Nonequilibrium scale selection mechanism for columnar jointing

Abstract

Crack patterns in laboratory experiments on thick samples of drying cornstarch are geometrically similar to columnar joints in cooling lava found at geological sites such as the Giant's Causeway. We present measurements of the crack spacing from both laboratory and geological investigations of columnar jointing, and show how these data can be collapsed onto a single master scaling curve. This is due to the underlying mathematical similarity between theories for the cracking of solids induced by differential drying or by cooling. We use this theory to give a simple quantitative explanation of how these geometrically similar crack patterns arise from a single dynamical law rooted in the nonequilibrium nature of the phenomena. We also give scaling relations for the characteristic crack spacing in other limits consistent with our experiments and observations, and discuss the implications of our results for the control of crack patterns in thin and thick solid films.pattern formation | fracture | geomorphology | volcanology | faulting D rying solids lose moisture from their exposed surfaces and shrink as a consequence. Similarly, cooling solids lose heat from their exposed surfaces and shrink as a consequence. In either case, this differential shrinkage of one part of the solid relative to another leads to stresses that can eventually lead to cracking (1-3). Although much is known about the nucleation, growth, dynamics, and stability of a single crack in an elastic solid, most questions associated with the patterns of multiple cracks due to stresses that arise from nonequilibrium processes such as drying and cooling (4-7) remain wide open. The resulting polygonal planform patterns can arise in a variety of situations, from the mundane cracks in drying mud, to the deliberately artistic cracks in ceramics and pottery, to the spectacular columnar joint formations of the Giant's Causeway in Northern Ireland, Fingal's Cave on Staffa, in Scotland, and the Devil's Postpile in California. The latter formations have fascinated casual observers, artists, and scientists for centuries (8-10), but no comprehensive physical theory for their form or scale exists. Indeed, it is only in the past decade or so that careful laboratory experiments have started to address the dependence of any of these crack patterns on such quantities as the rate of drying or cooling, the thickness of the layers, and their mechanical properties (4-7, 11-13). For example, recent experiments show that crack formation and propagation in drying thin films leads to length scales and patterns that can be strongly timedependent; cracks in directionally drying films grow diffusively at short times, and can advance intermittently via stick-slip-like motion over longer times (11,12). The patterns formed by these cracks depend in detail on the spatiotemporal dynamics of drying, substrate adhesion, and thickness variations (4,6,13,14). This immediately suggests a nonequilibrium origin to these crack patterns, one that couples the heterogeneous elastic stresses in th...

Pattern formation and coarsening dynamics in three-dimensional convective mixing in porous media

“…We are currently investigating how the detailed three-dimensional simulations and theory presented here may guide the development of non-equilibrium two-dimensional models of the pattern-forming process, in the spirit of surface-growth models [36,37]. This will inform our ability to model and predict the properties of other pattern-forming processes that lead to cellular structures [38], such as foams [39], elastocapillary assembly [40], desiccation cracks [41], columnar jointing [42,43] and mantle dynamics [44].…”

Morphometric analysis of polygonal cracking patterns in desiccated starch slurries

“…More precisely, it was proposed that a pentagon be preferred as the columnar shape at a higher cooling rate, whereas a hexagon is preferred at a lower cooling rate. From the analogy in the formation mechanism between the columnar joints and dried starch cracking 36 , it is therefore natural to deduce that a pentagon is preferred in the polygonal cracking of dried starch when the rate of drying is sufficiently large. This conclusion is consistent qualitatively with the starch-water experiment reported by Ref.…”

“…The difference in the preferred type of polygons is known to be due to the difference in the cooling rate of lava. The cooling starts from the top and bottom surfaces of the pooled lava, generating the variation in the internal temperature at different depths 36,40 . This is similar to the case of desiccation cracking, where the drying generates a variation in the water volume fraction.…”

“…Of the many candidates, corn starch slurry (i.e., a dense mixture of corn starch powder with distilled water) [33][34][35][36][37][38] can exhibit heterogeneous crack network characteristics consisting of two distinct types, namely primary cracks and secondary cracks. Primary cracks are long and nearly straight cracks, and they emerge in advance of the appearance of secondary cracks.…”

Cracking mud, freezing dirt, and breaking rocks

“…The crack network advances into a stressed region near the front whose thickness is determined by the speed v at which the whole planar network advances. 13 That speed, in turn, is proportional to the overall heat flux extracted by the water reflux to the surface.…”