reflective of experimental observations.

For adsorption kinetics, the resulting model produced physically sensible predictions for the measurable quantities, such as surface electric potential, without the need for the introduction of ad hoc a posteriori model modifications. An additional result of this theory was to provide a uniformly valid model for the adsorption behaviour of chemicals such as fatty acids following variations in pH.

The Future

This project not only provided successful mathematical models for adsorption and breakdown kinetics in surfactant systems, but uncovered additional physical insights that could not have been predicted without such modelling. The results of this work will provide fundamental models that capture the system physics to enable tailoring the use of surfactants within their many practical applications.