MSE4: Mathematical modelling of drying colloidal suspensions

Project completed September 30, 2011

Background

Understanding the drying and freezing of colloidal suspensions is of the utmost importance in soil science. Peeling and flaking drying mud can negatively affect plant growth and erosion, and freezing soil can cause the ground to uplift and damage structures.

In arid environments, mud dries rapidly as water evaporates from it, forming a crust that cracks and peels. This breaks up the soil, which impacts water retention, nutrient content and other factors affecting plant growth. Areas of drying mud are also especially susceptible to erosion.

Understanding the mechanism behind freezing of porous materials is important economically, as, for example, the freezing of soil can create ice lenses underground that push the soil upwards. This is known as frost heave and causes billions of dollars of damage annually in the USA alone. Despite being such a problem, there is still a lack of understanding of how ice lenses form and grow.

Researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) have modelled the drying and freezing of colloidal suspensions using a simple fracture-mechanics-based approach.

Techniques and Challenges

Crust formation: Mud is a colloidal suspension of particles in water. As water evaporates from the suspension, a crust forms. On the particle level, the concentration of particles at the surface increases as water evaporates, bringing the particles closer together. As the concentration continues to increase, eventually a critical point is reached where the concentration is high enough that short-range attractive forces between particles cause them to aggregate together to form a crust (or gel). Further evaporation causes the crust to grow downwards into the suspension.

In order to model the system, the researchers assumed an initially-uniform suspension of colloidal particles. The equations describing particle concentration in the crust were coupled with the equations for particle diffusion in the ungelled suspension.

Peeling and flaking of drying material: In a crust, cracks may form. There are two types of cracks that form in drying mud: shrinkage cracks and peeling cracks. The shrinkage cracks form first and are a result of the mud contracting as it dries. These are the cracks responsible for segmenting the drying mud into polygonal sections. Peeling cracks form after the shrinkage cracks have formed and run parallel to the surface of the mud, which can cause the drying mud layer to peel and curl. These cracks are shown in Figure 1.

Equations for peel thickness and peel time were derived from the stress, strain, pressure and evaporation in the mud. The shrinkage cracks form as a result of stresses from clay particle consolidation as the water evaporates. The peeling cracks are driven by the relaxation of stresses in the peel, reducing the stored strain energy in the system as the cracks propagate.

Ice lens formation and growth: Ice lenses are bands of pure frozen water that form in freezing soil. They grow because water is sucked up from the unfrozen soil below to the freezing front. The water then freezes, causing it to exert pressure on the surrounding soil. As the pressure increases, it eventually reaches a critical pressure where the ice suddenly expands horizontally through the soil, creating a new ice lens.

The model the OCCAM researchers derived for ice lens formation is simpler than existing models. It does not depend on empirical observations and is computationally tractable. To extend the usefulness of the new model, the ice lens growth rate was also examined. Existing models over predict the ice lens growth and frost heave rate, leading to unrealistic growth possibilities.

Results

Crust formation: The results showed several interesting insights. The particle concentration in the crust and in the ungelled suspension is constant except in a very thin boundary layer between the two. This significantly simplifies the system of equations and allowed the researchers to derive a simple model for the time that it takes for a suspension to fully gelate. It also allowed them to analyse the stress development in the material as a function of time and to determine the maximum tensile stress that can arise, which governs whether cracking can occur in the material during the drying process.

Peeling and flaking of drying material: Using the new model with typical parameter values, the resulting peel thickness estimates and times to peel were shown to be in line with existing experimental data and observations.

Ice lens formation and growth: The researchers have shown that the process by which water attaches to the existing ice lens determines the growth rate. Incorporating this into the model led to more realistic predictions of ice lens growth in various types of soils.

The Future

The work done is not only applicable to mud drying and ice lens formation, but lends itself readily to several other areas including applications in food and materials sciences.

Frost heave in compressible soils remains an active area of the project. This model gives macro-scale predictions of the rate of frost heave, and in the future will lead to a model describing the growth of individual ice lenses in soil.