Nature provides an infinite number of complex patterns -- clouds in the sky, ice crystals on the window, corrosion pits on the surface of a steel beam, geological layers underground, mud cracks, animal skins... Many of these patterns are formed far from equilibrium -- the formation processes are not reversible. This means that the systematic and generally well understood techniques of equilibrium statistical mechanics cannot be applied to most pattern formation processes.
The understanding of such processes addresses many of today's energy and environmental concerns, and is important in numerous practical applications in industry and technology. For instance, the patterns formed when a fluid displaces another immiscible fluid in a porous medium is of relevance in oil reservoir engineering, groundwater hydrology, soil science, nuclear waste management, biosciences, and other branches of technology. In this context ``patterns'' refer to the complex shape of the fluid-fluid interface in the pore space.
Observation of a pattern obtained in the slow displacement of a wetting fluid (clear) by a non-wetting fluid (dark) in a three-dimensional transparent porous medium.
Pattern formation far from equilibrium occurs also during the breaking and rupturing of material -- a process of vital importance to most branches of technology. Here, patterns refer to the complex shapes of cracks and fracture surfaces in the material, respectively.
And how do powders flow? In flow and transport of granular materials, segregation and ``bridging'' (clogging due to particle interactions) are just two of the phenomena that affect the flow pattern, and must be taken into account in the processing of pharmaceuticals, the design of silos, the manufacture of ceramic materials, and many other important applications.
Blobs of air (light) observed during the slow injection of air into a water-filled fracture model.
In the Cooperative Phenomena Group, miscible and immiscible multi-phase flow in porous media has been a major research activity in the last decade. In recent years, the focus has shifted towards the study of fluid-rock interactions and the distribution of faults and fractures in rocks. The systematic study of fluid-rock interactions is, at present, the largest single research enterprise in the Cooperative Phenomena Group, in collaboration with researchers at the Institute of Geology.
Flow in fractures is another example of a non-linear process far from thermodynamic equilibrium. In a fracture filled with fluid, the fracture profile controls the flow pattern; but wherever the fluid is flowing, it can, through erosion or deposition, change the profile -- eventually causing a new flow pattern.
Irregular cluster of air (light) formed during the slow displacement of water-glycerol (dark) in a porous model.
To approach these problems, a combination of experiments, computer modeling, and theory is used. In systematic experiments, the slow displacement of one fluid by another, immiscible fluid has been explored, using transparent two- and three-dimensional porous models. These processes often lead to fractal displacement patterns. Patterns that have fractal properties are characteristic ``fingerprints'' of non-equilibrium processes. Fractal patterns lack characteristic scales. In the displacement of one fluid by another, for instance, ``fluid blobs'' of all sizes up to the total size of the system can be formed, and it is not possible to characterize the blobs by a ``typical'' size.
Click here to read more about fractals. Click here to read more about fractals, scaling, and correlation functions.
The complex physics of such displacement processes is now being clarified. Two-phase flow in laboratory-scale porous models has been investigated and visualized in the presence of buoyancy forces and, most recently, viscous forces. Click here to read more about experimenting with porous media. Recent work also includes a study of the formation and stability of water coning in model reservoirs, and the construction of porous models using sintered glass. Slow evaporation processes and rapid displacements of immiscible fluids in macroscopic porous structures were studied extensively. At microscopic scales, the flow of particles through single pores was measured, and the stability and coalescence of fluid droplets was explored.
Miscible displacement processes lead to dispersion, mixing and spreading due to the interplay between diffusion and convection. Fractal dispersion fronts in porous models were observed and quantified, and tracer dispersion at a single stagnation point was studied in detail.
Rough tracer front observed in a dispersion experiment.
The experimental studies are closely accompanied by computational work. Fluid flow in porous media, flow of electric current in conductors, and the movement of random walkers in space are governed by similar differential equations. As a result, simple computer models can describe a wide range of different phenomena. One example is the diffusion-limited aggregation model (DLA) which describes the clustering of diffusive particles.
Computer-generated DLA cluster consisting of 10.000.000 particles.
This model serves also to represent fast immiscible displacements in porous media. Slow displacements can be simulated using the invasion percolation model (IP) which represents the effect of random capillary forces controlling displacement. These and similar models of non-equilibrium processes focus on ``universal'' phenomena, rather than attempting to represent the experimental observations in every detail.
Related to this research activity is the construction of process-based geophysical models for the description of sedimentation, compaction, erosion, and the dynamics of meandering rivers. These processes are controlled by and form the landscape in which they occur, during thousands to millions of years. The flow of water is the source of energy which generates complex patterns and shapes -- eroded mountains, deep valleys, winding rivers -- of enormous scales.
Visualization of the effects of compaction of sedimentary layers below a meander belt.
Fault and fracture systems have been studied in the Cooperative Phenomena program for more than a decade. Fracture processes are import in many fields of physics and in daily life. Mechanical failure due to fracturing has resulted in major catastrophes. For examples, earthquakes are often a result of faulting processes due to the plate tectonics of the Earth's crust. Understanding fracture systems is therefore of great economic and environmental relevance. For instance, in many oil and gas reservoirs, aquifers, and waste storage sites, fluid flow through fractures dominates the transport properties. Knowledge about fracture patterns is thus crucial for managing these systems.
Scanning electron microscope image of a fracture surface in polyethylene.
Theories for single fracture growth in crystals and other homogeneous materials have been known for some time. Much less is understood about inhomogeneous systems where multiple fractures cooperate and compete in growth and coalescence. Advances in computer technology and the advent of fractal geometry have made it possible to study such systems quantitatively.
Fracture research in the Cooperative Phenomena Group during the last few years focused on studies of fracture patterns in geological systems. In geology, models constructed from wax, clay, sand, pitch and plaster of Paris have been studied for almost 200 years. Laboratory-scale models have been used to study large scale faulting and fracturing in the Earth's crust. These qualitative models have been used in geology to better understand fracturing. The main contributions of the Cooperative Phenomena Group consist of more quantitative laboratory experiments and the development of computer models to describe experimental results. In the laboratory, clay bodies are subjected to uniform extension and shear deformation, or more complex deformations. Different loading conditions lead to different characteristic fracture patterns. In a typical experiment, geometrical properties of fractures (length, width, area), typical distance between fractures, and roughness characterization of fracture surfaces are determined. Click here to read more about self-affine fractals. Click here to read more about studying geologic fracturing. The study of fracturing systems also includes polymers with a broad range of applications in industry.
Part (a) shows a digitzed picture of fractures observed in an experiment on uniform extension in clay at 18\% extension in the horizontal direction. The experimental pattern is compared to patterns simulated using a stochastic computer model. A simulation result is shown in part (b).
Among the most familiar irregular patterns are corrosion patterns. Corrosion fronts can be described and characterized using fractal geometry and scaling concepts. In the Cooperative Phenomena Group, studies of corrosion morphology were carried out experimentally and by means of stochastic computer models. Computer simulations provided insight in the mechanism of phenomena such as pitting corrosion in which corrosion occurs locally and large pits are formed in initially massive metal.
Corrosion front with islands of aluminium (black) observed in a corrosion cell containing a sheet of aluminium that was immersed in an aggressive electrolyte.
Irreversible, non-linear processes do not occur only in dead matter -- they are the essence of life. In the Cooperative Phenomena Group, work in biophysics started about a decade ago with a theoretical and experimental investigation of the adsorption of globular protein molecules on flat surfaces. Since then, the properties of other biomaterials, such as DNA, has interested us. Significant advances over the past few years have been made in developing experimental techniques that provide qualitative and quantitative information of biological systems. Our current interest focuses on cellular processes, with emphasis on oscillations in living systems. Protein adsorption on surfaces is under investigation, using atomic force microscopy. The formation of spatial and temporal patterns in biological systems has lately attracted much attention. How does a snake get its coloration-pattern and why is a flower usually symmetrical? Deviations from the normal biological patterns are often associated with genetic mutations or infectious diseases. In our research, biological pattern formation is investigated from a physics point of view. Combining time-lapse video techniques and computer models, we have studied growth-related phenomena, like the temporal patterns and distinctive shapes developed by cellular or bacterial colonies, and the regular and irregular structures formed by the slime mold Physarum . Click here to read more about cooperative phenomena in biophysics.
The slime mold Physarum polycephalum growing on an inhomogeneous substrate consisting of agar based nutrient drops measuring 0.9 cm in diameter.